Research ArticleBistatic High Frequency Radar Ocean Surface Cross Section foran FMCW Source with an Antenna on a Floating Platform

Yue Ma Weimin Huang and Eric W Gill

Faculty of Engineering and Applied Science Memorial University of Newfoundland St Johnrsquos NL Canada A1B 3X5

Correspondence should be addressed to Yue Ma ym4428munca

Received 5 February 2016 Revised 13 May 2016 Accepted 29 May 2016

Academic Editor Atsushi Mase

Copyright copy 2016 Yue Ma et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The first- and second-order bistatic high frequency radar cross sections of the ocean surface with an antenna on a floating platformare derived for a frequency-modulated continuous wave (FMCW) source Based on previous work the derivation begins with thegeneral bistatic electric field in the frequency domain for the case of a floating antenna Demodulation and range transformationare used to obtain the range information distinguishing the process from that used for a pulsed radar After Fourier-transformingthe autocorrelation and comparing the result with the radar range equation the radar cross sections are derivedThe new first- andsecond-order antenna-motion-incorporated bistatic radar cross section models for an FMCW source are simulated and comparedwith those for a pulsed source Results show that for the same radar operating parameters the first-order radar cross section forthe FMCWwaveform is a little lower than that for a pulsed source The second-order radar cross section for the FMCWwaveformreduces to that for the pulsed waveform when the scattering patch limit approaches infinity The effect of platform motion on theradar cross sections for an FMCW waveform is investigated for a variety of sea states and operating frequencies and in general isfound to be similar to that for a pulsed waveform

1 Introduction

The derivation of high frequency radar ocean surface crosssections has been studied for over four decades The first-order high frequency radar scatter cross section was devel-oped and analysed in [1] Later Walsh and Gill [2] analysedthe scattering of high frequency electromagnetic radiation ofthe ocean surface for a pulse radar Then Gill and Walsh [3]developed the bistatic radar cross section of the ocean surfaceFollowing these analyses the work was extended to the highfrequency monostatic radar cross sections for an antenna ona floating platform [4 5]

All of the models mentioned above were developedspecifically for pulsed radar However there are inherentdisadvantages to using pulsed radar systems For examplethe detectable range capability is determined by the averagetransmitted power In a pulsed radar system both the rangeresolution and the average transmitted power are dependenton the pulse width Narrower pulses bringing better rangeresolution require large peak powers to be useful at long

range Compared to this FMCW radar systems are able toachieve satisfactory range resolution and long range withmoderate peak power due to a 100 duty cycle Thus inrecent years FMCW radars have been widely used in oceanremote sensing applications

A good summary of the digital processing of an FMCWsignal for radar systems has been reported by Barrick [6] Twoprocessing techniqueswere analysed and compared Based onthat work the design of a frequency-modulated interruptedcontinuous wave radar was described and implemented in[7] Then techniques for range and unambiguous velocitymeasurement for an FMCW radar were outlined in [8] Morerecently Walsh et al [9] developed the first- and second-order monostatic radar ocean surface cross sections for anFMCW waveform Also the first-order FMCW radar crosssectionmodel formixed-path ionosphere-ocean propagationhas been established and simulated in [10]

In this paper the first- and second-order bistatic radarocean surface cross sections for an antenna on a floatingplatform and incorporating an FMCW source are presented

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 8675964 9 pageshttpdxdoiorg10115520168675964

2 International Journal of Antennas and Propagation

y

x

Scattering point(x1 y1)

rarr1205882

(x y)Receiver

(R)

120575rarr1205880

Transmitter(T)

1205791205791

rarr1205881

rarr120588

(a)

y

x

(x1 y1)(x2 y2)rarr

12058812

rarr12058820

(x y)Receiver

(R)

120575rarr1205880

Transmitter(T)

1205791205791 1205792

rarr1205881 rarr

1205882

rarr120588

(b)

Figure 1 General (a) first-order and (b) second-order bistatic scatter geometry with antenna motion

In Section 2 the derivation process of the first- and second-order received electric field is reviewed Then a methodsimilar to that in [3] is used to obtain the first- and second-order radar cross section in Section 3 Section 4 containsmodel simulations and comparisons with the pulsed wave-form Section 5 provides conclusions

2 Radar Received Field Equations

21 General First- and Second-Order Electric Field EquationBy using a small displacement vector 120575

0= (120575120588

0 1205790) to

represent the sway motion of the platform (see Figure 1)and adding this small displacement in the source term thefirst-order bistatic scattered field for the case of a floatingtransmitter and a fixed receiver appearing in [11] may bewritten for a vertical dipole source as

(119864119899)1=

1198961198620

(2120587)32sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881) 119865 (120588

2)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 1198901198951198961205751205880[cos120601 cos(120579119870minus1205790)+sin120601 sin(120579119870minus1205790)]

sdot 119890119895120588119904[119870 cos120601minus2119896]

119889120588119904

(1)

Here 1198620= 119868Δ119897119896

2119895120596120576

0is the dipole constant with 119868 being

the current on the dipole of length Δ119897 120596 and 119896 are theradian frequency and wavenumber of the dipole currentrespectively in a space with permittivity of 120576

0 119875

represents

the Fourier coefficient of a surface component whose wavevector has magnitude 119870 and direction 120579

119870(ie = (119870 120579

119870))

With reference to Figure 1(a) = (120588 120579) is a distancevector pointing from the transmitter to the receiver withoutincorporating the platform motion 120588

119904= (120588

1+ 120588

2)2 and 120601

is the bistatic angle 119865 represents the Sommerfeld attenuationfunction

The second-order bistatic received electric field corre-sponding to the first-order found in (1) appears in [12] as

(119864119899)2=minus119896119862

0

(2120587)32sum

1

sum

2

11987511198752

radic119870119890119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

(minus119896120594) 119865 (1205882) 119865 (120588

20)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 1198901198951198961205751205880[cos120601 cos(120579119870minus1205790)+sin120601 sin(120579119870minus1205790)]

sdot 119890119895120588119904[119870 cos120601minus2119896]

119889120588119904

(2)

This expression accounts for the electric field arising due tothe transmitted signal being scattered twice by the roughsurface

1and

2are the first and the second scattering

wave vectors of the rough surface The rough surface may berepresented by a Fourier series with 119875

1and 119875

2being the

Fourier coefficients associated with 1and

2 respectively

Here = 1+

2 Again with reference to Figure 1 defining

119904(2

1) = 119896

2minus

1allows 120594 to be expressed as [12]

120594 = 119895 (1sdot

2) (

119904sdot

2) sdot 119866 [119870

119904(

2

1)] (3)

where

119866 [119870119904(

2

1)] =

1

119870119904

1 minus 119895119896 (1 + Δ)

radic1198702119904minus 1198962 + 119895119896Δ

(4)

with Δ being the intrinsic impedance of the surface At thisstage the current waveformon the dipole source has not beenspecified

International Journal of Antennas and Propagation 3

22 Applications to an FMCW Radar Following a similaranalysis as in [3 11] (1) may be inversely Fourier-transformedto give the received electric field in the time domain as

Fminus1[(119864

119899)1] (119905) =

1

(2120587)32

sdotFminus1[minus119895

1205780Δ119897

11988821205962119868 (120596)]

119905

lowast Fminus1

sum

119875

sdot radic119870119890119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881) 119865 (120588

2)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 1198901198951198961205751205880[cos120601 cos(120579119870minus1205790)+sin120601 sin(120579119870minus1205790)]

sdot 119890119895120588119904[119870 cos120601minus2119896]

119889120588119904

(5)

The current waveform of an FMCW radar may be writtenas [6 9]

119894 (119905) = 1198680119890119895(1205960119905+120572120587119905

2)ℎ [119905 +

119879119903

2] minus ℎ [119905 minus

119879119903

2] (6)

where 1198680is the peak current and120596

0= 2120587119891

0is the center radian

frequency of the sweep waveform 119879119903represents the sweep

interval and the sweep rate may be expressed as 120572 = 119861119879119903

where 119861 is the sweep bandwidth ℎ is the Heaviside functionIt is known from [9] that for an FMCW waveform

Fminus1[minus119895

1205780Δ119897

11988821205962119868 (120596)]

= minus1198951198680

1205780Δ119897120596

2

0

1198882119890119895(1205960119905+120572120587119905

2)

sdot ℎ [119905 +119879119903

2] minus ℎ [119905 minus

119879119903

2]

(7)

By direct comparison with the corresponding first-order casefor a pulsed dipole [11] the first-order time domain electricfield for an FMCW source may be written as

(119864119899)1(119905119903) =

minus11989511986801205780Δ119897119896

2

0

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601119890119895120588119904119870 cos120601

sdot 119890119895(1205960119905119903+120572120587119905

2

119903)119890minus1198951198960(2120588119904minus1205751205881199040)119890

minus119895(2120587120572(2120588119904minus1205751205881199040)119888)119905119903

sdot 119890119895(120587120572(2120588119904minus1205751205881199040)

21198882)ℎ [119905

119903+119879119903

2minus2120588

119904minus 120575120588

1199040

119888]

minus ℎ [119905119903minus119879119903

2minus2120588

119904minus 120575120588

1199040

119888] 119889120588

119904

(8)

where 1205751205881199040= 120575120588

0[cos120601 cos(120579

119870minus 120579

0) + sin120601 sin(120579

119870minus 120579

0)] 119905

is renamed as 119905119903to indicate that the time is within a sweep

repetition interval ((2120588119904minus 120575120588

1199040)119888 minus 119879

1199032 (2120588

119904minus 120575120588

1199040)119888 +

1198791199032) As stated in [6 9] the frequency difference between

the transmitted waveform and the received waveformmay beFourier-transformed within this interval to obtain the rangeinformationThis is the so-called ldquorange transformrdquo Becausethe received signals in the given time interval reflect theinformation for an extremely large range of ocean surfacehere range transformation is taken to accurately acquire apatch of ocean surface to analyse The frequency differencemay be obtained by the demodulation process in which thetransmitted signals and the received signals are mixed andthen low-pass-filtered

After the demodulation preprocess the exponential fac-tor 119890119895(1205960119905119903+120572120587119905

2

119903) in (8) will be eliminated Then Fourier-

transforming with respect to 119905119903gives

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601119890119895120588119904119870 cos120601

sdot 119890minus1198951198960(2120588119904minus1205751205881199040)119890

119895120596119903(2120588119904minus1205751205881199040)119888119890minus119895(120587120572(2120588119904minus1205751205881199040)

21198882)

sdot Sa[119879119903

2(120596

119903minus2120587120572 (2120588

119904minus 120575120588

1199040)

119888)] 119889120588

119904

(9)

where 120596119903is the transform variable in the frequency domain

Here it is helpful to define

1205881015840

119904= 120588

119904minus120575120588

1199040

2 (10)

Changing the integration variable from 120588119904to 1205881015840

119904and ignoring

the 12057512058811990402 factor in the magnitude terms give

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 119890119895(119870 cos120601minus21198960+2120596119903119888)1205881015840119904 119890minus119895(4120587120572(120588

1015840

119904)21198882)1198901198951205751205881199040119870 cos1206012

sdot Sa [119879119903

2(120596

119903minus4120587120572

1198881205881015840

119904)] 119889120588

1015840

119904

(11)

Since the maximum of the sampling function Sa(119909) occurs at119909 = 0 a representative range 120588

119903may be defined as

120588119903=119888120596

119903

4120587120572 (12)

4 International Journal of Antennas and Propagation

Based on the representative range defining the correspond-ing range variable

12058810158401015840

119904= 120588

1015840

119904minus 120588

119903(13)

and changing the integration variable from 1205881015840

119904to 12058810158401015840

119904 (11)

becomes

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

12058810158401015840

119904max

12058810158401015840

119904min

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 119890119895(minus21198960+119896119903)120588119903119890

119895(minus21198960)12058810158401015840

119904 119890119895120588119904119870 cos120601

sdot 119890minus119895(119896119903120588119903)(120588

10158401015840

119904)2

Sa [11989611986112058810158401015840

119904] 119889120588

10158401015840

119904

(14)

where 119896119861= 2120587119861119888 and 119896

119903= 120596

119903119888 A process similar to that

in [11] is used to simplify the terms in the integral Then (14)reduces to

(119864119899)1(120596

119903) =

minus11989511986801205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890minus119895(1205874)

radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot int

12058810158401015840

119904max

12058810158401015840

119904min

119890119895(119870 cos1206010minus21198960)12058810158401015840119904

sdot 119890minus119895(119896119903120588119903)(120588

10158401015840

119904)2

Sa [11989611986112058810158401015840

119904] 119889120588

10158401015840

119904

(15)

where 12058801and 120588

02are representative values of 120588

1and 120588

2(see

Figure 1) respectively By directly comparing (15) with (24)in [9] the first-order bistatic received electric field for anFMCWwaveformwith an antenna on a floating platformmaybe expressed as

(119864119899)1(120596

119903) =

minus11989511986801205780Δ119897119896

2

0

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890minus119895(1205874)

radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(16)

plusmnΔ119903 are the symmetrical limits of the integral in (15) where asampling function dominates this integral If only the valuesof 12058810158401015840

119904within the main lobe of the sampling function are

considered in the integral that is minus1205872 lt 11989611986112058810158401015840

119904lt 1205872 it

can be deduced as in [9] that Δ119903 = Δ1205882 = 1198884119861 Consider

Sm (119870 cos1206010 119896119861 Δ119903)

=1

120587Si [( 119870

cos1206010

minus 21198960+ 119896

119861)Δ119903]

minus Si [( 119870

cos1206010

minus 21198960minus 119896

119861)Δ119903]

(17)

where Si(119909) = int1199090(sin(119905)119905) 119889119905

Following a similar procedure to the first-order case thesecond-order bistatic received electric field with a transmitteron a floating platform for an FMCW waveform may bewritten as

(119864119899)2119864(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1

sum

2

11987511198752

radic119870119890minus119895(1205874)

sdot 119890119895(1205881198702) cos(120579119870minus120579)radiccos120601

0119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdotSEΓ119875119865 (12058802 1205960) 119865 (120588020 1205960)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(18)

where 12058802

and 120588020

are the representative values of 1205882and

12058820 respectively The symmetrical electromagnetic coupling

coefficient SEΓ119875 may be expressed as [12]

SEΓ119875 (1 2) =119895119896

0

2119870 cos1206010

(1sdot

2)

sdot [119904(

2

1) sdot

2] 119866 [119870

119904(

2

1)]

+ (2sdot

2) [

119904(

2

2) sdot

1] 119866 [119870

119904(

2

2)]

(19)

with 119904(2

2) = 119896

2minus

2and

119866 [119870119904(

2

1)] =

1

119870119904

1 minus 119895119896 (1 + Δ)

radic1198702119904minus 1198962 + 119895119896Δ

(20)

International Journal of Antennas and Propagation 5

3 Radar Cross Sections

31 First-Order Radar Cross Section In developing the oceanradar cross section a time-varying ocean surface representedas 120585( 119905) = sum

119905119875120596119890119895sdot119890119895120596119905 is used to replace the time-

invariant case 120585() = sum119875119890119895sdot This gives the time-varying

received electric field corresponding to (16) as

(119864119899)1(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

120596

119875120596radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890minus119895(1205874)radiccos120601

0119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(21)

A technique similar to that in [9 11] is used to obtain theradar cross section from the received electric field equationThe initial step of the approach is to write the autocorrelation119877(120591) as

119877 (120591) =119860119903

21205780

1

1198792119903

⟨(119864119899)1(120596

119903 119905 + 120591) (119864

119899)lowast

1(120596

119903 119905)⟩ (22)

where 119860119903= (120582

2

04120587)119866

119903 with 119866

119903being the gain of the

receiving array Here 120596119903represents the fixed patch over two

sweep intervals whose time interval is 120591After Fourier-transforming the autocorrelation and com-

paring directly with the radar range equation the radar crosssection 120590

1(120596

119889) may be written as

1205901(120596

119889) = 2

3120587119896

2

0Δ120588 sum

119898=plusmn1

int119870

1198781(119898)119870

2 cos1206010

sdot Sm2(119870 cos120601

0 119896119861 Δ119903)

sdot 1198692

0[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot 120575 (120596

119889+ 119898radic119892119870)

+

infin

sum

119899=1

1198692

119899[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot [120575 (120596

119889+ 119898radic119892119870

minus 119899120596119901) + 120575 (120596

119889+ 119898radic119892119870 + 119899120596

119901)]119889119870

(23)

where 119869119899represents the 119899th-order Bessel functions For

simulation purposes (see Section 4) and in keeping with [411] it will be assumed that the antenna motion is causedby the dominant ocean waves 119886 120596

119901 and 120579

119870119901represent the

antenna platform sway amplitude frequency and directionrespectively

32 Second-Order Radar Cross Section It is known that thesecond-order radar cross section contains two portions anhydrodynamic contribution and an electromagnetic contri-bution Using the Fourier coefficient for the second-orderocean waves sum

1sum2 119867

Γ11987511198752

to replace the first-ordercase sum

119875in (16) the hydrodynamic second-order electric

field may be written as

(119864119899)2119867(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1

sum

2

11987511198752

radic119870119890minus119895(1205874)

sdot 119890119895(1205881198702) cos(120579119870minus120579)radiccos120601

0119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119867Γ119865 (12058801 1205960) 119865 (12058802 1205960)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(24)

where119867Γ is the hydrodynamic coupling coefficient [13]

accounting for the coupling of two first-order ocean waveswhose wavenumbers are 119870

1and 119870

2 respectively Adding

the electromagnetic contribution (18) and the hydrodynamiccontribution (24) together and using the time-varying oceanwave surface to replace the time-invariant case the totalsecond-order bistatic electric field for an FMCW source withan antenna on a floating platform may be expressed as

(119864119899)2(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1 1205961

sum

2 1205962

1198751 1205961

1198752 1205962

sdot119878Γ119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)119890minus119895(1205874)

sdot radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(25)

where119878Γ119875= SEΓ119875 +119867Γ

Following the same procedure as for the first-ordercase based on the total second-order time-varying receivedelectric field (25) the corresponding second-order radarcross section 120590

2(120596

119889) may be obtained as

6 International Journal of Antennas and Propagation

1205902(120596

119889) = 2

3120587119896

2

0Δ120588 sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

int

infin

0

1198781(119898

11) 119878

1(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198702 cos120601

01198701Sm2

(119870 cos1206010 119896119861 Δ119903)

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816 sdot 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901) + 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)] 119889119870

1119889120579

1119889119870

(26)

4 Simulation and Discussion

Based on a Pierson-Moskowitz (PM) ocean spectral model[14] the newly derived radar cross sections accounting forantenna sway can be simulated to illustrate the differencesin the FMCW and pulsed waveform cases The sweepbandwidth of the FMCW waveform is chosen as 50 kHzThe operating frequency defined as the central frequency ofthe FMCW waveform is taken to be 25MHz The bistaticangle is 30∘ and the wind speed is 20 knots The scatteringellipse normal and the wind direction are 90∘ and 180∘respectively as measured from the positive 119909-axis (the lineconnecting the transmitter with the receiver) The swayamplitude and frequency depend on the wind velocity andin keeping with an example used earlier [4] for the purposeof illustration here these values are taken as 1228m and0127Hz respectively The sway direction is chosen to be thesame as the wind direction

41 First-Order Radar Cross Section Figure 2 shows a com-parison of the first-order radar cross section for a pulsedsource and that for an FMCW source For the FMCWwaveform Δ119903 = 1500m which equals half the width of thescattering patch (Δ120588 = 3000m) for the pulsed waveformin order to keep the same bandwidth for both waveformsA hamming window is used to smooth the curve andreduce the oscillations From this figure it can be observedthat additional peaks caused by the antenna motion appearsymmetrically in the Doppler spectrum with respect to theBragg peaks A detailed description and properties of thesemotion-induced peaks were discussed in [11] It can alsobe seen that the magnitude of the radar cross sections forthe FMCW waveform is a little lower than that for thecorresponding pulsed waveform which may be caused bythe value of Δ119903 Δ119903 is the limit value of the integral where asampling function dominates this integralΔ119903 is usually takento be Δ119903 = Δ1205882 which means only the contributions inthe main lobe of the sampling function are considered andno interaction between the range bins is assumed in the idealcase

It is clear that the first-order radar cross section has acertain relationship with the integral limit Δ119903 In Section 2it may be observed that there is no mathematical limit forthe parameter Δ119903 By varying Δ119903 the effect on the radar crosssection can be examined Keeping the value of Δ120588 = 3000m

Δ119903 = 05Δ120588 and Δ119903 = 10Δ120588 are simulated in Figures 3(a) and3(b) respectively It should be mentioned that the hammingwindow smoothing process is not used in Figure 3 in orderto clearly show the side lobe levels of the first-order radarcross sections The side lobe structure appears in the radarDoppler spectra due to the side lobes of the Sm functionfor the FMCW waveform By comparing Figures 3(a) and3(b) the magnitude of the side lobes for FMCW source isfound to decrease with increasing Δ119903 and the main lobe levelis a little raised with increasing Δ119903 due to the propertiesof the Sm function This seems to indicate an advantage ofan FMCW system When the value of Δ119903 is taken to belarger thanΔ1205882 the interactions between the range bins (thecontributions in the side lobe of the sampling function) areconsidered and appear in the received electric field at a fixeddistance Increasing Δ119903means the received signal is scatteredfrom a larger ocean surface region When Δ119903 approachesinfinity the radar cross section for the FMCW waveformbecomes a rectangular function whose width is determinedby119861(2119891

0120596119861) However when the patchwidthΔ120588 approaches

infinity the sampling functions in the first-order pulse radarocean cross section reduce to delta functions

By varying the radar bandwidth and keeping the relation-ships Δ120588 = 1198882119861 and Δ119903 = Δ1205882 the effect of the bandwidthon the radar cross sections is illustrated in Figure 4 Fromthis figure it can be seen that with increased bandwidththe magnitudes of the Bragg peaks and the motion-inducedpeaks are found to be reduced while the rest of the radar crosssection increases In addition the width of the Bragg peaksand the motion-induced peaks is also broadened Thereforeif a large radar bandwidth is used for ocean remote sensingthe Bragg peaks may be significantly contaminated by themotion-induced peaks

42 Second-Order Radar Cross Section A similar technique isused to simplify and simulate the second-order radar oceancross section for the FMCW waveform as that for the pulsedwaveform in [12 15] For the case of large Δ119903 it can be shownthat

limΔ119903rarrinfin

[Δ120588Sm2(119870 cos120601

0 119896119861 Δ119903)] asymp Δ120588 cos120601

0

sdot ℎ [119870 minus cos1206010(2119896

0minus 119896

119861)]

minus ℎ [119870 minus cos1206010(2119896

0+ 119896

119861)]

(27)

International Journal of Antennas and Propagation 7

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

FMCWPulsed

Figure 2 Comparison of the first-order radar cross sections for the FMCW waveform with that for the pulsed waveform

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

Doppler frequency f (Hz)

FMCWPulsed

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(a)

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

FMCWPulsed

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(b)Figure 3 Comparison of the side lobe levels of the first-order radar cross sections for the pulsed and FMCWwaveform (a) Δ119903 = 05Δ120588 and(b) Δ119903 = 10Δ120588

Assuming that the other terms in (26) are slowly varyingwithin the interval

cos1206010(2119896

0minus 119896

119861) lt 119870 lt cos120601

0(2119896

0+ 119896

119861) (28)

and carrying out the 119870 integration (26) reduces to

1205902(120596

119889) = 64120587

21198964

0cos4120601

0sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

1198781(119898

11)

sdot 1198781(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198701

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot 120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901)

+120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)]

sdot 1198891198701119889120579

1

(29)

8 International Journal of Antennas and Propagation

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

100 kHz50kHz

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 4 The effect of the bandwidth on the first-order radar crosssections

minus15 minus1 minus05 0 05 1 15minus70

minus60

minus50

minus40

minus30

minus20

minus10

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 5 Second-order bistatic radar cross section with a transmit-ter on a floating platform

Equation (29) is exactly the same as the correspondingmodel for the pulsed waveform when the scattering patch Δ120588approaches infinityTherefore the second-order cross sectionmodel for the FMCW waveform shows the same features inthe Doppler spectra as the model for the pulsed waveformin [12] for a given sea state radar operating parameters andplatform motion An example of the second-order bistaticradar cross section with a transmitter on a floating platformand a fixed receiver is shown in Figure 5 for the case of thescattering patch being assumed to be infinite in extent

5 Conclusion

Thefirst- and second-order bistatic radar ocean cross sectionsfor an antenna on a floating platform have been presentedfor the case of an FMCW waveform In the derivationprocess the first- and second-order models begin with

the bistatically received electric field equations derived in[11 12] Subsequently the derivation is carried out for anFMCW radar which is different from [11 12] where a pulsedradar is considered In particular the distinguishing featurebetween the current work and that presented earlier is thatdemodulation and range transformation must be used toobtain the range information Based on the new modelssimulations are made to compare the radar cross sectionsfor the FMCW waveform with that for the pulsed waveformIt is found that the first-order radar cross section for theFMCWwaveform is a little lower than that for a pulsed sourcewith the same simulation parameters With increased radaroperating bandwidth the magnitude and width of Braggpeaks and motion-induced peaks are found to be reducedand broadened respectively For an FMCW waveform thereis no definite mathematical limit for a patch width whichis different from that for a pulsed waveform Therefore themagnitude of the range bin is varied to examine the effecton the radar cross section The side lobe level is found to bereduced with increasing magnitude of the range bin Whenthe range bin approaches infinity the first-order radar crosssection for an FMCW waveform approaches a rectangularfunction and the second-order radar cross section modelfor the FMCW waveform is reduced to that of the pulsedwaveform

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work was supported in part by Natural Sciences andEngineering Research Council of Canada (NSERC) underDiscovery Grants to Weimin Huang (NSERC 402313-2012)and Eric W Gill (NSERC 238263-2010 and RGPIN-2015-05289) and by an Atlantic Innovation Fund Award (Eric WGill principal investigator)

References

[1] D E Barrick ldquoFirst-order theory and analysis of MFHFVHFscatter from the seardquo IEEE Transactions on Antennas andPropagation vol 20 no 1 pp 2ndash10 1972

[2] J Walsh and E W Gill ldquoAn analysis of the scattering of high-frequency electromagnetic radiation from rough surfaces withapplication to pulse radar operating in backscattermoderdquoRadioScience vol 35 no 6 pp 1337ndash1359 2000

[3] E W Gill and J Walsh ldquoHigh-frequency bistatic cross sectionsof the ocean surfacerdquoRadio Science vol 36 no 6 pp 1459ndash14752001

[4] J WalshW Huang and E Gill ldquoThe first-order high frequencyradar ocean surface cross section for an antenna on a floatingplatformrdquo IEEE Transactions on Antennas and Propagation vol58 no 9 pp 2994ndash3003 2010

[5] J Walsh W Huang and E Gill ldquoThe second-order highfrequency radar ocean surface cross section for an antenna

International Journal of Antennas and Propagation 9

on a floating platformrdquo IEEE Transactions on Antennas andPropagation vol 60 no 10 pp 4804ndash4813 2012

[6] D E Barrick ldquoFMCW radar signals and digital processingrdquoTech Rep ERL 283-WPL 26 NOAA 1973

[7] R Khan B Gamberg D Power et al ldquoTarget detection andtracking with a high frequency ground wave radarrdquo IEEEJournal of Oceanic Engineering vol 19 no 4 pp 540ndash548 1994

[8] A Wojthiewicz J Misiurewicz M Nalecz K Jedrzejewskiand K Kulpa ldquoTwo dimensional signal processing in FMCWradarsrdquo in Proceedings of the Conference on Circuit Theory andElectronics Circuits pp 474ndash480 Kolobrzeg Poland October1997

[9] J Walsh J Zhang and E W Gill ldquoHigh-frequency radar crosssection of the ocean surface for an FMCW waveformrdquo IEEEJournal of Oceanic Engineering vol 36 no 4 pp 615ndash626 2011

[10] S Chen EWGill andWHuang ldquoA first-orderHF radar cross-section model for mixed-path ionosphere-ocean propagationwith an FMCW sourcerdquo IEEE Journal of Oceanic Engineering2016

[11] Y Ma E Gill and W Huang ldquoThe first-order bistatic highfrequency radar ocean surface cross section for an antenna ona floating platformrdquo IET Radar Sonar amp Navigation vol 10 no6 pp 1136ndash1144 2016

[12] Y Ma W Huang and E Gill ldquoThe second-order bistatic highfrequency radar ocean surface cross section for an antenna on afloating platformrdquo Canadian Journal of Remote Sensing vol 42no 4 pp 332ndash343 2016

[13] K Hasselmann ldquoOn the non-linear energy transfer in a gravity-wave spectrum part 1 General theoryrdquo Journal of FluidMechan-ics vol 12 pp 481ndash500 1962

[14] W J Pierson and L Moskowitz ldquoA proposed spectral form forfully developed wind seas based on the similarity theory of S AKitaigorodskiirdquo Journal of Geophysical Research vol 69 no 24pp 5181ndash5190 1964

[15] B J Lipa and D E Barrick ldquoExtraction of sea state fromHF radar sea echo mathematical theory and modelingrdquo RadioScience vol 21 no 1 pp 81ndash100 1986

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

2 International Journal of Antennas and Propagation

y

x

Scattering point(x1 y1)

rarr1205882

(x y)Receiver

(R)

120575rarr1205880

Transmitter(T)

1205791205791

rarr1205881

rarr120588

(a)

y

x

(x1 y1)(x2 y2)rarr

12058812

rarr12058820

(x y)Receiver

(R)

120575rarr1205880

Transmitter(T)

1205791205791 1205792

rarr1205881 rarr

1205882

rarr120588

(b)

Figure 1 General (a) first-order and (b) second-order bistatic scatter geometry with antenna motion

In Section 2 the derivation process of the first- and second-order received electric field is reviewed Then a methodsimilar to that in [3] is used to obtain the first- and second-order radar cross section in Section 3 Section 4 containsmodel simulations and comparisons with the pulsed wave-form Section 5 provides conclusions

2 Radar Received Field Equations

21 General First- and Second-Order Electric Field EquationBy using a small displacement vector 120575

0= (120575120588

0 1205790) to

represent the sway motion of the platform (see Figure 1)and adding this small displacement in the source term thefirst-order bistatic scattered field for the case of a floatingtransmitter and a fixed receiver appearing in [11] may bewritten for a vertical dipole source as

(119864119899)1=

1198961198620

(2120587)32sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881) 119865 (120588

2)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 1198901198951198961205751205880[cos120601 cos(120579119870minus1205790)+sin120601 sin(120579119870minus1205790)]

sdot 119890119895120588119904[119870 cos120601minus2119896]

119889120588119904

(1)

Here 1198620= 119868Δ119897119896

2119895120596120576

0is the dipole constant with 119868 being

the current on the dipole of length Δ119897 120596 and 119896 are theradian frequency and wavenumber of the dipole currentrespectively in a space with permittivity of 120576

0 119875

represents

the Fourier coefficient of a surface component whose wavevector has magnitude 119870 and direction 120579

119870(ie = (119870 120579

119870))

With reference to Figure 1(a) = (120588 120579) is a distancevector pointing from the transmitter to the receiver withoutincorporating the platform motion 120588

119904= (120588

1+ 120588

2)2 and 120601

is the bistatic angle 119865 represents the Sommerfeld attenuationfunction

The second-order bistatic received electric field corre-sponding to the first-order found in (1) appears in [12] as

(119864119899)2=minus119896119862

0

(2120587)32sum

1

sum

2

11987511198752

radic119870119890119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

(minus119896120594) 119865 (1205882) 119865 (120588

20)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 1198901198951198961205751205880[cos120601 cos(120579119870minus1205790)+sin120601 sin(120579119870minus1205790)]

sdot 119890119895120588119904[119870 cos120601minus2119896]

119889120588119904

(2)

This expression accounts for the electric field arising due tothe transmitted signal being scattered twice by the roughsurface

1and

2are the first and the second scattering

wave vectors of the rough surface The rough surface may berepresented by a Fourier series with 119875

1and 119875

2being the

Fourier coefficients associated with 1and

2 respectively

Here = 1+

2 Again with reference to Figure 1 defining

119904(2

1) = 119896

2minus

1allows 120594 to be expressed as [12]

120594 = 119895 (1sdot

2) (

119904sdot

2) sdot 119866 [119870

119904(

2

1)] (3)

where

119866 [119870119904(

2

1)] =

1

119870119904

1 minus 119895119896 (1 + Δ)

radic1198702119904minus 1198962 + 119895119896Δ

(4)

with Δ being the intrinsic impedance of the surface At thisstage the current waveformon the dipole source has not beenspecified

International Journal of Antennas and Propagation 3

22 Applications to an FMCW Radar Following a similaranalysis as in [3 11] (1) may be inversely Fourier-transformedto give the received electric field in the time domain as

Fminus1[(119864

119899)1] (119905) =

1

(2120587)32

sdotFminus1[minus119895

1205780Δ119897

11988821205962119868 (120596)]

119905

lowast Fminus1

sum

119875

sdot radic119870119890119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881) 119865 (120588

2)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 1198901198951198961205751205880[cos120601 cos(120579119870minus1205790)+sin120601 sin(120579119870minus1205790)]

sdot 119890119895120588119904[119870 cos120601minus2119896]

119889120588119904

(5)

The current waveform of an FMCW radar may be writtenas [6 9]

119894 (119905) = 1198680119890119895(1205960119905+120572120587119905

2)ℎ [119905 +

119879119903

2] minus ℎ [119905 minus

119879119903

2] (6)

where 1198680is the peak current and120596

0= 2120587119891

0is the center radian

frequency of the sweep waveform 119879119903represents the sweep

interval and the sweep rate may be expressed as 120572 = 119861119879119903

where 119861 is the sweep bandwidth ℎ is the Heaviside functionIt is known from [9] that for an FMCW waveform

Fminus1[minus119895

1205780Δ119897

11988821205962119868 (120596)]

= minus1198951198680

1205780Δ119897120596

2

0

1198882119890119895(1205960119905+120572120587119905

2)

sdot ℎ [119905 +119879119903

2] minus ℎ [119905 minus

119879119903

2]

(7)

By direct comparison with the corresponding first-order casefor a pulsed dipole [11] the first-order time domain electricfield for an FMCW source may be written as

(119864119899)1(119905119903) =

minus11989511986801205780Δ119897119896

2

0

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601119890119895120588119904119870 cos120601

sdot 119890119895(1205960119905119903+120572120587119905

2

119903)119890minus1198951198960(2120588119904minus1205751205881199040)119890

minus119895(2120587120572(2120588119904minus1205751205881199040)119888)119905119903

sdot 119890119895(120587120572(2120588119904minus1205751205881199040)

21198882)ℎ [119905

119903+119879119903

2minus2120588

119904minus 120575120588

1199040

119888]

minus ℎ [119905119903minus119879119903

2minus2120588

119904minus 120575120588

1199040

119888] 119889120588

119904

(8)

where 1205751205881199040= 120575120588

0[cos120601 cos(120579

119870minus 120579

0) + sin120601 sin(120579

119870minus 120579

0)] 119905

is renamed as 119905119903to indicate that the time is within a sweep

repetition interval ((2120588119904minus 120575120588

1199040)119888 minus 119879

1199032 (2120588

119904minus 120575120588

1199040)119888 +

1198791199032) As stated in [6 9] the frequency difference between

the transmitted waveform and the received waveformmay beFourier-transformed within this interval to obtain the rangeinformationThis is the so-called ldquorange transformrdquo Becausethe received signals in the given time interval reflect theinformation for an extremely large range of ocean surfacehere range transformation is taken to accurately acquire apatch of ocean surface to analyse The frequency differencemay be obtained by the demodulation process in which thetransmitted signals and the received signals are mixed andthen low-pass-filtered

After the demodulation preprocess the exponential fac-tor 119890119895(1205960119905119903+120572120587119905

2

119903) in (8) will be eliminated Then Fourier-

transforming with respect to 119905119903gives

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601119890119895120588119904119870 cos120601

sdot 119890minus1198951198960(2120588119904minus1205751205881199040)119890

119895120596119903(2120588119904minus1205751205881199040)119888119890minus119895(120587120572(2120588119904minus1205751205881199040)

21198882)

sdot Sa[119879119903

2(120596

119903minus2120587120572 (2120588

119904minus 120575120588

1199040)

119888)] 119889120588

119904

(9)

where 120596119903is the transform variable in the frequency domain

Here it is helpful to define

1205881015840

119904= 120588

119904minus120575120588

1199040

2 (10)

Changing the integration variable from 120588119904to 1205881015840

119904and ignoring

the 12057512058811990402 factor in the magnitude terms give

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 119890119895(119870 cos120601minus21198960+2120596119903119888)1205881015840119904 119890minus119895(4120587120572(120588

1015840

119904)21198882)1198901198951205751205881199040119870 cos1206012

sdot Sa [119879119903

2(120596

119903minus4120587120572

1198881205881015840

119904)] 119889120588

1015840

119904

(11)

Since the maximum of the sampling function Sa(119909) occurs at119909 = 0 a representative range 120588

119903may be defined as

120588119903=119888120596

119903

4120587120572 (12)

4 International Journal of Antennas and Propagation

Based on the representative range defining the correspond-ing range variable

12058810158401015840

119904= 120588

1015840

119904minus 120588

119903(13)

and changing the integration variable from 1205881015840

119904to 12058810158401015840

119904 (11)

becomes

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

12058810158401015840

119904max

12058810158401015840

119904min

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 119890119895(minus21198960+119896119903)120588119903119890

119895(minus21198960)12058810158401015840

119904 119890119895120588119904119870 cos120601

sdot 119890minus119895(119896119903120588119903)(120588

10158401015840

119904)2

Sa [11989611986112058810158401015840

119904] 119889120588

10158401015840

119904

(14)

where 119896119861= 2120587119861119888 and 119896

119903= 120596

119903119888 A process similar to that

in [11] is used to simplify the terms in the integral Then (14)reduces to

(119864119899)1(120596

119903) =

minus11989511986801205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890minus119895(1205874)

radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot int

12058810158401015840

119904max

12058810158401015840

119904min

119890119895(119870 cos1206010minus21198960)12058810158401015840119904

sdot 119890minus119895(119896119903120588119903)(120588

10158401015840

119904)2

Sa [11989611986112058810158401015840

119904] 119889120588

10158401015840

119904

(15)

where 12058801and 120588

02are representative values of 120588

1and 120588

2(see

Figure 1) respectively By directly comparing (15) with (24)in [9] the first-order bistatic received electric field for anFMCWwaveformwith an antenna on a floating platformmaybe expressed as

(119864119899)1(120596

119903) =

minus11989511986801205780Δ119897119896

2

0

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890minus119895(1205874)

radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(16)

plusmnΔ119903 are the symmetrical limits of the integral in (15) where asampling function dominates this integral If only the valuesof 12058810158401015840

119904within the main lobe of the sampling function are

considered in the integral that is minus1205872 lt 11989611986112058810158401015840

119904lt 1205872 it

can be deduced as in [9] that Δ119903 = Δ1205882 = 1198884119861 Consider

Sm (119870 cos1206010 119896119861 Δ119903)

=1

120587Si [( 119870

cos1206010

minus 21198960+ 119896

119861)Δ119903]

minus Si [( 119870

cos1206010

minus 21198960minus 119896

119861)Δ119903]

(17)

where Si(119909) = int1199090(sin(119905)119905) 119889119905

Following a similar procedure to the first-order case thesecond-order bistatic received electric field with a transmitteron a floating platform for an FMCW waveform may bewritten as

(119864119899)2119864(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1

sum

2

11987511198752

radic119870119890minus119895(1205874)

sdot 119890119895(1205881198702) cos(120579119870minus120579)radiccos120601

0119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdotSEΓ119875119865 (12058802 1205960) 119865 (120588020 1205960)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(18)

where 12058802

and 120588020

are the representative values of 1205882and

12058820 respectively The symmetrical electromagnetic coupling

coefficient SEΓ119875 may be expressed as [12]

SEΓ119875 (1 2) =119895119896

0

2119870 cos1206010

(1sdot

2)

sdot [119904(

2

1) sdot

2] 119866 [119870

119904(

2

1)]

+ (2sdot

2) [

119904(

2

2) sdot

1] 119866 [119870

119904(

2

2)]

(19)

with 119904(2

2) = 119896

2minus

2and

119866 [119870119904(

2

1)] =

1

119870119904

1 minus 119895119896 (1 + Δ)

radic1198702119904minus 1198962 + 119895119896Δ

(20)

International Journal of Antennas and Propagation 5

3 Radar Cross Sections

31 First-Order Radar Cross Section In developing the oceanradar cross section a time-varying ocean surface representedas 120585( 119905) = sum

119905119875120596119890119895sdot119890119895120596119905 is used to replace the time-

invariant case 120585() = sum119875119890119895sdot This gives the time-varying

received electric field corresponding to (16) as

(119864119899)1(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

120596

119875120596radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890minus119895(1205874)radiccos120601

0119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(21)

A technique similar to that in [9 11] is used to obtain theradar cross section from the received electric field equationThe initial step of the approach is to write the autocorrelation119877(120591) as

119877 (120591) =119860119903

21205780

1

1198792119903

⟨(119864119899)1(120596

119903 119905 + 120591) (119864

119899)lowast

1(120596

119903 119905)⟩ (22)

where 119860119903= (120582

2

04120587)119866

119903 with 119866

119903being the gain of the

receiving array Here 120596119903represents the fixed patch over two

sweep intervals whose time interval is 120591After Fourier-transforming the autocorrelation and com-

paring directly with the radar range equation the radar crosssection 120590

1(120596

119889) may be written as

1205901(120596

119889) = 2

3120587119896

2

0Δ120588 sum

119898=plusmn1

int119870

1198781(119898)119870

2 cos1206010

sdot Sm2(119870 cos120601

0 119896119861 Δ119903)

sdot 1198692

0[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot 120575 (120596

119889+ 119898radic119892119870)

+

infin

sum

119899=1

1198692

119899[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot [120575 (120596

119889+ 119898radic119892119870

minus 119899120596119901) + 120575 (120596

119889+ 119898radic119892119870 + 119899120596

119901)]119889119870

(23)

where 119869119899represents the 119899th-order Bessel functions For

simulation purposes (see Section 4) and in keeping with [411] it will be assumed that the antenna motion is causedby the dominant ocean waves 119886 120596

119901 and 120579

119870119901represent the

antenna platform sway amplitude frequency and directionrespectively

32 Second-Order Radar Cross Section It is known that thesecond-order radar cross section contains two portions anhydrodynamic contribution and an electromagnetic contri-bution Using the Fourier coefficient for the second-orderocean waves sum

1sum2 119867

Γ11987511198752

to replace the first-ordercase sum

119875in (16) the hydrodynamic second-order electric

field may be written as

(119864119899)2119867(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1

sum

2

11987511198752

radic119870119890minus119895(1205874)

sdot 119890119895(1205881198702) cos(120579119870minus120579)radiccos120601

0119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119867Γ119865 (12058801 1205960) 119865 (12058802 1205960)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(24)

where119867Γ is the hydrodynamic coupling coefficient [13]

accounting for the coupling of two first-order ocean waveswhose wavenumbers are 119870

1and 119870

2 respectively Adding

the electromagnetic contribution (18) and the hydrodynamiccontribution (24) together and using the time-varying oceanwave surface to replace the time-invariant case the totalsecond-order bistatic electric field for an FMCW source withan antenna on a floating platform may be expressed as

(119864119899)2(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1 1205961

sum

2 1205962

1198751 1205961

1198752 1205962

sdot119878Γ119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)119890minus119895(1205874)

sdot radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(25)

where119878Γ119875= SEΓ119875 +119867Γ

Following the same procedure as for the first-ordercase based on the total second-order time-varying receivedelectric field (25) the corresponding second-order radarcross section 120590

2(120596

119889) may be obtained as

6 International Journal of Antennas and Propagation

1205902(120596

119889) = 2

3120587119896

2

0Δ120588 sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

int

infin

0

1198781(119898

11) 119878

1(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198702 cos120601

01198701Sm2

(119870 cos1206010 119896119861 Δ119903)

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816 sdot 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901) + 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)] 119889119870

1119889120579

1119889119870

(26)

4 Simulation and Discussion

Based on a Pierson-Moskowitz (PM) ocean spectral model[14] the newly derived radar cross sections accounting forantenna sway can be simulated to illustrate the differencesin the FMCW and pulsed waveform cases The sweepbandwidth of the FMCW waveform is chosen as 50 kHzThe operating frequency defined as the central frequency ofthe FMCW waveform is taken to be 25MHz The bistaticangle is 30∘ and the wind speed is 20 knots The scatteringellipse normal and the wind direction are 90∘ and 180∘respectively as measured from the positive 119909-axis (the lineconnecting the transmitter with the receiver) The swayamplitude and frequency depend on the wind velocity andin keeping with an example used earlier [4] for the purposeof illustration here these values are taken as 1228m and0127Hz respectively The sway direction is chosen to be thesame as the wind direction

41 First-Order Radar Cross Section Figure 2 shows a com-parison of the first-order radar cross section for a pulsedsource and that for an FMCW source For the FMCWwaveform Δ119903 = 1500m which equals half the width of thescattering patch (Δ120588 = 3000m) for the pulsed waveformin order to keep the same bandwidth for both waveformsA hamming window is used to smooth the curve andreduce the oscillations From this figure it can be observedthat additional peaks caused by the antenna motion appearsymmetrically in the Doppler spectrum with respect to theBragg peaks A detailed description and properties of thesemotion-induced peaks were discussed in [11] It can alsobe seen that the magnitude of the radar cross sections forthe FMCW waveform is a little lower than that for thecorresponding pulsed waveform which may be caused bythe value of Δ119903 Δ119903 is the limit value of the integral where asampling function dominates this integralΔ119903 is usually takento be Δ119903 = Δ1205882 which means only the contributions inthe main lobe of the sampling function are considered andno interaction between the range bins is assumed in the idealcase

It is clear that the first-order radar cross section has acertain relationship with the integral limit Δ119903 In Section 2it may be observed that there is no mathematical limit forthe parameter Δ119903 By varying Δ119903 the effect on the radar crosssection can be examined Keeping the value of Δ120588 = 3000m

Δ119903 = 05Δ120588 and Δ119903 = 10Δ120588 are simulated in Figures 3(a) and3(b) respectively It should be mentioned that the hammingwindow smoothing process is not used in Figure 3 in orderto clearly show the side lobe levels of the first-order radarcross sections The side lobe structure appears in the radarDoppler spectra due to the side lobes of the Sm functionfor the FMCW waveform By comparing Figures 3(a) and3(b) the magnitude of the side lobes for FMCW source isfound to decrease with increasing Δ119903 and the main lobe levelis a little raised with increasing Δ119903 due to the propertiesof the Sm function This seems to indicate an advantage ofan FMCW system When the value of Δ119903 is taken to belarger thanΔ1205882 the interactions between the range bins (thecontributions in the side lobe of the sampling function) areconsidered and appear in the received electric field at a fixeddistance Increasing Δ119903means the received signal is scatteredfrom a larger ocean surface region When Δ119903 approachesinfinity the radar cross section for the FMCW waveformbecomes a rectangular function whose width is determinedby119861(2119891

0120596119861) However when the patchwidthΔ120588 approaches

infinity the sampling functions in the first-order pulse radarocean cross section reduce to delta functions

By varying the radar bandwidth and keeping the relation-ships Δ120588 = 1198882119861 and Δ119903 = Δ1205882 the effect of the bandwidthon the radar cross sections is illustrated in Figure 4 Fromthis figure it can be seen that with increased bandwidththe magnitudes of the Bragg peaks and the motion-inducedpeaks are found to be reduced while the rest of the radar crosssection increases In addition the width of the Bragg peaksand the motion-induced peaks is also broadened Thereforeif a large radar bandwidth is used for ocean remote sensingthe Bragg peaks may be significantly contaminated by themotion-induced peaks

42 Second-Order Radar Cross Section A similar technique isused to simplify and simulate the second-order radar oceancross section for the FMCW waveform as that for the pulsedwaveform in [12 15] For the case of large Δ119903 it can be shownthat

limΔ119903rarrinfin

[Δ120588Sm2(119870 cos120601

0 119896119861 Δ119903)] asymp Δ120588 cos120601

0

sdot ℎ [119870 minus cos1206010(2119896

0minus 119896

119861)]

minus ℎ [119870 minus cos1206010(2119896

0+ 119896

119861)]

(27)

International Journal of Antennas and Propagation 7

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

FMCWPulsed

Figure 2 Comparison of the first-order radar cross sections for the FMCW waveform with that for the pulsed waveform

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

Doppler frequency f (Hz)

FMCWPulsed

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(a)

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

FMCWPulsed

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(b)Figure 3 Comparison of the side lobe levels of the first-order radar cross sections for the pulsed and FMCWwaveform (a) Δ119903 = 05Δ120588 and(b) Δ119903 = 10Δ120588

Assuming that the other terms in (26) are slowly varyingwithin the interval

cos1206010(2119896

0minus 119896

119861) lt 119870 lt cos120601

0(2119896

0+ 119896

119861) (28)

and carrying out the 119870 integration (26) reduces to

1205902(120596

119889) = 64120587

21198964

0cos4120601

0sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

1198781(119898

11)

sdot 1198781(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198701

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot 120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901)

+120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)]

sdot 1198891198701119889120579

1

(29)

8 International Journal of Antennas and Propagation

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

100 kHz50kHz

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 4 The effect of the bandwidth on the first-order radar crosssections

minus15 minus1 minus05 0 05 1 15minus70

minus60

minus50

minus40

minus30

minus20

minus10

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 5 Second-order bistatic radar cross section with a transmit-ter on a floating platform

Equation (29) is exactly the same as the correspondingmodel for the pulsed waveform when the scattering patch Δ120588approaches infinityTherefore the second-order cross sectionmodel for the FMCW waveform shows the same features inthe Doppler spectra as the model for the pulsed waveformin [12] for a given sea state radar operating parameters andplatform motion An example of the second-order bistaticradar cross section with a transmitter on a floating platformand a fixed receiver is shown in Figure 5 for the case of thescattering patch being assumed to be infinite in extent

5 Conclusion

Thefirst- and second-order bistatic radar ocean cross sectionsfor an antenna on a floating platform have been presentedfor the case of an FMCW waveform In the derivationprocess the first- and second-order models begin with

the bistatically received electric field equations derived in[11 12] Subsequently the derivation is carried out for anFMCW radar which is different from [11 12] where a pulsedradar is considered In particular the distinguishing featurebetween the current work and that presented earlier is thatdemodulation and range transformation must be used toobtain the range information Based on the new modelssimulations are made to compare the radar cross sectionsfor the FMCW waveform with that for the pulsed waveformIt is found that the first-order radar cross section for theFMCWwaveform is a little lower than that for a pulsed sourcewith the same simulation parameters With increased radaroperating bandwidth the magnitude and width of Braggpeaks and motion-induced peaks are found to be reducedand broadened respectively For an FMCW waveform thereis no definite mathematical limit for a patch width whichis different from that for a pulsed waveform Therefore themagnitude of the range bin is varied to examine the effecton the radar cross section The side lobe level is found to bereduced with increasing magnitude of the range bin Whenthe range bin approaches infinity the first-order radar crosssection for an FMCW waveform approaches a rectangularfunction and the second-order radar cross section modelfor the FMCW waveform is reduced to that of the pulsedwaveform

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work was supported in part by Natural Sciences andEngineering Research Council of Canada (NSERC) underDiscovery Grants to Weimin Huang (NSERC 402313-2012)and Eric W Gill (NSERC 238263-2010 and RGPIN-2015-05289) and by an Atlantic Innovation Fund Award (Eric WGill principal investigator)

References

[1] D E Barrick ldquoFirst-order theory and analysis of MFHFVHFscatter from the seardquo IEEE Transactions on Antennas andPropagation vol 20 no 1 pp 2ndash10 1972

[2] J Walsh and E W Gill ldquoAn analysis of the scattering of high-frequency electromagnetic radiation from rough surfaces withapplication to pulse radar operating in backscattermoderdquoRadioScience vol 35 no 6 pp 1337ndash1359 2000

[3] E W Gill and J Walsh ldquoHigh-frequency bistatic cross sectionsof the ocean surfacerdquoRadio Science vol 36 no 6 pp 1459ndash14752001

[4] J WalshW Huang and E Gill ldquoThe first-order high frequencyradar ocean surface cross section for an antenna on a floatingplatformrdquo IEEE Transactions on Antennas and Propagation vol58 no 9 pp 2994ndash3003 2010

[5] J Walsh W Huang and E Gill ldquoThe second-order highfrequency radar ocean surface cross section for an antenna

International Journal of Antennas and Propagation 9

on a floating platformrdquo IEEE Transactions on Antennas andPropagation vol 60 no 10 pp 4804ndash4813 2012

[6] D E Barrick ldquoFMCW radar signals and digital processingrdquoTech Rep ERL 283-WPL 26 NOAA 1973

[7] R Khan B Gamberg D Power et al ldquoTarget detection andtracking with a high frequency ground wave radarrdquo IEEEJournal of Oceanic Engineering vol 19 no 4 pp 540ndash548 1994

[8] A Wojthiewicz J Misiurewicz M Nalecz K Jedrzejewskiand K Kulpa ldquoTwo dimensional signal processing in FMCWradarsrdquo in Proceedings of the Conference on Circuit Theory andElectronics Circuits pp 474ndash480 Kolobrzeg Poland October1997

[9] J Walsh J Zhang and E W Gill ldquoHigh-frequency radar crosssection of the ocean surface for an FMCW waveformrdquo IEEEJournal of Oceanic Engineering vol 36 no 4 pp 615ndash626 2011

[10] S Chen EWGill andWHuang ldquoA first-orderHF radar cross-section model for mixed-path ionosphere-ocean propagationwith an FMCW sourcerdquo IEEE Journal of Oceanic Engineering2016

[11] Y Ma E Gill and W Huang ldquoThe first-order bistatic highfrequency radar ocean surface cross section for an antenna ona floating platformrdquo IET Radar Sonar amp Navigation vol 10 no6 pp 1136ndash1144 2016

[12] Y Ma W Huang and E Gill ldquoThe second-order bistatic highfrequency radar ocean surface cross section for an antenna on afloating platformrdquo Canadian Journal of Remote Sensing vol 42no 4 pp 332ndash343 2016

[13] K Hasselmann ldquoOn the non-linear energy transfer in a gravity-wave spectrum part 1 General theoryrdquo Journal of FluidMechan-ics vol 12 pp 481ndash500 1962

[14] W J Pierson and L Moskowitz ldquoA proposed spectral form forfully developed wind seas based on the similarity theory of S AKitaigorodskiirdquo Journal of Geophysical Research vol 69 no 24pp 5181ndash5190 1964

[15] B J Lipa and D E Barrick ldquoExtraction of sea state fromHF radar sea echo mathematical theory and modelingrdquo RadioScience vol 21 no 1 pp 81ndash100 1986

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 3

22 Applications to an FMCW Radar Following a similaranalysis as in [3 11] (1) may be inversely Fourier-transformedto give the received electric field in the time domain as

Fminus1[(119864

119899)1] (119905) =

1

(2120587)32

sdotFminus1[minus119895

1205780Δ119897

11988821205962119868 (120596)]

119905

lowast Fminus1

sum

119875

sdot radic119870119890119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881) 119865 (120588

2)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 1198901198951198961205751205880[cos120601 cos(120579119870minus1205790)+sin120601 sin(120579119870minus1205790)]

sdot 119890119895120588119904[119870 cos120601minus2119896]

119889120588119904

(5)

The current waveform of an FMCW radar may be writtenas [6 9]

119894 (119905) = 1198680119890119895(1205960119905+120572120587119905

2)ℎ [119905 +

119879119903

2] minus ℎ [119905 minus

119879119903

2] (6)

where 1198680is the peak current and120596

0= 2120587119891

0is the center radian

frequency of the sweep waveform 119879119903represents the sweep

interval and the sweep rate may be expressed as 120572 = 119861119879119903

where 119861 is the sweep bandwidth ℎ is the Heaviside functionIt is known from [9] that for an FMCW waveform

Fminus1[minus119895

1205780Δ119897

11988821205962119868 (120596)]

= minus1198951198680

1205780Δ119897120596

2

0

1198882119890119895(1205960119905+120572120587119905

2)

sdot ℎ [119905 +119879119903

2] minus ℎ [119905 minus

119879119903

2]

(7)

By direct comparison with the corresponding first-order casefor a pulsed dipole [11] the first-order time domain electricfield for an FMCW source may be written as

(119864119899)1(119905119903) =

minus11989511986801205780Δ119897119896

2

0

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601119890119895120588119904119870 cos120601

sdot 119890119895(1205960119905119903+120572120587119905

2

119903)119890minus1198951198960(2120588119904minus1205751205881199040)119890

minus119895(2120587120572(2120588119904minus1205751205881199040)119888)119905119903

sdot 119890119895(120587120572(2120588119904minus1205751205881199040)

21198882)ℎ [119905

119903+119879119903

2minus2120588

119904minus 120575120588

1199040

119888]

minus ℎ [119905119903minus119879119903

2minus2120588

119904minus 120575120588

1199040

119888] 119889120588

119904

(8)

where 1205751205881199040= 120575120588

0[cos120601 cos(120579

119870minus 120579

0) + sin120601 sin(120579

119870minus 120579

0)] 119905

is renamed as 119905119903to indicate that the time is within a sweep

repetition interval ((2120588119904minus 120575120588

1199040)119888 minus 119879

1199032 (2120588

119904minus 120575120588

1199040)119888 +

1198791199032) As stated in [6 9] the frequency difference between

the transmitted waveform and the received waveformmay beFourier-transformed within this interval to obtain the rangeinformationThis is the so-called ldquorange transformrdquo Becausethe received signals in the given time interval reflect theinformation for an extremely large range of ocean surfacehere range transformation is taken to accurately acquire apatch of ocean surface to analyse The frequency differencemay be obtained by the demodulation process in which thetransmitted signals and the received signals are mixed andthen low-pass-filtered

After the demodulation preprocess the exponential fac-tor 119890119895(1205960119905119903+120572120587119905

2

119903) in (8) will be eliminated Then Fourier-

transforming with respect to 119905119903gives

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601119890119895120588119904119870 cos120601

sdot 119890minus1198951198960(2120588119904minus1205751205881199040)119890

119895120596119903(2120588119904minus1205751205881199040)119888119890minus119895(120587120572(2120588119904minus1205751205881199040)

21198882)

sdot Sa[119879119903

2(120596

119903minus2120587120572 (2120588

119904minus 120575120588

1199040)

119888)] 119889120588

119904

(9)

where 120596119903is the transform variable in the frequency domain

Here it is helpful to define

1205881015840

119904= 120588

119904minus120575120588

1199040

2 (10)

Changing the integration variable from 120588119904to 1205881015840

119904and ignoring

the 12057512058811990402 factor in the magnitude terms give

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

infin

1205882

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 119890119895(119870 cos120601minus21198960+2120596119903119888)1205881015840119904 119890minus119895(4120587120572(120588

1015840

119904)21198882)1198901198951205751205881199040119870 cos1206012

sdot Sa [119879119903

2(120596

119903minus4120587120572

1198881205881015840

119904)] 119889120588

1015840

119904

(11)

Since the maximum of the sampling function Sa(119909) occurs at119909 = 0 a representative range 120588

119903may be defined as

120588119903=119888120596

119903

4120587120572 (12)

4 International Journal of Antennas and Propagation

Based on the representative range defining the correspond-ing range variable

12058810158401015840

119904= 120588

1015840

119904minus 120588

119903(13)

and changing the integration variable from 1205881015840

119904to 12058810158401015840

119904 (11)

becomes

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

12058810158401015840

119904max

12058810158401015840

119904min

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 119890119895(minus21198960+119896119903)120588119903119890

119895(minus21198960)12058810158401015840

119904 119890119895120588119904119870 cos120601

sdot 119890minus119895(119896119903120588119903)(120588

10158401015840

119904)2

Sa [11989611986112058810158401015840

119904] 119889120588

10158401015840

119904

(14)

where 119896119861= 2120587119861119888 and 119896

119903= 120596

119903119888 A process similar to that

in [11] is used to simplify the terms in the integral Then (14)reduces to

(119864119899)1(120596

119903) =

minus11989511986801205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890minus119895(1205874)

radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot int

12058810158401015840

119904max

12058810158401015840

119904min

119890119895(119870 cos1206010minus21198960)12058810158401015840119904

sdot 119890minus119895(119896119903120588119903)(120588

10158401015840

119904)2

Sa [11989611986112058810158401015840

119904] 119889120588

10158401015840

119904

(15)

where 12058801and 120588

02are representative values of 120588

1and 120588

2(see

Figure 1) respectively By directly comparing (15) with (24)in [9] the first-order bistatic received electric field for anFMCWwaveformwith an antenna on a floating platformmaybe expressed as

(119864119899)1(120596

119903) =

minus11989511986801205780Δ119897119896

2

0

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890minus119895(1205874)

radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(16)

plusmnΔ119903 are the symmetrical limits of the integral in (15) where asampling function dominates this integral If only the valuesof 12058810158401015840

119904within the main lobe of the sampling function are

considered in the integral that is minus1205872 lt 11989611986112058810158401015840

119904lt 1205872 it

can be deduced as in [9] that Δ119903 = Δ1205882 = 1198884119861 Consider

Sm (119870 cos1206010 119896119861 Δ119903)

=1

120587Si [( 119870

cos1206010

minus 21198960+ 119896

119861)Δ119903]

minus Si [( 119870

cos1206010

minus 21198960minus 119896

119861)Δ119903]

(17)

where Si(119909) = int1199090(sin(119905)119905) 119889119905

Following a similar procedure to the first-order case thesecond-order bistatic received electric field with a transmitteron a floating platform for an FMCW waveform may bewritten as

(119864119899)2119864(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1

sum

2

11987511198752

radic119870119890minus119895(1205874)

sdot 119890119895(1205881198702) cos(120579119870minus120579)radiccos120601

0119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdotSEΓ119875119865 (12058802 1205960) 119865 (120588020 1205960)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(18)

where 12058802

and 120588020

are the representative values of 1205882and

12058820 respectively The symmetrical electromagnetic coupling

coefficient SEΓ119875 may be expressed as [12]

SEΓ119875 (1 2) =119895119896

0

2119870 cos1206010

(1sdot

2)

sdot [119904(

2

1) sdot

2] 119866 [119870

119904(

2

1)]

+ (2sdot

2) [

119904(

2

2) sdot

1] 119866 [119870

119904(

2

2)]

(19)

with 119904(2

2) = 119896

2minus

2and

119866 [119870119904(

2

1)] =

1

119870119904

1 minus 119895119896 (1 + Δ)

radic1198702119904minus 1198962 + 119895119896Δ

(20)

International Journal of Antennas and Propagation 5

3 Radar Cross Sections

31 First-Order Radar Cross Section In developing the oceanradar cross section a time-varying ocean surface representedas 120585( 119905) = sum

119905119875120596119890119895sdot119890119895120596119905 is used to replace the time-

invariant case 120585() = sum119875119890119895sdot This gives the time-varying

received electric field corresponding to (16) as

(119864119899)1(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

120596

119875120596radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890minus119895(1205874)radiccos120601

0119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(21)

A technique similar to that in [9 11] is used to obtain theradar cross section from the received electric field equationThe initial step of the approach is to write the autocorrelation119877(120591) as

119877 (120591) =119860119903

21205780

1

1198792119903

⟨(119864119899)1(120596

119903 119905 + 120591) (119864

119899)lowast

1(120596

119903 119905)⟩ (22)

where 119860119903= (120582

2

04120587)119866

119903 with 119866

119903being the gain of the

receiving array Here 120596119903represents the fixed patch over two

sweep intervals whose time interval is 120591After Fourier-transforming the autocorrelation and com-

paring directly with the radar range equation the radar crosssection 120590

1(120596

119889) may be written as

1205901(120596

119889) = 2

3120587119896

2

0Δ120588 sum

119898=plusmn1

int119870

1198781(119898)119870

2 cos1206010

sdot Sm2(119870 cos120601

0 119896119861 Δ119903)

sdot 1198692

0[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot 120575 (120596

119889+ 119898radic119892119870)

+

infin

sum

119899=1

1198692

119899[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot [120575 (120596

119889+ 119898radic119892119870

minus 119899120596119901) + 120575 (120596

119889+ 119898radic119892119870 + 119899120596

119901)]119889119870

(23)

where 119869119899represents the 119899th-order Bessel functions For

simulation purposes (see Section 4) and in keeping with [411] it will be assumed that the antenna motion is causedby the dominant ocean waves 119886 120596

119901 and 120579

119870119901represent the

antenna platform sway amplitude frequency and directionrespectively

32 Second-Order Radar Cross Section It is known that thesecond-order radar cross section contains two portions anhydrodynamic contribution and an electromagnetic contri-bution Using the Fourier coefficient for the second-orderocean waves sum

1sum2 119867

Γ11987511198752

to replace the first-ordercase sum

119875in (16) the hydrodynamic second-order electric

field may be written as

(119864119899)2119867(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1

sum

2

11987511198752

radic119870119890minus119895(1205874)

sdot 119890119895(1205881198702) cos(120579119870minus120579)radiccos120601

0119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119867Γ119865 (12058801 1205960) 119865 (12058802 1205960)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(24)

where119867Γ is the hydrodynamic coupling coefficient [13]

accounting for the coupling of two first-order ocean waveswhose wavenumbers are 119870

1and 119870

2 respectively Adding

the electromagnetic contribution (18) and the hydrodynamiccontribution (24) together and using the time-varying oceanwave surface to replace the time-invariant case the totalsecond-order bistatic electric field for an FMCW source withan antenna on a floating platform may be expressed as

(119864119899)2(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1 1205961

sum

2 1205962

1198751 1205961

1198752 1205962

sdot119878Γ119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)119890minus119895(1205874)

sdot radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(25)

where119878Γ119875= SEΓ119875 +119867Γ

Following the same procedure as for the first-ordercase based on the total second-order time-varying receivedelectric field (25) the corresponding second-order radarcross section 120590

2(120596

119889) may be obtained as

6 International Journal of Antennas and Propagation

1205902(120596

119889) = 2

3120587119896

2

0Δ120588 sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

int

infin

0

1198781(119898

11) 119878

1(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198702 cos120601

01198701Sm2

(119870 cos1206010 119896119861 Δ119903)

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816 sdot 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901) + 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)] 119889119870

1119889120579

1119889119870

(26)

4 Simulation and Discussion

Based on a Pierson-Moskowitz (PM) ocean spectral model[14] the newly derived radar cross sections accounting forantenna sway can be simulated to illustrate the differencesin the FMCW and pulsed waveform cases The sweepbandwidth of the FMCW waveform is chosen as 50 kHzThe operating frequency defined as the central frequency ofthe FMCW waveform is taken to be 25MHz The bistaticangle is 30∘ and the wind speed is 20 knots The scatteringellipse normal and the wind direction are 90∘ and 180∘respectively as measured from the positive 119909-axis (the lineconnecting the transmitter with the receiver) The swayamplitude and frequency depend on the wind velocity andin keeping with an example used earlier [4] for the purposeof illustration here these values are taken as 1228m and0127Hz respectively The sway direction is chosen to be thesame as the wind direction

41 First-Order Radar Cross Section Figure 2 shows a com-parison of the first-order radar cross section for a pulsedsource and that for an FMCW source For the FMCWwaveform Δ119903 = 1500m which equals half the width of thescattering patch (Δ120588 = 3000m) for the pulsed waveformin order to keep the same bandwidth for both waveformsA hamming window is used to smooth the curve andreduce the oscillations From this figure it can be observedthat additional peaks caused by the antenna motion appearsymmetrically in the Doppler spectrum with respect to theBragg peaks A detailed description and properties of thesemotion-induced peaks were discussed in [11] It can alsobe seen that the magnitude of the radar cross sections forthe FMCW waveform is a little lower than that for thecorresponding pulsed waveform which may be caused bythe value of Δ119903 Δ119903 is the limit value of the integral where asampling function dominates this integralΔ119903 is usually takento be Δ119903 = Δ1205882 which means only the contributions inthe main lobe of the sampling function are considered andno interaction between the range bins is assumed in the idealcase

It is clear that the first-order radar cross section has acertain relationship with the integral limit Δ119903 In Section 2it may be observed that there is no mathematical limit forthe parameter Δ119903 By varying Δ119903 the effect on the radar crosssection can be examined Keeping the value of Δ120588 = 3000m

Δ119903 = 05Δ120588 and Δ119903 = 10Δ120588 are simulated in Figures 3(a) and3(b) respectively It should be mentioned that the hammingwindow smoothing process is not used in Figure 3 in orderto clearly show the side lobe levels of the first-order radarcross sections The side lobe structure appears in the radarDoppler spectra due to the side lobes of the Sm functionfor the FMCW waveform By comparing Figures 3(a) and3(b) the magnitude of the side lobes for FMCW source isfound to decrease with increasing Δ119903 and the main lobe levelis a little raised with increasing Δ119903 due to the propertiesof the Sm function This seems to indicate an advantage ofan FMCW system When the value of Δ119903 is taken to belarger thanΔ1205882 the interactions between the range bins (thecontributions in the side lobe of the sampling function) areconsidered and appear in the received electric field at a fixeddistance Increasing Δ119903means the received signal is scatteredfrom a larger ocean surface region When Δ119903 approachesinfinity the radar cross section for the FMCW waveformbecomes a rectangular function whose width is determinedby119861(2119891

0120596119861) However when the patchwidthΔ120588 approaches

infinity the sampling functions in the first-order pulse radarocean cross section reduce to delta functions

By varying the radar bandwidth and keeping the relation-ships Δ120588 = 1198882119861 and Δ119903 = Δ1205882 the effect of the bandwidthon the radar cross sections is illustrated in Figure 4 Fromthis figure it can be seen that with increased bandwidththe magnitudes of the Bragg peaks and the motion-inducedpeaks are found to be reduced while the rest of the radar crosssection increases In addition the width of the Bragg peaksand the motion-induced peaks is also broadened Thereforeif a large radar bandwidth is used for ocean remote sensingthe Bragg peaks may be significantly contaminated by themotion-induced peaks

42 Second-Order Radar Cross Section A similar technique isused to simplify and simulate the second-order radar oceancross section for the FMCW waveform as that for the pulsedwaveform in [12 15] For the case of large Δ119903 it can be shownthat

limΔ119903rarrinfin

[Δ120588Sm2(119870 cos120601

0 119896119861 Δ119903)] asymp Δ120588 cos120601

0

sdot ℎ [119870 minus cos1206010(2119896

0minus 119896

119861)]

minus ℎ [119870 minus cos1206010(2119896

0+ 119896

119861)]

(27)

International Journal of Antennas and Propagation 7

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

FMCWPulsed

Figure 2 Comparison of the first-order radar cross sections for the FMCW waveform with that for the pulsed waveform

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

Doppler frequency f (Hz)

FMCWPulsed

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(a)

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

FMCWPulsed

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(b)Figure 3 Comparison of the side lobe levels of the first-order radar cross sections for the pulsed and FMCWwaveform (a) Δ119903 = 05Δ120588 and(b) Δ119903 = 10Δ120588

Assuming that the other terms in (26) are slowly varyingwithin the interval

cos1206010(2119896

0minus 119896

119861) lt 119870 lt cos120601

0(2119896

0+ 119896

119861) (28)

and carrying out the 119870 integration (26) reduces to

1205902(120596

119889) = 64120587

21198964

0cos4120601

0sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

1198781(119898

11)

sdot 1198781(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198701

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot 120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901)

+120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)]

sdot 1198891198701119889120579

1

(29)

8 International Journal of Antennas and Propagation

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

100 kHz50kHz

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 4 The effect of the bandwidth on the first-order radar crosssections

minus15 minus1 minus05 0 05 1 15minus70

minus60

minus50

minus40

minus30

minus20

minus10

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 5 Second-order bistatic radar cross section with a transmit-ter on a floating platform

Equation (29) is exactly the same as the correspondingmodel for the pulsed waveform when the scattering patch Δ120588approaches infinityTherefore the second-order cross sectionmodel for the FMCW waveform shows the same features inthe Doppler spectra as the model for the pulsed waveformin [12] for a given sea state radar operating parameters andplatform motion An example of the second-order bistaticradar cross section with a transmitter on a floating platformand a fixed receiver is shown in Figure 5 for the case of thescattering patch being assumed to be infinite in extent

5 Conclusion

Thefirst- and second-order bistatic radar ocean cross sectionsfor an antenna on a floating platform have been presentedfor the case of an FMCW waveform In the derivationprocess the first- and second-order models begin with

the bistatically received electric field equations derived in[11 12] Subsequently the derivation is carried out for anFMCW radar which is different from [11 12] where a pulsedradar is considered In particular the distinguishing featurebetween the current work and that presented earlier is thatdemodulation and range transformation must be used toobtain the range information Based on the new modelssimulations are made to compare the radar cross sectionsfor the FMCW waveform with that for the pulsed waveformIt is found that the first-order radar cross section for theFMCWwaveform is a little lower than that for a pulsed sourcewith the same simulation parameters With increased radaroperating bandwidth the magnitude and width of Braggpeaks and motion-induced peaks are found to be reducedand broadened respectively For an FMCW waveform thereis no definite mathematical limit for a patch width whichis different from that for a pulsed waveform Therefore themagnitude of the range bin is varied to examine the effecton the radar cross section The side lobe level is found to bereduced with increasing magnitude of the range bin Whenthe range bin approaches infinity the first-order radar crosssection for an FMCW waveform approaches a rectangularfunction and the second-order radar cross section modelfor the FMCW waveform is reduced to that of the pulsedwaveform

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work was supported in part by Natural Sciences andEngineering Research Council of Canada (NSERC) underDiscovery Grants to Weimin Huang (NSERC 402313-2012)and Eric W Gill (NSERC 238263-2010 and RGPIN-2015-05289) and by an Atlantic Innovation Fund Award (Eric WGill principal investigator)

References

[1] D E Barrick ldquoFirst-order theory and analysis of MFHFVHFscatter from the seardquo IEEE Transactions on Antennas andPropagation vol 20 no 1 pp 2ndash10 1972

[2] J Walsh and E W Gill ldquoAn analysis of the scattering of high-frequency electromagnetic radiation from rough surfaces withapplication to pulse radar operating in backscattermoderdquoRadioScience vol 35 no 6 pp 1337ndash1359 2000

[3] E W Gill and J Walsh ldquoHigh-frequency bistatic cross sectionsof the ocean surfacerdquoRadio Science vol 36 no 6 pp 1459ndash14752001

[4] J WalshW Huang and E Gill ldquoThe first-order high frequencyradar ocean surface cross section for an antenna on a floatingplatformrdquo IEEE Transactions on Antennas and Propagation vol58 no 9 pp 2994ndash3003 2010

[5] J Walsh W Huang and E Gill ldquoThe second-order highfrequency radar ocean surface cross section for an antenna

International Journal of Antennas and Propagation 9

on a floating platformrdquo IEEE Transactions on Antennas andPropagation vol 60 no 10 pp 4804ndash4813 2012

[6] D E Barrick ldquoFMCW radar signals and digital processingrdquoTech Rep ERL 283-WPL 26 NOAA 1973

[7] R Khan B Gamberg D Power et al ldquoTarget detection andtracking with a high frequency ground wave radarrdquo IEEEJournal of Oceanic Engineering vol 19 no 4 pp 540ndash548 1994

[8] A Wojthiewicz J Misiurewicz M Nalecz K Jedrzejewskiand K Kulpa ldquoTwo dimensional signal processing in FMCWradarsrdquo in Proceedings of the Conference on Circuit Theory andElectronics Circuits pp 474ndash480 Kolobrzeg Poland October1997

[9] J Walsh J Zhang and E W Gill ldquoHigh-frequency radar crosssection of the ocean surface for an FMCW waveformrdquo IEEEJournal of Oceanic Engineering vol 36 no 4 pp 615ndash626 2011

[10] S Chen EWGill andWHuang ldquoA first-orderHF radar cross-section model for mixed-path ionosphere-ocean propagationwith an FMCW sourcerdquo IEEE Journal of Oceanic Engineering2016

[11] Y Ma E Gill and W Huang ldquoThe first-order bistatic highfrequency radar ocean surface cross section for an antenna ona floating platformrdquo IET Radar Sonar amp Navigation vol 10 no6 pp 1136ndash1144 2016

[12] Y Ma W Huang and E Gill ldquoThe second-order bistatic highfrequency radar ocean surface cross section for an antenna on afloating platformrdquo Canadian Journal of Remote Sensing vol 42no 4 pp 332ndash343 2016

[13] K Hasselmann ldquoOn the non-linear energy transfer in a gravity-wave spectrum part 1 General theoryrdquo Journal of FluidMechan-ics vol 12 pp 481ndash500 1962

[14] W J Pierson and L Moskowitz ldquoA proposed spectral form forfully developed wind seas based on the similarity theory of S AKitaigorodskiirdquo Journal of Geophysical Research vol 69 no 24pp 5181ndash5190 1964

[15] B J Lipa and D E Barrick ldquoExtraction of sea state fromHF radar sea echo mathematical theory and modelingrdquo RadioScience vol 21 no 1 pp 81ndash100 1986

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

4 International Journal of Antennas and Propagation

Based on the representative range defining the correspond-ing range variable

12058810158401015840

119904= 120588

1015840

119904minus 120588

119903(13)

and changing the integration variable from 1205881015840

119904to 12058810158401015840

119904 (11)

becomes

(119864119899)1(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot int

12058810158401015840

119904max

12058810158401015840

119904min

119865 (1205881 120596

0) 119865 (120588

2 120596

0)

radic120588119904[1205882119904minus (1205882)

2]

119890minus119895(1205874)

radiccos120601

sdot 119890119895(minus21198960+119896119903)120588119903119890

119895(minus21198960)12058810158401015840

119904 119890119895120588119904119870 cos120601

sdot 119890minus119895(119896119903120588119903)(120588

10158401015840

119904)2

Sa [11989611986112058810158401015840

119904] 119889120588

10158401015840

119904

(14)

where 119896119861= 2120587119861119888 and 119896

119903= 120596

119903119888 A process similar to that

in [11] is used to simplify the terms in the integral Then (14)reduces to

(119864119899)1(120596

119903) =

minus11989511986801205780Δ119897119896

2

0119879119903

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890minus119895(1205874)

radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot int

12058810158401015840

119904max

12058810158401015840

119904min

119890119895(119870 cos1206010minus21198960)12058810158401015840119904

sdot 119890minus119895(119896119903120588119903)(120588

10158401015840

119904)2

Sa [11989611986112058810158401015840

119904] 119889120588

10158401015840

119904

(15)

where 12058801and 120588

02are representative values of 120588

1and 120588

2(see

Figure 1) respectively By directly comparing (15) with (24)in [9] the first-order bistatic received electric field for anFMCWwaveformwith an antenna on a floating platformmaybe expressed as

(119864119899)1(120596

119903) =

minus11989511986801205780Δ119897119896

2

0

(2120587)32

sum

119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890minus119895(1205874)

radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(16)

plusmnΔ119903 are the symmetrical limits of the integral in (15) where asampling function dominates this integral If only the valuesof 12058810158401015840

119904within the main lobe of the sampling function are

considered in the integral that is minus1205872 lt 11989611986112058810158401015840

119904lt 1205872 it

can be deduced as in [9] that Δ119903 = Δ1205882 = 1198884119861 Consider

Sm (119870 cos1206010 119896119861 Δ119903)

=1

120587Si [( 119870

cos1206010

minus 21198960+ 119896

119861)Δ119903]

minus Si [( 119870

cos1206010

minus 21198960minus 119896

119861)Δ119903]

(17)

where Si(119909) = int1199090(sin(119905)119905) 119889119905

Following a similar procedure to the first-order case thesecond-order bistatic received electric field with a transmitteron a floating platform for an FMCW waveform may bewritten as

(119864119899)2119864(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1

sum

2

11987511198752

radic119870119890minus119895(1205874)

sdot 119890119895(1205881198702) cos(120579119870minus120579)radiccos120601

0119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdotSEΓ119875119865 (12058802 1205960) 119865 (120588020 1205960)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(18)

where 12058802

and 120588020

are the representative values of 1205882and

12058820 respectively The symmetrical electromagnetic coupling

coefficient SEΓ119875 may be expressed as [12]

SEΓ119875 (1 2) =119895119896

0

2119870 cos1206010

(1sdot

2)

sdot [119904(

2

1) sdot

2] 119866 [119870

119904(

2

1)]

+ (2sdot

2) [

119904(

2

2) sdot

1] 119866 [119870

119904(

2

2)]

(19)

with 119904(2

2) = 119896

2minus

2and

119866 [119870119904(

2

1)] =

1

119870119904

1 minus 119895119896 (1 + Δ)

radic1198702119904minus 1198962 + 119895119896Δ

(20)

International Journal of Antennas and Propagation 5

3 Radar Cross Sections

31 First-Order Radar Cross Section In developing the oceanradar cross section a time-varying ocean surface representedas 120585( 119905) = sum

119905119875120596119890119895sdot119890119895120596119905 is used to replace the time-

invariant case 120585() = sum119875119890119895sdot This gives the time-varying

received electric field corresponding to (16) as

(119864119899)1(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

120596

119875120596radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890minus119895(1205874)radiccos120601

0119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(21)

A technique similar to that in [9 11] is used to obtain theradar cross section from the received electric field equationThe initial step of the approach is to write the autocorrelation119877(120591) as

119877 (120591) =119860119903

21205780

1

1198792119903

⟨(119864119899)1(120596

119903 119905 + 120591) (119864

119899)lowast

1(120596

119903 119905)⟩ (22)

where 119860119903= (120582

2

04120587)119866

119903 with 119866

119903being the gain of the

receiving array Here 120596119903represents the fixed patch over two

sweep intervals whose time interval is 120591After Fourier-transforming the autocorrelation and com-

paring directly with the radar range equation the radar crosssection 120590

1(120596

119889) may be written as

1205901(120596

119889) = 2

3120587119896

2

0Δ120588 sum

119898=plusmn1

int119870

1198781(119898)119870

2 cos1206010

sdot Sm2(119870 cos120601

0 119896119861 Δ119903)

sdot 1198692

0[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot 120575 (120596

119889+ 119898radic119892119870)

+

infin

sum

119899=1

1198692

119899[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot [120575 (120596

119889+ 119898radic119892119870

minus 119899120596119901) + 120575 (120596

119889+ 119898radic119892119870 + 119899120596

119901)]119889119870

(23)

where 119869119899represents the 119899th-order Bessel functions For

simulation purposes (see Section 4) and in keeping with [411] it will be assumed that the antenna motion is causedby the dominant ocean waves 119886 120596

119901 and 120579

119870119901represent the

antenna platform sway amplitude frequency and directionrespectively

32 Second-Order Radar Cross Section It is known that thesecond-order radar cross section contains two portions anhydrodynamic contribution and an electromagnetic contri-bution Using the Fourier coefficient for the second-orderocean waves sum

1sum2 119867

Γ11987511198752

to replace the first-ordercase sum

119875in (16) the hydrodynamic second-order electric

field may be written as

(119864119899)2119867(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1

sum

2

11987511198752

radic119870119890minus119895(1205874)

sdot 119890119895(1205881198702) cos(120579119870minus120579)radiccos120601

0119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119867Γ119865 (12058801 1205960) 119865 (12058802 1205960)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(24)

where119867Γ is the hydrodynamic coupling coefficient [13]

accounting for the coupling of two first-order ocean waveswhose wavenumbers are 119870

1and 119870

2 respectively Adding

the electromagnetic contribution (18) and the hydrodynamiccontribution (24) together and using the time-varying oceanwave surface to replace the time-invariant case the totalsecond-order bistatic electric field for an FMCW source withan antenna on a floating platform may be expressed as

(119864119899)2(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1 1205961

sum

2 1205962

1198751 1205961

1198752 1205962

sdot119878Γ119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)119890minus119895(1205874)

sdot radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(25)

where119878Γ119875= SEΓ119875 +119867Γ

Following the same procedure as for the first-ordercase based on the total second-order time-varying receivedelectric field (25) the corresponding second-order radarcross section 120590

2(120596

119889) may be obtained as

6 International Journal of Antennas and Propagation

1205902(120596

119889) = 2

3120587119896

2

0Δ120588 sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

int

infin

0

1198781(119898

11) 119878

1(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198702 cos120601

01198701Sm2

(119870 cos1206010 119896119861 Δ119903)

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816 sdot 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901) + 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)] 119889119870

1119889120579

1119889119870

(26)

4 Simulation and Discussion

Based on a Pierson-Moskowitz (PM) ocean spectral model[14] the newly derived radar cross sections accounting forantenna sway can be simulated to illustrate the differencesin the FMCW and pulsed waveform cases The sweepbandwidth of the FMCW waveform is chosen as 50 kHzThe operating frequency defined as the central frequency ofthe FMCW waveform is taken to be 25MHz The bistaticangle is 30∘ and the wind speed is 20 knots The scatteringellipse normal and the wind direction are 90∘ and 180∘respectively as measured from the positive 119909-axis (the lineconnecting the transmitter with the receiver) The swayamplitude and frequency depend on the wind velocity andin keeping with an example used earlier [4] for the purposeof illustration here these values are taken as 1228m and0127Hz respectively The sway direction is chosen to be thesame as the wind direction

41 First-Order Radar Cross Section Figure 2 shows a com-parison of the first-order radar cross section for a pulsedsource and that for an FMCW source For the FMCWwaveform Δ119903 = 1500m which equals half the width of thescattering patch (Δ120588 = 3000m) for the pulsed waveformin order to keep the same bandwidth for both waveformsA hamming window is used to smooth the curve andreduce the oscillations From this figure it can be observedthat additional peaks caused by the antenna motion appearsymmetrically in the Doppler spectrum with respect to theBragg peaks A detailed description and properties of thesemotion-induced peaks were discussed in [11] It can alsobe seen that the magnitude of the radar cross sections forthe FMCW waveform is a little lower than that for thecorresponding pulsed waveform which may be caused bythe value of Δ119903 Δ119903 is the limit value of the integral where asampling function dominates this integralΔ119903 is usually takento be Δ119903 = Δ1205882 which means only the contributions inthe main lobe of the sampling function are considered andno interaction between the range bins is assumed in the idealcase

It is clear that the first-order radar cross section has acertain relationship with the integral limit Δ119903 In Section 2it may be observed that there is no mathematical limit forthe parameter Δ119903 By varying Δ119903 the effect on the radar crosssection can be examined Keeping the value of Δ120588 = 3000m

Δ119903 = 05Δ120588 and Δ119903 = 10Δ120588 are simulated in Figures 3(a) and3(b) respectively It should be mentioned that the hammingwindow smoothing process is not used in Figure 3 in orderto clearly show the side lobe levels of the first-order radarcross sections The side lobe structure appears in the radarDoppler spectra due to the side lobes of the Sm functionfor the FMCW waveform By comparing Figures 3(a) and3(b) the magnitude of the side lobes for FMCW source isfound to decrease with increasing Δ119903 and the main lobe levelis a little raised with increasing Δ119903 due to the propertiesof the Sm function This seems to indicate an advantage ofan FMCW system When the value of Δ119903 is taken to belarger thanΔ1205882 the interactions between the range bins (thecontributions in the side lobe of the sampling function) areconsidered and appear in the received electric field at a fixeddistance Increasing Δ119903means the received signal is scatteredfrom a larger ocean surface region When Δ119903 approachesinfinity the radar cross section for the FMCW waveformbecomes a rectangular function whose width is determinedby119861(2119891

0120596119861) However when the patchwidthΔ120588 approaches

infinity the sampling functions in the first-order pulse radarocean cross section reduce to delta functions

By varying the radar bandwidth and keeping the relation-ships Δ120588 = 1198882119861 and Δ119903 = Δ1205882 the effect of the bandwidthon the radar cross sections is illustrated in Figure 4 Fromthis figure it can be seen that with increased bandwidththe magnitudes of the Bragg peaks and the motion-inducedpeaks are found to be reduced while the rest of the radar crosssection increases In addition the width of the Bragg peaksand the motion-induced peaks is also broadened Thereforeif a large radar bandwidth is used for ocean remote sensingthe Bragg peaks may be significantly contaminated by themotion-induced peaks

42 Second-Order Radar Cross Section A similar technique isused to simplify and simulate the second-order radar oceancross section for the FMCW waveform as that for the pulsedwaveform in [12 15] For the case of large Δ119903 it can be shownthat

limΔ119903rarrinfin

[Δ120588Sm2(119870 cos120601

0 119896119861 Δ119903)] asymp Δ120588 cos120601

0

sdot ℎ [119870 minus cos1206010(2119896

0minus 119896

119861)]

minus ℎ [119870 minus cos1206010(2119896

0+ 119896

119861)]

(27)

International Journal of Antennas and Propagation 7

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

FMCWPulsed

Figure 2 Comparison of the first-order radar cross sections for the FMCW waveform with that for the pulsed waveform

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

Doppler frequency f (Hz)

FMCWPulsed

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(a)

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

FMCWPulsed

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(b)Figure 3 Comparison of the side lobe levels of the first-order radar cross sections for the pulsed and FMCWwaveform (a) Δ119903 = 05Δ120588 and(b) Δ119903 = 10Δ120588

Assuming that the other terms in (26) are slowly varyingwithin the interval

cos1206010(2119896

0minus 119896

119861) lt 119870 lt cos120601

0(2119896

0+ 119896

119861) (28)

and carrying out the 119870 integration (26) reduces to

1205902(120596

119889) = 64120587

21198964

0cos4120601

0sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

1198781(119898

11)

sdot 1198781(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198701

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot 120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901)

+120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)]

sdot 1198891198701119889120579

1

(29)

8 International Journal of Antennas and Propagation

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

100 kHz50kHz

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 4 The effect of the bandwidth on the first-order radar crosssections

minus15 minus1 minus05 0 05 1 15minus70

minus60

minus50

minus40

minus30

minus20

minus10

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 5 Second-order bistatic radar cross section with a transmit-ter on a floating platform

Equation (29) is exactly the same as the correspondingmodel for the pulsed waveform when the scattering patch Δ120588approaches infinityTherefore the second-order cross sectionmodel for the FMCW waveform shows the same features inthe Doppler spectra as the model for the pulsed waveformin [12] for a given sea state radar operating parameters andplatform motion An example of the second-order bistaticradar cross section with a transmitter on a floating platformand a fixed receiver is shown in Figure 5 for the case of thescattering patch being assumed to be infinite in extent

5 Conclusion

Thefirst- and second-order bistatic radar ocean cross sectionsfor an antenna on a floating platform have been presentedfor the case of an FMCW waveform In the derivationprocess the first- and second-order models begin with

the bistatically received electric field equations derived in[11 12] Subsequently the derivation is carried out for anFMCW radar which is different from [11 12] where a pulsedradar is considered In particular the distinguishing featurebetween the current work and that presented earlier is thatdemodulation and range transformation must be used toobtain the range information Based on the new modelssimulations are made to compare the radar cross sectionsfor the FMCW waveform with that for the pulsed waveformIt is found that the first-order radar cross section for theFMCWwaveform is a little lower than that for a pulsed sourcewith the same simulation parameters With increased radaroperating bandwidth the magnitude and width of Braggpeaks and motion-induced peaks are found to be reducedand broadened respectively For an FMCW waveform thereis no definite mathematical limit for a patch width whichis different from that for a pulsed waveform Therefore themagnitude of the range bin is varied to examine the effecton the radar cross section The side lobe level is found to bereduced with increasing magnitude of the range bin Whenthe range bin approaches infinity the first-order radar crosssection for an FMCW waveform approaches a rectangularfunction and the second-order radar cross section modelfor the FMCW waveform is reduced to that of the pulsedwaveform

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work was supported in part by Natural Sciences andEngineering Research Council of Canada (NSERC) underDiscovery Grants to Weimin Huang (NSERC 402313-2012)and Eric W Gill (NSERC 238263-2010 and RGPIN-2015-05289) and by an Atlantic Innovation Fund Award (Eric WGill principal investigator)

References

[1] D E Barrick ldquoFirst-order theory and analysis of MFHFVHFscatter from the seardquo IEEE Transactions on Antennas andPropagation vol 20 no 1 pp 2ndash10 1972

[2] J Walsh and E W Gill ldquoAn analysis of the scattering of high-frequency electromagnetic radiation from rough surfaces withapplication to pulse radar operating in backscattermoderdquoRadioScience vol 35 no 6 pp 1337ndash1359 2000

[3] E W Gill and J Walsh ldquoHigh-frequency bistatic cross sectionsof the ocean surfacerdquoRadio Science vol 36 no 6 pp 1459ndash14752001

[4] J WalshW Huang and E Gill ldquoThe first-order high frequencyradar ocean surface cross section for an antenna on a floatingplatformrdquo IEEE Transactions on Antennas and Propagation vol58 no 9 pp 2994ndash3003 2010

[5] J Walsh W Huang and E Gill ldquoThe second-order highfrequency radar ocean surface cross section for an antenna

International Journal of Antennas and Propagation 9

on a floating platformrdquo IEEE Transactions on Antennas andPropagation vol 60 no 10 pp 4804ndash4813 2012

[6] D E Barrick ldquoFMCW radar signals and digital processingrdquoTech Rep ERL 283-WPL 26 NOAA 1973

[7] R Khan B Gamberg D Power et al ldquoTarget detection andtracking with a high frequency ground wave radarrdquo IEEEJournal of Oceanic Engineering vol 19 no 4 pp 540ndash548 1994

[8] A Wojthiewicz J Misiurewicz M Nalecz K Jedrzejewskiand K Kulpa ldquoTwo dimensional signal processing in FMCWradarsrdquo in Proceedings of the Conference on Circuit Theory andElectronics Circuits pp 474ndash480 Kolobrzeg Poland October1997

[9] J Walsh J Zhang and E W Gill ldquoHigh-frequency radar crosssection of the ocean surface for an FMCW waveformrdquo IEEEJournal of Oceanic Engineering vol 36 no 4 pp 615ndash626 2011

[10] S Chen EWGill andWHuang ldquoA first-orderHF radar cross-section model for mixed-path ionosphere-ocean propagationwith an FMCW sourcerdquo IEEE Journal of Oceanic Engineering2016

[11] Y Ma E Gill and W Huang ldquoThe first-order bistatic highfrequency radar ocean surface cross section for an antenna ona floating platformrdquo IET Radar Sonar amp Navigation vol 10 no6 pp 1136ndash1144 2016

[12] Y Ma W Huang and E Gill ldquoThe second-order bistatic highfrequency radar ocean surface cross section for an antenna on afloating platformrdquo Canadian Journal of Remote Sensing vol 42no 4 pp 332ndash343 2016

[13] K Hasselmann ldquoOn the non-linear energy transfer in a gravity-wave spectrum part 1 General theoryrdquo Journal of FluidMechan-ics vol 12 pp 481ndash500 1962

[14] W J Pierson and L Moskowitz ldquoA proposed spectral form forfully developed wind seas based on the similarity theory of S AKitaigorodskiirdquo Journal of Geophysical Research vol 69 no 24pp 5181ndash5190 1964

[15] B J Lipa and D E Barrick ldquoExtraction of sea state fromHF radar sea echo mathematical theory and modelingrdquo RadioScience vol 21 no 1 pp 81ndash100 1986

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International Journal of

International Journal of Antennas and Propagation 5

3 Radar Cross Sections

31 First-Order Radar Cross Section In developing the oceanradar cross section a time-varying ocean surface representedas 120585( 119905) = sum

119905119875120596119890119895sdot119890119895120596119905 is used to replace the time-

invariant case 120585() = sum119875119890119895sdot This gives the time-varying

received electric field corresponding to (16) as

(119864119899)1(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

120596

119875120596radic119870119890

119895(1205881198702) cos(120579119870minus120579)

sdot 119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890minus119895(1205874)radiccos120601

0119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(21)

A technique similar to that in [9 11] is used to obtain theradar cross section from the received electric field equationThe initial step of the approach is to write the autocorrelation119877(120591) as

119877 (120591) =119860119903

21205780

1

1198792119903

⟨(119864119899)1(120596

119903 119905 + 120591) (119864

119899)lowast

1(120596

119903 119905)⟩ (22)

where 119860119903= (120582

2

04120587)119866

119903 with 119866

119903being the gain of the

receiving array Here 120596119903represents the fixed patch over two

sweep intervals whose time interval is 120591After Fourier-transforming the autocorrelation and com-

paring directly with the radar range equation the radar crosssection 120590

1(120596

119889) may be written as

1205901(120596

119889) = 2

3120587119896

2

0Δ120588 sum

119898=plusmn1

int119870

1198781(119898)119870

2 cos1206010

sdot Sm2(119870 cos120601

0 119896119861 Δ119903)

sdot 1198692

0[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot 120575 (120596

119889+ 119898radic119892119870)

+

infin

sum

119899=1

1198692

119899[119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901)

+ tan1206010sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816] sdot [120575 (120596

119889+ 119898radic119892119870

minus 119899120596119901) + 120575 (120596

119889+ 119898radic119892119870 + 119899120596

119901)]119889119870

(23)

where 119869119899represents the 119899th-order Bessel functions For

simulation purposes (see Section 4) and in keeping with [411] it will be assumed that the antenna motion is causedby the dominant ocean waves 119886 120596

119901 and 120579

119870119901represent the

antenna platform sway amplitude frequency and directionrespectively

32 Second-Order Radar Cross Section It is known that thesecond-order radar cross section contains two portions anhydrodynamic contribution and an electromagnetic contri-bution Using the Fourier coefficient for the second-orderocean waves sum

1sum2 119867

Γ11987511198752

to replace the first-ordercase sum

119875in (16) the hydrodynamic second-order electric

field may be written as

(119864119899)2119867(120596

119903)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1

sum

2

11987511198752

radic119870119890minus119895(1205874)

sdot 119890119895(1205881198702) cos(120579119870minus120579)radiccos120601

0119890119895(119870 cos1206010minus21198960+119896119903)120588119903

sdot119867Γ119865 (12058801 1205960) 119865 (12058802 1205960)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(24)

where119867Γ is the hydrodynamic coupling coefficient [13]

accounting for the coupling of two first-order ocean waveswhose wavenumbers are 119870

1and 119870

2 respectively Adding

the electromagnetic contribution (18) and the hydrodynamiccontribution (24) together and using the time-varying oceanwave surface to replace the time-invariant case the totalsecond-order bistatic electric field for an FMCW source withan antenna on a floating platform may be expressed as

(119864119899)2(120596

119903 119905)

=minus119895119868

01205780Δ119897119896

2

0

(2120587)32

sum

1 1205961

sum

2 1205962

1198751 1205961

1198752 1205962

sdot119878Γ119875radic119870119890

119895(1205881198702) cos(120579119870minus120579)119890minus119895(1205874)

sdot radiccos1206010119890119895(119870 cos1206010minus21198960+119896119903)120588119903119890119895120596119905

sdot119865 (120588

01 120596

0) 119865 (120588

02 120596

0)

radic120588119903[1205882119903minus (1205882)

2]

119890119895(12057512058811990402)(119870 cos1206010)

sdot (119879119903Δ120588) Sm (119870 cos120601

0 119896119861 Δ119903)

(25)

where119878Γ119875= SEΓ119875 +119867Γ

Following the same procedure as for the first-ordercase based on the total second-order time-varying receivedelectric field (25) the corresponding second-order radarcross section 120590

2(120596

119889) may be obtained as

6 International Journal of Antennas and Propagation

1205902(120596

119889) = 2

3120587119896

2

0Δ120588 sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

int

infin

0

1198781(119898

11) 119878

1(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198702 cos120601

01198701Sm2

(119870 cos1206010 119896119861 Δ119903)

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816 sdot 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901) + 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)] 119889119870

1119889120579

1119889119870

(26)

4 Simulation and Discussion

Based on a Pierson-Moskowitz (PM) ocean spectral model[14] the newly derived radar cross sections accounting forantenna sway can be simulated to illustrate the differencesin the FMCW and pulsed waveform cases The sweepbandwidth of the FMCW waveform is chosen as 50 kHzThe operating frequency defined as the central frequency ofthe FMCW waveform is taken to be 25MHz The bistaticangle is 30∘ and the wind speed is 20 knots The scatteringellipse normal and the wind direction are 90∘ and 180∘respectively as measured from the positive 119909-axis (the lineconnecting the transmitter with the receiver) The swayamplitude and frequency depend on the wind velocity andin keeping with an example used earlier [4] for the purposeof illustration here these values are taken as 1228m and0127Hz respectively The sway direction is chosen to be thesame as the wind direction

41 First-Order Radar Cross Section Figure 2 shows a com-parison of the first-order radar cross section for a pulsedsource and that for an FMCW source For the FMCWwaveform Δ119903 = 1500m which equals half the width of thescattering patch (Δ120588 = 3000m) for the pulsed waveformin order to keep the same bandwidth for both waveformsA hamming window is used to smooth the curve andreduce the oscillations From this figure it can be observedthat additional peaks caused by the antenna motion appearsymmetrically in the Doppler spectrum with respect to theBragg peaks A detailed description and properties of thesemotion-induced peaks were discussed in [11] It can alsobe seen that the magnitude of the radar cross sections forthe FMCW waveform is a little lower than that for thecorresponding pulsed waveform which may be caused bythe value of Δ119903 Δ119903 is the limit value of the integral where asampling function dominates this integralΔ119903 is usually takento be Δ119903 = Δ1205882 which means only the contributions inthe main lobe of the sampling function are considered andno interaction between the range bins is assumed in the idealcase

It is clear that the first-order radar cross section has acertain relationship with the integral limit Δ119903 In Section 2it may be observed that there is no mathematical limit forthe parameter Δ119903 By varying Δ119903 the effect on the radar crosssection can be examined Keeping the value of Δ120588 = 3000m

Δ119903 = 05Δ120588 and Δ119903 = 10Δ120588 are simulated in Figures 3(a) and3(b) respectively It should be mentioned that the hammingwindow smoothing process is not used in Figure 3 in orderto clearly show the side lobe levels of the first-order radarcross sections The side lobe structure appears in the radarDoppler spectra due to the side lobes of the Sm functionfor the FMCW waveform By comparing Figures 3(a) and3(b) the magnitude of the side lobes for FMCW source isfound to decrease with increasing Δ119903 and the main lobe levelis a little raised with increasing Δ119903 due to the propertiesof the Sm function This seems to indicate an advantage ofan FMCW system When the value of Δ119903 is taken to belarger thanΔ1205882 the interactions between the range bins (thecontributions in the side lobe of the sampling function) areconsidered and appear in the received electric field at a fixeddistance Increasing Δ119903means the received signal is scatteredfrom a larger ocean surface region When Δ119903 approachesinfinity the radar cross section for the FMCW waveformbecomes a rectangular function whose width is determinedby119861(2119891

0120596119861) However when the patchwidthΔ120588 approaches

infinity the sampling functions in the first-order pulse radarocean cross section reduce to delta functions

By varying the radar bandwidth and keeping the relation-ships Δ120588 = 1198882119861 and Δ119903 = Δ1205882 the effect of the bandwidthon the radar cross sections is illustrated in Figure 4 Fromthis figure it can be seen that with increased bandwidththe magnitudes of the Bragg peaks and the motion-inducedpeaks are found to be reduced while the rest of the radar crosssection increases In addition the width of the Bragg peaksand the motion-induced peaks is also broadened Thereforeif a large radar bandwidth is used for ocean remote sensingthe Bragg peaks may be significantly contaminated by themotion-induced peaks

42 Second-Order Radar Cross Section A similar technique isused to simplify and simulate the second-order radar oceancross section for the FMCW waveform as that for the pulsedwaveform in [12 15] For the case of large Δ119903 it can be shownthat

limΔ119903rarrinfin

[Δ120588Sm2(119870 cos120601

0 119896119861 Δ119903)] asymp Δ120588 cos120601

0

sdot ℎ [119870 minus cos1206010(2119896

0minus 119896

119861)]

minus ℎ [119870 minus cos1206010(2119896

0+ 119896

119861)]

(27)

International Journal of Antennas and Propagation 7

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

FMCWPulsed

Figure 2 Comparison of the first-order radar cross sections for the FMCW waveform with that for the pulsed waveform

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

Doppler frequency f (Hz)

FMCWPulsed

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(a)

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

FMCWPulsed

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(b)Figure 3 Comparison of the side lobe levels of the first-order radar cross sections for the pulsed and FMCWwaveform (a) Δ119903 = 05Δ120588 and(b) Δ119903 = 10Δ120588

Assuming that the other terms in (26) are slowly varyingwithin the interval

cos1206010(2119896

0minus 119896

119861) lt 119870 lt cos120601

0(2119896

0+ 119896

119861) (28)

and carrying out the 119870 integration (26) reduces to

1205902(120596

119889) = 64120587

21198964

0cos4120601

0sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

1198781(119898

11)

sdot 1198781(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198701

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot 120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901)

+120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)]

sdot 1198891198701119889120579

1

(29)

8 International Journal of Antennas and Propagation

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

100 kHz50kHz

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 4 The effect of the bandwidth on the first-order radar crosssections

minus15 minus1 minus05 0 05 1 15minus70

minus60

minus50

minus40

minus30

minus20

minus10

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 5 Second-order bistatic radar cross section with a transmit-ter on a floating platform

Equation (29) is exactly the same as the correspondingmodel for the pulsed waveform when the scattering patch Δ120588approaches infinityTherefore the second-order cross sectionmodel for the FMCW waveform shows the same features inthe Doppler spectra as the model for the pulsed waveformin [12] for a given sea state radar operating parameters andplatform motion An example of the second-order bistaticradar cross section with a transmitter on a floating platformand a fixed receiver is shown in Figure 5 for the case of thescattering patch being assumed to be infinite in extent

5 Conclusion

Thefirst- and second-order bistatic radar ocean cross sectionsfor an antenna on a floating platform have been presentedfor the case of an FMCW waveform In the derivationprocess the first- and second-order models begin with

the bistatically received electric field equations derived in[11 12] Subsequently the derivation is carried out for anFMCW radar which is different from [11 12] where a pulsedradar is considered In particular the distinguishing featurebetween the current work and that presented earlier is thatdemodulation and range transformation must be used toobtain the range information Based on the new modelssimulations are made to compare the radar cross sectionsfor the FMCW waveform with that for the pulsed waveformIt is found that the first-order radar cross section for theFMCWwaveform is a little lower than that for a pulsed sourcewith the same simulation parameters With increased radaroperating bandwidth the magnitude and width of Braggpeaks and motion-induced peaks are found to be reducedand broadened respectively For an FMCW waveform thereis no definite mathematical limit for a patch width whichis different from that for a pulsed waveform Therefore themagnitude of the range bin is varied to examine the effecton the radar cross section The side lobe level is found to bereduced with increasing magnitude of the range bin Whenthe range bin approaches infinity the first-order radar crosssection for an FMCW waveform approaches a rectangularfunction and the second-order radar cross section modelfor the FMCW waveform is reduced to that of the pulsedwaveform

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work was supported in part by Natural Sciences andEngineering Research Council of Canada (NSERC) underDiscovery Grants to Weimin Huang (NSERC 402313-2012)and Eric W Gill (NSERC 238263-2010 and RGPIN-2015-05289) and by an Atlantic Innovation Fund Award (Eric WGill principal investigator)

References

[1] D E Barrick ldquoFirst-order theory and analysis of MFHFVHFscatter from the seardquo IEEE Transactions on Antennas andPropagation vol 20 no 1 pp 2ndash10 1972

[2] J Walsh and E W Gill ldquoAn analysis of the scattering of high-frequency electromagnetic radiation from rough surfaces withapplication to pulse radar operating in backscattermoderdquoRadioScience vol 35 no 6 pp 1337ndash1359 2000

[3] E W Gill and J Walsh ldquoHigh-frequency bistatic cross sectionsof the ocean surfacerdquoRadio Science vol 36 no 6 pp 1459ndash14752001

[4] J WalshW Huang and E Gill ldquoThe first-order high frequencyradar ocean surface cross section for an antenna on a floatingplatformrdquo IEEE Transactions on Antennas and Propagation vol58 no 9 pp 2994ndash3003 2010

[5] J Walsh W Huang and E Gill ldquoThe second-order highfrequency radar ocean surface cross section for an antenna

International Journal of Antennas and Propagation 9

on a floating platformrdquo IEEE Transactions on Antennas andPropagation vol 60 no 10 pp 4804ndash4813 2012

[6] D E Barrick ldquoFMCW radar signals and digital processingrdquoTech Rep ERL 283-WPL 26 NOAA 1973

[7] R Khan B Gamberg D Power et al ldquoTarget detection andtracking with a high frequency ground wave radarrdquo IEEEJournal of Oceanic Engineering vol 19 no 4 pp 540ndash548 1994

[8] A Wojthiewicz J Misiurewicz M Nalecz K Jedrzejewskiand K Kulpa ldquoTwo dimensional signal processing in FMCWradarsrdquo in Proceedings of the Conference on Circuit Theory andElectronics Circuits pp 474ndash480 Kolobrzeg Poland October1997

[9] J Walsh J Zhang and E W Gill ldquoHigh-frequency radar crosssection of the ocean surface for an FMCW waveformrdquo IEEEJournal of Oceanic Engineering vol 36 no 4 pp 615ndash626 2011

[10] S Chen EWGill andWHuang ldquoA first-orderHF radar cross-section model for mixed-path ionosphere-ocean propagationwith an FMCW sourcerdquo IEEE Journal of Oceanic Engineering2016

[11] Y Ma E Gill and W Huang ldquoThe first-order bistatic highfrequency radar ocean surface cross section for an antenna ona floating platformrdquo IET Radar Sonar amp Navigation vol 10 no6 pp 1136ndash1144 2016

[12] Y Ma W Huang and E Gill ldquoThe second-order bistatic highfrequency radar ocean surface cross section for an antenna on afloating platformrdquo Canadian Journal of Remote Sensing vol 42no 4 pp 332ndash343 2016

[13] K Hasselmann ldquoOn the non-linear energy transfer in a gravity-wave spectrum part 1 General theoryrdquo Journal of FluidMechan-ics vol 12 pp 481ndash500 1962

[14] W J Pierson and L Moskowitz ldquoA proposed spectral form forfully developed wind seas based on the similarity theory of S AKitaigorodskiirdquo Journal of Geophysical Research vol 69 no 24pp 5181ndash5190 1964

[15] B J Lipa and D E Barrick ldquoExtraction of sea state fromHF radar sea echo mathematical theory and modelingrdquo RadioScience vol 21 no 1 pp 81ndash100 1986

International Journal of

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Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

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Electrical and Computer Engineering

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Volume 2014

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SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 International Journal of Antennas and Propagation

1205902(120596

119889) = 2

3120587119896

2

0Δ120588 sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

int

infin

0

1198781(119898

11) 119878

1(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198702 cos120601

01198701Sm2

(119870 cos1206010 119896119861 Δ119903)

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816 sdot 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0sin (120579

119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901) + 120575 (120596

119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)] 119889119870

1119889120579

1119889119870

(26)

4 Simulation and Discussion

Based on a Pierson-Moskowitz (PM) ocean spectral model[14] the newly derived radar cross sections accounting forantenna sway can be simulated to illustrate the differencesin the FMCW and pulsed waveform cases The sweepbandwidth of the FMCW waveform is chosen as 50 kHzThe operating frequency defined as the central frequency ofthe FMCW waveform is taken to be 25MHz The bistaticangle is 30∘ and the wind speed is 20 knots The scatteringellipse normal and the wind direction are 90∘ and 180∘respectively as measured from the positive 119909-axis (the lineconnecting the transmitter with the receiver) The swayamplitude and frequency depend on the wind velocity andin keeping with an example used earlier [4] for the purposeof illustration here these values are taken as 1228m and0127Hz respectively The sway direction is chosen to be thesame as the wind direction

41 First-Order Radar Cross Section Figure 2 shows a com-parison of the first-order radar cross section for a pulsedsource and that for an FMCW source For the FMCWwaveform Δ119903 = 1500m which equals half the width of thescattering patch (Δ120588 = 3000m) for the pulsed waveformin order to keep the same bandwidth for both waveformsA hamming window is used to smooth the curve andreduce the oscillations From this figure it can be observedthat additional peaks caused by the antenna motion appearsymmetrically in the Doppler spectrum with respect to theBragg peaks A detailed description and properties of thesemotion-induced peaks were discussed in [11] It can alsobe seen that the magnitude of the radar cross sections forthe FMCW waveform is a little lower than that for thecorresponding pulsed waveform which may be caused bythe value of Δ119903 Δ119903 is the limit value of the integral where asampling function dominates this integralΔ119903 is usually takento be Δ119903 = Δ1205882 which means only the contributions inthe main lobe of the sampling function are considered andno interaction between the range bins is assumed in the idealcase

It is clear that the first-order radar cross section has acertain relationship with the integral limit Δ119903 In Section 2it may be observed that there is no mathematical limit forthe parameter Δ119903 By varying Δ119903 the effect on the radar crosssection can be examined Keeping the value of Δ120588 = 3000m

Δ119903 = 05Δ120588 and Δ119903 = 10Δ120588 are simulated in Figures 3(a) and3(b) respectively It should be mentioned that the hammingwindow smoothing process is not used in Figure 3 in orderto clearly show the side lobe levels of the first-order radarcross sections The side lobe structure appears in the radarDoppler spectra due to the side lobes of the Sm functionfor the FMCW waveform By comparing Figures 3(a) and3(b) the magnitude of the side lobes for FMCW source isfound to decrease with increasing Δ119903 and the main lobe levelis a little raised with increasing Δ119903 due to the propertiesof the Sm function This seems to indicate an advantage ofan FMCW system When the value of Δ119903 is taken to belarger thanΔ1205882 the interactions between the range bins (thecontributions in the side lobe of the sampling function) areconsidered and appear in the received electric field at a fixeddistance Increasing Δ119903means the received signal is scatteredfrom a larger ocean surface region When Δ119903 approachesinfinity the radar cross section for the FMCW waveformbecomes a rectangular function whose width is determinedby119861(2119891

0120596119861) However when the patchwidthΔ120588 approaches

infinity the sampling functions in the first-order pulse radarocean cross section reduce to delta functions

By varying the radar bandwidth and keeping the relation-ships Δ120588 = 1198882119861 and Δ119903 = Δ1205882 the effect of the bandwidthon the radar cross sections is illustrated in Figure 4 Fromthis figure it can be seen that with increased bandwidththe magnitudes of the Bragg peaks and the motion-inducedpeaks are found to be reduced while the rest of the radar crosssection increases In addition the width of the Bragg peaksand the motion-induced peaks is also broadened Thereforeif a large radar bandwidth is used for ocean remote sensingthe Bragg peaks may be significantly contaminated by themotion-induced peaks

42 Second-Order Radar Cross Section A similar technique isused to simplify and simulate the second-order radar oceancross section for the FMCW waveform as that for the pulsedwaveform in [12 15] For the case of large Δ119903 it can be shownthat

limΔ119903rarrinfin

[Δ120588Sm2(119870 cos120601

0 119896119861 Δ119903)] asymp Δ120588 cos120601

0

sdot ℎ [119870 minus cos1206010(2119896

0minus 119896

119861)]

minus ℎ [119870 minus cos1206010(2119896

0+ 119896

119861)]

(27)

International Journal of Antennas and Propagation 7

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

FMCWPulsed

Figure 2 Comparison of the first-order radar cross sections for the FMCW waveform with that for the pulsed waveform

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

Doppler frequency f (Hz)

FMCWPulsed

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(a)

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

FMCWPulsed

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(b)Figure 3 Comparison of the side lobe levels of the first-order radar cross sections for the pulsed and FMCWwaveform (a) Δ119903 = 05Δ120588 and(b) Δ119903 = 10Δ120588

Assuming that the other terms in (26) are slowly varyingwithin the interval

cos1206010(2119896

0minus 119896

119861) lt 119870 lt cos120601

0(2119896

0+ 119896

119861) (28)

and carrying out the 119870 integration (26) reduces to

1205902(120596

119889) = 64120587

21198964

0cos4120601

0sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

1198781(119898

11)

sdot 1198781(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198701

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot 120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901)

+120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)]

sdot 1198891198701119889120579

1

(29)

8 International Journal of Antennas and Propagation

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

100 kHz50kHz

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 4 The effect of the bandwidth on the first-order radar crosssections

minus15 minus1 minus05 0 05 1 15minus70

minus60

minus50

minus40

minus30

minus20

minus10

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 5 Second-order bistatic radar cross section with a transmit-ter on a floating platform

Equation (29) is exactly the same as the correspondingmodel for the pulsed waveform when the scattering patch Δ120588approaches infinityTherefore the second-order cross sectionmodel for the FMCW waveform shows the same features inthe Doppler spectra as the model for the pulsed waveformin [12] for a given sea state radar operating parameters andplatform motion An example of the second-order bistaticradar cross section with a transmitter on a floating platformand a fixed receiver is shown in Figure 5 for the case of thescattering patch being assumed to be infinite in extent

5 Conclusion

Thefirst- and second-order bistatic radar ocean cross sectionsfor an antenna on a floating platform have been presentedfor the case of an FMCW waveform In the derivationprocess the first- and second-order models begin with

the bistatically received electric field equations derived in[11 12] Subsequently the derivation is carried out for anFMCW radar which is different from [11 12] where a pulsedradar is considered In particular the distinguishing featurebetween the current work and that presented earlier is thatdemodulation and range transformation must be used toobtain the range information Based on the new modelssimulations are made to compare the radar cross sectionsfor the FMCW waveform with that for the pulsed waveformIt is found that the first-order radar cross section for theFMCWwaveform is a little lower than that for a pulsed sourcewith the same simulation parameters With increased radaroperating bandwidth the magnitude and width of Braggpeaks and motion-induced peaks are found to be reducedand broadened respectively For an FMCW waveform thereis no definite mathematical limit for a patch width whichis different from that for a pulsed waveform Therefore themagnitude of the range bin is varied to examine the effecton the radar cross section The side lobe level is found to bereduced with increasing magnitude of the range bin Whenthe range bin approaches infinity the first-order radar crosssection for an FMCW waveform approaches a rectangularfunction and the second-order radar cross section modelfor the FMCW waveform is reduced to that of the pulsedwaveform

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work was supported in part by Natural Sciences andEngineering Research Council of Canada (NSERC) underDiscovery Grants to Weimin Huang (NSERC 402313-2012)and Eric W Gill (NSERC 238263-2010 and RGPIN-2015-05289) and by an Atlantic Innovation Fund Award (Eric WGill principal investigator)

References

[1] D E Barrick ldquoFirst-order theory and analysis of MFHFVHFscatter from the seardquo IEEE Transactions on Antennas andPropagation vol 20 no 1 pp 2ndash10 1972

[2] J Walsh and E W Gill ldquoAn analysis of the scattering of high-frequency electromagnetic radiation from rough surfaces withapplication to pulse radar operating in backscattermoderdquoRadioScience vol 35 no 6 pp 1337ndash1359 2000

[3] E W Gill and J Walsh ldquoHigh-frequency bistatic cross sectionsof the ocean surfacerdquoRadio Science vol 36 no 6 pp 1459ndash14752001

[4] J WalshW Huang and E Gill ldquoThe first-order high frequencyradar ocean surface cross section for an antenna on a floatingplatformrdquo IEEE Transactions on Antennas and Propagation vol58 no 9 pp 2994ndash3003 2010

[5] J Walsh W Huang and E Gill ldquoThe second-order highfrequency radar ocean surface cross section for an antenna

International Journal of Antennas and Propagation 9

on a floating platformrdquo IEEE Transactions on Antennas andPropagation vol 60 no 10 pp 4804ndash4813 2012

[6] D E Barrick ldquoFMCW radar signals and digital processingrdquoTech Rep ERL 283-WPL 26 NOAA 1973

[7] R Khan B Gamberg D Power et al ldquoTarget detection andtracking with a high frequency ground wave radarrdquo IEEEJournal of Oceanic Engineering vol 19 no 4 pp 540ndash548 1994

[8] A Wojthiewicz J Misiurewicz M Nalecz K Jedrzejewskiand K Kulpa ldquoTwo dimensional signal processing in FMCWradarsrdquo in Proceedings of the Conference on Circuit Theory andElectronics Circuits pp 474ndash480 Kolobrzeg Poland October1997

[9] J Walsh J Zhang and E W Gill ldquoHigh-frequency radar crosssection of the ocean surface for an FMCW waveformrdquo IEEEJournal of Oceanic Engineering vol 36 no 4 pp 615ndash626 2011

[10] S Chen EWGill andWHuang ldquoA first-orderHF radar cross-section model for mixed-path ionosphere-ocean propagationwith an FMCW sourcerdquo IEEE Journal of Oceanic Engineering2016

[11] Y Ma E Gill and W Huang ldquoThe first-order bistatic highfrequency radar ocean surface cross section for an antenna ona floating platformrdquo IET Radar Sonar amp Navigation vol 10 no6 pp 1136ndash1144 2016

[12] Y Ma W Huang and E Gill ldquoThe second-order bistatic highfrequency radar ocean surface cross section for an antenna on afloating platformrdquo Canadian Journal of Remote Sensing vol 42no 4 pp 332ndash343 2016

[13] K Hasselmann ldquoOn the non-linear energy transfer in a gravity-wave spectrum part 1 General theoryrdquo Journal of FluidMechan-ics vol 12 pp 481ndash500 1962

[14] W J Pierson and L Moskowitz ldquoA proposed spectral form forfully developed wind seas based on the similarity theory of S AKitaigorodskiirdquo Journal of Geophysical Research vol 69 no 24pp 5181ndash5190 1964

[15] B J Lipa and D E Barrick ldquoExtraction of sea state fromHF radar sea echo mathematical theory and modelingrdquo RadioScience vol 21 no 1 pp 81ndash100 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

FMCWPulsed

Figure 2 Comparison of the first-order radar cross sections for the FMCW waveform with that for the pulsed waveform

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

Doppler frequency f (Hz)

FMCWPulsed

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(a)

0475 048 0485minus100

minus80

minus60

minus40

minus20

0

FMCWPulsed

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

(b)Figure 3 Comparison of the side lobe levels of the first-order radar cross sections for the pulsed and FMCWwaveform (a) Δ119903 = 05Δ120588 and(b) Δ119903 = 10Δ120588

Assuming that the other terms in (26) are slowly varyingwithin the interval

cos1206010(2119896

0minus 119896

119861) lt 119870 lt cos120601

0(2119896

0+ 119896

119861) (28)

and carrying out the 119870 integration (26) reduces to

1205902(120596

119889) = 64120587

21198964

0cos4120601

0sum

1198981=plusmn1

sum

1198982=plusmn1

int

infin

0

int

120587

minus120587

1198781(119898

11)

sdot 1198781(119898

22)1003816100381610038161003816 119878Γ119875

100381610038161003816100381621198701

sdot 1198692

0119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot 120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2)

+

infin

sum

119899=1

1198692

119899119886119870

2

100381610038161003816100381610038161003816cos (120579

119870minus 120579

119870119901) + tan120601

0

sdot sin (120579119870minus 120579

119870119901)100381610038161003816100381610038161003816

sdot [120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2minus 119899120596

119901)

+120575 (120596119889+ 119898

1radic119892119870

1+ 119898

2radic119892119870

2+ 119899120596

119901)]

sdot 1198891198701119889120579

1

(29)

8 International Journal of Antennas and Propagation

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

100 kHz50kHz

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 4 The effect of the bandwidth on the first-order radar crosssections

minus15 minus1 minus05 0 05 1 15minus70

minus60

minus50

minus40

minus30

minus20

minus10

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 5 Second-order bistatic radar cross section with a transmit-ter on a floating platform

Equation (29) is exactly the same as the correspondingmodel for the pulsed waveform when the scattering patch Δ120588approaches infinityTherefore the second-order cross sectionmodel for the FMCW waveform shows the same features inthe Doppler spectra as the model for the pulsed waveformin [12] for a given sea state radar operating parameters andplatform motion An example of the second-order bistaticradar cross section with a transmitter on a floating platformand a fixed receiver is shown in Figure 5 for the case of thescattering patch being assumed to be infinite in extent

5 Conclusion

Thefirst- and second-order bistatic radar ocean cross sectionsfor an antenna on a floating platform have been presentedfor the case of an FMCW waveform In the derivationprocess the first- and second-order models begin with

the bistatically received electric field equations derived in[11 12] Subsequently the derivation is carried out for anFMCW radar which is different from [11 12] where a pulsedradar is considered In particular the distinguishing featurebetween the current work and that presented earlier is thatdemodulation and range transformation must be used toobtain the range information Based on the new modelssimulations are made to compare the radar cross sectionsfor the FMCW waveform with that for the pulsed waveformIt is found that the first-order radar cross section for theFMCWwaveform is a little lower than that for a pulsed sourcewith the same simulation parameters With increased radaroperating bandwidth the magnitude and width of Braggpeaks and motion-induced peaks are found to be reducedand broadened respectively For an FMCW waveform thereis no definite mathematical limit for a patch width whichis different from that for a pulsed waveform Therefore themagnitude of the range bin is varied to examine the effecton the radar cross section The side lobe level is found to bereduced with increasing magnitude of the range bin Whenthe range bin approaches infinity the first-order radar crosssection for an FMCW waveform approaches a rectangularfunction and the second-order radar cross section modelfor the FMCW waveform is reduced to that of the pulsedwaveform

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work was supported in part by Natural Sciences andEngineering Research Council of Canada (NSERC) underDiscovery Grants to Weimin Huang (NSERC 402313-2012)and Eric W Gill (NSERC 238263-2010 and RGPIN-2015-05289) and by an Atlantic Innovation Fund Award (Eric WGill principal investigator)

References

[1] D E Barrick ldquoFirst-order theory and analysis of MFHFVHFscatter from the seardquo IEEE Transactions on Antennas andPropagation vol 20 no 1 pp 2ndash10 1972

[2] J Walsh and E W Gill ldquoAn analysis of the scattering of high-frequency electromagnetic radiation from rough surfaces withapplication to pulse radar operating in backscattermoderdquoRadioScience vol 35 no 6 pp 1337ndash1359 2000

[3] E W Gill and J Walsh ldquoHigh-frequency bistatic cross sectionsof the ocean surfacerdquoRadio Science vol 36 no 6 pp 1459ndash14752001

[4] J WalshW Huang and E Gill ldquoThe first-order high frequencyradar ocean surface cross section for an antenna on a floatingplatformrdquo IEEE Transactions on Antennas and Propagation vol58 no 9 pp 2994ndash3003 2010

[5] J Walsh W Huang and E Gill ldquoThe second-order highfrequency radar ocean surface cross section for an antenna

International Journal of Antennas and Propagation 9

on a floating platformrdquo IEEE Transactions on Antennas andPropagation vol 60 no 10 pp 4804ndash4813 2012

[6] D E Barrick ldquoFMCW radar signals and digital processingrdquoTech Rep ERL 283-WPL 26 NOAA 1973

[7] R Khan B Gamberg D Power et al ldquoTarget detection andtracking with a high frequency ground wave radarrdquo IEEEJournal of Oceanic Engineering vol 19 no 4 pp 540ndash548 1994

[8] A Wojthiewicz J Misiurewicz M Nalecz K Jedrzejewskiand K Kulpa ldquoTwo dimensional signal processing in FMCWradarsrdquo in Proceedings of the Conference on Circuit Theory andElectronics Circuits pp 474ndash480 Kolobrzeg Poland October1997

[9] J Walsh J Zhang and E W Gill ldquoHigh-frequency radar crosssection of the ocean surface for an FMCW waveformrdquo IEEEJournal of Oceanic Engineering vol 36 no 4 pp 615ndash626 2011

[10] S Chen EWGill andWHuang ldquoA first-orderHF radar cross-section model for mixed-path ionosphere-ocean propagationwith an FMCW sourcerdquo IEEE Journal of Oceanic Engineering2016

[11] Y Ma E Gill and W Huang ldquoThe first-order bistatic highfrequency radar ocean surface cross section for an antenna ona floating platformrdquo IET Radar Sonar amp Navigation vol 10 no6 pp 1136ndash1144 2016

[12] Y Ma W Huang and E Gill ldquoThe second-order bistatic highfrequency radar ocean surface cross section for an antenna on afloating platformrdquo Canadian Journal of Remote Sensing vol 42no 4 pp 332ndash343 2016

[13] K Hasselmann ldquoOn the non-linear energy transfer in a gravity-wave spectrum part 1 General theoryrdquo Journal of FluidMechan-ics vol 12 pp 481ndash500 1962

[14] W J Pierson and L Moskowitz ldquoA proposed spectral form forfully developed wind seas based on the similarity theory of S AKitaigorodskiirdquo Journal of Geophysical Research vol 69 no 24pp 5181ndash5190 1964

[15] B J Lipa and D E Barrick ldquoExtraction of sea state fromHF radar sea echo mathematical theory and modelingrdquo RadioScience vol 21 no 1 pp 81ndash100 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

minus15 minus1 minus05 0 05 1 15minus80

minus60

minus40

minus20

0

20

Doppler frequency f (Hz)

100 kHz50kHz

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 4 The effect of the bandwidth on the first-order radar crosssections

minus15 minus1 minus05 0 05 1 15minus70

minus60

minus50

minus40

minus30

minus20

minus10

Doppler frequency f (Hz)

Rada

r cro

ss se

ctio

n (1

Hz)

(dB)

Figure 5 Second-order bistatic radar cross section with a transmit-ter on a floating platform

Equation (29) is exactly the same as the correspondingmodel for the pulsed waveform when the scattering patch Δ120588approaches infinityTherefore the second-order cross sectionmodel for the FMCW waveform shows the same features inthe Doppler spectra as the model for the pulsed waveformin [12] for a given sea state radar operating parameters andplatform motion An example of the second-order bistaticradar cross section with a transmitter on a floating platformand a fixed receiver is shown in Figure 5 for the case of thescattering patch being assumed to be infinite in extent

5 Conclusion

Thefirst- and second-order bistatic radar ocean cross sectionsfor an antenna on a floating platform have been presentedfor the case of an FMCW waveform In the derivationprocess the first- and second-order models begin with

the bistatically received electric field equations derived in[11 12] Subsequently the derivation is carried out for anFMCW radar which is different from [11 12] where a pulsedradar is considered In particular the distinguishing featurebetween the current work and that presented earlier is thatdemodulation and range transformation must be used toobtain the range information Based on the new modelssimulations are made to compare the radar cross sectionsfor the FMCW waveform with that for the pulsed waveformIt is found that the first-order radar cross section for theFMCWwaveform is a little lower than that for a pulsed sourcewith the same simulation parameters With increased radaroperating bandwidth the magnitude and width of Braggpeaks and motion-induced peaks are found to be reducedand broadened respectively For an FMCW waveform thereis no definite mathematical limit for a patch width whichis different from that for a pulsed waveform Therefore themagnitude of the range bin is varied to examine the effecton the radar cross section The side lobe level is found to bereduced with increasing magnitude of the range bin Whenthe range bin approaches infinity the first-order radar crosssection for an FMCW waveform approaches a rectangularfunction and the second-order radar cross section modelfor the FMCW waveform is reduced to that of the pulsedwaveform

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work was supported in part by Natural Sciences andEngineering Research Council of Canada (NSERC) underDiscovery Grants to Weimin Huang (NSERC 402313-2012)and Eric W Gill (NSERC 238263-2010 and RGPIN-2015-05289) and by an Atlantic Innovation Fund Award (Eric WGill principal investigator)

References

[1] D E Barrick ldquoFirst-order theory and analysis of MFHFVHFscatter from the seardquo IEEE Transactions on Antennas andPropagation vol 20 no 1 pp 2ndash10 1972

[2] J Walsh and E W Gill ldquoAn analysis of the scattering of high-frequency electromagnetic radiation from rough surfaces withapplication to pulse radar operating in backscattermoderdquoRadioScience vol 35 no 6 pp 1337ndash1359 2000

[3] E W Gill and J Walsh ldquoHigh-frequency bistatic cross sectionsof the ocean surfacerdquoRadio Science vol 36 no 6 pp 1459ndash14752001

[4] J WalshW Huang and E Gill ldquoThe first-order high frequencyradar ocean surface cross section for an antenna on a floatingplatformrdquo IEEE Transactions on Antennas and Propagation vol58 no 9 pp 2994ndash3003 2010

[5] J Walsh W Huang and E Gill ldquoThe second-order highfrequency radar ocean surface cross section for an antenna

International Journal of Antennas and Propagation 9

on a floating platformrdquo IEEE Transactions on Antennas andPropagation vol 60 no 10 pp 4804ndash4813 2012

[6] D E Barrick ldquoFMCW radar signals and digital processingrdquoTech Rep ERL 283-WPL 26 NOAA 1973

[7] R Khan B Gamberg D Power et al ldquoTarget detection andtracking with a high frequency ground wave radarrdquo IEEEJournal of Oceanic Engineering vol 19 no 4 pp 540ndash548 1994

[8] A Wojthiewicz J Misiurewicz M Nalecz K Jedrzejewskiand K Kulpa ldquoTwo dimensional signal processing in FMCWradarsrdquo in Proceedings of the Conference on Circuit Theory andElectronics Circuits pp 474ndash480 Kolobrzeg Poland October1997

[9] J Walsh J Zhang and E W Gill ldquoHigh-frequency radar crosssection of the ocean surface for an FMCW waveformrdquo IEEEJournal of Oceanic Engineering vol 36 no 4 pp 615ndash626 2011

[10] S Chen EWGill andWHuang ldquoA first-orderHF radar cross-section model for mixed-path ionosphere-ocean propagationwith an FMCW sourcerdquo IEEE Journal of Oceanic Engineering2016

[11] Y Ma E Gill and W Huang ldquoThe first-order bistatic highfrequency radar ocean surface cross section for an antenna ona floating platformrdquo IET Radar Sonar amp Navigation vol 10 no6 pp 1136ndash1144 2016

[12] Y Ma W Huang and E Gill ldquoThe second-order bistatic highfrequency radar ocean surface cross section for an antenna on afloating platformrdquo Canadian Journal of Remote Sensing vol 42no 4 pp 332ndash343 2016

[13] K Hasselmann ldquoOn the non-linear energy transfer in a gravity-wave spectrum part 1 General theoryrdquo Journal of FluidMechan-ics vol 12 pp 481ndash500 1962

[14] W J Pierson and L Moskowitz ldquoA proposed spectral form forfully developed wind seas based on the similarity theory of S AKitaigorodskiirdquo Journal of Geophysical Research vol 69 no 24pp 5181ndash5190 1964

[15] B J Lipa and D E Barrick ldquoExtraction of sea state fromHF radar sea echo mathematical theory and modelingrdquo RadioScience vol 21 no 1 pp 81ndash100 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

on a floating platformrdquo IEEE Transactions on Antennas andPropagation vol 60 no 10 pp 4804ndash4813 2012

[6] D E Barrick ldquoFMCW radar signals and digital processingrdquoTech Rep ERL 283-WPL 26 NOAA 1973

[7] R Khan B Gamberg D Power et al ldquoTarget detection andtracking with a high frequency ground wave radarrdquo IEEEJournal of Oceanic Engineering vol 19 no 4 pp 540ndash548 1994

[8] A Wojthiewicz J Misiurewicz M Nalecz K Jedrzejewskiand K Kulpa ldquoTwo dimensional signal processing in FMCWradarsrdquo in Proceedings of the Conference on Circuit Theory andElectronics Circuits pp 474ndash480 Kolobrzeg Poland October1997

[9] J Walsh J Zhang and E W Gill ldquoHigh-frequency radar crosssection of the ocean surface for an FMCW waveformrdquo IEEEJournal of Oceanic Engineering vol 36 no 4 pp 615ndash626 2011

[10] S Chen EWGill andWHuang ldquoA first-orderHF radar cross-section model for mixed-path ionosphere-ocean propagationwith an FMCW sourcerdquo IEEE Journal of Oceanic Engineering2016

[11] Y Ma E Gill and W Huang ldquoThe first-order bistatic highfrequency radar ocean surface cross section for an antenna ona floating platformrdquo IET Radar Sonar amp Navigation vol 10 no6 pp 1136ndash1144 2016

[12] Y Ma W Huang and E Gill ldquoThe second-order bistatic highfrequency radar ocean surface cross section for an antenna on afloating platformrdquo Canadian Journal of Remote Sensing vol 42no 4 pp 332ndash343 2016

[13] K Hasselmann ldquoOn the non-linear energy transfer in a gravity-wave spectrum part 1 General theoryrdquo Journal of FluidMechan-ics vol 12 pp 481ndash500 1962

[14] W J Pierson and L Moskowitz ldquoA proposed spectral form forfully developed wind seas based on the similarity theory of S AKitaigorodskiirdquo Journal of Geophysical Research vol 69 no 24pp 5181ndash5190 1964

[15] B J Lipa and D E Barrick ldquoExtraction of sea state fromHF radar sea echo mathematical theory and modelingrdquo RadioScience vol 21 no 1 pp 81ndash100 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of