Research Article Transmit TACAN Bearing Information with a

Research ArticleTransmit TACAN Bearing Information with a Circular Array

W Mark Dorsey Jeffrey O Coleman and William R Pickles

Radar Division Naval Research Laboratory Washington DC 20375 USA

Correspondence should be addressed to W Mark Dorsey markdorseynrlnavymil

Received 6 August 2015 Accepted 30 September 2015

Academic Editor Toni Bjorninen

Copyright copy 2015 W Mark Dorsey et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Using TACAN and array fundamentals we derive an architecture for transmitting TACAN bearing information from a circulararray with time-varying weights We evaluate performance for a simulated example array of Vivaldi elements

1 Introduction

The Tactical Air Navigational (TACAN) system providesdistance and bearing information to aircraft from groundstations and is widely used in military settings Traditionallya ground stationrsquos physically rotating transmit antenna createsbearing-dependent amplitude modulation from which air-craft can determine their bearings from that ground stationWhere space for such a dedicated special-purpose transmitantenna is difficult to obtain such as onNaval vessels sharinga multifunction array with other systems is an option In thatcase the TACAN application would use time-varying arrayweights to approximate a rotating pattern

Replacing the rotating antenna with a circular arraywould have benefits beyond facilitating the consolidation ofapertures Certainly these would include simplified mainte-nance [1] and the potential for elevation beam shaping andoroperation onlywithin desired azimuth ranges [2] In additionan array could be given an operational bandwidth coveringnot only the current TACAN bands of 962ndash1024MHz and1025ndash1087MHz [3] but also future TACAN bands consideredlikely to result from revised spectrum allocations [4]

With those motivations this paper derives time-varyingTACAN array weights for a uniform cylindrical array WhileTACAN specifications [5] address both the static elevationpattern and the dynamic azimuth pattern here we focus onthe latterOur design example assumes an array ofVivaldi ele-ments characterized by embedded element patterns obtainedthrough HFSS simulations To evaluate the design we use abearing-error metric that falls naturally out of the derivation

The standard TACAN ground transmitter of interestslowly amplitude-modulates a fast pulse signal with anantenna pattern that rotates at 15Hz and that is designed toyield sinusoidal AM components in the pulse amplitudesat the aircraft receiver at 15Hz and 9 times 15Hz = 135Hz Areference burst transmitted as the rotating main lobe passesnorth enables an aircraft to obtain a coarse bearing from thetransmitter as the phase of the 15Hz modulation componentrelative to a zero time marked by burst receptionThat coarsebearing and the phase of the 135Hz component then togetheryield a fine bearing measurement Here we focus on creatinga time-varying array pattern that permits accurate bearingestimation at the receiver using this process The fast pulsemodulation and reference bursts are independent of theantenna andpattern used and are not considered further here

This paper presents the initial study into the developmentof the time-harmonic weights required for transmitting theTACAN waveform from a circular array A discussion on thetheory is provided and validated using simulations

2 Theory

Thenext section derives the array structure and time-varyingarray weights Performance is then derived as a function ofthose weights and the complex embedded array patterns

21 Deriving the Array Time-varying weights for a circulararray of 119873 elements are derived below with the goal of pro-viding accurate TACAN bearing measurement in receivers atarbitrary bearings

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 846720 6 pageshttpdxdoiorg1011552015846720

2 International Journal of Antennas and Propagation

There are several steps Formally assuming the array to becircularly symmetric and requiring its pattern sampled at 119873equally spaced bearings to smoothly rotate in space with timeturns outmdashno surprisemdashto formally imply that the weightsmust also rotate so that only one weight requires explicitdesign That design follows from the desired temporal mod-ulation of the array-pattern amplitude along a single direc-tion The pattern modulation between the 119873 bearings thusaddressed explicitly takes the desired general form automat-ically with only pattern magnitude and signal modulationindices free to vary modestly (given reasonable assumptions)with bearing

211 The Array Center the 119873-element array on the originwith symmetry about the vertical axis and with element indi-ces increasing with bearing Align element 0 with bearing0∘ (any bearing can be made the new zero by changing thereference-burst timing) and interpret element indices mod-ulo119873 so that the elements adjacent to element 0 for examplecan be indexed with plusmn1 or 1119873 minus 1 In the developmentbelow each summation sum over index 119899 should be read asa sum over element indices 119899 = 0 119873 minus 1 and eachsummation sum over index ℓ should be read as the doublyinfinite sum over ℓ = minusinfin infin

Let k designate the real wavenumber vector of a trans-mitted signal and let complex vector-valued function

119891119899(k)

be the origin-referenced embedded far-field complex patternof element 119899 We assume elements are identical in the sensethat

119891119899(k) = R119899

1198910(Rminus119899k) (1)

for all k of interest where linear operation k 997891rarr Rk rotatesreal vector k about the vertical by 2120587119873 to increase bearingIdentity R119873 = R will be used freely

In practice imperfect array construction will result innonidentical embedded element patterns so the transmittedTACANwaveformwill vary somewhat from the ideal derivedhere We have yet to study such errors but hope to eventually

212 One Weight Implies the Others Write the time-varyingfar-field complex array pattern as

119891 (k 119905) = sum

119899

119908119899(119905)

119891119899(k) = sum

119899

119908119899(119905)R119899

1198910(Rminus119899k) (2)

using array symmetry (1) on the right A classic TACANsystemrsquos pattern rotates spatially at frequency 1119879 = 15Hzbut here we require that behavior only at 119873 equally spacedbearings Period 119879 rotation over 2120587119873 in angle is given by

119891 (Rk 119905) = R

119891 (k 119905 minus 119879

119873

) (3)

Substituting Rk for k in (2) and a change of index yield

119891 (Rk 119905) = sum

119899

119908119899(119905)R119899

1198910(R1minus119899k) (4)

119891 (Rk 119905) = sum

119899

119908119899+1

(119905)R119899+1 1198910(Rminus119899k) (5)

Likewise applying (2) to the right side of (3) yields

R 119891 (k 119905 minus 119879

119873

) = sum

119899

119908119899(119905 minus

119879

119873

)R119899+1 1198910(Rminus119899k) (6)

Substituting (5) and (6) into (3) and comparing terms thenformally show that 119908

119899+1(119905) = 119908

119899(119905 minus 119879119873) for all 119899 so

119908119899(119905) = 119908

0(119905 minus

119899119879

119873

) (7)

A rotating bearing-sampled pattern thus implies weightperiodicity 119908

0(119905) = 119908

119873(119905) = 119908

0(119905 minus 119879) This will not produce

rotation for all bearings but we will preserve property (7) forsimplicity of structure and in order to obtain nearly rotatingbehavior

213 DesiredModulation Thedesired complex array patternis an arbitrary constant complex amplitude modulated by

119898(119905 120601) = 1 + 2119886 cos(2120587119905119879

minus 120601)

+ 2119887 cos(21205879119905119879

minus 9120601)

(8)

where 120601 is bearing Positive real modulation indices 2119886 and2119887 are kept small enough that |119898(119905 120601)| = 119898(119905 120601) for simplereceiver demodulationThe terms at frequencies 1119879 and 9119879are respectively used for coarse and fine bearing measure-ment

The 120601 = 0 array pattern should be using arbitrary scaling

119898(119905 0) = 1 + 2119886 cos(2120587119905119879

) + 2119887 cos(21205879119905119879

) (9)

214 Determining Weight 1199080(119905) Let wavenumber vector

kimp and complex polarization unit vector 119888 govern co-polpropagation at 120601 = 0 at the most important elevation Usingsuperscripts to index coefficients the Fourier series of asso-ciated pattern sample ⟨

119891(kimp 119905) 119888⟩ and weight 1199080(119905) take

forms

⟨

119891 (kimp 119905) 119888⟩ = sumℓ

119888ℓe1198952120587ℓ119905119879 (10)

1199080(119905) = sum

ℓ

119908ℓe1198952120587ℓ119905119879 (11)

The co-pol array pattern at k = kimp is by (2) and (7)

⟨

119891 (kimp 119905) 119888⟩ = sum119899

⟨

119891119899(kimp) 119888⟩1199080 (119905 minus

119899119879

119873

) (12)

Fourier-series forms (10) and (11) and simple algebra thenyield

sum

ℓ

119888ℓe1198952120587ℓ119905119879 = sum

ℓ

119908ℓ

ℎℓe1198952120587ℓ119905119879 (13)

after defining DFT sum (periodically extended in ℓ)

ℎℓ

= sum

119899

⟨

119891119899(kimp) 119888⟩ e

minus1198952120587ℓ119899119873

(14)

International Journal of Antennas and Propagation 3

which allows ℎℓ to be computed from the embedded complexelement patterns The 120601 = 0 pattern (9)

⟨

119891 (kimp 119905) 119888⟩ = 119898 (119905 0)

= 1 + 119886e1198952120587119905119879 + 119886eminus1198952120587119905119879 + 119887e11989521205879119905119879

+ 119887eminus11989521205879119905119879

(15)

yields coefficients 119888ℓ From these ℎℓ and 119888ℓ we can obtain 119908ℓusing the uniqueness of Fourier series and (13) which imply

119888ℓ

= 119908ℓ

ℎℓ (16)

for integer ℓ Thus Fourier series (11) can be written as

1199080(119905) =

1

ℎ0

+

119886

ℎ1

e1198952120587119905119879 + 119886

ℎminus1

eminus1198952120587119905119879 + 119887

ℎ9

e11989521205879119905119879

+

119887

ℎminus9

eminus11989521205879119905119879(17)

This and (7) specify weights that fix the co-pol array patternfor the119873wavenumber vectors of formR119899kimp to ideal valuesThe pattern in other directionspolarizations cannot be inde-pendently specified and depends on the element patterns

22 Performance

221 The Received Signalrsquos Overall Amplitude ModulationMuch of the above can be generalized to arbitrary polariza-tion unit vector and wavenumber vector k GeneralizingFourier series (10) along with (13) and (14)

⟨

119891 (k 119905) ⟩ = sumℓ

119901ℓe1198952120587ℓ119905119879 (18)

119901ℓ

= 119908ℓ

sum

119899

⟨

119891119899(k) ⟩ eminus1198952120587ℓ119899119873 (19)

Using (17) for 119908ℓ the nonzero Fourier coefficients are

1199010

=

1

ℎ0

sum

119899

⟨

119891119899(k) ⟩

119901plusmn1

=

119886

ℎplusmn1

sum

119899

⟨

119891119899(k) ⟩ e∓1198952120587119899119873

119901plusmn9

=

119887

ℎplusmn9

sum

119899

⟨

119891119899(k) ⟩ e∓11989521205879119899119873

(20)

Fourier sum (18) is a complex constant times a real modula-tion function if each of 119901plusmn11199010 and 119901plusmn91199010 is a conjugate pairTo distinguish desired and undesired pair behaviors we candefine sum and difference coefficients For each 119896 isin 1 9 let

119901119896Σ=

119901119896

1199010

+ (

119901minus119896

1199010

)

lowast

119901119896Δ=

119901119896

1199010

minus (

119901minus119896

1199010

)

lowast

(21)

so that

119901119896

1199010

=

119901119896Σ+ 119901119896Δ

2

119901minus119896

1199010

=

(119901119896Σminus 119901119896Δ)lowast

2

(22)

Fourier sum (18) then becomes

⟨

119891 (k 119905) ⟩ = 1199010 (1 + 1

2

(1199011Σe1198952120587119905119879 + 119901lowast

1Σ

eminus1198952120587119905119879

+ 1199011Δe1198952120587119905119879 minus 119901lowast

1Δ

eminus1198952120587119905119879 + 1199019Σe11989521205879119905119879

+ 119901lowast

9Σ

eminus11989521205879119905119879 + 1199019Δe11989521205879119905119879 minus 119901lowast

9Δ

eminus11989521205879119905119879))

(23)

Combining sums and differences of conjugate pairs yields

⟨

119891 (k 119905) ⟩ = 1199010 (1 + Re 1199011Σe1198952120587119905119879

+ Re 1199019Σe11989521205879119905119879 + 119895 Im 119901

1Δe1198952120587119905119879

+ 119895 Im 1199019Δe11989521205879119905119879)

(24)

or

⟨

119891 (k 119905) ⟩ = 1199010 (1 + 10038161003816100381610038161199011Σ

1003816100381610038161003816cos(2120587119905

119879

+ ang1199011Σ)

+10038161003816100381610038161199019Σ

1003816100381610038161003816cos(21205879119905

119879

+ ang1199019Σ)

+ 11989510038161003816100381610038161199011Δ

1003816100381610038161003816sin(2120587119905

119879

+ ang1199011Δ)

+ 11989510038161003816100381610038161199019Δ

1003816100381610038161003816sin(21205879119905

119879

+ ang1199019Δ))

(25)

Ideally |1199011Δ| and |119901

9Δ| are negligibly small so that

10038161003816100381610038161003816⟨

119891 (k 119905) ⟩1003816100381610038161003816

1003816=

10038161003816100381610038161003816119901010038161003816100381610038161003816

1003816100381610038161003816100381610038161003816

1 +10038161003816100381610038161199011Σ

1003816100381610038161003816cos(2120587119905

119879

+ ang1199011Σ)

+10038161003816100381610038161199019Σ

1003816100381610038161003816cos(21205879119905

119879

+ ang1199019Σ)

1003816100381610038161003816100381610038161003816

(26)

In analogy to (8) magnitudes |1199011Σ| and |119901

9Σ| are the

modulation indices and angles

ang1199011Σ

mod 2120587= minus120601coarse (27)

ang1199019Σ

mod 2120587= minus9120601fine (28)

relate to coarse and fine bearing estimates 120601coarse and 120601fine

222 The Fine Bearing Measurement The receiver can com-pute 120601coarse from (27) directly but computing 120601fine requiresresolving the ninefold ambiguity in (28) The key is to let

120601diffmod 2120587= 120601fine minus 120601coarse (29)


and assume that since 120601coarse and 120601fine are close as angles120601diff isin [minus1205879 1205879) Scaling (29) by 92120587 and using (27) and(28)

9120601diff2120587

+ 119898 =

9ang1199011Σminus ang1199019Σ

2120587

(30)

for some integer 119898 Since 9120601diff2120587 isin [minus12 12) roundingyields the first of the three needed computational steps and(30) and (29) yield the other two

119898 = round(9ang1199011Σminus ang1199019Σ

2120587

)

120601diff = ang1199011Σ minusang1199019Σ+ 2120587119898

9

120601fine = (120601coarse + 120601diff) mod 2120587

(31)

223 Intrinsic Bearing Measurement Error The measuredbearing generally contains some error even when 119901

1Δ=

1199019Δ

= 0 and when the receiver measures angles ang1199011Σ

andang1199019Σ

perfectly To derive the intrinsic residual fine bearingerror relative to actual bearing 120601 add 9120601 to each side of (28)and apply angle-folding map 119909 997891rarr ((119909+120587) mod 2120587)minus120587 Thisyields ((ang119901

9Σ+9120601+120587) mod 2120587)minus120587 = minus9(120601fineminus120601) where the

right side is unchanged because the map is an identity whenminus120587 le 119909 lt 120587 The intrinsic fine bearing error is therefore

120601fine minus 120601 =(120587 minus ((ang119901

9Σ+ 9120601 + 120587) mod 2120587))

9

(32)

Replace 9 by unity to derive intrinsic coarse bearing error

120601coarse minus 120601 = (120587 minus ((ang1199011Σ + 120601 + 120587) mod 2120587)) (33)

3 Simulation

We tested the approach using weights and performance mea-sures computed from simulated vertical-polarization elementpatterns ⟨

119891119899(k) 119888⟩ of 45 Vivaldi radiators embedded in

the uniform circular array of Figure 1 The 1GHz carrierfrequency and 229 cm (110 in) array radius used were con-venient but have no TACAN significance HFSS array sim-ulation with one element driven and the others terminatedyielded one embedded element pattern and (1) provided therest Time-varying array excitations are from (7) and (17) Weaimedwavenumber vector kimp at the north horizon for a zeroldquomost important elevationrdquo Modulation indices 2119886 and 2119887

were each set to 02 per Shestag [2]The embedded co-pol element pattern ⟨

119891119899(k) 119888⟩ of the

Vivaldi radiator appears in Figure 2 Essentially all of thesamples used in DFT (14) were significant in magnitude

Figure 3 shows that the co-pol array pattern obtainedapproximates 15Hz rotation and the Figure 4 slice at 119905 = 0

of that pattern hews closely to desired form (8) from Shestag[2] In both figures gain is normalized to the 119905 = 0 peak

Section 22 discussion assumed that for 119899 isin 1 9 thehypotenuse of a right trianglewith side lengths |119901

119899Δ| and |119901

119899Σ|

559 cm

145 cm

159

cm

Figure 1 Simulated uniform circular array of 45 Vivaldi radiators

Co-

pol e

mbe

dded

elem

ent p

atte

rn 1

05

0

minus1

minus05

Bearing 120601

minus180∘ minus90∘ 0∘ 90∘ 180∘

Figure 2 Real (solid) and imaginary (knobby) parts andmagnitude(dashed) of embedded co-pol pattern ⟨

1198910

(k) 119888⟩ of element zero ona linear scale with k at bearing 120601 and the elevation of kimp Theelements are identical so the knobs every 8∘ mark those 120601 wherethe curves also yield the ⟨

119891119899

(kimp) 119888⟩ for 119899 = 0 119873 minus 1 used inDFT (14)

minus180∘ minus90∘

Bearing 120601

0∘ 90∘ 180∘

minus6

minus4

minus2

(dB)

0

05

tT

1

Figure 3 Co-pol array gain as a function of bearing and time overperiod 119879 = 1(15Hz)

was essentially of the latter length because |119901119899Δ|was relatively

tiny This is verified in Figure 5Figure 6 shows that time-average array gain 20 log

10

|1199010

|

and modulation indices |1199011Σ| and |119901

9Σ| vary little with bear-

ing Average gain is consistent with Figure 4 and the modu-lation indices approximate the 20 desired value

The most important quantities computed in this systemsimulation are undoubtedly the intrinsic errors (32) and (33)in the coarse and fine bearing measurements respectively


DesiredSynthesized

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601

minus8

minus6

minus4

minus2

0

Nor

mal

ized

gai

n (d

Bp)

Figure 4 This paperrsquos synthesized 119905 = 0 co-pol array gain plottedover the desired pattern of (8) and [2]

0006

0003

00

0002

0001

00

|p1Δ||p1Σ|

|p9Δ||p9Σ|

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601

Figure 5 Approximating (25) by (26) is validated by the small errorratio |119901

119899Δ

||119901119899Σ

| shown here for 119899 = 1 (solid curve) and 119899 = 9

(dashed curve)

Tim

e Avg

gai

n (d

B) minus285

minus29

minus295

Mod

ulat

ion

indi

ces (

)

205

200

195

Bearing 120601

minus180∘ minus90∘ 0∘ 90∘ 180∘

Figure 6 Time-averaged co-pol array gain (dashed curve usingsame normalization as Figures 3 and 4) and modulation indices forthe 15Hz (solid curve) and 135Hz (knobby curve) components ofthe AM signal

intrinsic because they assume noise-free reception at the air-craft Those are shown in Figure 7 The intrinsic errors in thefine bearing measurement never exceed 01∘ in magnitudewhile the magnitudes of the coarse errors never exceed 002∘While this appears to suggest that coarse measurement ismore accurate this is somewhat illusory as the error com-ponent due to signal noise not included here will generallydominate and be substantially greater for the coarsemeasure-ment than for the fine measurement Certainly the Figure 7numbers leave plenty of room for those noise-related errorsbefore the TACAN system error limits of 10∘ and 2∘ for thecoarse and fine readings respectively [5] are breached

01∘

0∘

minus01∘

Fine

erro

r

0∘

002∘

minus002∘Coa

rse e

rror

minus90∘ 0∘ 90∘ 180∘minus180∘

Bearing 120601

Figure 7 Bearingmeasurement errors computed from the phases of1199011Σ

and 1199019Σ

for coarse (solid curve) and fine (dashed curve) bearinginformation respectively

4 Conclusions

In this preliminary study we developed time-harmonicweights to allow a uniform circular array to support TACANtransmission of bearing information We have shown howthose time-varying weights can be determined from theembedded element pattern Design and error calculations foran example circular array of Vivaldi elements suggest thatacceptable accuracy is feasible with reasonable arrays

Appropriate future work to expand upon these begin-nings includes examining performance over an appropriateelevation interval considering other array dimensions andnumbers of elements exploring other element geometriesand of course validating the theoretical development viameasurements Probably most important however is toexplore the effects of imperfect knowledge of the embeddedelement patterns

Disclosure

Jeffrey O Coleman collaborated with former colleagues atthe Naval Research Laboratory for this work he has actuallyretired from that laboratory

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the InTop program of the Office ofNaval Research

References

[1] E J Christopher ldquoElectronically scanned TACAN antennardquoIEEE Transactions on Antennas and Propagation vol 22 no 1pp 12ndash16 1974

[2] L N Shestag ldquoA cylindrical array for the TACAN systemrdquo IEEETransactions on Antennas and Propagation vol 22 no 1 pp 17ndash25 1974

[3] A Casabona ldquoAntenna for the ANURN-3 Tacan beaconrdquoElectrical Communications vol 33 pp 35ndash59 1956

[4] G W Hein J-A Avila-Rodriguez S Wallner B Eissfeller MIrsigler and J-L Issler ldquoA vision onnew frequencies signals and


concepts for future GNSS systemsrdquo in Proceedings of the 20thInternational Technical Meeting of the Satellite Division of theInstitute of Navigation (IONGNSS rsquo07) pp 517ndash534 FortWorthTex USA September 2007

[5] S J Foti M W Shelley R Cahill et al ldquoAn intelligent elec-tronically spinning TACAN antennardquo in Proceedings of the 19thEuropean Microwave Conference pp 959ndash965 IEEE LondonUK September 1989

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There are several steps Formally assuming the array to becircularly symmetric and requiring its pattern sampled at 119873equally spaced bearings to smoothly rotate in space with timeturns outmdashno surprisemdashto formally imply that the weightsmust also rotate so that only one weight requires explicitdesign That design follows from the desired temporal mod-ulation of the array-pattern amplitude along a single direc-tion The pattern modulation between the 119873 bearings thusaddressed explicitly takes the desired general form automat-ically with only pattern magnitude and signal modulationindices free to vary modestly (given reasonable assumptions)with bearing

211 The Array Center the 119873-element array on the originwith symmetry about the vertical axis and with element indi-ces increasing with bearing Align element 0 with bearing0∘ (any bearing can be made the new zero by changing thereference-burst timing) and interpret element indices mod-ulo119873 so that the elements adjacent to element 0 for examplecan be indexed with plusmn1 or 1119873 minus 1 In the developmentbelow each summation sum over index 119899 should be read asa sum over element indices 119899 = 0 119873 minus 1 and eachsummation sum over index ℓ should be read as the doublyinfinite sum over ℓ = minusinfin infin

Let k designate the real wavenumber vector of a trans-mitted signal and let complex vector-valued function

119891119899(k)

be the origin-referenced embedded far-field complex patternof element 119899 We assume elements are identical in the sensethat

119891119899(k) = R119899

1198910(Rminus119899k) (1)

for all k of interest where linear operation k 997891rarr Rk rotatesreal vector k about the vertical by 2120587119873 to increase bearingIdentity R119873 = R will be used freely

In practice imperfect array construction will result innonidentical embedded element patterns so the transmittedTACANwaveformwill vary somewhat from the ideal derivedhere We have yet to study such errors but hope to eventually

212 One Weight Implies the Others Write the time-varyingfar-field complex array pattern as

119891 (k 119905) = sum

119899

119908119899(119905)

119891119899(k) = sum

119899

119908119899(119905)R119899

1198910(Rminus119899k) (2)

using array symmetry (1) on the right A classic TACANsystemrsquos pattern rotates spatially at frequency 1119879 = 15Hzbut here we require that behavior only at 119873 equally spacedbearings Period 119879 rotation over 2120587119873 in angle is given by

119891 (Rk 119905) = R

119891 (k 119905 minus 119879

119873

) (3)

Substituting Rk for k in (2) and a change of index yield

119891 (Rk 119905) = sum

119899

119908119899(119905)R119899

1198910(R1minus119899k) (4)

119891 (Rk 119905) = sum

119899

119908119899+1

(119905)R119899+1 1198910(Rminus119899k) (5)

Likewise applying (2) to the right side of (3) yields

R 119891 (k 119905 minus 119879

119873

) = sum

119899

119908119899(119905 minus

119879

119873

)R119899+1 1198910(Rminus119899k) (6)

Substituting (5) and (6) into (3) and comparing terms thenformally show that 119908

119899+1(119905) = 119908

119899(119905 minus 119879119873) for all 119899 so

119908119899(119905) = 119908

0(119905 minus

119899119879

119873

) (7)

A rotating bearing-sampled pattern thus implies weightperiodicity 119908

0(119905) = 119908

119873(119905) = 119908

0(119905 minus 119879) This will not produce

rotation for all bearings but we will preserve property (7) forsimplicity of structure and in order to obtain nearly rotatingbehavior

213 DesiredModulation Thedesired complex array patternis an arbitrary constant complex amplitude modulated by

119898(119905 120601) = 1 + 2119886 cos(2120587119905119879

minus 120601)

+ 2119887 cos(21205879119905119879

minus 9120601)

(8)

where 120601 is bearing Positive real modulation indices 2119886 and2119887 are kept small enough that |119898(119905 120601)| = 119898(119905 120601) for simplereceiver demodulationThe terms at frequencies 1119879 and 9119879are respectively used for coarse and fine bearing measure-ment

The 120601 = 0 array pattern should be using arbitrary scaling

119898(119905 0) = 1 + 2119886 cos(2120587119905119879

) + 2119887 cos(21205879119905119879

) (9)

214 Determining Weight 1199080(119905) Let wavenumber vector

kimp and complex polarization unit vector 119888 govern co-polpropagation at 120601 = 0 at the most important elevation Usingsuperscripts to index coefficients the Fourier series of asso-ciated pattern sample ⟨

119891(kimp 119905) 119888⟩ and weight 1199080(119905) take

forms

⟨

119891 (kimp 119905) 119888⟩ = sumℓ

119888ℓe1198952120587ℓ119905119879 (10)

1199080(119905) = sum

ℓ

119908ℓe1198952120587ℓ119905119879 (11)

The co-pol array pattern at k = kimp is by (2) and (7)

⟨

119891 (kimp 119905) 119888⟩ = sum119899

⟨

119891119899(kimp) 119888⟩1199080 (119905 minus

119899119879

119873

) (12)

Fourier-series forms (10) and (11) and simple algebra thenyield

sum

ℓ

119888ℓe1198952120587ℓ119905119879 = sum

ℓ

119908ℓ

ℎℓe1198952120587ℓ119905119879 (13)

after defining DFT sum (periodically extended in ℓ)

ℎℓ

= sum

119899

⟨

119891119899(kimp) 119888⟩ e

minus1198952120587ℓ119899119873

(14)



⟨

119891 (kimp 119905) 119888⟩ = 119898 (119905 0)

= 1 + 119886e1198952120587119905119879 + 119886eminus1198952120587119905119879 + 119887e11989521205879119905119879

+ 119887eminus11989521205879119905119879

(15)


119888ℓ

= 119908ℓ

ℎℓ (16)


1199080(119905) =

1

ℎ0

+

119886

ℎ1

e1198952120587119905119879 + 119886

ℎminus1

eminus1198952120587119905119879 + 119887

ℎ9

e11989521205879119905119879

+

119887

ℎminus9

eminus11989521205879119905119879(17)


22 Performance


⟨

119891 (k 119905) ⟩ = sumℓ

119901ℓe1198952120587ℓ119905119879 (18)

119901ℓ

= 119908ℓ

sum

119899

⟨

119891119899(k) ⟩ eminus1198952120587ℓ119899119873 (19)


1199010

=

1

ℎ0

sum

119899

⟨

119891119899(k) ⟩

119901plusmn1

=

119886

ℎplusmn1

sum

119899

⟨

119891119899(k) ⟩ e∓1198952120587119899119873

119901plusmn9

=

119887

ℎplusmn9

sum

119899

⟨

119891119899(k) ⟩ e∓11989521205879119899119873

(20)


119901119896Σ=

119901119896

1199010

+ (

119901minus119896

1199010

)

lowast

119901119896Δ=

119901119896

1199010

minus (

119901minus119896

1199010

)

lowast

(21)

so that

119901119896

1199010

=

119901119896Σ+ 119901119896Δ

2

119901minus119896

1199010

=

(119901119896Σminus 119901119896Δ)lowast

2

(22)


⟨

119891 (k 119905) ⟩ = 1199010 (1 + 1

2

(1199011Σe1198952120587119905119879 + 119901lowast

1Σ

eminus1198952120587119905119879

+ 1199011Δe1198952120587119905119879 minus 119901lowast

1Δ

eminus1198952120587119905119879 + 1199019Σe11989521205879119905119879

+ 119901lowast

9Σ


9Δ

eminus11989521205879119905119879))

(23)


⟨

119891 (k 119905) ⟩ = 1199010 (1 + Re 1199011Σe1198952120587119905119879

+ Re 1199019Σe11989521205879119905119879 + 119895 Im 119901

1Δe1198952120587119905119879

+ 119895 Im 1199019Δe11989521205879119905119879)

(24)

or

⟨

119891 (k 119905) ⟩ = 1199010 (1 + 10038161003816100381610038161199011Σ

1003816100381610038161003816cos(2120587119905

119879

+ ang1199011Σ)

+10038161003816100381610038161199019Σ

1003816100381610038161003816cos(21205879119905

119879

+ ang1199019Σ)

+ 11989510038161003816100381610038161199011Δ

1003816100381610038161003816sin(2120587119905

119879

+ ang1199011Δ)

+ 11989510038161003816100381610038161199019Δ

1003816100381610038161003816sin(21205879119905

119879

+ ang1199019Δ))

(25)

Ideally |1199011Δ| and |119901


10038161003816100381610038161003816⟨

119891 (k 119905) ⟩1003816100381610038161003816

1003816=

10038161003816100381610038161003816119901010038161003816100381610038161003816

1003816100381610038161003816100381610038161003816

1 +10038161003816100381610038161199011Σ

1003816100381610038161003816cos(2120587119905

119879

+ ang1199011Σ)

+10038161003816100381610038161199019Σ

1003816100381610038161003816cos(21205879119905

119879

+ ang1199019Σ)

1003816100381610038161003816100381610038161003816

(26)


9Σ| are the


ang1199011Σ


ang1199019Σ

mod 2120587= minus9120601fine (28)






9120601diff2120587

+ 119898 =


2120587

(30)



2120587

)


9


(31)


1Δ=

1199019Δ


andang1199019Σ





9Σ+ 9120601 + 120587) mod 2120587))

9

(32)



3 Simulation





119891119899(k) 119888⟩ of the





119899Δ| and |119901

119899Σ|

559 cm

145 cm

159

cm


Co-

pol e

mbe

dded

elem

ent p

atte

rn 1

05

0

minus1

minus05

Bearing 120601

minus180∘ minus90∘ 0∘ 90∘ 180∘


1198910


119891119899



Bearing 120601

0∘ 90∘ 180∘

minus6

minus4

minus2

(dB)

0

05

tT

1




10

|1199010

|






DesiredSynthesized

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601

minus8

minus6

minus4

minus2

0

Nor

mal

ized

gai

n (d

Bp)


0006

0003

00

0002

0001

00

|p1Δ||p1Σ|

|p9Δ||p9Σ|

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601


119899Δ

||119901119899Σ


(dashed curve)

Tim

e Avg

gai

n (d

B) minus285

minus29

minus295

Mod

ulat

ion

indi

ces (

)

205

200

195

Bearing 120601

minus180∘ minus90∘ 0∘ 90∘ 180∘



01∘

0∘

minus01∘

Fine

erro

r

0∘

002∘

minus002∘Coa

rse e

rror

minus90∘ 0∘ 90∘ 180∘minus180∘

Bearing 120601


and 1199019Σ


4 Conclusions



Disclosure




Acknowledgment


References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











⟨

119891 (kimp 119905) 119888⟩ = 119898 (119905 0)

= 1 + 119886e1198952120587119905119879 + 119886eminus1198952120587119905119879 + 119887e11989521205879119905119879

+ 119887eminus11989521205879119905119879

(15)


119888ℓ

= 119908ℓ

ℎℓ (16)


1199080(119905) =

1

ℎ0

+

119886

ℎ1

e1198952120587119905119879 + 119886

ℎminus1

eminus1198952120587119905119879 + 119887

ℎ9

e11989521205879119905119879

+

119887

ℎminus9

eminus11989521205879119905119879(17)


22 Performance


⟨

119891 (k 119905) ⟩ = sumℓ

119901ℓe1198952120587ℓ119905119879 (18)

119901ℓ

= 119908ℓ

sum

119899

⟨

119891119899(k) ⟩ eminus1198952120587ℓ119899119873 (19)


1199010

=

1

ℎ0

sum

119899

⟨

119891119899(k) ⟩

119901plusmn1

=

119886

ℎplusmn1

sum

119899

⟨

119891119899(k) ⟩ e∓1198952120587119899119873

119901plusmn9

=

119887

ℎplusmn9

sum

119899

⟨

119891119899(k) ⟩ e∓11989521205879119899119873

(20)


119901119896Σ=

119901119896

1199010

+ (

119901minus119896

1199010

)

lowast

119901119896Δ=

119901119896

1199010

minus (

119901minus119896

1199010

)

lowast

(21)

so that

119901119896

1199010

=

119901119896Σ+ 119901119896Δ

2

119901minus119896

1199010

=

(119901119896Σminus 119901119896Δ)lowast

2

(22)


⟨

119891 (k 119905) ⟩ = 1199010 (1 + 1

2

(1199011Σe1198952120587119905119879 + 119901lowast

1Σ

eminus1198952120587119905119879

+ 1199011Δe1198952120587119905119879 minus 119901lowast

1Δ

eminus1198952120587119905119879 + 1199019Σe11989521205879119905119879

+ 119901lowast

9Σ


9Δ

eminus11989521205879119905119879))

(23)


⟨

119891 (k 119905) ⟩ = 1199010 (1 + Re 1199011Σe1198952120587119905119879

+ Re 1199019Σe11989521205879119905119879 + 119895 Im 119901

1Δe1198952120587119905119879

+ 119895 Im 1199019Δe11989521205879119905119879)

(24)

or

⟨

119891 (k 119905) ⟩ = 1199010 (1 + 10038161003816100381610038161199011Σ

1003816100381610038161003816cos(2120587119905

119879

+ ang1199011Σ)

+10038161003816100381610038161199019Σ

1003816100381610038161003816cos(21205879119905

119879

+ ang1199019Σ)

+ 11989510038161003816100381610038161199011Δ

1003816100381610038161003816sin(2120587119905

119879

+ ang1199011Δ)

+ 11989510038161003816100381610038161199019Δ

1003816100381610038161003816sin(21205879119905

119879

+ ang1199019Δ))

(25)

Ideally |1199011Δ| and |119901


10038161003816100381610038161003816⟨

119891 (k 119905) ⟩1003816100381610038161003816

1003816=

10038161003816100381610038161003816119901010038161003816100381610038161003816

1003816100381610038161003816100381610038161003816

1 +10038161003816100381610038161199011Σ

1003816100381610038161003816cos(2120587119905

119879

+ ang1199011Σ)

+10038161003816100381610038161199019Σ

1003816100381610038161003816cos(21205879119905

119879

+ ang1199019Σ)

1003816100381610038161003816100381610038161003816

(26)


9Σ| are the


ang1199011Σ


ang1199019Σ

mod 2120587= minus9120601fine (28)






9120601diff2120587

+ 119898 =


2120587

(30)



2120587

)


9


(31)


1Δ=

1199019Δ


andang1199019Σ





9Σ+ 9120601 + 120587) mod 2120587))

9

(32)



3 Simulation





119891119899(k) 119888⟩ of the





119899Δ| and |119901

119899Σ|

559 cm

145 cm

159

cm


Co-

pol e

mbe

dded

elem

ent p

atte

rn 1

05

0

minus1

minus05

Bearing 120601

minus180∘ minus90∘ 0∘ 90∘ 180∘


1198910


119891119899



Bearing 120601

0∘ 90∘ 180∘

minus6

minus4

minus2

(dB)

0

05

tT

1




10

|1199010

|






DesiredSynthesized

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601

minus8

minus6

minus4

minus2

0

Nor

mal

ized

gai

n (d

Bp)


0006

0003

00

0002

0001

00

|p1Δ||p1Σ|

|p9Δ||p9Σ|

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601


119899Δ

||119901119899Σ


(dashed curve)

Tim

e Avg

gai

n (d

B) minus285

minus29

minus295

Mod

ulat

ion

indi

ces (

)

205

200

195

Bearing 120601

minus180∘ minus90∘ 0∘ 90∘ 180∘



01∘

0∘

minus01∘

Fine

erro

r

0∘

002∘

minus002∘Coa

rse e

rror

minus90∘ 0∘ 90∘ 180∘minus180∘

Bearing 120601


and 1199019Σ


4 Conclusions



Disclosure




Acknowledgment


References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











9120601diff2120587

+ 119898 =


2120587

(30)



2120587

)


9


(31)


1Δ=

1199019Δ


andang1199019Σ





9Σ+ 9120601 + 120587) mod 2120587))

9

(32)



3 Simulation





119891119899(k) 119888⟩ of the





119899Δ| and |119901

119899Σ|

559 cm

145 cm

159

cm


Co-

pol e

mbe

dded

elem

ent p

atte

rn 1

05

0

minus1

minus05

Bearing 120601

minus180∘ minus90∘ 0∘ 90∘ 180∘


1198910


119891119899



Bearing 120601

0∘ 90∘ 180∘

minus6

minus4

minus2

(dB)

0

05

tT

1




10

|1199010

|






DesiredSynthesized

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601

minus8

minus6

minus4

minus2

0

Nor

mal

ized

gai

n (d

Bp)


0006

0003

00

0002

0001

00

|p1Δ||p1Σ|

|p9Δ||p9Σ|

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601


119899Δ

||119901119899Σ


(dashed curve)

Tim

e Avg

gai

n (d

B) minus285

minus29

minus295

Mod

ulat

ion

indi

ces (

)

205

200

195

Bearing 120601

minus180∘ minus90∘ 0∘ 90∘ 180∘



01∘

0∘

minus01∘

Fine

erro

r

0∘

002∘

minus002∘Coa

rse e

rror

minus90∘ 0∘ 90∘ 180∘minus180∘

Bearing 120601


and 1199019Σ


4 Conclusions



Disclosure




Acknowledgment


References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










DesiredSynthesized

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601

minus8

minus6

minus4

minus2

0

Nor

mal

ized

gai

n (d

Bp)


0006

0003

00

0002

0001

00

|p1Δ||p1Σ|

|p9Δ||p9Σ|

0∘ 90∘ 180∘minus90∘minus180∘

Bearing 120601


119899Δ

||119901119899Σ


(dashed curve)

Tim

e Avg

gai

n (d

B) minus285

minus29

minus295

Mod

ulat

ion

indi

ces (

)

205

200

195

Bearing 120601

minus180∘ minus90∘ 0∘ 90∘ 180∘



01∘

0∘

minus01∘

Fine

erro

r

0∘

002∘

minus002∘Coa

rse e

rror

minus90∘ 0∘ 90∘ 180∘minus180∘

Bearing 120601


and 1199019Σ


4 Conclusions



Disclosure




Acknowledgment


References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Transmit TACAN Bearing Information with a