FORWARD-LOOKING IMAGING OF SCANNING PHAS- ED ARRAY …

Progress In Electromagnetics Research, Vol. 143, 575–604, 2013

FORWARD-LOOKING IMAGING OF SCANNING PHAS-ED ARRAY RADAR BASED ON THE COMPRESSEDSENSING

Xiaoyang Wen*, Gangyao Kuang, Jiemin Hu, Ronghui Zhan,and Jun Zhang

College of Electronic Science and Engineering, National University ofDefense Technology, Changsha 410073, P. R. China

Abstract—In this paper, a novel forward-looking imaging methodbased on the compressed sensing is proposed for scanning phased arrayradar (PAR) in order to improve the azimuth resolution. Firstly,the echo of targets is modeled according to the principle of PAR.Then, it is analyzed why some of the former methods as multi-channeldeconvolution are ineffective based on the signal model. Using awidely accepted assumption that dominant scatterers in an interestingarea are sparse or compressible, an imaging algorithm based onthe compressed sensing is proposed and investigated. This methodobtains its high range resolution by transmitting and compressingchirp pulse signal, and improves its azimuth resolution by utilizingthe compressed sensing technique. The effectiveness of the proposedmethod is illustrated and analyzed with simulations data.

1. INTRODUCTION

Since synthetic aperture radar (SAR) has the characteristics of imagingan interesting area with high resolution in all weather conditions, itmoves radar technology from crude detection and estimation to fineresolution imaging and feature extraction [1–5]. Nowadays, it hasbecome an important detecting tool in the military reconnaissanceand civil remote sensing fields and has been widely used on airborneor missile-borne platforms. However, due to the limitations of itsprinciple, the conventional SAR can work well only when the radaris side looking or squint looking with a small squint angle. If the radar

Received 18 October 2013, Accepted 25 November 2013, Scheduled 6 December 2013* Corresponding author: Xiaoyang Wen ([email protected]).

576 Wen et al.

works with a large enough squint angle even under the forward-lookingcondition, it cannot provide images with high azimuth resolution. Sothe conventional SAR cannot be applied in certain scenarios, such asaircraft landing, which require the radar to image a forward swathrather than a sideward swath like a vehicle’s headlight [4, 6]. It meansthat the SAR exists an imaging blind zone which locates at the forwardof the platform.

In order to overcome this problem, many methods have beenproposed and these methods can be divided into two categoriesroughly. The first is that the platform maneuvers along a pre-designedtrajectory and lets the target area locate at the obliquely forward ofthe platform. The radar illuminates the target area with a high squintangle and synthetic aperture processing can be used to image the area.This method avoids the forward looking imaging problem. However,it can reduce but cannot eliminate the imaging blind zone. Especiallyat the end of the trajectory, the problem still exists. And this methodincreases demands for maneuverability of the platform. The secondcategory includes various systematic or numerical methods, which havebeen proposed to enable the radar to image the forward swath with anazimuth resolution to some extent. Commonly, these methods can becategorized into the following classes.

(1). Forward imaging SAR using a forward looking array as thesector imaging radar for enhanced vision (SIREV) which has beendeveloped at the German Aerospace Center [7–15]. The problem ofthis approach is that it has so complicated a system that when ahigh azimuth resolution is required, there will be an expensive andlarge-scale transformation to the existing systems. Even so, accordingto SAR, its azimuth resolution is constrained by the length of thesynthetic aperture, which is equal to the real aperture length of thearray, so this method is only suitable for landing of airplanes on whicha large scale antenna can be achieved, and unsuitable if the apertureis not large enough, such as a missile-borne radar.

(2). Monopulse forward imaging based on monopulse anglemeasurement. This method is investigated in many literatures [16, 17]and it has been used in some systems [18]. This method uses theadvantage of monopulse angle measurement which has high precision,but it cannot distinguish scatterers in the same range resolution cell.

(3). Forward imaging based on the technology of direction ofarrival (DOA) [19, 20]. If this method is used, multiple channelsare required, and the scatterers’ number must be smaller than thechannels’. It can be acquired that more scatterers mean more channelsand more complicated system. And more important is that high-volume data should be processed. If a broadband signal is used, the

Progress In Electromagnetics Research, Vol. 143, 2013 577

impact of bandwidth should be considered, which makes the algorithmmore complicated.

(4). Scanning radar forward-looking imaging based on deconvolu-tion [21–23]. Research of this method can be traced back to 1980s,and a lot of papers have been presented. Recently, because of theincreasingly urgent requirement for the forward-looking imaging, itattracts people’s attention again. Because the antenna pattern is alow-pass system and the single-channel deconvolution in the frequencydomain cannot work well in the singular region, a multi-channel de-volution method [24] in the frequency domain is proposed, which uti-lizes the characteristic of monopulse radar which has a sum and adifference channel. This method also includes some modified forms,such as deconvolution in the time domain [25], least square estima-tion (LSE) [26, 27], et al. Limitations of these approaches are that theimaging result is sensitive to the signal-to-noise ratio (SNR) and whatmatters more is that absolution of the antenna pattern is used in theformer researches. But according to antenna theory, it is not the fact. Ifthe phase of the antenna pattern is considered, its bandwidth in whicharea its spectrum is not equal to zero is finite and approximately equalto the electrical length of the antenna, so the azimuth resolution isapproximately equal to the beam width of the antenna pattern, whichwill be investigated in detail in the paper. When deconvolution intime domain or LSE is used, the difficulty will be encountered becausethe sensing matrix is of a great condition number even ill-conditioned.The rank of the measurement matrix is much smaller than the azimuthresolution cells’ number, so the equations will have many solutions. Ifsome regularization technologies are adopted, the rank of the measure-ment matrix will be increased and its singularity will be eliminated.However, the equations will be changed seriously and the correct solu-tion cannot be obtained.

In addition, there are some other forward-looking imagingmethods, such as scanned angle/time correlation (SATC) [28] andbistatic forward-looking SAR imaging [29–37]. The former is not givendetailed in the literature, and an important precondition of this methodwhich is difficult to achieve is that the antenna pattern should maintaina fixed phase relationship with the transmitted signal during theazimuth scanning process, and what is more important is that echoesfrom different directions do not satisfy the translation invariance,These problems constrain the use of the method for forward-lookingimaging. The latter is a promising method for forward-looking imagingwith high resolution, but the difficulty is the synchronization of thereceiver and the transmitter.

Now, more and more phased array radar (PAR) systems are

578 Wen et al.

used on airborne or missile-borne platform, and sometimes, becauseof the limitations of the platform size, data storage capacity andcosts, multiple channels cannot be achieved and the sensor has onlytwo channels — the sum and the difference channel, which is namedas monopulse PAR, so in this paper, the forward-looking imagingof monopulse PAR is considered. Signal model of an azimuth-scanning PAR is established according to its principle. Limitations ofdeconvolution forward-looking imaging algorithm are investigated indetail through theoretical analysis and experiments. A novel forward-looking imaging method is proposed based on compressed sensing(CS), which obtains high range resolution by pulse compression ofa broadband signal and azimuth discrimination in the same rangeresolution cell by an optimum estimation based on CS for a sparsescene. After pulse compression, data of the same range cell atdifferent azimuth scanning angles constitute the azimuth echo signal,and they are the sums of multiple targets’ echoes from different azimuthdirections which are weighted by the antenna pattern. Therefore,estimation of the targets’ amplitudes and azimuths from the azimuthecho signal is an inverse problem. In most cases, inverse problemsare underdetermined and ill-posed. And it is difficult to obtain anaccurate and stable solution. Tikhonov regularization method [38, 39]can eliminate isolated singular points of the inverse problem. However,if the system function is equal to zero in a large continuous area of thefrequency domain, such as an antenna pattern, it is difficult to getthe correct answer. Compressed sensing can be used to resolve inverseproblems [40], and it has been applied in radar imaging [41–44]. Butit is mainly used to reduce the sampling pressure. If the sparsity ofthe scene is used as the regularization condition and CS is applied toforward-looking imaging, image resolution can be improved.

This paper is organized as follows. In Section 2, the signalmodel is established based on the principle of monopulse PAR. InSection 3, some deconvolution forward-looking imaging methods areanalyzed in detail, which include the multi-channel deconvolution andthe LSE method. The inefficiency of these methods is verified throughtheoretical analysis and simulations. Based on the prior informationthat the target area is sparse or compressible, a novel method basedon compressed sensing is proposed, and performance of the methodis established. In Section 4, some numerical simulation results arepresented, and performance of the novel method is compared with thedeconvolution imaging methods, and system parameters’ impact alsois analyzed.


2. ESTABLISHMENT OF THE SIGNAL MODEL

Suppose that the operating frequency is f0 and correspondingwavelength λ0. A forward-looking uniform linear PAR is constitutedby Mx elements. The space between two adjacent elements is dx. Achirp pulse is emitted during a single pulse repetition interval, itsinstantaneous bandwidth is Br and its time duration is Γ. In orderto simplify the problem, PAR remains stationary during an imagingperiod, thus impact of the motion can be ignored, and the chirp’sbandwidth is so small that effects of aperture filled phenomenon canbe ignored and azimuth scanning is achieved through phase-shiftedmethod. Signals from the sum and the difference channels are obtainedthrough weighting the signal from each element. The monopulse PAR’sstructure is shown in Fig. 1.

s s

Transmitter

Σ

Σ

Σ ∆

Figure 1. Structure diagram of a monopulse PAR.

Let the antenna phase center (APC) locate at the geometry centerof the array. When the radar is working for forward-looking imaging,the beam of the antenna will scan the target area. If the beam’sdirection is θ0

i , the signal emitted from the mx-th element can beexpressed as

smx(i, tf )=rect(

tfΓ

)exp

{j2πf0tf +jπγt2f

}exp

{−j2πf0

mxdx sin θ0i

c

}

(1)where i is the index number of a probe, and tf is the fast timewhose origin is located at the start of a pulse repeated interval as

580 Wen et al.

many literatures have been explained about synthetic aperture radar.γ = Br/Γ is the chirp rate. If the k-th scatterer locates at (Rk, θk)and its amplitude is Ak, the distance between the scatter and themx-th element is equal to Rk,mx = Rk − mxdx sin θk approximately,where Rk is the distance between the scatterer and the APC, θk is itsazimuth angle. Time delay between the scatterer and the element isτk,mx = Rk,mx/c = Rk/c−mxdx sin θk/c = τk− τmx , so the echo whichis collected by the p-th element is given as follows

sk,px(i, tf ) = Ak exp{−j2πf0

pxdx sin θ0i

c

}∑mx

exp{−j2πf0

mxdx sin θ0i

c

}

rect(

tf − τk,mx−τk,px

Γ

)exp

{j2πf0 (tf−τk,mx−τk,px)+jπγ (tf−τk,mx−τk,px)2

}(2)

where τpx = pxdx sin θk/c, τk,px = Rk,px/c = Rk/c − pxdx sin θk/c =τk − τpx . In order to form the sum and difference antenna patternswith lower sidelobes, Taylor and Bayliss series are used to weightsignals from these elements [45]. Supposing that the weighting seriesare aΣ = [aΣpx ] and a∆ = [a∆px ] respectively, the signals which arecollected by the radar are the weighted summation of the signal fromeach element and can be written as

sΣk (i, tf ) =∑px

{aΣpxsk,px (i, tf )} (3)

s∆k (i, tf ) =∑px

{a∆pxsk,px (i, tf )} (4)

If there are K scatterers in the target area, the signal can beexpressed as

sΣ (i, tf ) =∑

k

sΣk (i, tf ) (5)

s∆ (i, tf ) =∑

k

s∆k (i, tf ) (6)

Based on the previous assumption, effects of the aperture filledphenomenon are ignored, and approximately, the signal from the sumchannel can be rewritten as

sΣ(i, tf ) ≈∑

k

AkgΣ

(θ0i , θk

)rect

(tf − 2τk

Γ

)

exp{j2πf0(tf − 2τk) + jπγ(tf − 2τk)2

}(7)


where gΣ(θ0i , θk) is the gain of the sum channel antenna pattern which

is given as

gΣ

(θ0i , θk

)

= gΣR

(θ0i , θk

)gT

(θ0i , θk

)=

∑px

aΣpx exp

{j2πf0

pxdx

(sin θk − sin θ0

i

)

c

}

∑mx

exp

{j2πf0

mxdx


i

)

c

}(8)

Similarly, the signal from the difference channel can be expressed as

s∆ (i, tf ) ≈∑

k

Akg∆

(θ0i , θk

)rect

(tf − 2τk

Γ

)

exp{

j2πf0 (tf − 2τk) + jπγ (tf − 2τk)2}

(9)

where g∆(θ0i , θk) is the gain of the difference channel antenna pattern

which is given as

g∆

(θ0i , θk

)

= g∆R

(θ0i , θk

)gT

(θ0i , θk

)=

∑px

a∆px exp

{j2πf0

pxdx

(sin θk−sin θ0

i

)

c

}

∑mx

exp

{j2πf0

mxdx


i

)

c

}(10)

Since signal processing of the sum and the difference channel arethe same, only processing of the sum channel is given below. Afterdown-conversion and pulse compression, the signal of the sum channelcan be expressed as

sΣ(i, tf )≈∑

k

Ak exp {−j4πf0τk} sinc {Br (tf−2τk)} gΣ

(θ0i , θk

)(11)

Then azimuth resolution is obtained through processing the signalin the same range resolution cell which is formed by multiple probes.Herein, the number of probes is I. The echo collection model isillustrated in Fig. 2.

The length of the antenna pattern is defined as the region in whichthe gain is significantly larger than zero, and here is N = 2K+1, whichis shown in Fig. 2(a) as a bold solid line. Echo collection process isillustrated in Fig. 2(b). It shows that the scanning process is a one ofweighted summation to the echoes from different targets in the same

582 Wen et al.

range resolution cell. Thus, after pulse compression the total signalalso can be expressed in matrix form as

SΣ = GΣX (12)

If noise is considered, it can be rewritten as

SΣ = GΣX + WΣ (13)

If sampling points’ number along the fast time is L, dimensionof SΣ is I × L. Each row is a high range resolution profile of eachprobe, and each column has the same radial distance and constitutesthe echo of the ring. Rows’ number of GΣ is I, and the i-th row isthe sampling of the antenna pattern of the i-th probe. It should benoted that the antenna pattern is changing with the beam’s direction,which is different from mechanical rotating radar. Sampling range isfrom θ0

0 − Kδθ to θ0I−1 + Kδθ, which is the total antenna patterns

coverage of the whole scanning process and the number of samplingpoints is Q. Here, sampling interval of the antenna pattern is assumedas δθ. It should be noted that, the sampling interval of the antennapattern is not necessarily equal to the step between two adjacentprobes which is denoted as ∆θ. Thus, the i-th row of GΣ can bewritten as a row vector gΣi = [gΣ(θ0

i , θq)], where θq is a sampling pointin [θ0

0 −Kδθ, θ0I−1 + Kδθ], and the length of gΣi is Q. The rows’

number of X is equal to Q, which corresponds to the length of gΣi,and its columns’ number is equal to L, which corresponds to the rangesampling number. Expressions of SΣ, GΣ, X are given as follows

SΣ =

sΣ (0, tf,0) sΣ (0, tf,1) . . . sΣ (0, tf,L−1)sΣ (1, tf,0) sΣ (1, tf,1) . . . sΣ (1, tf,L−1)

......

. . ....

sΣ (I − 1, tf,0) sΣ (I − 1, tf,1) . . . sΣ (I − 1, tf,L−1)

GΣ =

gΣ

(θ00, θ0

)gΣ

(θ00, θ1

). . . gΣ

(θ00, θQ−1

)

gΣ

(θ01, θ0

)gΣ

(θ01, θ1

). . . gΣ

(θ01, θQ−1

)...

.... . .

...gΣ

(θ0I−1, θ0

)gΣ

(θ0I−1, θ1

). . . gΣ

(θ0I−1, θQ−1

)

X =

A (θ0, tf,0) A (θ0, tf,1) . . . A (θ0, tf,L−1)A (θ1, tf,0) A (θ1, tf,1) . . . A (θ1, tf,L−1)

......

. . ....

A (θQ−1, tf,0) A (θQ−1, tf,1) . . . A (θQ−1, tf,L−1)

where, tf,l (l = 0, 1, . . . , L − 1) is each sampling instant along thefast time, A(θq, tf,l) is the amplitude of the scatterer at (θq, ctf,l/2)


g

0

i

0, i

-K K

00

Scanning DirectionStart Line

End

Lin

e

Interesting Area

Scanning Edge

Antenna Pattern

00

0

1I

00

K

0 1I

K

Scanning Edge

Antenna Pattern

-K

(θ θ )

θ θ

θ< >

−

θ

θ

θ

δθ

+−

θ

δθ

θ

(a) (b)

-θ

Figure 2. Signal received process. (a) Truncated antenna pattern.(b) Signal collection process of the scanning radar.

with the impact of the pulse compression, which contains a delayedphase determined by the radial distance. Obviously, it can be seenthat X is a mesh dissection of the target area. In case that X couldbe recovered through signal processing, finer dissection means higherresolution. Because the antenna pattern has a certain beam width,beam coverage is larger than the scanning area, which is to say thatthe range from θ0

0 − Kδθ to θ0I−1 + Kδθ is larger than that from θ0

0

to θ0I−1, which is illustrated in Fig. 2(b), and commonly in order to

improve the resolution, δθ is smaller than ∆θ, so Q > I, and it meansthat Equations (12) and (13) are underdetermined, and the number ofunknowns is larger than the equations’ number.

3. IMAGING ALGORITHM

If Equations (11), (12) and (13) can be solved, a high resolutionimage can be achieved. The antenna pattern of the PAR changeswith the beam’s direction which is different from the mechanicalrotating radar, and from the point of signal and system, it meansthat the antenna pattern is a linear time-varying system. The azimuthecho is a result of the target area information passing through thislinear time-varying system. We cannot use the linear deconvolutionmethod to solve this problem. However, if the scanning range is smallenough, the antenna pattern can be considered as a constant andapproximately time-invariant system. Deconvolution in the frequencydomain is highly time-efficient, which is benefit for achieving real-timeimplementation. If this algorithm could be used to obtain high azimuth

584 Wen et al.

resolution, it was a promising method for forward-looking imaging, andperformance of the scanning radar would be improved significantly.Because the antenna pattern of a single channel has singular areas,multi-channel deconvolution has been proposed. This method utilizesthe characteristic that a monopulse radar has two channels, the sumand the difference channel. Unfortunately, it is inefficient for azimuthresolution improvement, which will be verified below. In addition, twochannels can double the number of the equations while the numberof unknowns keeps constant, so LSE has been proposed for solvingequations. Some results of LES is presented, which shows that this isinefficient, and the reason is analyzed. Based on the analysis about theecho collected by the PAR, a novel forward-looking imaging methodusing compressed sensing is proposed.

3.1. Analysis of the Two-channel DeconvolutionForward-looking Imaging Method

For a scanning monopulse radar, the received echo can be modeled as

sΣ (θ, tf ) = x (θ, tf )⊗ gΣ (−θ) + wΣ (θ, tf ) (14)s∆ (θ, tf ) = x (θ, tf )⊗ g∆ (−θ) + w∆ (θ, tf ) (15)

where x(θ, tf ) is the targets’ distribution on the ring distance fromwhich to the APC is ctf/2, gΣ(θ) and g∆(θ) are the sum andthe difference channel antenna pattern respectively. wΣ(θ, tf ) andw∆(θ, tf ) are noise, and the symbol ‘⊗’ means the convolution of twofunctions. Result of two-channel deconvolution is given as

X(Θ, tf )

=G∗

Σ(−Θ)SΣ(Θ, tf )/W 2Σ(Θ, tf )+G∗

∆(−Θ)S∆(Θ, tf )/W 2∆(Θ, tf )

G∗Σ(−Θ)GΣ(−Θ)/W 2

Σ(Θ, tf )+G∗∆(−Θ)G∆(−Θ)/W 2

∆(Θ, tf )(16)

If noise is not taken into account or the noise’s spectrums of twochannels are the same, Equation (16) can be rewritten as

X (Θ, tf ) =G∗

Σ (−Θ)SΣ (Θ, tf ) + G∗∆ (−Θ)S∆ (Θ, tf )

G∗Σ (−Θ)GΣ (−Θ) + G∗

∆ (−Θ)G∆ (−Θ)(17)

where X(Θ, tf ), SΣ(Θ, tf ), S∆(Θ, tf ), GΣ(−Θ), G∆(−Θ) are theFourier transformations of x(θ, tf ), sΣ(θ, tf ), s∆(θ, tf ), gΣ(−θ) andg∆(−θ) respectively. W 2

Σ(Θ, tf ) and W 2∆(Θ, tf ) are the power

spectral density functions of wΣ(θ, tf ) and w∆(θ, tf ) respectively.The superscript ‘*’ means complex conjugate. The inverse Fouriertransformation of X(Θ, tf ) is the scatterers’ distribution. From theupper analysis, some conclusions can be obtained. Singular areas ofX(Θ, tf ) are determined by GΣ(−Θ) and G∆(−Θ), i.e., zero points


of GΣ(Θ) and G∆(Θ). According to the principle of sum-differenceamplitude-comparison monopulse technology, gΣ(θ) and g∆(θ) can beexpressed as

gΣ (θ) = g (θ −∆θ) + g (θ + ∆θ) (18)g∆ (θ) = g (θ −∆θ)− g (θ + ∆θ) (19)

where g (θ −∆θ) and g (θ + ∆θ) are two sub-beams’ antenna patternswhich are used to form the sum channel antenna pattern and thedifference channel antenna pattern, respectively. ∆θ is half of theangular interval between these two sub-beams. Fourier transformationsof gΣ (θ) and g∆ (θ) are given by

GΣ(Θ) = 2 cos(2πΘ∆θ)G(Θ) (20)G∆(Θ) = −j2 sin(2πΘ∆θ)G(Θ) (21)

It shows that zero points of G(θ) also are zero points of GΣ(Θ)and G∆(Θ). If the antenna surface current intensity is I(u) and theaperture length is L, the antenna pattern can be expressed as

g (θ) =

L/2∫

−L/2

I (u) exp{−j2πf0

u sin θ

c

}du (22)

Generally, L is much larger than λ0 = c/f0 and the beam is sonarrow that Equation (22) can be approximated as follows

g (θ) ≈L/2∫

−L/2

I (u) exp{−j2πf0

uθ

c

}du (23)

So G(Θ) can be expressed as

G(Θ)=

π∫

−π

L/2∫

−L/2

I(u) exp{−j2πf0

uθ

c

}exp {−j2πΘθ} dudθ

=I

(−cΘ

f0

)(24)

Since I(u) is non-zero only in [−L/2, L/2], G(Θ) will be non-zero only under the condition that |Θ| ≤ Lf0/(2c), which meansthat the value of X(Θ, tf ) can be recovered only when it is in[−Lf0/(2c), Lf0/(2c)], and the length of this area is L/(c/f0), so theazimuth resolution is (c/f0)/L which is equal to the beam width of one-way antenna pattern. Similarly, the azimuth resolution of the round-trip antenna pattern is (c/f0)/(2L) which is equal to the beam width

586 Wen et al.

of the round-trip antenna pattern. Exact Fourier transformation ofg(θ) can be expressed as

G(Θ) =

π∫

−π

g(θ) exp {−j2πΘθ} dθ

= 2πλ0

L/(2λ0)∫

−L/(2λ0)

I(v)J2πΘ(−2πv)dv (25)

where J2πΘ(−2πv) is the first kind Bessel function. It shows that, G(Θ)is a Bessel integral and it is difficult to obtain its accurate expression,but its change rule can be found by numerical analysis. The numericalintegral of G(Θ) can be expressed as

G (Θ) = 2πλ0

L/(2λ0)∑

−L/(2λ0)

I (v) J2πΘ (−2πv)∆v (26)

Suppose that the antenna current distribution is uniform andelectrical length of the aperture L/λ0 equal to 20, 40 or 80, whichmeans v = [−10, 10], [−20, 20] or [−40, 40], respectively. Thenumerical integral is illustrated in Fig. 3. It shows that, if Θ is largerthan L/λ0, the value of G(Θ) will decline to zero rapidly, so multi-channel deconvolution in the frequency domain cannot be used to theforward-looking imaging.

The function |sinc(·)| is often used as an approximation of theantenna pattern for forward-looking imaging in the existing literatures.An instance is given below. Supposing that every sub-beam’s width is4◦, and 2∆θ = 0.4◦, rotating frequency of the antenna is 6 rounds perminute and the scanning area ranges from −15◦ to 15◦. A chirp pulseis emitted in each pulse repeated interval, the operating frequencyis 10 GHz, Br = 150 MHz, its pulse time duration is 5µs and thepulse repeated frequency is 1000 Hz. Distribution of the scatterers isillustrated in Fig. 4, and these amplitudes are all equal to 1. Imagingresult with high resolution of multi-channel deconvolution is shownin Fig. 5(a). Every scatterer can be discriminated. However, if thefunction sinc(·) which is closer to the real situation is used, the imagingresult is shown in Fig. 5(b). It shows that high azimuth resolutioncannot be achieved. The reason is that |sinc(·)| has a wider and richerfrequency domain information than sinc(·) and the bandwidth of sinc(·)is limited. However, it can be deduced from above analysis that |sinc(·)|cannot be realized.

Above theoretical analysis and simulations prove that multi-channel deconvolution cannot be used to improve azimuth resolution of


0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45G

()/

(2o)

Half of the Electrical Length

0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


0 20 40 60 800

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


(a) (b)

(c)

Θπλ

G(

)/(2

o)

Θπλ

G(

)/(2

o)

Θπλ

Θ Θ

Θ

Figure 3. Bessel integration with different electrical length.(a) Electrical length equal to 20. (b) Electrical length equal to 40.(c) Electrical length equal to 80.

1900 1950 2000 2050 2100-100

-50

0

50

100

Azimuth Range (m)

Sla

nt

Ra

ng

e (

m)

Figure 4. Distribution of the scatterers (illustrated as black dots).

the image formed by the antenna-rotating radar. In fact this derivationfor phased array radar is also true. If scanning area is not large enough,the antenna pattern can be considered approximately as a constant.

588 Wen et al.o

1900 1950 2000 2050 2100

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

1900 1950 2000 2050 2100

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Slant Range (m) Slant Range (m)

Azim

uth

An

gle

(

)

oA

zim

uth

An

gle

( )

(a) (b)

Figure 5. Imaging result using different antenna patterns.(a) Imaging result using the |sinc| antenna pattern. (b) Imaging resultusing the sinc antenna pattern.

-10 -5 0 5 10-1

-0.5

0

0.5

1

1.5

Azimuth Angle (o)

No

rma

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Figure 6. Antenna pattern of the monopulse phased array radar.

Although the principle of monopulse PAR has little difference with themonopulse mechanical rotating radar, the essence is the same, Fouriertransformation of the PAR antenna pattern is also equal to the antennacurrent distribution density approximately, and its bandwidth also isdetermined by the electrical length of the aperture. For an instance,supposing that the operating frequency f0 is 35 GHz and that thenumber of elements is 25, the distance between two adjacent elementsis 0.6 times of the wavelength λ0; chirp signal is used, Br = 80 MHz,Γ = 5µs; scanning area ranges from −15.6◦ to 15.6◦ with a constantstep; Taylor and Bayliss series are used to form the sum and thedifference channel antenna pattern, respectively. The antenna patternsare shown in Fig. 6 when the beams are pointing along the normal lineof the array.

Antenna patterns at all the time and these Fourier transformations


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Figure 7. Antenna pattern and these spectrums. (a) Antennapatterns of the sum channel. (b) Antenna patterns’ spectrums ofthe sum channel. (c) Antenna patterns of the difference channel.(d) Antenna patterns’ spectrums of the difference channel.

are shown in Fig. 7. It shows that bandwidths of the sum channelantenna patterns and the difference channel antenna patterns areextremely limited, therefore, two-channel deconvolution in frequencydomain is inefficient when it is used to improve azimuth resolutionof the forward-looking image. The reason is that the Fouriertransformations of the antenna patterns have a finite bandwidth. Inorder to overcome this shortcoming, the existing methods use someregularized measurement for deconvolution, such as supposing that theamplitudes of scatterers is positive. However, after pulse compressionalong the fast time, the amplitudes include the phases which areformed by the fixed distance between the targets and the radar, andthese phases cannot be compensated because these positions cannotbe obtained accurately enough. It can be seen that these simpleregularized measurements are inefficient.

590 Wen et al.

3.2. Analysis of the LSE Forward-looking Imaging Method

If the scanning area is so large that the change of the antennapattern cannot be ignored, deconvolution method cannot be utilizedfor forward-looking imaging. So another methods based on the matrixform of the echo which is given by Equations (12) and (13) areproposed, and one of these methods is the LSE, which means thatX is estimated from SΣ according to the least square criterion. Fromthe above description about the model, we know that the number ofunknowns is larger than the number of equations, and LSE can beused to solve the problem and achieved an optimum solution. If theradar is a monopulse one which has two channels, LSE also can beused to image. At this time, S = GX + W, where S =

[ST

Σ ST∆

]T ,G =

[GT

Σ GT∆

]T and the superscript ‘T ’ means transpose operation.As described above, the number of equations will be doubled while thenumber of unknowns keeps constant, and the number of equations willbe larger than that of unknowns. Imaging results of the LSE methodcan be expressed as

X =(GHG

)−1GHS (27)

where the superscript ‘H’ indicates conjugate transpose. Dimension ofGHG is Q ×Q. Since strong correlations between any two equationsmay be presented, the rank of G will be much smaller than the columnnumber, which implies that most of the eigenvalues of GHG are equalto zeros, and GHG is irreversible. In order to eliminate its singularity,Tikhonov regularization method is adopted. And the result of themodified LSE is given by

X =(GHG + λE

)−1GHS (28)

where λ is a small positive value, and E is an identity matrix whosedimension is Q×Q. The singularity is eliminated while the equationsare changed significantly and the real solution cannot be achieved.For an instance, if the radar parameters are the same with the abovesimulation which is used in the last of Section 3.1, the number ofscanning steps is 625 from −15.6◦ to 15.6◦, and the scanning stepbetween two adjacent probes is 0.05◦. The antenna pattern coveragearea is from −15.6− 8.5 = 24.1◦ to 15.6 + 8.5 = 24.1◦, that is, targetsfrom −24.1◦ to 24.1◦ are considered. If the sampling interval of theantenna pattern also is 0.05◦, the sampling number of the antennapattern will be 965, and the number of unknowns is 965. If two channelsare used, there will be 625 × 2 = 1250 equations. It shows that thenumber of the equations is larger than that of unknowns. Scatterers’distribution is illustrated in Table 1.


Table 1. Scatterers distribution in the area.

No. Slant Range/m Azimuth/◦ Amplitude1 1995 0.5 12 1995 0 13 1995 −0.5 14 2000 0.5 15 2000 0 16 2000 −0.5 17 2005 0.5 18 2005 0 19 2005 −0.5 1

The condition number of G which is calculated using Matlab isequal to 1.7944×1018, and the rank of GHG is 30. If λ = 1×10−10, realbeam scanning result and imaging result based on the LSE are shownin Fig. 8. Spots in the figure are the real positions of the scatterers.It shows that the image resolution is improved but the extent is non-significant.

Several simulations validate the conclusion. The reason is thatthe scanning step is so small that two adjacent rows of GΣ or G∆ areapproximately equal to each other. And the LSE method cannot solvethe problem efficiently.

3.3. forward-looking Imaging Based on Compressed Sensing

Above theoretical analysis and numerical simulations show that neithertwo-channel deconvolution nor the LSE can improve azimuth resolutionof the real beam scanning image efficiently. The reason is thatin order to solve the inverse problem and improve the azimuthresolution of the image, priori information is essential. If it couldbe used efficiently, forward-looking imaging with high resolution maybe possible. An important priori information is that scatteringcharacteristic of the target area in the optical area is equivalent tothe coherent superposition of a number of strong scatterers, thesescatterers are only a small part of the area, and amplitudes of theother area are equal to zero approximately. That is to say, the sceneis sparse or compressible. Specifically, most of elements of X are equalto zero, and only few of them have non-zero values, which implies thatthe inverse problem is a one about sparse signal reconstruction andcompressed sensing method could be able to solve the problem. After

592 Wen et al.

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Figure 8. Imaging result of the LSE method. (a) Result of the sumchannel. (b) Result of the difference channel. (c) Image by the leastsquare estimation. (d) Azimuth resolution of a target at 2000 m alongthe slant range.

pulse compression, the u-th column could be abstracted from SΣ andX, and a novel equation is obtained which is given by

su = GΣxu + wu (29)

where wu indicates the impact of noise, su the u-th column of SΣ, andxu the u-th column of X which indicates the targets’ distribution onthe u-th isometric ring, and its length is Q. Suppose that there are kscatterers on the ring. Because of the sparsity of the scene, k is muchsmaller than Q, and xu is a sparse vector with the order equal to k. Letθ0 = θ0

0−Kδθ, θQ−1 = θ0I−1 +Kδθ, θq = θ0 +qδθ (q = 0, 1, . . . , Q−1).

Without loss of generality, we can assume that the relationship betweenthe scanning step ∆θ and the sampling interval of the antenna patternδθ is ∆θ = Jδθ, so θ0

i = θ00 + i∆θ = θ0

0 + iJδθ, i = 0, 1, . . . , I − 1,


and GΣ can be expressed as GΣ = [gΣ0 gΣ1 . . . gΣQ−1 ], wheregΣq =

[gΣ

(θ00, θq

)gΣ

(θ01, θq

). . . gΣ

(θ0I−1, θq

)]T , and in order tobring (29) to a CS scheme, a new vector is constructed which is givenby

g′Σq =[gΣ

(θ00, θq

), gΣ

(θ00 + δθ, θq

), . . . , gΣ

(θ00 + (J − 1) δθ, θq

),

gΣ

(θ00 + Jδθ, θq

), . . . , gΣ

(θ00 + (I − 1)Jδθ, θq

)]T

Thus we have a dense dictionary as follows:G′

Σ =[g′Σ0 g′Σ1 . . . g′ΣQ−1

]

It shows that GΣ = DG′Σ, where D is given by

D =

1 0 . . . 0 0 . . . 00 0 . . . 1 0 . . . 0...

.... . .

......

. . ....

0 0 . . . 0 0 . . . 1

Dimension of D is I × [(I − 1)J + 1], only the iJ-th element ofthe i-th row is equal to 1, and the others are all equal to 0, and thatis to say, the matrix D chooses the columns of G′

Σ whose columnsindex number is 0, J , iJ , . . . , or (I − 1)J . So a new equation can beobtained which is written as follows:

su = DG′Σxu + wu = GΣxu + wu (30)

It shows that su is the sparse sampling of G′Σxu which is impacted

by noise. GΣ = DG′Σ is the sparse measurement matrix of a

compressed sensing. Forward-looking Imaging based on compressedsensing is to estimate xu based on the Equation (30) under thecondition that the scene is sparse or compressible. Above analysisshows that xu is a sparse representation on the overredundant basis{gΣq| q = 0, 1, . . . , Q− 1}, where gΣq is named as an atom. Accordingto the compressed sensing, xu can be recovered by solving the followingminimization problems which is expressed as:

(P0) min ‖xu‖0 s.t. ‖su −GΣxu‖2 ≤ ρ (31)where, ‖x‖0 indicates the number of non-zero values in the vector,i.e., the sparsity order of the vector. ρ is the error parameterwhich indicates the impact of noise. Because the minimization isan NP-hard problem, its approximate solution is considered and theminimization problem will be relaxed. It has been verified that undersome conditions, the problem can be converted from the nonconvexoptimization to a convex one which can be solved using the linearprogramming technique. And the problem can be relaxed as follows:

(P0) min ‖xu‖1 s.t. ‖su −GΣxu‖2 ≤ ρ (32)

594 Wen et al.

Using the famous Laplace method, the problem can be rewritten as(P λ

1

)xu = arg min {‖xu‖1 + λ ‖su −GΣxu‖2} (33)

where λ is a positive value, and it is determined by the powerof noise. In order to obtain correct reconstruction results, somerequirements are placed on the sensing matrix. As mentioned above,one sufficient condition is known as the restricted isometric property(RIP) of the sensing matrix. If the sensing matrix GΣ satisfies RIPand I ≥ O(k · log Q), xu can be recovered exactly with overwhelmingprobability. A matrix GΣ is said to satisfy the RIP of order k if thereexists a constant εk ∈ (0, 1) such that

(1− εk) ‖x‖22 ≤ ‖GΣx‖2

2 ≤ (1 + εk) ‖x‖22 (34)

holds for all ‖x‖0 ≤ k. εk is named as restricted isometric constant(RIC). If the sensing matrix satisfies the RIP with appropriate orderand constant, the solution of the (P1) problem in Equation (31) willbe equivalent to that of the (P0) problem in Equation (32), and the(P λ

1 ) problem can also recover sparse signal in noise stably. RIPis broadly viewed as a sufficient criterion for evaluating whether adictionary behaves like an isometry system in CS recovery. However,for a special dictionary, in general, it is not easy to test whether GΣ

satisfies RIP. Herein, a method based on eigenvalue statistics which hasbeen described in [46] is adopted. After scaling, columns of GΣ haveunit norm, and it is explicit that GΣ obeys-the RIP of order k whenGT

ΣkGΣk has eigenvalues sufficiently within (1− δk, 1 + δk). Herein,GΣk is constructed by abstracting k columns from the scaled GΣ, andthe index number of columns are random, so there will be many sets.Each set will have its own maximum eigenvalue ek max and minimumeigenvalue ek min. The means of ek max and ek min can be obtained whichare denoted as ek max and ek min. If ek max and ek min are all within(1− εk, 1 + εk), we can speculate that GΣ satisfies the RIP of orderk. In the experiment, supposing that the radar scans from −15.6◦ to15.6◦ with a uniform step, coverage of the scanning processing rangesfrom −24.1◦ to 24.1◦. The sampling interval of the antenna patternis 0.05◦, and the number of the sampling points is 965, and spacingbetween two adjacent elements keeps 0.6 times of the wavelength. kvaries from 1 to 30. Fig. 9(a) shows the means of the maximum andthe minimum eigenvalues of GΣ with Mx = 25, 35, 45, 55. Fig. 9(b)shows the means of the maximum and the minimum eigenvalues of GΣ

if there are I = 40, 79, 157, 313, 625 samplings available.Some conclusions can be deduced from above experiment:

(1) Under the condition that Mx = 25, the means of the maximumand the minimum eigenvalues are within (0, 2) only when the


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Figure 9. Analyzing RIP of GΣ. (a) Relationship between RIC andMx. (b) Relationship between RIC and sampling number.

order of sparsity k is less than 5, which indicates that there existsa positive value εk ∈ (0, 1). RIP is satisfied and the sparse vectorcan be recovered exactly by solving the optimum problem. But ifk > 5, the mean of these maximum eigenvalues will be larger thantwo, RIP cannot be satisfied, and the problem cannot be solvedby the convex optimization. In order to overcome this difficulty,performance of the sensing matrix GΣ should be improved throughsome methods, such as more complicated antenna pattern design.

(2) If Mx increases and the sparsity order of the vector keeps constant,RIC will decrease and xu will be recovered more easily. This isconsistent with the intuition. More elements mean a narrowerbeam width and better azimuth resolution. But the cost, volumeand weight of the radar will increase.

(3) Under the condition that I ≥ O(k · log Q), the maximum andminimum eigenvalues do not change with the number of thescanning steps, which is shown in Fig. 9(c) and indicates that wecan obtain the high azimuth resolution with the fewest number ofscanning steps.In summary, the proposed forward-looking imaging method based

on compressed sensing includes the following steps:(1) A train of pulses with a large time-bandwidth product are emitted

to the target area which is located at the front of the monopulsePAR at the same time the radar is scanning along the azimuthdirection, and echoes are collected by the radar.

(2) High range resolution profiles are obtained after pulse compressionto echoes of each pulse.

596 Wen et al.

(3) Echoes in each range cell from the sum channel is used to estimatethe distribution of the scatterers based on compressed sensingrespectively, and the azimuth discrimination is achieved.

4. SIMULATION RESULTS

In order to validate the proposed method and compare it with theimaging results of the real beam scanning and the LSE method, somesimulations’ results are presented. Herein, point scattering model isused. Since the imaging in three dimensional space is equivalent tothe imaging in the two dimensional space which is determined bythe spindle of the beam, these simulations are performed in the two-dimensional space. Parameters used in the simulations are given below:the operating frequency is 35 GHz, the space between two adjacentelements is 0.6 times of the wavelength, and there exists 25 elementsin the array. The beam scans from −15.6◦ to 15.6◦, the number ofthe scanning steps is 79, and the scanning step is 0.4◦. The targetarea locates at the front of the radar from −24.1◦ to 24.1◦. Chirppulses are transmitted, and the instantaneous bandwidth is equal to80MHz, which means that the range resolution is 1.875 meters. Threesimulations are carried out as follows.

4.1. Validity Simulation

In order to validate the method, suppose that there are nine scatterersin the target area and that their polar coordinates are [1995m, 0.4◦],[1995m, 0◦], [1995m, −0.4◦], [2000 m, 0.4◦], [2000m, 0◦], [2000m,−0.4◦], [2005 m, 0.4◦], [2005 m, 0◦], and [2005 m, −0.4◦], respectively.Amplitudes of the scatterers are all uniform. Noise impact is ignored.Figs. 10(a) and (b) show the imaging results of the target area basedon the real beam scanning and the LSE method. Spots in the figuresare the real positions of the scatterers. It can be seen that neitherof them can discriminate two adjacent scatterers in the same rangeresolution cell. Fig. 10(c) shows the imaging result of the proposedmethod. Compared with above figures, it can be seen that theazimuth resolution is improved significantly, and the scatterers canbe discriminated easily. Fig. 10(d) shows the imaging results of thescatterers whose range are 2000 meters, which indicates that azimuthresolution based on the LSE method is better than that of the realbeam scanning, but cannot discriminate two adjacent scatterers, andazimuth resolution based on compressed sensing is the best. Validationof the method is verified.


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Figure 10. Efficiency of the imaging method by the compressivesensing. (a) Scanning result of the sum channel. (b) Imaging resultafter the least square estimation. (c) Imaging result of the compressivesensing method. (d) Results’ comparison of different methods.

4.2. Robustness Test

To verify the robustness of the algorithm, we designed the followingexperiment: radar operating parameters have been mentioned above,and assuming there is a scatterer whose range is 2000 meters andazimuth angle is −0.5◦. Gaussian white noise model is assumed, andMonte Carlo experiments are carried out and the signal-to-noise ratio(SNR) ranges from 0 dB to 45 dB. Five hundred experiments are carriedout under each SNR. Herein, the SNR is defined as the ratio of theecho’s total energy to noise power, which is given by

SNR = 10 log10

sHu su

/I

σ2

598 Wen et al.

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Figure 12. Estimation meansand variance of compressive sens-ing under different SNRs.

Figure 11 shows the imaging results along azimuth dimension basedon the real beam scanning and the compressed sensing. Herein, noiseimpact is not considered. It indicates that azimuth resolution of theproposed method is better. Since the target occupies only one or twopoints and it is difficult to define the azimuth resolution of the imagingresult, the mean and variance of the optimum estimation are selectedas a measure of imaging result.

It shows that even at low SNR, high azimuth resolution can beachieved. However, it should be noted that lower SNR means longertime for estimation. Fig. 12 shows the change rules of the meanand variance via the SNR, which indicates that if the SNR increases,estimation of the target position will be more accurate and the varianceof the estimation will be lower. It can be seen that improvement of theSNR is benefit for estimation.

4.3. Impact Analysis of RIC

If the elements’ number Mx keeps constant, a greater order of sparsityk means a larger RIC. And if the order of sparsity k keeps constant,a greater Mx means a smaller RIC. Supposing that Mx = 25 andnoise does not exist, imaging results are illustrated in Fig. 13 withk = 2, 3, 4. Herein, the azimuth angle interval between two adjacentscatterers is 0.4◦. It shows that the correct recovery can be achievedif k < 4. Otherwise, we cannot achieve the correct recovery.

If k keeps constant which is equal to 3 and the azimuth angleinterval between two adjacent scatterers is 0.3◦, imaging results aregiven in Fig. 14 with Mx = 25, 35, 45. It shows that if Mx increase,results of the estimation will be better. So a conclusion can be achieved


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Figure 14. Relationship bet the estimation result and the elementsnumber. (a) Mx = 25. (b) Mx = 35. (c) Mx = 45.

that RIC is important for the forward-looking imaging based on thecompressed sensing. In order to obtain smaller RIC, the numberof elements in the array should be increased, but the cost increasesdramatically, and a comprise needs to be done.

5. CONCLUSION

In order to improve the azimuth resolution of the image based onthe real beam scanning under the forward-looking condition, a novelmethod based on the compressed sensing is proposed. Simulationsverify the method and images with high resolution are achieved undercertain conditions. The algorithm does not require a large antennaaperture or large-scale transformation, especially, relative tangentialmotion between targets and the radar. The azimuth resolution canbe improved through signal processing, which should be a promisingmethod for forward-looking imaging.

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