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Signal Processing

Signal Processing 90 (2010) 1402–1414

0165-16

doi:10.1

� Tel.

E-m

journal homepage: www.elsevier.com/locate/sigpro

On compressive sensing applied to radar

Joachim H.G. Ender �

Fraunhofer Institute for High Frequency Physics and Radar Techniques FHR, Neuenahrer Str. 20, D-53343 Wachtberg, Germany

a r t i c l e i n f o

Article history:

Received 29 May 2009

Received in revised form

25 October 2009

Accepted 6 November 2009

Dedicated to J.F. Bohme on the occasion

of his 70th birthday

antennas. Often only a small amount of target parameters is the final output, arising the

question, if CS could not be a good mean to reduce data size, complexity, weight, power

Available online 3 December 2009

Keywords:

Compressive sensing

Radar

Sparse arrays

Pulse compression

Radar imaging

ISAR

Airspace surveillance

DOA estimation

84/$ - see front matter & 2009 Elsevier B.V. A

016/j.sigpro.2009.11.009

: +49 228 9435 226; fax: +49 228 9435 627.

ail address: [email protected]

a b s t r a c t

Compressive sensing (CS) techniques offer a framework for the detection and allocation

of sparse signals with a reduced number of samples. Today, modern radar systems

operate with high bandwidths—demanding high sample rates according to the

Shannon–Nyquist theorem—and a huge number of single elements for phased array

consumption and costs of radar systems. There is only a small number of publications

addressing the application of CS to radar, leaving several open questions. This paper

addresses some aspects as a further step to CS-radar by presenting generic system

architectures and implementation considerations. It is not the aim of this paper to

investigate numerically efficient algorithms but to point to promising applications as

well as arising problems.

Three possible applications are considered: pulse compression, radar imaging, and

air space surveillance with array antennas. Some simulation results are presented and

enriched by the evaluation of real data acquired by an experimental radar system of

Fraunhofer FHR.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

State of the art radar systems apply a large bandwidthand an increasing number of channels produce hugeamount of data. Often the data handling is the mostcrucial matter of design. On the other hand, large phasedarray systems are equipped with many thousands ofsingle elements and expansive T/R modules. Often thereseems to be a distinct imbalance between the scene—

which often is rather sparse, e.g. only some airplanes inthe wide sky—and the big machinery of the phased arraysystem, the huge amount of generated data, and the finalresults delivered by the system, which again may be very‘sparse’ e.g. with only a few coordinates and RCS values ofthe airplanes.

ll rights reserved.

There are several approaches to reduce this imbalance.Sparse arrays have been early investigated to overcomethe Nyquist criterium for spatial sampling [1]. A seconddevelopment with the ability to reduce costs and com-plexity is multi-input–multi-output (MIMO) radar [2],which uses NTx independent transmit antennas and NRx

receive channels yielding NTx � NRx different paths to thetarget and back and increases the degrees of freedom inthis way, thereby increasing the angular estimationperformance.

The third remarkable and quite new research field iscompressive sensing (CS, e.g. [3–5]) proposing sparsesampling for sparse scenes. One of the main algorithmsdeveloped in this field is the conditioned minimization ofthe ‘1�norm of the vector describing the amplitudedistribution of the scene under the condition that themeasurements are compatible with the signal model. Thistheory is applicable for temporal as well as for spatialsampling.

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2 The expression ‘incoherent’ is not related to incoherent processing

J.H.G. Ender / Signal Processing 90 (2010) 1402–1414 1403

The aim of this paper is to investigate some aspects ofCS applied to radar concerning the reduction of temporalsamples as well as the reduction of spatial samples (i.e.antenna elements). In the literature some papers on radarwith CS can be found, but several questions have not yetbeen answered sufficiently, e.g. the impact of deviationsfrom the model on the performance. Finally a realisticanalysis of the whole radar system comparing benefitsand shortcomings of CS-radar has not been published tothe knowledge of the authors.

We will consider three possible applications of CS toradar. The first two of them address temporal sampling,namely pulse compression and radar imaging. The thirdconcerns spatial sampling: we will treat DOA-estimationwith a reduced number of antenna elements.

Our paper is organized as follows: in Section 2 thebasics of CS as needed for this paper are summarized. InSection 3 we will give some general considerations of CSapplied to radar. In Section 3.5 the treatment of clutter isdiscussed. The mentioned three applications will betreated in Section 4. Section 5 considers the informationloss due to sparse sampling by regarding the Fisherinformation matrix for two approaches of thinning the setof samples. In Section 6 several remaining open questionsare discussed, followed by a summary.

2. Background of compressive sensing

This section gives a short review of CS, as introduced in[5]. A good overview can be found in [6]. To getinformation about a scene f 2 RN , M measurementsum; m¼ 1; . . . ;M are applied to f with the resultsym ¼ut

mf. If the sensing waveforms are collected in anM � N sensing matrix ~U ¼ ðu1; . . . ;uMÞ

t , the whole sensingprocess may be expressed by the matrix equation y¼ ~Uf.Now a representation basis wn; n¼ 1; . . . ;N for the scene isintroduced: f ¼

PNn ¼ 1 xnwn, in matrix notation f ¼Wx

with the coefficient vector x and the N � N matrixW¼ ðw1; . . . ;wNÞ. The measurements can be expressed independance on x:

y¼ Ax with A¼ ~UW: ð1Þ

In the theory of CS the undersampling situation isconsidered where the number M of measurements ismuch smaller than the dimension N of the signal. In thiscase Eq. (1) is underdetermined. Nevertheless, if a sparse

scene is assumed, the number of measurements may besufficient to determine x. A scene is called S-sparse, if onlyS or less of the coefficients xn are non-zero. A successfulway to estimate the coefficient vector in this situation isto solve the optimization problem

x ¼ argminfJx0J1 : x0 2 RN ;y¼Ax0g ð2Þ

for given measurements y. In Eq. (2), Jx0J1 means the ‘1norm, the sum of the absolute values of xn

0 . This equationcan be solved by linear programming, e.g. using thesimplex algorithm.1

1 The ‘1�type of minimization tends to produce zeros at the

positions where the true coefficients xn vanish in contrary to the same

optimization task then using the ‘2 norm.

Under some conditions it was shown in [5] that thisapproach is very efficient in a certain sense: first, weassume that the representation basis is orthonormal, i.e.WtW¼ IN where IN denotes the N � N unit matrix. For thesecond, let the sensing matrix ~U be a part of a largerN � N matrix U which is also assumed to be orthonormal:~U ¼ JU, where J is an M � N matrix consisting only of

zeros and ones, selecting M rows out of the N rows of U.Ideally, the bases pair should be incoherent2 (see [6]).

The coherency mðU;WÞ of the two bases is defined by

mðU;WÞ :¼ffiffiffiffiNp

maxfjutmwnj : 1rm;nrNg: ð3Þ

The minimum coherency is 1, this applies for instance forthe ‘spike’ basis W¼ IN as representation basis and theFourier basis as sensing.3 A remarkable property of CS isgiven by the:

Theorem of Candes and Romberg [5]: Let an S-sparsescene f with the coefficient vector x be given with respectto an orthonormal representation basis W. With uniformprobability let M sensor waveforms be drawn from theorthonormal sensing basis W, forming the thinned sensingmatrix ~U. Then the solution to Eq. (2) is exact withprobability PZ1�d, if the following condition holds:MZCm2ðU;WÞSlnðN=dÞ with an appropriate coefficient C.

Taking noise into account, the modified optimizationproblem is proposed under the name noisy basis pursuit

[7]:

x ¼ argminfJx0J1 : x0 2 RN ; Jy�Ax0J22reg ð4Þ

which can be transformed to the minimization task

x ¼ argminfJy�Ax0J22þlJx0J1 : x0 2 RN

g: ð5Þ

3. Compressive sensing applied to radar

There are two different tasks of radar: the first is todetect and localize distinct targets. Here, the resolutioncells in range, Doppler, and angle are designed coarseenough to assume that the target is contained in a singlecell. The second radar operation is imaging; here the aimis to have many fine resolution cells covering the target(inverse synthetic aperture radar (ISAR), range-profiling)or the scene (synthetic aperture radar (SAR)) to get image-like information. In both situations, CS can be applied inprinciple. For imaging, the sparsity property can bejustified if there are only a few dominant point-likescattering centers in the scene/on the target, which isoften true especially for targets like vehicles, airplanesand so on. A spiky reconstruction of the reflectivity—as CSoffers—has the potential to be of high value for automatictarget recognition.

There are only a few papers concerning the applicationof CS to radar [8–15]. The paper [8], which is one of thefirst publications on CS-radar, investigates an A/D

as known in the radar world!3 The representation basis for all of the following radar applications

will be the unit matrix, since the original reflectivity distribution is

already assumed to be sparse. So, x and f will always be identically in the

scope of this paper.

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J.H.G. Ender / Signal Processing 90 (2010) 1402–14141404

converter that operates at a sampling frequency lowerthan the Nyquist rate, and presents a rather preliminaryapproach to CS SAR imaging. In [9] a ‘stylized’ CS-radar isconsidered exploiting the assumed sparsity in the time–frequency plane by usage of suitable waveforms (i.e. theAlltop sequence [16]). The paper [10] sketches the CSapplication for distributed radar systems of MIMO-type[2] and investigates waveforms optimized to this envir-onment. In [11] a multi-static SAR system of MIMO-typeis modelled; chirp signals are transmitted, the echoes arede-ramped with respect to the reference signal from theorigin of the scene. This approach is generalized tomoving targets in [12]. The paper [13] regards CS rangefocusing by taking only a fraction of the temporal signalsequence; the same technique is applied to real SAR datain the azimuth domain, after conventional range com-pression. In [14] CS is investigated as a SAR focusingtechnique after discarding a large portion of the data inthe ðo; kÞ domain. As expected, the obtained SAR images,presented in comparison to classical SAR focusing, are oflower quality since the total signal-to-noise ratio (SNR)suffers from discarding data. The paper [15] proposes theapplication of CS to passive radar nets (‘coherent passivelocation’) using sources emitting orthogonal frequencydivision multiplexing (OFDM) signals.

Although really interesting ideas are presented, some-times the proposals are rather far away from real liferadar operation. For instance, a radar engineer wouldhardly throw away samples containing significant con-tributions to the cumulative SNR. Sparing samplesshould always be in accordance to the overall powerbudget, such that the design of waveforms and samplingtechniques should consider this as an important condi-tion. Further, up to now only a very few papersinvestigated real data.

3.1. Coherent radar signals from point scatterers

For a coherent radar working at the radio frequency(RF) reference frequency f0, a quadrature modulatortransfers the complex baseband signal sðtxÞðtÞ to the RFsignal RfsðtxÞðtÞej2pf0tg which is emitted by the antenna. Apoint reflector at distance r scatters the wave back and isreceived by the same antenna after a time delay t¼ 2r=c0

(c0 denotes the velocity of light). The RF receive signal isdown-converted by a quadrature demodulator resultingin the normalized baseband receive signal stðt; rÞ ¼sðtxÞðt�tÞexpf�j2pf0tg. Using the wavenumber k0 ¼

2p=l0, where l0 is the wavelength with respect to thereference frequency f0, the received signal can also bewritten as stðt; rÞ ¼ sðtxÞðt�2r=c0Þexpf�j2k0rg.

If the baseband receive signal is Fourier-transformedand the frequency axis scaled and shifted to the RFwavenumber domain, and inverse filtering according tothe Fourier-transform of the transmit signal is applied, theresult is the ‘standard form’ skðk; rÞ ¼ expf�j2krg over theset of k with non-zero transmit signal power withoutexplicit dependance on the shape of the transmittedwaveform.

3.2. General target states and measurement sets

The state of a scatterer (this may be a whole target forlow resolution radar, or a dominant scatterer on the targetor the scene for imaging radar) may be characterized by aparameter vector ! containing e.g. position, velocity, andother parameters. On the other hand, the radar systemsprovide certain measurement dimensions (e.g. frequency,slow time, fast time, channel number, etc.) described by avector n. The echo of a scatterer with normalizedreflectivity at state ! produces a complex valued modelsignal sðn;!Þ; if S scatterers are in the visibility range ofthe radar system, the undisturbed measurement is givenby

zðnÞ ¼XS

n ¼ 1

ansðn;!nÞ; ð6Þ

where an is the complex amplitude of the nth scatterer. Amore realistic model takes into account also interference:

~Z ðnÞ ¼ zðnÞþMðnÞ; ð7Þ

where MðnÞ ¼NðnÞþCðnÞ. NðnÞ and CðnÞ are randomvariables modelling the receiver noise and the cluttercontribution. ‘Clutter’ can be defined as the superpositionof echoes from all unwanted objects, like fields, wood,buildings, clouds, sea surface and so on, if the targets ofinterests are e.g. airplanes. In contrary to this, syntheticaperture radar (SAR) aims to image just these objects, sotheir echoes are regarded as useful signals and are notintroduced as ‘clutter’.

This model is very simple as it does not regard target-induced variations of xn over the set of measured variablesn; however, in the context of this paper this will besufficient. We will only regard scatterer states ! in a set Ycomposed of possible and visible states; i.e. firstly realscatterers can reach this state, and secondly this stateproduces non-zero signals at the radar.

Since the measurements are done at discrete points,the M vectors of variables can be counted: n1; . . . ; nM andthe model signal as well as the measurement z can becollected in vectors:

sð!Þ ¼ ðsðn1;!Þ; . . . ; sðnM;!ÞÞt ; ð8Þ

z¼ ðzðn1Þ; . . . ; zðnMÞÞt ; ð9Þ

and M; ~Z defined by analogy yielding ~Z ¼ zþM.If there are S scatterers at the states ~!1 ; . . . ;

~!S with thecomplex amplitudes ~a1; . . . ; ~aS, we get

z¼XS

n ¼ 1

~ansð ~!nÞ: ð10Þ

Also the possible and visible state vectors of thescatterers have to be expressed discretely to match theenvironment of the CS-theory. We create a ‘grid’ of N

representative states in Y counted as !1; . . . ;!N (alsocalled dictionary).

If only scatterers are present with a state vector exactlyat a grid point, Eq. (10) is modified to

z¼ Sa ð11Þ

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with the M � N matrix

S¼ ðsð!1Þ; . . . ; sð!NÞÞ ð12Þ

and the ‘scene’ vector a with an ¼ ~an, if scatterer n occupiesthe grid point n and zero else. Since we assume sparsity ina, most of the coefficients of this vector will vanish.

3.3. Transform to CS environment

Now we are close to Eq. (1), with the exception that theradar vectors and matrices are complex valued in contraryto the original CS environment. But it is easy to transfer itto real variables (see [11]): with

A¼RfSg �IfSg

IfSg RfSg

" #; y¼

Rfzg

Ifzg

" #; x¼

Rfag

Ifag

" #: ð13Þ

Eq. (11) becomes

y¼ Ax: ð14Þ

The contribution of interference can be considered by

~y ¼Axþ ~M with ~M ¼RfMg

IfMg

" #: ð15Þ

Note that the dimensions have doubled: Nre ¼ 2N;

Mre ¼ 2M. Finally, for this general radar scenario, asmentioned, the ‘spike’ representation basis W¼ IN seems tobe appropriate, so there is no difference between the scene fand the coefficient vector x, further in this case A¼ ~U.

Eq. (14) is the basis for the solution of Eq. (2). A slightdisadvantage of this method may be that because real partand imaginary part of the original coefficients aredecoupled. Since the result of the optimization shallproduce zeros at the positions of vanishing signal power,non-zero real and imaginary components should alwaysoccur at the same index.

3.4. Transform into the standard of the simplex-algorithm

The simplex algorithm—which is a standard procedurein the most mathematical software libraries—solves thebasic problem

w ¼ argminfctw : w 2 RP ;wZ0; ~Awryg: ð16Þ

We can transform the original minimization problemto this standard by defining

w¼maxfx;0g

maxf�x;0g

!; ~A ¼ ðA;�AÞ; c¼ 1: ð17Þ

The solution w is then transformed according tox ¼ wþ�w�, where wþ is the first half of w and w�the second.

3.5. Treatment of clutter

Depending on the application, clutter echoes canviolate the assumptions for CS severely. For instance, inthe application of array processing assuming a lownumber of distinct target directions, clutter often spreadscontinuously over the observed angular sector destroyingthe CS performance completely if not suppressed before.

In the simplest case, for stationary radars clutter can bemodelled as a random vector which is constant over time.This is very close to reality for echoes from stable objectslike buildings. In classical processing, clutter cancellationfor this type of echoes is often performed simply by thesubtraction of the temporal mean from the samples—-

which is mathematically represented as a projection tothe clutter free subspace.

The same procedure can be performed as pre-proces-sing for the subsequent CS processing. After pre-proces-sing, the clutter has ideally been suppressed completelyand the CS assumption of a low number S of useful signalsis again valid. But also a part of the useful signal is lost byfiltering. This has to be taken into account by modificationof the signal matrix S in Eq. (11). Since the clutter filter isdeterministic and known, the modelling of filtered signalsis no problem.

Also more complicated linear clutter filters can behandled in the same way. Generally, the filter can berepresented by an M �M filter matrix F applied to ~Z. Inthis model it is assumed that temporal sampling oversome pulses is included in the sampling scheme. Thesignal matrix is modified to ~S ¼ FS and the filteredinterference ~M ¼ FM contains only noise if the cluttercontribution is totally removed.

In our opinion, the approach of clutter cancellation bypre-processing is a practical way to preserve the favorableproperties of CS.

4. Three radar applications of compressive sensing

4.1. Pulse compression

The most obvious application to radar is pulsecompression which has been investigated also in someof the cited CS-radar papers and in [17,18]. We do notproceed from an usual radar waveform like a chirp, butassume the transmission of M distinct frequenciesrepresented by the wave numbers k1; . . . ; kM . These maybe distributed along a sequence of pulses or transmittedsimultaneously. In the receive branch, the individualfrequencies can be isolated by the use of appropriatebandpass filters. At this stage we assume that a phasemodulation effected by target motion (Doppler effect) canbe neglected. The normalized received signals of a pointscatterer at range r are modelled by

smðrÞ ¼ skðkm; rÞ; m¼ 1; . . . ;M with skðk; rÞ ¼ expf�j2krg

ð18Þ

and the noise-free signal received assuming a reflectivitydistribution as; s¼ 1; . . . ; S at the ranges rs is given by

zm ¼XS

s ¼ 1

~assmðrsÞ: ð19Þ

Pulse compression using the classical matched filterwould result in

aðrÞ ¼1

M

XMm ¼ 1

s�mðrÞzm: ð20Þ

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Provided the elementary reflectors are distributedwithin a range interval ½r1; r2� with Dr ¼ r2�r1, a selectionof N wavenumbers kn ¼ k0þnDk; n¼ 1; . . . ;N admits anunambiguous reconstruction of reflectivity distributionsconcentrated to the range points rn ¼ r1þnDr; n¼ 1; . . .Nwith Dr¼Dr=N and Dk¼ 4p=Dr . In this case, the sensingwave forms

un ¼1ffiffiffiffiNp ðs1ðrnÞ; . . . ; sNðrnÞÞ ð21Þ

related to the discrete Fourier transform build anorthonormal sensing basis according to the Candes–Romberg theorem and the random selection of M of thesewaveforms can be used for the illumination of the target.The underlying complex matrix equation (11) is fulfilledwith

Smn ¼ ðsnðmÞðrnÞÞ; m¼ 1; . . . ;M; n¼ 1; . . . ;N ð22Þ

with nðmÞ being the mth selected index of the N sensingsignals.

4.1.1. Simulation results

This situation was simulated with various parametersets. Fig. 1 shows an example with a sparse reflectivitydistribution ðS¼ 15Þ and sensing waveforms with N¼ 500frequencies, from which M¼ 100 were drawn by random.To test the robustness against noise, simulated noise wasadded with several values of SNR. As expected, thealgorithm works perfectly if the SNR is extremely high(e.g. 50 dB). It degenerates (for the assumed parameters),if the SNR is 20 dB or lower.4

4.1.2. Discussion of the discrete grid model

In reality, the dominant scattering centers cannot beassumed to be positioned exactly at the selected rangepoints. Simulations showed that the CS-algorithm breaksdown if the positions are between these grid points.Obviously the corresponding signals violate the originalsignal model. Consequently, the model has to be extendedby decreasing the spacing Dr, which in turn leads to anincreased number N of scene points without increasingthe bandwidth of the set of sensing frequencies. It appearsto be decisive that the signal for each r 2 ½r1; r2� can beapproximated sufficiently by a linear combination of theneighboring signals:��smðrÞ�

Xn

gnmsmðrnÞ

��51; ð23Þ

where the sum extends over a few values of n with rn

close to r. In this case, the ‘1�minimization leads tosparse solutions where the coefficients of x are distributedaround the close neighborhood of the true r which isacceptable. On the other hand, for this finer grid theorthogonality of sensing waveforms is violated, so theCandes–Romberg theorem can no longer be applieddirectly, and due to the larger number N and thedistribution of the signal energy to adjacent grid-points

4 Algorithms aiming on robustness against noise have not been

investigated in this paper.

(larger S) also the number of selected frequencies M has tobe increased.

Nevertheless, the application of ‘1 minimizations leadsto good results, if the grid spacing is refined by a factortwo to four.

4.1.3. Compressive range profiling applied to real data

To test the CS approach to pulse compression a set ofraw data was used obtained with the FHR high powerradar installation TIRA (tracking and imaging radar) froman aircraft in flight. This radar uses the de-ramping

technique, i.e. a linear chirp is transmitted and thereceived echoes are down-converted by a linearly fre-quency modulated ‘ramp’; in this way, different rangesare transferred to different constant frequencies. Conse-quently, the raw data can already be interpreted in the k-space representation.

The sampling frequency of the down-converted dataled to 1024 range frequencies. Classically, a simpleFourier-transform is applied to achieve the final range-profiles. A sequence of 1024 such profiles is shown inFig. 2 left. Obviously, the reflectivity of the airplane showsa few dominant scatterers in front of a background ofmore or less continuous contributions. From the 1024range frequencies M¼ 100 were selected randomlyaccording to a uniform distribution serving as themeasurement set for the CS-technique as describedbefore. The resulting range profiles are depicted in Fig. 2on the right. Clearly the dominant scatterers are visibleshowing also a tendency to super-resolution (which wasalready mentioned in [9]). Compared to the classicalmethod the SNR-reduction caused by the selection of onlyabout 1

10 of the data needs consideration; if a systemtailored to CS had been used, the transmit energy wouldhave been concentrated to the M selected sensingwaveforms without SNR loss!

For the 100 selected range frequencies also theclassical matched filter (beamformer) was applied; theresult, shown in the center part of Fig. 2, suffers asexpected by high statistical sidelobes concealing weakerechoes.

4.1.4. Radar architecture for high range resolution via

compressive sensing

High range resolution (HRR)—used for obtaining rangeprofile signatures—is a splendid technique for the classifica-tion of radar targets. If the sparsity of the reflectivitydistribution is valid, the use of a limited set of frequenciescombined with CS data processing, may serve as an adequatetool for this purpose. The radar architecture may beoptimized to this special situation with respect to data rateand SNR.

It is important to note that in an operational system,frequency samples would never be thrown away. Instead,unnecessary frequencies would not be generated at all.The available energy is concentrated on the usedfrequencies preserving the total SNR.

Fig. 3 shows two types of such CS-radar compared tothe classical pulse compression radar. For the latterusually a chirp waveform of bandwidth b and duration T

is used to obtain profiles with a range resolution of

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Fig. 1. Application of compressive sensing to pulse compression. The simulation was performed for four different SNR-values with the parameters

N¼ 500;M¼ 100; S¼ 15 assuming that the scattering centers are positioned exactly at the grid points. Positions and complex amplitudes were drawn by

random, also the used frequencies. The true reflectivity amplitudes are marked by green stars with the real part depicted at positive abscissa values and

the real part at negative values, respectively. The quadrates render the estimated values exceeding a given small threshold. (For interpretation of the

references to colour in this figure legend, the reader is referred to the web version of this article.)


dr¼ c=ð2bÞ. The sampling rate has to be at least b complexsamples per second according to the Nyquist condition.For the application of CS only M frequencies are required.This could be achieved either by transmission of a non-uniform stepped frequency pulse (bottom left) or bysimultaneously transmitted frequencies (bottom right).The individual frequencies can be filtered out in theanalogue part of the receive branch. Provided the samebandwidth and the same duration of the pulse resulting inthe same energy content and the same maximum range,for the first waveform the subpulses have a bandwidthM=T, such that M filter outputs have to be sampled at arate of at least M=T complex samples per second resultingin an overall data rate of M2=T .

Let Q ¼ Tb denote the time-bandwidth product of theoriginal chirp. Then the minimum data rate for the firstCS-architecture can be expressed by bM2=Q instead of b

complex samples per second. Hence, a saving in data rate

can only be obtained, if M2=Q o1. The second architecturewith M simultaneous frequencies obtains a much bettersaving: each frequency is transmitted with the duration T,so the sampling rate has to be larger or equal to 1=T andthe overall data rate will be at least M=T ¼ bM=Q with thesaving factor M=Q leading to considerable values forrealistic parameter assumptions.

4.2. Radar imaging

While we do not see proper scenarios for the applica-tion of CS to synthetic aperture radar (SAR) imaging ofextended scenes, CS might be valuable for imagingselected objects in a low reflecting surrounding [11].Especially, inverse SAR (ISAR) imaging of flying aircrafts orsatellites showing dominating point like scattering cen-ters may profit from this technique.

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Fig. 2. Application of compressive sensing to pulse compression for real data. A sequence of range profiles for an aircraft in flight is shown with the

vertical axis ‘range’ and the horizontal axis ‘pulse-to-pulse’. Left: obtained with a Fourier transform along 1024 range-frequencies. Center: obtained with

a beamformer for randomly selected 100 range-frequencies. Right: obtained with CS applied to the same set of range-frequencies. (By courtesy of Dr.

Leushacke and Dr. Schiller.)


4.2.1. ISAR imaging using CS-techniques

After compensation of the translational motion of areference point on the target, the normalized signal of ascatterer at the coordinates ðx; yÞ measured in an objectcoordinate system with its origin at the reference pointand the z-axis aligned to the axis of relative rotation withrespect to the radar, is given by

sðk;j; x; yÞ ¼ expf�j2kðxcosjþysinjÞg; ð24Þ

where j is the angle between the x-axis and the directionto the radar. Assuming that the reflectivity is concentratedto grid points ðxn; ynÞ in the image plane, the signalmeasured at radar state ðk;jÞ is

zðk;jÞ ¼XN

n ¼ 1

ansðk;j; xn; ynÞ: ð25Þ

Measuring at the pairs ðkm;jmÞ; m¼ 1; . . . ;M; we getthe vector

z¼ ðzðkm;jmÞÞMm ¼ 1 ¼ Sa ð26Þ

with the reflectivity distribution vector a¼ ða1; . . . ; aNÞt

and the model signal matrix

Smn ¼ sðkm;jm; xn; ynÞ; m¼ 1; . . . ;M; n¼ 1; . . . ;N: ð27Þ

Again, to cover also scatterers at continuous positions,the grid has to be chosen finer than the Nyquist rate.

A classical imaging technique is polar reformatting [19].Here, a variable substitution ðk;jÞ-ðkx; kyÞ is performed bykx ¼ kcosj; ky ¼ ksinj and by interpolation to a rectangulargrid in the ðkx; kyÞ plane. Subsequently a two-dimensionalinverse Fourier transform yields the image.

Inspecting Eq. (26), CS may be applied to the problem. Asparse sampling in the ðkx; kyÞ plane can be obtained byselection of a few frequencies and pulse emission timescorresponding to the aspect angles jm. The frequency setsmay vary from pulse to pulse and the pulse emission timesmust not be equi-spaced; intermediate pulses might be used

for the illumination of other targets achieving simultaneousimaging of several targets in time multiplex.

4.2.2. CS ISAR imaging applied to real data

With the mentioned TIRA system satellites can beimaged on-orbit using the ISAR technique. The data areobtained by the de-ramping technique as describedbefore. Experimental data for a satellite on a low-earthorbit were available to the author. It is noteworthy that inthis application no clutter appears. The reason is that dueto the large distance to the object there are no terrestrialreflectors visible with concurring echoes in the samereceive window.

To avoid too large dimensions of the signal matrix S, ahybrid matched filter/CS-technique has been applied:first, the data were transformed to the ðkx; kyÞ plane inthe classical manor. Then, an inverse Fourier transform(using a Hamming window) was applied in x-direction.The achieved compression in this direction implies thatthe data for fixed x and running y can be regarded assparse, since the probability is low, that for a dominantscatterer at ðx; yÞ there are more than a few further strongscatterers at the same x- but different y-values. From1024� 1697¼ 1;737;728 samples in the k-space, 1024�70¼ 71;680 were randomly selected for hybrid matchedfilter/CS-imaging. Again we point to the fact that for anoperational scene the transmission power would beconcentrated to the used k-values from the beginning. InFig. 4 the classical ISAR image using all samples iscompared to matched filter and CS-imaging with thereduced sample number.

4.3. Radar DOA estimation

Assume an airspace surveillance radar system consist-ing of a transmitter illuminating with a broad beam a

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Fig. 3. Architectures of pulse compression radar. Top: classical structure with matched filtering. Bottom left: CS architecture with consecutively

transmitted frequencies. Bottom right: CS architecture with frequencies transmitted in parallel.


spatial angular area O and a receiving array with its phasecenters at the positions ðxm; ymÞ; m¼ 1; . . . ;M and adirectivity covering the same spatial angular area. Targetsmay appear at directions represented by unit vectors uwith the first two components u and v and the thirdcomponent w¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�u2�v2p

. In this environment, theaperture coordinates ðx; yÞ represent the radar measure-ment dimension, the directional cosines ðu;vÞ the targetstates. The task of signal processing is to estimate thedirections to the targets and the corresponding complexamplitudes (DOA estimation, see [20]). We assume thatthe other parameters like range and Doppler frequencyhave been isolated before by appropriate processing.Hence, we are searching only for those targets whose

range and Doppler fall in the same range–Dopplerresolution cell which justifies the assumption of sparsity.Only a small number S of airplanes will be present in thisresolution cell which makes the application of CS feasible.

4.3.1. Signal model and sparse sampling

From a target in direction ðu;vÞ we will receive at theaperture point ðx; yÞ the signal

sðx; y;u;vÞ ¼Dðu;vÞexpfjk0ðuxþvyÞg; ð28Þ

where D is the two-way characteristics of transmitantenna and receive element and k0 the center wave-number. Analogue to the preceding applications weintroduce a raster ðun;vnÞ; n¼ 1; . . . ;N of target directions

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Fig. 4. Application of compressive sensing to ISAR imaging for real data. Left: image of a satellite obtained with the classical polar-reformatting algorithm

(1.737.728 samples in the k-space). Center: obtained with a beamformer for 70 randomly selected ky�values resulting in 71.680 samples in k-space.

Right: obtained with the same samples by a hybrid beamformer-CS technique. (By courtesy of Dr. Leushacke.)


being taken into account and the aperture samplingðxm; ymÞ; m¼ 1; . . . ;M given by the element positions.With

Smn ¼ sðxm; ym;un;vnÞ; m¼ 1; . . . ;M; n¼ 1; . . . ;N; ð29Þ

we get a similar situation as in the preceding applicationwith the vector z¼ Sa measured at the element outputs.The amplitudes of the targets are represented by thevector a with most of its coefficients being zero in a sparsesituation.

If D is constant over the upper half sphere and zeroelse, a l=2 raster of the element positions (the wavelengthl corresponds to the wavenumber k¼ 2p=l) will yield N

sensing ‘waveforms’ sð:; :;un;vnÞ; n¼ 1; . . . ;N which arenearly orthogonal over a grid of directions with its spacingtuned to the aperture size according to the spatial Nyquistcondition. Drawing by random a subset of these ‘wave-forms’ means here to thin the array statistically.

4.3.2. A simulation of CS applied to radar DOA estimation

To test the validity of this approach, a simulation of CSDOA estimation was performed. A circular aperture withthe radius 6.3 wavelengths was assumed, resulting in anaperture area of 125 l2. Consequently, for a fully filledrectangular l=2�grid the necessary number of elementswould be equal to 500, which equals the number ofnecessary beams to cover the whole half space. To thinthis array, M¼ 40 elements were randomly distributedover the circular aperture, see Fig. 5. The surveillance areawas defined by limiting the angle between antennanormal and look direction to 301. In the ðu;vÞ�plane thisdefines a circular search area with radius 0.5. Since thesurface measure of this is 1

4 of the whole unit circle, aminimum number of about 500

4 ¼ 125 beams would benecessary to sample the search area. As indicated inSection 4.1.2, the sampling in the target state space has tobe finer if the targets are allowed to assume statesbetween the grid points. The simulation trials indicatedthat an oversampling by the factor 4 in each direction is afeasible value; so, N¼ 16 � 125¼ 2000 ðu;vÞ�values on arectangular grid in the search area were specified. Thesefigures are summarized in Table 1 once more.

The scenario for simulation consists of five aircrafts(modelled as point scatterers with constant amplitude)flying in different directions with different velocities andRCS’s through the observed angular area. To the simulatedsuperposition of echoes noise was added resulting in amean SNR of 30 dB for each target, related to the output ofconventional beams focused to the target directions. Fromthis simulation a movie was created showing on the onehand the spatial matched filter output (i.e. the outputs ofthe 2000 beams formed in parallel) and on the other handthe results of the CS algorithm during the simulatedairplanes crossing the observation area. So, the behaviorof the two techniques could be studied under varioustarget constellations.

In Fig. 6 one frame of this movie is shown. As expected,the classical beamformer produces high sidelobes causedby the sparse aperture sampling. The sidelobescorresponding to the five targets often superposeeffecting amplitudes in the same order as the mainlobes.A detector based on the beamformer technique wouldproduce many false alarms; for this application, classicalbeamforming does not seem to be the adequate solution.In comparison, the CS-estimator shows promising results:the super-resolved angular spectrum exhibits peaks closeto the true directions of the targets. Due to theoversampling in the ðu;vÞ�plane, the energy isdistributed over the neighboring grid points dependingon the position of the true direction with respect to thegrid; but no peaks appear in target-free regions.

In contrary to common super-resolution methods theCS-estimator does not need any order estimation andworks also for a single ‘snapshot’. The output represents areconstruction of the amplitude distribution which is inthe noise-free case perfect with a large probability. Theseare strong advantages of the CS-technique.

On the other hand, the experience with the DOAestimation simulator showed that problems immediatelyarise if there appear signals which do not belong to thespace spanned by the model signals related to the given gridof target states. For example, if a DOA vector for a directionoutside the defined observation area is present with a non-negligible amplitude, the CS-estimate fails immediately.

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Fig. 5. Element distribution for a thinned array used for CS DOA estimation.

Table 1Parameters for the simulation of DOA estimation.

Radius 6.3 lSearch area

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2þv2p

r0:5

Grid spacing in u;v�plane 0.02

Signal-to-noise ratio 30 dB

Number of elements M ¼ 40

Number of elements for l=2�array 500

Number of grid points N ¼ 2000

Fig. 6. Frame of a movie on DOA estimation of five simulated airplanes

crossing the observation area. Left: results of conventional beamforming.

Right: amplitude distribution estimated by the CS technique. The true

positions of the targets are marked by violet ‘�’ � symbols. (For

interpretation of the references to color in this figure legend, the reader

is referred to the web version of this article.)


5. Information loss by sparse sampling

If we descend from a Nyquist-sampled to a sparselysampled data set by taking away samples, clearlyinformation about the scene is lost. The philosophy ofCS implicitly claims that the information loss will betolerable if the scene itself is sparse.

To examine this, a measure of information is necessary.In this section, we will investigate the Fisher informationmatrix for parameters like frequency, range or directionand the amplitudes and phases of distinct scatterers. Thederived Cramer–Rao bounds (CRB) for the variances ofestimators may serve as one possibility to measureinformation loss induced by sparsity.

A complex valued random vector of the form

Z¼ qð!ÞþM ð30Þ

with a deterministic part qð!Þ depending on someparameter vector ! and a random part M which isGaussian distributed with expectation 0 and covariancematrix R is described by the probability density function

p!ðzÞ ¼1

pndetRexpf�ðz�qð!ÞÞ�R�1

ðz�qð!ÞÞg: ð31Þ

From this, the Fisher information matrix for theestimation of ! is derived to

Jnm ¼ 2Rfq�Wn ð!ÞR�1qWm ð!Þg; ð32Þ

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where the subscript Wn indicates the derivative withrespect to Wn.

If S point sources are present, the deterministic partcan be expressed by

qðu1; . . . ;uS;a1; . . . ;aS;j1; . . . ;jSÞ ¼XS

s ¼ 1

asexpfjjsgsðusÞ:

ð33Þ

u1; . . . ;uS here are real-valued unknown parameters, forinstance the directional cosines, if a linear array isregarded, or the range, if the signal is sampled in thewave number domain. a1; . . . ;aS are the magnitudes andj1; . . . ;jS the phases of the complex amplitudesas ¼ asexpfjjsg; s¼ 1; . . . ; S which are unknown, too. Sothe parameter vector ! is composed of three parametersets:

!¼ ðu1; . . . ;uS;a1; . . . ;aS;j1; . . . ;jSÞ: ð34Þ

sðuÞ is the normalized model signal depending only onu. To derive the Fisher information matrix, we need all thederivatives:

qus¼ assuðusÞ; ð35Þ

qas¼

as

jasjsðusÞ; ð36Þ

qjs¼ jassðusÞ ð37Þ

which can be combined to the matrices

Q u :¼ ða1suðu1Þ; . . . ; aSsuðuSÞÞ; ð38Þ

Q a :¼a1

ja1jsðu1Þ; . . . ;

aS

jaSjsðuSÞ

� �; ð39Þ

Qj :¼ ðja1sðu1Þ; . . . ; jaSsðuSÞÞ; ð40Þ

Q :¼ ðQ u;Q a;QjÞ: ð41Þ

Now, the Fisher information matrix is given by

J¼ 2RfQ �R�1Q g: ð42Þ

The inverse of J yields the CRBs for the commonestimation of the 3S unknown real valued parameters.

Based on this evaluation, the performance of severalsituations of sparse sampling was investigated. The scenewas simulated randomly but consisting of a fixed numberS of sources. White noise with the covariance matrixR¼ s2I was assumed.

For the comparison between dense and sparse sam-pling, different conditions have to be considered:

�
Starting with equidistant Nyquist sampling,samples are removed one by one. This is close e.g. tothe real situation, where an array antenna is thinned tospare complexity, weight and money. The SNR of asingle sample remains unchanged while the cumula-tive SNR decreases proportionally to the number ofsamples. � Same as the situation before, but the cumulative SNR is
fixed. This situation is given e.g. if the number offrequencies is reduced, but the whole illuminationtime remains the same, see Section 4.1.

�
The number of temporal (or spatial) samples remainsfixed, but the bandwidth (aperture, respectively) isexpanded by changing the measured frequencies(positions, respectively). In this case, the informationabout the unknown scatterer parameter range (direc-tion, respectively) will even increase, if measured withthe Fisher information matrix.
Based on Eq. (42), we investigated a one-dimensionallinear sensing task starting with Nyquist sampling. In asimulation sequence, the unknown scene parameterswere randomly drawn according to a uniform distributionover the visible interval. Realizations with the minimumparameter distance smaller than a beamwidth werediscarded to exclude super-resolution situations. Alsothe phases of the individual signals were chosen byrandom, uniformly over ½0;2p�. For each scene realization,the number of samples M was reduced one by one,drawing M sampling indices randomly from the completenumber determined by Nyquist sampling.

Fig. 7 shows the evaluated CRBs for the estimation ofscatterer parameters, amplitudes and phases for the firsttwo mentioned situations. In the first situation, theperformance degrades as expected, first due to thedecreasing SNR, later, if M approaches the number S ofscatterers, the estimation problem becomes illconditioned indicated by steeply increasing CRBs. In thesecond situation, the performance obviously is hold over aconsiderable number of removed samples; a variation ofvalues showed that instability occurs approximately atM¼ 3

2 S.This little simulation study gave a view on the

information loss by thinning; it is decoupled from theapplied estimation method. Maximum likelihood estima-tors will tend asymptotically the CRB; a comparison withthe estimation performance by minimizing the ‘1�normwould be beyond the scope of this paper.

6. Open questions

There remain several questions not yet answeredsufficiently. The results of CS-processing applied to airspace surveillance as in Section 4.3.2 have to be insertedto a target detector. Is it sufficient to compare the CS-results with a threshold? We need a suitable detectionalgorithm, based on the probability distribution of the CS-results. How can the signal and noise power budget of thewhole system with the CS-processor at the end of thechain be derived, what is the maximum range of a CS-radar? Another question: Will it be possible to post-process the CS-cells e.g. from pulse to pulse applying aFourier transform (Doppler processing)? Which propertieswill the processed data have? Some additional theory isstill necessary to understand and model the spread ofenergy to neighboring cells.

7. Summary

In this paper, the sparse nature of many radarsituations suggests the application of CS. It was first

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Fig. 7. Standard deviations according to the Cramer–Rao Bounds. S¼ 10, SNR¼ 20 dB accumulated to the complete number of samples Ncomplete ¼ 100.

The black line shows the median value over 30 simulations. For comparison the blue (horizontal) line marks the value obtained for S¼ 1 at the full

number of samples. Left: the SNR decreases according to the reduction of samples. From top to bottom: direction, amplitude and phase estimators. Right:

the same figures and simulations, but the accumulated SNR is fixed. (For interpretation of the references to color in this figure legend, the reader is

referred to the web version of this article.)


shown how the complex signals of coherent radars can behandled in the CS-environment, followed by the formula-tion of a general model and the discussion of severalapplications of CS to radar. CS pulse compression yieldsHRR profiles of distinct targets, usable for classification ifa few dominant scatterers determine the reflectivity ofthe target. For this purpose, a radar architecture wasproposed tailored to the CS-approach using a few constantfrequencies instead of a chirp waveform. ISAR imaging ofisolated targets is possible showing super-resolutionproperties. HRR profiling and ISAR imaging were demon-

strated with real data. An airspace surveillance radarusing a broad beam transmitter and a sparse receivingarray was proposed to apply CS to DOA estimation.Simulations showed the feasibility of this approach. Tosuppress clutter, it was proposed that a clutter cancella-tion filter is applied as pre-processing. The informationloss by thinning Nyquist sampling has been studied bymeans of Cramer–Rao Bounds. Several questions havebeen formulated waiting for further investigations. Thepreliminary investigations on the topic CS-radar encou-rage to proceed in the analysis and development of

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suitable architectures. Possibly, specialized radars orradar processors will benefit from these new ideas inthe near future.

Acknowledgments

This work is funded by the German Federal Ministry ofDefense (BMVg). Especially, I express my thanks to mycolleagues Mrs. Rustemeyer and Dr. Rosebrock for thepreparation of the real radar data and Dr. Prunte for fruitfuldiscussions and providing me with some recent literature.

References

[1] J. Schwartz, B. Steinberg, Ultrasparse, ultrawideband arrays, IEEETransactions on Ultrasonics, Ferroelectrics and Frequency Control45 (2) (1998) 376–393.

[2] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, R.Valenzuela, Mimo radar: an idea whose time has come, in: IEEERadar Conference, 2004, pp. 71–78.

[3] E. Candes, J. Romberg, T. Tao, Robust uncertainty principles:exact signal reconstruction from highly incomplete frequencyinformation, IEEE Transactions on Information Theory 52 (2)(2006) 489–509.

[4] D. Donoho, Compressed sensing, IEEE Transactions on InformationTheory 52 (4) (2006) 1289–1306.

[5] E.J. Candes, J. Romberg, Sparsity and incoherence in compressivesampling, Inverse Problems 23 (3) (2007) 969–985.

[6] E. Candes, M. Wakin, An introduction to compressive sampling [asensing/sampling paradigm that goes against the common knowl-edge in data acquisition], IEEE Signal Processing Magazine 25 (2)(2008) 21–30.

[7] S. Chen, D. Donoho, M. Saunders, Atomic decomposition by basispursuit, SIAM Journal on Scientific Computing 20 (1998) 33–61.

[8] R. Baraniuk, P. Steeghs, Compressive radar imaging, in: IEEE RadarConference, 2007, pp. 128–133.

[9] M. Herman, T. Strohmer, Compressed sensing radar, in: IEEE RadarConference, 2008, pp. 1–6.

[10] N. Subotic, B. Thelen, K. Cooper, W. Buller, J. Parker, J. Browning, H.Beyer, Distributed radar waveform design based on compressivesensing considerations, 2008, pp. 1–6.

[11] I. Stojanovic, W. Karl, M. Cetin, Compressed sensing of mono-staticand multi-static SAR, in: SPIE Defense and Security Symposium,Algorithms for Synthetic Aperture Radar Imagery XVI, Orlando, FL.

[12] I. Stojanovic, W.C. Karl, Imaging of moving targets withmulti-static SAR using an overcomplete dictionary, Selected Topicsin Signal Processing, IEEE Journal of, Vol. 4, no. 1, 2010.

[13] M. Tello, P. Lopez-Dekker, J. Mallorqui, A novel strategy for radarimaging based on compressive sensing, in: IEEE InternationalGeoscience and Remote Sensing Symposium, 2008. IGARSS 2008,vol. 2, 2008, pp. II-213–II-216.

[14] S. Bhattacharya, T. Blumensath, B. Mulgrew, M. Davies, Fastencoding of synthetic aperture radar raw data using compressedsensing, 2007, pp. 448–452.

[15] C. Berger, S. Zhou, P. Willett, Signal extraction using compressedsensing for passive radar with OFDM signals, 2008, pp. 1–6.

[16] W. Alltop, Complex sequences with low periodic correlation, IEEETransactions on Information Theory 26 (3) (1980) 350–354.

[17] J. Laska, S. Kirolos, M. Duarte, T. Ragheb, R. Baraniuk, Y. Massoud,Theory and implementation of an analog-to-information converterusing random demodulation, 2007, pp. 1959–1962.

[18] L. Applebaum, S.D. Howard, S. Searle, R. Calderbank, Chirp sensingcodes: deterministic compressed sensing measurements for fastrecovery, Applied and Computational Harmonic Analysis 26 (2008)283–290.

[19] D.R. Wehner, High-resolution Radar, Artech House, 1995.[20] A. Gurbuz, J. McClellan, V. Cevher, A compressive beamforming

method, in: IEEE International Conference on Acoustics, Speech andSignal Processing, 2008. ICASSP 2008, 2008, pp. 2617–2620.

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