TF MTI Barbarossa Farina

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    Detection and imaging of moving objects withsynthetic aperture radarPart 2 : Joint time-frequency analysis by Wigner-VilledistributionS . Ba rba rossa , PhDProf. A . Farina

    Indexing terms: Wigner-Vi lle distribution, Synthetic aperture radar, Moving objects, Detection

    I 1Abstract: The aim of the work is to show howtime-frequency representation by Wigner-Villedistribution of the echoes received by a syntheticaperture radar provides a useful tool for detectionof moving objects and the estimation of theinstantaneous phase shift induced by relativeradar-object motion. The phase history is thenused to compensate the received signal and toform a synthetic aperture with respect to themoving object, necessary to produce a highresolution image.

    1 int roduct ionThe main problems related to the detection and imagingof moving targets with synthetic aperture radars (SAR)are analysed in the first part of the paper. This secondpart is entirely focused on the use of the Wigner-Villedistribution (WVD) in the SAR signal processing formoving target detection and imaging.

    The possibility of focusing a moving target observedby radar requires knowledge of the phase modulationinduced on the received echo by the radar-target relativemotion. The time behaviour and/or the spectrum of thetarget echoes alone are not sufficient to provide thisinformation. The knowledge of the phase history of theechoes received during the whole observation intervalallows us to form the synthetic aperture with respect tothe moving object, necessary to produce a high cross-range resolution image.A method for extracting the instantaneous phase canbe based on analysis of the time-frequency (TF) distribu-tion of the received signal. Several distributions are avail-able and an excellent review is given in a recent paper byCohen [ 3. The Wigner-Ville distribution has beenchosen in this work because it presents some importantfeatures concerning detection and estimation issues, asalready pointed out [2-51. There are simpler methods foranalysing signals in the TF domain, such as the short-Paper 8216F (ElS), first received 10th J uly 1990 and in revised form 8thApril 1991Dr . Barbarossa is with the Univerita di Roma L a Sapienza, INFO-COM Department, V ia E udossiana 18, 00184, Roma, I talyProf. Farina is with Alenia SpA., Radar Factory, Via Tiburtina, Km12400,00131 R oma, Italy

    time Fourier transform (STFT) [l], but they do notexhibit the same resolution capabilities in the TF domainas does the WVD. In particular, since the STFT is basedon a Fourier transform (FT) applied to a time windowedversion of the signal, with the window central instantvarying with time, the frequency resolution is inverselyproportional to the window duration. The narrower thewindow, the better is the time resolution, but the worse isthe frequency resolution and vice versa. Conversely, theWVD does not suffer from this shortcoming. On theother hand, the WVD poses other problems since it isnota linear transformation. This causes the appearance ofundesired cross-products when more than one signal ispresent.With respect to other T F distributions, such as Rihac-zeks (e.g. Reference l) , the WVD provides a higher con-centration of the signal energy in the TF plane, aroundthe curve of the signal instantaneous frequency (IF). Thisallows a better estimation of the IF in the presence ofnoise and this information is fundamental for the syn-thesis of the long aperture with respect to the movingobject.Mapping of the received signal in the TF plane pro-vides a tool for the synthesis of the optimal receiver filterwithout a priori knowledge of the useful signal, providedthat the signal-to-noise ratio be sufficiently large. The TFrepresentation provides a unique tool for exploiting oneof the most relevant differences between useful signalsand disturbances in the imaging of small moving objects,namely the instantaneous frequency and the bandwidth.In fact, it can be shown that, while the bandwidthoccupied by a target echo during the observation intervalnecessary to form the synthetic aperture mainly dependson radar-object motion, the instantaneous bandwidth isproportional to object size. Therefore the echo corres-ponding to a small target can occupy a large band duringthe overall observation time, but its instantaneous band-width is considerably narrower (i.e. the echo back-scattered by a point-like target has a zero instantaneousbandwidth but it may exhibit a large overall bandwidth).Conversely, the echo from the background and thereceiver noise have a large instantaneous bandwidth.Therefore, even if the useful signal and the disturbancemay have a large total band, the possibility of trackingthe instantaneous bandwidth, made available by the TFrepresentation allows a discrimination of the useful signalfrom the disturbance not possible by conventional pro-cessing.

    I E E P R OCE E D I N G S -F , Vol.139, NO . , F EB RUA RY 1992 89

    Authorized licensed use limited to: UNIVERSITAT POLITCNICA DE CATALUNYA. Downloaded on September 6, 2009 at 03:12 from IEEE Xplore. Restrictions apply.

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    Another important and unique advantage related touse of the WVD is that it allows the recovery of the echophase history even in the case of undersampling, asshown in Reference 6. This is particularly important inSAR applications since it allows us to work with pulserepetition frequencies (PRF) lower than the limitsimposed by the signal bandwidth occupied during theobservation interval. Owing to the target motion, thisbandwidth may be considerably larger than the band-width occupied by the background echo. According toconventional processing, we should then use a corre-spondingly higher PRF. Conversely, if the useful signalhas a large total bandwidth, but a narrow instantaneousbandwidth, the TF representation prevents superpositionof spectrum replicas created by undersampling because,even if the replicas occupy the same bandwidth, theyoccur at different times. This property allows us torecover the desired information even from undersampledsignals. Since the PRF value imposes a limit on the sizeof the monitorable area, due to time, and then range,ambiguities, the possibility of using a low PRF preventsthe reduction of the region to be imaged, as well as theincrease of the data rate.The WVD is first recalled in Section 2. In particular,its main properties in the presence of noise are derived.Section 3 is then concerned with recasting the optimaldetection and parameter estimation in presence of noisein the TF domain, based on the WVD. Use of the WVDfor the detection of chirp signals embedded in white noisehas already been considered [2-51. The approach is hereextended to different signal modulations and to the caseof correlated noise (the echo from the background). Onthe basis of the matched filter theory, transferred into theTF domain, it is shown that, in the presence of whitenoise, the received signal can be mapped directly onto theTF domain. In the presence of a correlated disturbance(the echo from the background), a cancellation must beperformed before evaluating the WVD. A possible algo-rithm for filtering the received signal in the TF plane isalso proposed. Finally, the application to the SAR case isconsidered in Section 4. Simulation results are shown forevaluating the capabilities of the proposed approach.2 Timefr equen cy analysis by means of theWigner-Vil le distr i but ion2.1 Definit ion and main properties of the W VDIn a recent paper [l], a thorough overview of TF dis-tributions is provided. An extensive analysis of theWigner-Ville distribution is also given in References 7-9.The WVD of a signal is defined as:

    W(t,f)=I tms(t +;)* -;) exp( - j2nf~) z (1)- m

    wheres ( t ) represents the analytic signal.Two properties, particularly important in detectionand estimation problems are: (a) conservation of thescalar or inner products; (b) estimation of the instantan-eous frequency. These properties are now briefly recalled.(a) Conseroation o the scalar products: Given twosignals x i ( t ) , i =1, 2, the square modulus of their scalarproduct is equal to the scalar product between their dis-tributions:I J - r x l ( t ) x m dtJ 2=~ ~ k ( t 9 f ) w 2 2 ( t 9 f )t df (2)

    having indicated with WJ t, o) he WVD of the signalsx i ( t ) , i =1, 2.90

    This property will provide the basis for the formula-(b) Estimation o the instantaneous frequency: If wetion of the detection scheme in the TF domain.express the signal in terms of its envelope and phase:s(t) = exp {jdt) j (3)

    it can be shown that the local or mean conditional fre-quency of the WVD distribution, defined as[I ]

    is equal to the signal IF:

    The estimate of the mean conditional frequency of theWVD then provides the information about the signal IF.This is exactly the information we need for rephasing thereceived signal in order to produce a focused image.

    2.2 W VD of signal plus noiseA problem which arises when using the WVD is that,since it is not a linear transformation, the distribution ofthe sum of more than one signal is not equal to the sumof the distributions of each signal, but contains all thecross-products.In general, if a signal s& is given by the sum of Ncontributions

    Ns J t ) =2 &

    i = 1its WVD is

    K is,(t,f)= - m@s i ( t+ t-I)xp(-j2nf7) d7In detection problems, it is important to analyse themain statistical properties of the WVD of deterministicsignals superimposed on stochastic processes (thedisturbance). Therefore it is useful to review the mainproperties of the WVD of a random process.In the case of a stationary random process c(t), theexpected value of the WVD is constant over the time axisand is equal to the power spectral density of the processover the frequency axis. In fact:

    E{W,(t,f)}= +$*(t -mx exp(-j2nf7) dT

    =