Equalization of multipath effects in radar...

Equalization of multipath effects in radar signals

Ajeesh P. Kurian, Henry Leung and Jim P.Y. Lee

Defence R&D Canada – Ottawa

Technical Memorandum

DRDC Ottawa TM 2010-014 February 2010


Ajeesh P. Kurian and Henry Leung Complex System Inc.

Jim P.Y. Lee DRDC Ottawa

Defence R&D Canada – Ottawa

Technical Memorandum DRDC Ottawa TM 2010-014 February 2010

Principal Author

Original signed by Jim P.Y. Lee

Jim P.Y. Lee

Defence Scientist

Approved by

Original signed by Jean-François Rivest

Jean-François Rivest

Head/Radar Electronic Warfare Section

Approved for release by

Original signed by Brian Eatock

Brian Eatock

Document Review Chair, DRDC Ottawa

© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2010

© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2010

Abstract

In electronic warfare (EW), signals emitted from different radars are collected and ana-lyzed for various purposes. The radar signals received often contain multiple returns dueto ground or sea surface reflections. These reflections are received at different delays andcan result in waveform distortions. This situation can be treated as a convolution of trans-mitted signal with an unknown channel. Coefficients of such channels are time varyingand dependant on the surface reflection characteristics. Waveform distortions can be com-pensated using appropriate equalization methods at the receiver. Since the amplitude andphase of the unwanted multipath component are unknown to the receiver, blind equalizationschemes are suitable. We consider blind equalization schemes, such as, constant modulusalgorithm (CMA), higher order statistics (HOS) based algorithms, subspace algorithms andmaximization of negative entropies (MNE), in our study. We first simulate a phase codedradar waveform with multipath properties using the sea surface reflection characteristicsand 4/3 earth model. Real data collected by DRDC were used to analyze each of the blindequalization schemes. We conclude that the best performance is obtained when trainingsequence based least mean square algorithm is used for the equalization. In blind equal-ization, HOS algorithm based on the 4th order cumulant gives the best performance suchthat it reconstructs the radar waveform more faithfully. We further conclude that the bestphilosophy for the equalization of radar waveforms is the semiblind identification method,where some prior information about the incoming waveform is used.

Resume

En guerre electronique (GE), les signaux emis par differents radars sont collectes et ana-lyses a diverses fins. Les signaux radar recus contiennent souvent de multiples echos acause des reflexions sur la surface de la terre ou de la mer. Ces reflexions sont recues avecdifferents delais et risquent de causer des distorsions de la forme d’onde. Cette situationpeut etre traitee comme une convolution du signal emis et d’un canal inconnu. Les coef-ficients de ces canaux varient dans le temps et dependent des caracteristiques de reflexionde la surface. Les distorsions de la forme d’onde peuvent etre compensees au moyen demethodes d’egalisation appropriees au niveau du recepteur. Puisque l’amplitude et la phasede la composante de trajets multiples indesirable sont inconnues du recepteur, des tech-niques d’egalisation aveugles sont adequates. Dans notre etude, nous abordons des tech-niques d’egalisation aveugles telles que l’algorithme de module constant (CMA), les algo-rithmes bases sur les statistiques d’ordre superieur (HOS), les algorithmes de sous espaceset de maximisation des entropies negatives (MNE). Nous simulons d’abord un signal radarcode en phase avec des proprietes de trajets multiples au moyen des caracteristiques dereflexion sur la surface de la mer et du modele de Terre 4/3. Des donnees reelles collecteespar RDDC ont ete utilisees pour analyser chacune des techniques d’egalisation aveugles.Nous concluons que les meilleures performances sont obtenues lorsque l’algorithme de

DRDC Ottawa TM 2010-014 i

moyenne quadratique minimale base sur la sequence d’apprentissage est utilise pour l’egalisation.En egalisation aveugle, l’algorithme HOS base sur le cumulant d’ordre quatre donne lesmeilleurs resultats de sorte qu’il reconstitue le signal radar plus fidelement. De plus, nousconcluons que le meilleur principe pour l’egalisation des signaux radar est la methoded’identification semi-aveugle, dans laquelle certains renseignements a priori sur le signald’arrivee sont utilises.

ii DRDC Ottawa TM 2010-014

Executive summary

Equalization of Multipath Effects in Radar SignalsAjeesh P. Kurian and Henry Leung, Jim P. Y. Lee ; DRDC Ottawa TM 2010-014;

Defence R&D Canada – Ottawa; February 2010.

Background: Multiple paths in radar returns are due to the sea or ground surface reflec-tions. Similar to the direct path signals, these reflected waves illuminate the target causingmultiple returns. These returns are received at the electronic warfare (EW) receiver at dif-ferent delays due to the difference in distance traveled by each wave. Ground or sea surfacereflections cause additional amplitude and phase changes. The resultant signal received atthe EW receiver is a distorted version of the original waveform which has different spectralproperties than the original signal. In other words, the original radar signal is convolvedwith an unknown channel and the resultant signal contaminated with noise is observedat the EW receiver. In this project, blind equalization techniques are used to retrieve theoriginal radar signal for further processing. We study the performance of different blindschemes originally proposed in digital communications for the equalization of receivedradar signals. Their merits and demerits are discussed from a radar signal processing view-point.

Principal results: Since the finite impulse response (FIR) coefficients that constitute theradar multipath are not known in advance, blind identification schemes are suitable. Thisreport examines the results of different multipath compensation algorithms on radar returns.A detailed analysis is performed on why certain algorithms fail in case of radar multipathcompensation.

Significance of results: Classification of radar signals is important in today’s operationalenvironment because of high signal density and the deployment and complexity of today’sradars. Since the spectrum of the signal is altered by the unknown multipath channel, com-pensation mechanisms are of utmost importance. This report presents the feasibility studyof blind equalization methods for radar multipath compensation. We perform qualitativeanalysis (i.e. we aim to achieve a multi-path compensated signal which is visually similarto the one without multipath) of the waveform equalized signal. Our objective here is toreconstruct a signal which has close resemblance to the one which does not have multipatheffect. Though, an exact reconstruction of the clean signal is not possible by any of theequalization techniques we considered, we have noticed that some of the blind equaliza-tion schemes (such as fourth order cumulant based higher order statistics equalizer) arecapable of giving certain levels of improvement.

DRDC Ottawa TM 2010-014 iii

Future work: It has been shown that, radar multipath can be compensated to certain extentby appropriate blind schemes. One of the future directions will be to utilize the informationthat we get as a result of high sampling rate and develop a better blind equalization scheme.Also, due to the special structure of the radar signal, estimating equalizer coefficients fromdifferent segments of data can be a good future direction. We believe that some priorintrinsic information that may be available from the radar can be used for building a newclass of equalizers.

iv DRDC Ottawa TM 2010-014

Sommaire

Equalization of Multipath Effects in Radar SignalsAjeesh P. Kurian and Henry Leung, Jim P. Y. Lee ; DRDC Ottawa TM 2010-014 ;

R & D pour la defense Canada – Ottawa ; fevrier 2010.

Introduction : La transmission par trajets multiples des echos radar est causee par lesreflexions sur la surface de la mer ou de la terre. Semblables aux signaux transmis partrajet direct, ces ondes reflechies illuminent la cible et produisent des echos multiples. Cesechos sont recus par un recepteur de guerre electronique (GE) avec differents delais a causede la difference de distance parcourue par chaque onde. Les reflexions sur la surface de laterre ou de la mer causent des variations additionnelles d’amplitude et de phase. Le signalresultant recu par le recepteur GE est une version deformee du signal d’origine et a desproprietes spectrales differentes de celles du signal d’origine. En d’autres termes, le signalradar d’origine est combine a un canal inconnu, et le signal resultant, contamine par dubruit, est observe au recepteur GE. Dans ce projet, des techniques d’egalisation aveuglessont utilisees pour recuperer le signal radar d’origine en vue d’un traitement ulterieur. Nousetudions les performances de diverses techniques aveugles proposees a l’origine en com-munications numeriques pour l’egalisation des signaux radar recus. Leurs avantages et in-convenients sont discutes au point de vue du traitement des signaux radar.

Resultats : Puisque les coefficients de reponse impulsionnelle finie (FIR) qui constituentla transmission radar par trajets multiples ne sont pas connus d’avance, les techniquesd’identification aveugles sont adequates. Le present rapport examine les resultats de l’ap-plication de differents algorithmes de compensation de la transmission par trajets multiplesaux echos radar. Une analyse detaillee est effectuee pour determiner pourquoi certains al-gorithmes echouent dans le cas de la compensation de la transmission radar par trajetsmultiples.

Portee : La classification des signaux radar est importante dans l’environnement operationnelde nos jours en raison de la grande densite des signaux et du deploiement et de la com-plexite des radars modernes. Comme le spectre du signal est modifie par le canal in-connu de transmission par trajets multiples, les mecanismes de compensation sont de laplus haute importance. Le present rapport renferme l’etude de faisabilite de methodesd’egalisation aveugles pour la compensation de la transmission radar par trajets multiples.Nous effectuons une analyse qualitative (c’est-a-dire que nous visons a realiser un signalde trajets multiples compense qui est visuellement semblable au signal sans trajets mul-tiples) du signal egalise en forme d’onde. Nous avons pour objectif de reconstituer unsignal qui ressemble de pres a un signal qui n’a pas ete transmis par trajets multiples. Bienqu’une reconstitution exacte du signal d’origine ne soit pas possible au moyen d’une destechniques d’egalisation que nous avons etudiees, nous avons constate que certaines destechniques d’egalisation aveugles (telles que l’egalisation a statistiques des harmoniques

DRDC Ottawa TM 2010-014 v

superieures basees sur le cumulant d’ordre quatre) sont capables d’atteindre certains ni-veaux d’amelioration.

Recherches futures : Il a ete demontre que la transmission radar par trajets multiplespeut etre compensee dans une certaine mesure au moyen de techniques aveugles appro-priees. Une des directions futures des recherches consistera a utiliser l’information ob-tenue au moyen d’un taux d’echantillonnage eleve et a developper une meilleure tech-nique d’egalisation aveugle. De plus, etant donnee la structure speciale du signal radar,l’estimation des coefficients d’egaliseur a partir de differents segments des donnees peutetre une bonne direction pour l’avenir. Nous croyons que certains renseignements intrinsequesanterieurs qui peuvent etre fournis par le radar peuvent etre utilises pour construire unenouvelle classe d’egaliseurs.

vi DRDC Ottawa TM 2010-014

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Blind Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Constant Modulus Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Initialization of CMA and Other Considerations . . . . . . . . . . 4

2.1.2 Results on QPSK and 16QAM modulation . . . . . . . . . . . . 5

2.2 Higher Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Comments about HOS algorithms . . . . . . . . . . . . . . . . . 6

2.3 Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Negentropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Simulation of Radar Pulses with Multipath . . . . . . . . . . . . . . . . . . . . . 11

3.1 Radar Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.1 Specular Reflection . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.2 Diffuse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.3 4/3 Earth Model . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Demodulation of Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Appendix – ASoftware Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

viii DRDC Ottawa TM 2010-014

List of figures

Figure 1. CMA cost function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Figure 2. Equalized QPSK constellation . . . . . . . . . . . . . . . . . . . . . . . 5

Figure 3. CMA square error in dB (QPSK) . . . . . . . . . . . . . . . . . . . . . 5

Figure 4. Equalized 16QAM constellation . . . . . . . . . . . . . . . . . . . . . . 5

Figure 5. CMA square error in dB (16QAM) . . . . . . . . . . . . . . . . . . . . 5

Figure 6. HOS3 Equalized QPSK constellation . . . . . . . . . . . . . . . . . . . 7

Figure 7. HOS4 Equalized QPSK constellation . . . . . . . . . . . . . . . . . . . 7

Figure 8. 16QAM constellation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Figure 9. Equalized constellation. . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Figure 10. Performance comparison of MNC-based kurtosis equalizer andproposed MNE equalizer: Equalizer length 20. . . . . . . . . . . . . . . 11

Figure 11. Performance comparison of MNC-based kurtosis equalizer andproposed MNE equalizer: Equalizer length 50. . . . . . . . . . . . . . . 11

Figure 12. Radar multipath model . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Figure 13. Simulated radar returns with and without noise. . . . . . . . . . . . . . . 15

Figure 14. Waveforms extracted file provided by DRDC . . . . . . . . . . . . . . . 16

Figure 15. Demodulated pulse: Pulse # 10 . . . . . . . . . . . . . . . . . . . . . . 16

Figure 16. Results provided by DRDC . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 17. Demodulated waveforms and scatter plots of DFT coefficients for pulsenumbers 10 and 43. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 18. Clean signal, signal with multipath and results of LMS, CMA, andsubspace & zero forcing algorithms. . . . . . . . . . . . . . . . . . . . . 18

Figure 19. Clean signal, signal with multipath and results of HOS–4 and HOS–3algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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Figure 20. Clean signal, signal with multipath and results of HOS with channeldistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 21. Clean signal, signal with multipath and results of negentropy algorithm. . 19

Figure 22. GUI of the MATLABⓇ software package . . . . . . . . . . . . . . . . . 21

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1 Introduction

Radar is an important sensing device for civilian and military applications. Electromag-netic waves transmitted from the radar travel through the free space, get reflected by mov-ing/static objects and are received at the radar receiver. In electronic warfare (EW), emis-sions from different radars need to be collected and processed for gaining intelligence.In addition to the channel noise, these receptions often contain multiple reflections fromground or sea surfaces. If there is no multipath, received radar signal will have only onecomponent which is due to the direct path. It will be a delayed replica of transmitted signalwith changes in amplitude and phase. On the other hand, strong ground reflections may re-sult in additional signal component. i.e. in addition to the direct path, there will be anotherpath due to the bounced ray from the ground or sea surface [1]. Each path has differentdelays, amplitudes and phase characteristics which is due to the difference in the path thateach waveform traverses. The complex return coefficients can be seen as the tap weightsof a complex channel and thus, this situation is transformed into a multipath receptionsituation which is very common in wireless communications [2].

Channel equalization is a very common signal processing operation in communication[3, 4]. There are three different classes of channel compensation algorithms: (i) blind, (ii)semi-blind and (iii) non-blind equalizations. Classifications of these algorithms are donebased on the availability of information about the input sequence at the receiver. Amongthem, the training sequence based non–blind identification is one of the most popular meth-ods. Its popularity is due to its accuracy and simplicity [5]. However, it requires precisesynchronization of input and output at the estimator along with the full knowledge of theinput sequence [6]. Naturally, blind system identification has attracted significant interestsince it does not assume any knowledge (other than some statistical properties) about theinput at the estimator [4]. semi-blind identification schemes fall in between the non–blindand blind identification, where some prior knowledge of the signal such as the shape of thepulse shaping waveform, binary pattern etc. are used for the system identification.

Since transmitted signal as well as the propagation channel are unknown to the receiver,blind identification schemes are appropriate for radar multipath compensation (in this re-port multipath compensation and equalization are used interchangeably). We consider fourclasses of equalization algorithms namely: (i) constant modulus algorithms (CMA), (ii)higher order statistics (HOS) based equalization, (iii) subspace based equalization and (iv)maximization of negative entropy (a.k.a negentropy) equalization (MNE). We will analyzethe properties of these algorithms for different signals and study how they enhance theclassification accuracy.

The report is organized as follows. Blind equalization schemes that are used in this reportare discussed in section 2. In section 3, we discuss simulation techniques for generat-ing multipath radar waveforms. Also, the performances of these schemes for QPSK and16QAM modulations are analyzed. The results are discussed in section 4. Some conclud-

DRDC Ottawa TM 2010-014 1

ing remarks are provided in section 5. MATLABⓇ codes and their usage are provided inAppendix–A.

2 Blind Equalization

Blind equalization is a highly active research area in digital communications [7, 8]. Incommunication and radar, the transmitted signal is convolved with an unknown propaga-tion channel. In communication, this convolution is a result of reflectors such as mountains,buildings, vehicles, etc. When omni–directional antennas are used, the number of taps inthe multipath is relatively large. On the other hand, in radar, the reflections are due to thesea or ground reflection and the number of taps is limited to two [1] due to the directionalnature of radar signals. Equalization is required for the exact reconstruction of the transmit-ted signal. Since the channel that needs to be estimated as well as the transmitted signalsare unknown, blind equalization methods are studied in this report. Typically, the onlyassumed knowledge at the equalizer is the statistics of the input sequence. We considerthree different classes of blind identification algorithms. They are (i) constant modulus al-gorithm, (ii) higher order statistics based equalization and (iii) subspace algorithms. In thenext few sections, we will discuss these algorithms in detail. We will assume the followingmodel for the channel.

In a typical communication system the convolved signal which is corrupted by the channelnoise can be represented as

y(t) =∞∑

i=−∞h(t − iT )x[i]+w(t), (1)

where h(t) is the combined channel and transmitter filter response, x[i] is the unknowninput sequence and w(t) is the additive channel noise. The transmitted signal can be real orcomplex. At the receiver, this signal is sampled at a rate of 1

T (in communication schemes,T is the symbol duration). Such sampling is called baud rate sampling. The resultant signalcan be written as

y(T n) =∞∑

i=−∞x[i]h(nT − iT )+w(nT ). (2)

The above expression can be simplified as

y[n] =∞∑

i=−∞x[i]h[n− i]+w[n], (3)

where y[n]≜ y(T n), h[n− i]≜ h(nT − iT ), and w[n]≜ w(nT ).

2 DRDC Ottawa TM 2010-014

In the recent past, fractionally spaced equalizer (FSE) has gained importance in channelequalization for its advantages over the conventional baud spaced equalizers. It has beenproven that when training data is available, FSE is capable of overcoming issues related totiming and phase synchronization. Typical noise amplification problem found in baud ratedequalizers is not observed in FSEs [8]. Also, in radar, the sampling rate is much higher thanthe pulse width of the radar pulse code and naturally, FSEs are best suited for equalization.We over sample the incoming signal and equalize it with an FSE i.e., let m = T/Δ be thenumber of samples in one symbol duration T where Δ is the new sampling interval. ThenEq. 2 can be rewritten as

y(Δn) =∞∑

i=−∞x[i]h(nΔ− imΔ)+w(nΔ). (4)

which leads to

yk[n] =∞∑

i=−∞x[i]hk[n− i]+wk[n], (5)

where yk[n]≜ y[nT +kΔ], hk[n− i]≜ h(nT +kΔ) and wk[n]≜ w(nT +kΔ) for k = 1, . . . ,m.This leads to m outputs of m stationary output channels.

In practice, the support for h is finite. Then the idea of equalization is to find a vectorv = v[0], . . . ,v[K −1] of length K such that a valid objective function, J(x[n], x[n]), is min-imized/maximized with respect to v, where x[n] is given by

x[n] =K−1∑i=0

y[n− i]v[i]. (6)

The algorithms which are considered in this report change according to the construction ofthe objective function J(.). For instance, CMA uses squared error between the modulus ofthe input signal and the equalized signal as the cost function while higher order statisticsuses the higher order cumulants of x[n] for constructing the cost function.

2.1 Constant Modulus AlgorithmOne of the simplest and most widely used blind equalization algorithm is the constantmodulus algorithm (CMA) [9]. When the transmitted signal has constant modulus, we canuse CMA for equalization. In fact, it has been shown that there are many other signalswhich do not have constant modulus but can be equalized by the CMA algorithm. Themathematical formulation of CMA can be done as follows. Let κ be the modulus of theinput signal x[n] which was estimated as

κ =𝔼[∣x[n]∣4]𝔼[∣x[n]∣2] (7)


CMA minimizes the objective function ∣x[n]∣2 − κ. With an equalizer length K, CMAutilizes the following sets of equations.

x[n] =K−1∑i=0

y[n− i]v(n)[i]

e[n] = ∣x[n]∣2 −κv(n+1) = v(n)−μe[n]y[n]x[n] (8)

where μ is the learning rate and v(i) = {v[0],v[1], . . . ,v[p−1]}(i) represents equalizer weightsat the ith iteration and p is order of the equalizer. It can be seen that, this algorithms is verymuch similar to the LMS algorithm.

2.1.1 Initialization of CMA and Other Considerations

The typical error surface of CMA is shown in Figure 1. We considered a simple case whereIt has two global minima (∗) and two local minima (+). Initialization of the equalizerweights plays an important role in fast convergence of the algorithm to its global minima.As a remedy, a center tap weight initialization is followed, i.e. we keep only the center tapof the equalizer to non–zero value and all other weights are set to zero.

v[0]

v[1]

CMA Error Surface

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1

−0.5

0

0.5

1

Figure 1. CMA cost function.


2.1.2 Results on QPSK and 16QAM modulation

Our experimental results show that CMA is able to successfully cancel the multipath effectsfor QPSK and 16-QAM signals. We use 10dB noise for QPSK and 20db noise for 16QAM(since the Euclidean distance is less in the later case). FSEs are used for the equalization ofthese signals. The results of this experiment are shown in Figures 2 to 5. We can see that theconstellations of both QPSK and 16QAM are reconstructed faithfully by CMA. The squareerror is also getting lower when the number of iterations are increased. Since bi–phasecoded signal has the constant modulus property (this will be clear when we discuss thesimulation of radar waveforms), we extend this method for radar multipath compensation.

−1.5 −1 −0.5 0 0.5 1 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

imag

inar

y

real

constellation diagram (last 20% of output data)

Figure 2. Equalized QPSK constellation Figure 3. CMA square error in dB (QPSK)

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

imag

inar

y

real

constellation diagram (last 20% of output data)

Figure 4. Equalized 16QAM constellation Figure 5. CMA square error in dB (16QAM)


2.2 Higher Order StatisticsThe advantage of using higher order statistics (HOS) for the blind equalization is that theHOS can retrieve both phase and amplitude [7]. We have seen from the previous sectionthat CMA has phase ambiguities and extra care needs to be taken to avoid them. Anotheradvantage of HOS based algorithms is its ability to estimate non-minimum phase systems[7, 10]. Depending on the order of the cumulant we use, the algorithm can be roughlydivided into two classes namely, HOS based on third order cumulant (HOS–3) and fourthorder cumulant (HOS–4). The basic idea of the HOS (3 and 4) is discussed next. The costfunction for HOS algorithm is defined as

Jp,q(v) = ∣γp,q{x[n]}∣= Cp,q{x[n]}(σx)(p+q)/2

, (9)

where p and q are nonnegative integers with p+ q ≥ 3 and γp,q{x[n]} is the normalized(p+ q) order cumulant of x[n]. In maximum normalized cumulant (MNC) equalizationalgorithm, Jp,q(v) is maximized with respect to the equalization coefficient. When p+q = 3, we get HOS–3 and when it is 4, we get HOS–4 algorithm. Steps involved aresummarized in Table 1 and the results for QPSK equalization are shown in Figure 6 and 7.

Table 1. Steps in HOS blind equalization [7]

Step–1 Set the iteration number i = 0 and initialize v0 with non zero entitiesStep–2 Compute the equalized signal x0[n] = (v0)Ty[n]Step–3 Generate a new approximation to the parameter vector v via{

vi+1 = vi +μi di

∣∣di∣∣vi+1 = vi+1

∣∣vi+1∣∣where an integer k ∈ [0,K] is determined so that μi = μ0/2k

is the maximum step size leading to Jp,q(vi+1)> Jp,q(v

i)Step–4 if

Jp,q(vi+1)−Jp,q(v

i)

Jp,q(vi)≥ ζ

then go to step 4; otherwise obtain a local maximum solutionv = vi+1 for the MNC equalizer vMNC[n]

Step–5 Compute the equalized signal xi+1[n] = (vi+1)Ty[n] andobtain the corresponding step direction

Step–6 Update the iteration number i by (i+1) and go to Step 3.

2.2.1 Comments about HOS algorithms

The maximum normalized cumulant equalization algorithm (MNC), is capable of identi-fying both non–minimum phase and minimum phase systems [11]. However, kurtosis (i.e.the fourth-order cumulant) also has some drawbacks in practice when its value has to be


Figure 6. HOS3 Equalized QPSK constella-tion

Figure 7. HOS4 Equalized QPSK constella-tion

estimated from a measured sample [12]. The main problem is that kurtosis can be verysensitive to outliers. Another problem is that HOS requires large number of data samples(in the range of thousands). This is because the higher order cumulant computation will beless accurate when the data sequences used are short.

2.3 SubspaceRecently, second order statistics based methods opened a new area of blind identification[13]. In equalization, it is obtained by sampling the incoming signal above the baud ratethus forming a single input multiple output system. The over sampled system is constructedas

y[n] =M−1∑i=0

x[i]h[n− i]+ w[n], for k = 1, . . . ,m, (10)

where, y[n], h[i], i = 0, . . . ,M−1, and w[n] are m dimensional vectors and M is the numbertaps in multipath channel. We have a total of L observations at the receiver. By forming an(Lm)× (L+M+1) block Toeplitz matrix

H =

⎛⎜⎜⎜⎝

h[0] . . . h[M−1] 0 . . . . . . 0

0 h[0] . . . h[M−1] 0 . . . 0...

......

......

......

0 . . . . . . 0 h[0] . . . h[M−1]

⎞⎟⎟⎟⎠ , (11)

The fractionally sampled channel output can be written as,

y[n] = H x[n]+w[n] (12)


where

y[n] =

⎡⎢⎢⎢⎣

y[n]y[n−1]...y[n−L]

⎤⎥⎥⎥⎦ , x[n] =

⎡⎢⎢⎢⎣

x[n]x[n−1]...x[n−L]

⎤⎥⎥⎥⎦ , w[n] =

⎡⎢⎢⎢⎣

w[n]w[n−1]...w[n−L]

⎤⎥⎥⎥⎦ . (13)

Let us assume that the channel input signal and the channel noise are white with zero mean(the first condition may not hold for the radar signal). As a result the input signal has thefollowing autocorrelation

Rx[n] = 𝔼[x[n]xT [n− k]

]=

{Jk k ≥ 0,(J∣k∣)′ k < 0

(14)

with J defined to be a shifting matrix

J=

⎡⎢⎢⎢⎣

0 0 . . . 0 01 0 . . . 0 0...

... . . . ......

0 0 . . . 1 0

⎤⎥⎥⎥⎦ . (15)

Since the signal and noise are uncorrelated, we get

Ry[n] = H Rx[n]H T +Rw[n]. (16)

For the noiseless scenario, Rw[n] = 0 and Ry[0] = H H T . Using the algorithm presentedin [14] we develop the subspace method for blind equalization. Let the singular valuedecomposition (SVD) of Ry[0] be

Ry[0] =U

⎡⎢⎢⎢⎢⎢⎢⎢⎣

σ21

. . .σ2

d0

. . .0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

Lm×Lm

UH (17)

and ui be the ith column of U. Let us construct a matrix Us such that

Us = [u1, . . . ,ud],

Σs =

⎡⎢⎣

σ1. . .

σd

⎤⎥⎦ (18)


where d = L+M+1 Since the channel input is uncorrelated, we can write

H =UsΣsV, (19)

with V as an unknown unitary matrix yet to be identified from Ry[1]. From a whitingmatrix

F= Σ−1s UT

s . (20)

Construct a new matrixD= FRy[1]FT =VJVT . (21)

From these values, H can be identified with certain phase ambiguities.

For the noisy system, the autocorrelation matrix is

Ry[n] = H Rx[n]H T +Rw[n]. (22)

where Rw[n] = σ2wJ

nm. Thus, the white noise variance need to be estimated before oncecan apply the above algorithm.

The result of subspace method applied to the equalization of 16QAM is shown in Figure 9.

Figure 8. 16QAM constellation. Figure 9. Equalized constellation.

2.4 NegentropyNegentropy is a better measure of nongaussianity which can overcome the drawback ofkurtosis. Compared with kurtosis, negentropy is robust but computationally complicated.


However, some approximations of negentropy can more or less combine the good proper-ties of both measures. Negentropy is based on the information-theoretic quantity of dif-ferential entropy, which in this report is simply called entropy. The entropy of a randomvariable is related to the information that the observation of the variable gives. The morerandom, i.e., unpredictable and unstructured the variable is, the larger its entropy will be.The (differential) entropy H of a random vector y with density py(η) is defined as

H(y) =−∫

py(η) log py(η)dη. (23)

A basic result of information theory is that, a Gaussian variable has the largest entropyamong all random variables of equal variance. To obtain a measure of non-Gaussianitythat is zero for a Gaussian variable and always nonnegative, one often uses a normalizedversion of differential entropy called negentropy. Negentropy J is defined as follows

J(y) = H(yguass)−H(y). (24)

The advantage of using negentropy, or equivalently differential entropy, as a measure ofnon-Gaussianity is that it is well justified by statistical theory. In fact, negentropy is insome sense the optimal estimator of non-Gaussianity, as far as the statistical performanceis concerned. The problem in using negentropy is, however, that it is computationallycomplex. Estimating negentropy using the definition would require an estimate (possiblynonparametric) of the pdf. Therefore, simple approximations of negentropy are useful. Aclassic method of approximating negentropy is using higher-order cumulants, using thepolynomial density expansions. This gives the approximation

Negentro(y) =112

[Cum3(y)]2 +

148

[Kurt(y)]2 , (25)

where y has a zero expectation, Cum3(y) is the third order cumulant of the random varianty, denoted by Cum3(y) = E

[y3] and Kurt(y) is the fourth order cumulant of the random

variant y denoted by Kurt(y) = E[y4]− 3E

[y2]. We replace the kurtosis with the negen-

tropy expression in equation (3) and get the maximum normalized cumulant equalizationalgorithm (MNE), the cost function of which is given by,

Jneg−equ (v[n]) = γneg−equ (x[n]) =∣Negentro(x[n])∣∣Cum2 (x[n])∣2

, (26)

where c(n) is the impulse response of the channel equalizer and the output signal of theequalizer e(n) is calculated by x[n] = v[n]⊗ y[n], x[n] is the observed signal. It’s proventhat the maximum of this objective function is approximately equal to the normalized ne-gentropy of the original signal x[n]. It can be shown that Jneg−equ (x[n]) ≤ γneg−equ (x[n]),if and only if, v[n] = αh[n− τ] where α is a real or complex constant and τ is an integerdelay. The time sequence with the length of 50000 samples is employed for testing and apart of the samples are shown in Figure 10 and 11, in order to clearly see the performanceimprovement. Two Figures are given for two different equalization lengths, 20 and 50. Wecan clearly see that the effect of multipath is minimized in case of the MNE algorithm.


Table 2. Steps in Maximum Normalized Negentropy Equalization [12]

Step–1 Let c denote the initial guess which we take to be ’center’ tap set toone and all the remaining taps set to zero.

Step–2 Set ρ = 1Step–3 Calculate v′ = v+ρ∂J4(v)

∂v∗ If v′ ∕= 0,calculate the resulting cost Jneg−equ(c′), else set ρ = ρ/2and repeat Step 3.

Step–4 If J4(v′)> J4, then accept v′′ = c′∣∣v′∣∣

as the new equalizer tap vector, set c v′′,and go to Step 2. Else set ρ = ρ/2 and go to Step 3.

Figure 10. Performance comparison ofMNC-based kurtosis equalizer and proposedMNE equalizer: Equalizer length 20.

Figure 11. Performance comparison ofMNC-based kurtosis equalizer and proposedMNE equalizer: Equalizer length 50.

3 Simulation of Radar Pulses with Multipath

In order to simulate radar multipath, we consider a phase coded radar with Barker codes.We use B13 codes for the simulation. Transmitter radar pulse can be written as

u(t) =1√T w

M∑m=1

umrect[

t − (m−1)TT

]exp( jωt), (27)

where Tw is the width of the radar pulse, M is the length of the phase code sequencesuch as Barker codes, rect represents a rectangular pulse, ω is the angular frequency andum = exp( jφm) is the phase coded signal. φm, m = 1, . . . ,M is the sequence such as theBarker or any other phase coding signal. Real part of u(t), [ur(t)], will be transmitted. Ifthere is no radar multipath, the received signal will be a delayed and attenuated copy of the


Radar (hr) EW (ht)

Figure 12. Radar multipath model

transmitted signal corrupted with radio frequency (RF) noise which is given by

r(t) = γ(t)ur(t − τ)+n(t), (28)

where γ(t) is the amplitude which depends on the distance between the radar and object,radar cross section, etc., τ is the delay and n(t) is the channel noise. We will next writemore formally the radar equation for multipath.

3.1 Radar MultipathWhen a signal is transmitted from the radar antenna, there will be at least two distinctpaths it can trace namely (i) direct and (ii) reflected paths as shown in Figure 12. Here, theelectronic support is assumed to be static and ht meters above from the sea level. Similarly,the radar is located hr meters above the sea level. Multipath interferences from a roughsurface generally involve two components, specular (coherent) and diffusive (incoherent)reflections which will be detailed in the next two subsections.

3.1.1 Specular Reflection

The specular reflection coefficient νs is given by

νs = ρ0Dρs (29)

where ρ0 is the Fresnel reflection coefficient, D is the divergence factor, and ρs specularscattering factor. The Fresnel reflection coefficient for a smooth surface is determined bythe electromagnetic properties of the surface [1]. It is a function of the radar waveformfrequency, the grazing angle of the incident ray, ψg, and the waveform polarization. ρ0 canbe represented as

ρ0 =εc sinψg −

√εc − cos2 ψg

εc sinψg +√

εc − cos2 ψg, (30)


for the vertical polarization and

ρ0 =sinψg −

√εc − cos2 ψg

sinψg +√

εc − cos2 ψg, (31)

for the horizontally polarized rays. The complex dielectric constant εc is given by

εc =εε0

− jλσ, (32)

with σ as the conductivity, λ as the wavelength and εε0

as the relative dielectric constantof the reflecting surface. Specular scattering factor, ρs, represents the amplitude reductiondue the surface roughness which is given by

ρs = exp(−2(2πΓ)2), Γ =σh sinψg

λ, (33)

where σh is the rms of the surface roughness. Divergence is another quantity which isincluded to account for the curvature of the earth. This causes attenuation in the powerdensity of the rays, which is approximated as

D ≈[

1+2d1d2

ad sinψg

], (34)

where d = d1 +d2, d1 and d2 are the ground distances from the incident point to the radarnormal projection and the target normal projection points respectively and a is the radiusof the earth.

3.1.2 Diffuse Scattering

Diffuse scattering is generated by the reflection of tilted section of the rough surface. Thediffuse reflection coefficient is given as

νd = ρ0ρd, (35)

where ρd is the diffuse scattering coefficient which is a function of grazing angle (ψg), rmssurface roughness (hrms) and wavelength (λ).

3.1.3 4/3 Earth Model

Using the above discussion, radar pulses are simulated as follows. By considering the two-ray model we have two separate delays for the received radar signal corresponding to thedistances traveled by the direct wave (R1) and reflected wave (R2) . Individual delays canbe computed as

τi =Ri

cp. (36)


where cp is the velocity of RF signal. With this, the radar equation for all the rays can bewritten as

Ai =PtGtGrλ2

(4π)3R2i, (37)

where Pt is the transmitter power, Gt is the transmitter antenna gain and Gr is the receiverantenna gain. Both antenna gains will be different for different rays based on the type ofantenna used. In addition to this, specular and diffuse scattering coefficient will also comeinto picture. Over all, the received signal can be written as

r(t) = A1ur(t − τ1)+A1r1ur(t − τ2)+n(t), (38)

Now, at the receiver, this signal is sampled with sampling interval Δ. This results in

r(nΔ) = α1ur(nΔ− k1Δ)+α2ur(nΔ− k2Δ)+n(nΔ)r[n] = α1ur[n− k1]+α2ur[n− k2]+n[n], (39)

where, α1 = A1, α2 = A1r1, and k1 ≤ k2. Equation (39) gives the justification for the mul-tipath compensation. The length of the radar buffer is decided by the maximum distancebetween the radar and EW establishment. We simulate the radar returns using Barker codeof length 13 (B13) and the resultant returns are shown in Figure 10. Later we will findthat the real radar returns are much different than this idealized simulation example. Inthe real radar, the RF circuit is band limited and its components can introduce additionalnonlinearities. However, it is fair enough to use these waveforms for multipath analysis.

4 Results and Discussion

In this section, we will discuss the performance of each blind identification algorithm onthe real radar data. Figure 14 is the original data obtained from the provided data set“pulse.mat.” From this plot we can see that there are 20 pulses in the data set. Thesepulses are numbered as 1,4,7,10,13,16,19,21,24,27,30,33,36,37,40,43,46,49,52, and55 according to the DRDC document. Pulses 1 to 33 belong to the 1st radar scan with“clean pulses,” and the rest belong to the 2nd radar scan with multipath.

4.1 Demodulation of Raw DataIt was given to us that the original radar frequency is down converted to 249.83 MHz andthe sampling frequency is set to 1 GHz. We perform standard demodulation followed by alow pass filter. The received signal is assumed to be of the form

s(t) = a(t)cos(ω0t +φ(t)). (40)


Figure 13. Simulated radar returns with and without noise.

To construct the analytical signal, we take the Hilbert transform of s(t), s(t) =ℍ(s(t)), andthen form

sa(t) = s(t)+ js(t). (41)

To demodulate the signal we multiply the analog signal with complex wave form generatedby the local oscillator with frequency ωloc

y(t) = sa(t)exp(− jωloct). (42)

ωloc = ω0 +Δω, where Δω accounts for the Doppler frequency and local oscillator drifts.This mismatch may introduce higher frequency components to the original baseband sig-nal. A low pass FIR filter with linear phase is used as the next stage to filter out these higherfrequency components. We used a low pass filter with a cut off frequency of 10 MHz andtransition width of 40 MHz. Thus the demodulation results in complex low–pass signal.An example of the demodulated signal is shown in Figure 15. After this demodulation weapply the equalization algorithms to these pulses.

Since multipath propagation results in convolution and hence a change in frequency re-sponse we look at the fourier transform coefficients of signals with and without multipath.For the analysis here, we take pulse number 10 and 43. Figure 17 shows demodulatedwaveforms and the scatter plot of discrete Fourier transform (DFT) coefficients. A sim-ple visual inspection clearly reveals the change in pulse shapes due to the radar multipath.


Figure 14. Waveforms extracted file provided by DRDC

0 1000 2000 3000 4000 5000 6000 7000−0.5

0

0.5

Time

Am

plitu

de

Real Signal

0 1000 2000 3000 4000 5000 6000 7000−0.5

0

0.5

Time

Am

plitu

de

Modulated Signal: Inphase

0 1000 2000 3000 4000 5000 6000 7000−0.5

0

0.5

Time

Am

plitu

de

Modulated Signal: Quadrature Phase

Figure 15. Demodulated pulse: Pulse # 10 Figure 16. Results provided by DRDC

By comparing DFT coefficients, we can see a change in pattern which is due to the radarmultipath.

Now, we study the effect of different algorithms for radar multipath compensation. Figures18 to 21 compare all the above equalization methods for in-phase plot of the scan 2 pulse43 data. Training signal based least mean square (LMS) algorithm is used for comparison.Among all these approaches, LMS shows the best performance. In other words, the non-blind equalization method has better performance than blind equalization methods. High


Figure 18. Clean signal, signal with multipath and results of LMS, CMA, and subspace &zero forcing algorithms.

Figure 19. Clean signal, signal with multipath and results of HOS–4 and HOS–3 algo-rithms.

though none of these algorithms give perfect equalization we anticipate that these can beadapted to radar signal processing by properly incorporating the properties of the wave-forms in the algorithm. This leads to a change in the philosophy of equalization: a methodwhich lies in between blind and non–blind equalization.


Figure 20. Clean signal, signal with multipath and results of HOS with channel distribu-tion.

Figure 21. Clean signal, signal with multipath and results of negentropy algorithm.

References

[1] E. Daeipour, W. D. Blair, and Y. Bar-Shalom, “Bias compensation and tracking withmonopulse radars in the presence of multi path,” vol. 33, no. 3, pp. 863–882, 1997.

[2] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. CambridgeUniversity Press, 2005.


[3] L. Ljung, System Identification: Theory for the users. Prentice Hall, 1999.

[4] K. Abed-Meraim, W. Qiu, and Y. Hua, “Blind system identification,” Proc. IEEE,vol. 85, no. 8, pp. 1310–1322, 1997.

[5] H. Vikalo, B. Hassibi, B. Hochwald, and T. Kailath, “Optimal training forfrequency-selective fading channels,” IEEE Intl Conf. Acoust., Speech, SignalProcessing, 2001, ICASSP-’01, vol. 4, pp. 2105–2108, vol.4, 2001.

[6] J. G. Proakis, Digital Communications, 4th ed. McGraw Hill, 2001.

[7] C. Chi, C. Feng, C. Chen, and C. Chen, Blind Equalization and SystemIdentification: Batch Processing Algorithms, Performance and Applications.Springer, 2006.

[8] D. Zhi and Y. G. Li, Blind Equalization and Identification. Marcel Dekker Inc.,2001.

[9] R. Johnson, Jr., P. Schniter, T. Endres, J. Behm, D. Brown, and R. Casas, “Blindequalization using the constant modulus criterion: a review,” Proc. IEEE, vol. 86,no. 10, pp. 1927–1950, 1998.

[10] J. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing andsystem theory: theoretical results and some applications,” Proc. IEEE, vol. 79, no. 3,pp. 278–305, 1991.

[11] C.-Y. Chi, C.-Y. Chen, C.-H. Chen, and C.-C. Feng, “Batch processing algorithmsfor blind equalization using higher-order statistics,” Signal Processing Magazine,IEEE, vol. 20, no. 1, pp. 25–49, 2003.

[12] S. Choi and T.-W. Lee, “A negentropy minimization approach to adaptiveequalization for digital communication systems,” Neural Networks, IEEETransactions on, vol. 15, no. 4, pp. 928–936, 2004.

[13] E. Moulines, P. Duhamel, J. F. Cardoso, and S. Mayrargue, “Subspace methods forthe blind identification of multichannel fir filters,” IEEE Trans. Signal Processing,vol. 43, no. 2, pp. 516–525, 1995.

[14] L. Tong, G. Xu; T. Kailath, “A new approach to blind identification and equalizationof multipath channels,” Conference Record of the Twenty-Fifth AsilomarConference on Signals, Systems and Computers, 1991. pp.856-860, vol.2, 4-6 Nov1991.


Appendix – ASoftware Package

As a final product of this work, we have developed a MATLAB package for the analysis.The purpose of different MATLAB files is presented in this section. Mainly there aretwo sets of files: (i) files related to the simulation of the radar pulse and (ii) equalizationmethods. These algorithms are put together in the form of a GUI, so that the individualprogram can be run easily. Screen capture of the GUI is shown in Figure 22. Left panel ofthe GUI gives the control for phase coded radar multipath simulation. On the right handside, parameters for various equalization algorithms can be set.

Figure 22. GUI of the MATLABⓇ software package .


Matalb File: CMA Equalization.mImplementation of Adaptive CMA algorithm.

Input ArgumentsPulse : Radar scan pulseMu : Learning rateN : Equalization lengthR : Modulus of the signal

Output Argumentssb : Equalized signale : Error estimate

function [sb, e] = CMA_Equalization(Pulse, mu, R,N)

T = length(Pulse);x = Pulse;% remove several first samples to avoid 0 or negative subscriptLp = T;% sample vectors (each column is a sample vector)X = zeros(N+1,Lp);x = [x; zeros(N,1)];

for i=1:LpX(:,i) = x(i+N:-1:i);

ende = zeros(1,Lp); % used to save instant error

f = 0.1*zeros(N+1,1);f(round(N/2)) = 1;

iter = 10;for k=1:iter

for i=1:Lpe(i)=abs(f.’*X(:,i))ˆ2-R; % instant error%e(i)=abs(abs(f’*X(:,i))-sqrt(R2));f=f-mu*2*e(i)*X(:,i)*X(:,i).’*f; % update equalizer

endend


%obtain the signals after equalization using the "f" from LMSsb=f.’*X;

end


MATLAB File: cumu3equalizer.mImplementation of equalization scheme based on the 3rd order cumulant.

Input Argumentsy : Signal corrupted by multipathL : Equalizer lengthExtended CL : Extended channel length, which is the

length of the estimated channel,usually much longer than the original one

Output Argumentsf : Estimated channel responsee : Equalized signalc : Equalizer

function [f, c, e] = cumu3equalizer(y, L, Extended_CL)y = y(:).’;T = length(y);if nargin == 2

MAX_CORRELATION_LENGTH=L;else

MAX_CORRELATION_LENGTH=floor(Extended_CL/2);endITERATION_TIMES=30%%% Initialization of the Equalizerc = zeros(1, L);c(1,floor(L/2)) = 1;rho=1;%%% Iteration BEGIN herefor iter=1:ITERATION_TIMES

iter;e = filter(c(1,:),1,y(1,:));CUM2c = sum(e.ˆ2)/T;CUM3c = sum(e.ˆ3)/T;J32 = CUM3c/(CUM2cˆ1.5);dCUM2c_dc = zeros(size(c));dCUM3c_dc = zeros(size(c));for jj=1:L

dCUM2c_dc(1,jj) = e(jj:T)*y(1,1:(T-jj+1)).’*2/T;dCUM3c_dc(1,jj) = (e(jj:T).ˆ2)*y(1,1:(T-jj+1)).’*3/T;


enddJ32_dc = CUM2c (-1.5)*dCUM3c_dc - 1.5*CUM3c*CUM2cˆ(-2.5)*dCUM2c_dc;dJ32_dc = dJ32_dc*sign(J32);c1 = c+rho*dJ32_dc;c1 = c1/sqrt(sum(sum(c1.ˆ2)));if max(max(abs(c1))) < 0.001

rho = rho/2;c1 = c+rho*dJ32_dc;

end%%% Calculate NEW Object functione = filter(c1(1,:),1,y(1,:));CUM2c = sum(e.ˆ2)/T;CUM3c = sum(e.ˆ3)/T;J32_new = CUM3c/(CUM2cˆ1.5);while(abs(J32_new) < abs(J32))

rho = rho/2;c1 = c+rho*dJ32_dc;%%% Calculate NEW Object function with reduced step size rhoe = filter(c1(1,:),1,y(1,:));CUM2c = sum(e.ˆ2)/T;CUM3c = sum(e.ˆ3)/T;J32_new = CUM3c/(CUM2cˆ1.5);

endJ32 = J32_new;c = c1;J32_his(iter,1)=J32;

endf=xcorr(e,y,MAX_CORRELATION_LENGTH)/T;

end


MATLAB File: cumu4equalizer.mImplementation of equalization scheme based on the 4th order cumulant.

Input Argumentsy : Signal corrupted by multipathL : Equalizer lengthExtended CL : Extended channel length, which is the

length of the estimated channel,usually much longer than the original one

Output Argumentsf : Estimated channel responsee : Equalized signalc : Equalizer

function [f, c, e] = cumulant4equalizer(y, L, Extended_CL)y=y(:).’;

T=length(y);

if nargin == 2MAX_CORRELATION_LENGTH=L;

elseMAX_CORRELATION_LENGTH=floor(Extended_CL/2);

end

ITERATION_TIMES=240

%%% Initialization of the Equalizerc=zeros(1, L);c(1,floor(L/2))=1;

rho=1;

%%% Iteration BEGIN herefor iter=1:ITERATION_TIMES

iter;

e=filter(c(1,:),1,y(1,:));


CUM2c=sum(e. 2)/T;MOM4c=sum(e. 4)/T;J42 =MOM4c/(CUM2c 2)-3;

dCUM2c_dc = zeros(size(c));dMOM4c_dc = zeros(size(c));

for jj=1:LdCUM2c_dc(1,jj) = e(jj:T)*y(1,1:(T-jj+1)).’*2/T;dMOM4c_dc(1,jj) = (e(jj:T).ˆ3)*y(1,1:(T-jj+1)).’*4/T;

enddJ42_dc = CUM2c (-2)*dMOM4c_dc - 2*MOM4c*CUM2cˆ(-3)*dCUM2c_dc;dJ42_dc = dJ42_dc*sign(J42);

c1 = c+rho*dJ42_dc;c1 = c1/sqrt(sum(sum(c1.ˆ2)));

if max(max(abs(c1))) < 0.001rho = rho/2;c1 = c+rho*dJ42_dc;

end

%%% Calculate NEW Object functione = filter(c1(1,:),1,y(1,:));CUM2c = sum(e. 2)/T;MOM4c = sum(e. 4)/T;J42_new = MOM4c/(CUM2cˆ2)-3;

while(abs(J42_new) < abs(J42))rho = rho/2;c1 = c+rho*dJ42_dc;

%%% Calculate NEW Object function with reduced step size rhoe = filter(c1(1,:),1,y(1,:));

CUM2c = sum(e.ˆ2)/T;MOM4c = sum(e.ˆ4)/T;J42_new = MOM4c/(CUM2cˆ2)-3;

end

J42 = J42_new;c = c1;


J42_his(iter,1)=J42;

end

f = xcorr(e,y,MAX_CORRELATION_LENGTH)/T;end


MATLAB File: LMS Equalizer.mImplementation of least mean square algorithm.

Input ArgumentsPulse Ref : Reference pulsePulse : Pulse with multipathmu : Learning rateN : Equalizer length

Output Argumentssb : Equalized signale : Error signal

function [sb,err] = LMS_Equalzer(Pulse_Ref,Pulse,mu,N)T = length(Pulse);x = Pulse;r = Pulse_Ref;x = (x-mean(x))/(max(x)-min(x));r = (r-mean(x))/(max(r)-min(r));P = round((N+Lh)/2); % equalization delay% remove several first samples to avoid 0 or negative subscriptLp = T-N;% sample vectors (each column is a sample vector)X = zeros(N+1,Lp);for i=1:Lp

X(:,i) = x(i+N:-1:i);ende = zeros(1,Lp); % used to save instant errorf = 0.1*randn(N+1,1);f(P) = 1;% parameter to adjust convergence and steady erroriters = 100;err = zeros(iters,1);for k=1:iters

for i=1:Lpe(i) = r(i)-(f.’*X(:,i));f = f+mu*2*e(i)*X(:,i);

enderr(k) = mean(e.ˆ2);

end


sb = f.’*X;end


MATLAB File: write lecroy.mImplementation of Lecroy file writing program.

Input ArgumentsFile Name : Output filenamef format : File format

(IEEE big endian/small endian)DataInfo : Structure that contains the values

Output Argumentsflag : Success/failure

Details of the DataInfo StructureDataInfo.WaveHeadder string 21 Char string with

the waveform headderDataInfo.descriptorName string 16 char Starts with

WAVEDESC terminates with NULLDataInfo.commType ushort Encoding formatDataInfo.userText char User text upto 160 charecterDataInfo.DataArray matrix Raw data (volt)

(each pulse is column vector)DataInfo.NumPulses scalar Number of pulsesDataInfo.NumPoints scalar Size of each pulseDataInfo.HorizInterval scalar Sampling period (second)DataInfo.SamplingRate scalar Sampling rate (MHz)DataInfo.VerticalGain scalar Vertical gainDataInfo.VerticalOffset scalar Vertical offset (volt)DataInfo.TimeStamp string Date and time for first triggerDataInfo.TriggerTime vector Trigger offset for all pulses

function [flag] = write_lecroy(file_name,f_format,DataInfo)

if ( strcmp(f_format, ’l’) && ˜strcmp(f_format, ’ieee-le’) && ˜strcmp(f_format, ’b’) ...&& strcmp(f_format, ’ieee-be’) )disp(’Unknown file encoding format’)disp(’Using default file format Little-endian’);f_format = ’l’;

end[fid message] = fopen(file_name, ’w’, f_format);if(fid < 0)


disp(’Error in opening the file’);disp(message);

ends_str = size(DataInfo.WaveHeadder);if (s_str(2) <21)

disp(’Size of the waveform header text should be equal to 21’);flag = 0;return;

endtem_str = DataInfo.WaveHeadder(1:21);tem_str = strcat(tem_str,char(0));

% Write the first waveform data to the file.fwrite(fid, tem_str, ’char’);

% Write the wave descriptions_str = size(DataInfo.DescriptorName);if (s_str(2) < 16)

disp(’Error in the descriptorName--- Default value will be added’)disp(’Default-- WAVEDESCWAVEDESC’)

DataInfo.DescriptorName = ’WAVEDESCWAVEDESC’;else

tem_str = [];tem_str = strcat(tem_str, ’WAVEDESC’) ;if strcmp(DataInfo.DescriptorName(1:8),tem_str)˜=1

DataInfo.DescriptorName(1:8) = tem_str;end

end

tem_str = DataInfo.DescriptorName(1:16);tem_str = strcat(tem_str, char(0));fwrite(fid, tem_str, ’char’);

% Template name ˆ Dummy ˆfwrite(fid,’WAVEDESCWAVEDESC’,’char’);

% Write the wave form encoding typeif (DataInfo.CommType == 1 || DataInfo.CommType == 0)

fwrite(fid,DataInfo.CommType,’ushort’);else

disp(’Unkonwn coding format’);


disp(’Using default value 0’);fwrite(fid,0,’uint16’);DataInfo.CommType = 0;

end

% Write the file format % ComOrderif (strcmp(f_format, ’l’) || strcmp(f_format, ’ieee-le’))

fwrite(fid,1,’ushort’);else

fwrite(fid,0,’ushort’);end

% Wave Descriptor ˆ Dummy ˆˆfwrite(fid,0,’long’);

% Write the user textif( DataInfo.UserText)

fwrite(fid,char(0),’long’);else

s_str = length(DataInfo.USERTEXT);

if ( strcpm(DataInfo.USERTEXT(1:8),’USERTEXT’))s_str = s_str +8;

end

if (s_str > 160)s_str = 160;

end

user_text_length = s_str;fwrite(fid,s_str,’long’);

end

% Resolution Descriptor ˆ Dummy ˆˆfwrite(fid,0,’long’);

% Write the length of Trigger time in array of bytes.

% Check if the array is in the right format.% If yes, do the computation of% the total size of the array. Many need a modification.


s_str = length(DataInfo.TriggerTime);

size_trig = s_str*2*8; % Size of the array in bytes.fwrite(fid, size_trig, ’long’);

%risTimeArray Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’long’);

% resArray Descriptor ˆ Dummy ˆˆfwrite(fid,0,’long’);

% waveArray1 Descriptor ˆ Dummy ˆˆfwrite(fid,0,’long’);

% waveArray2 Descriptor ˆ Dummy ˆˆfwrite(fid,0,’long’);

% resArray2 Descriptor ˆ Dummy ˆˆfwrite(fid,0,’long’);

% resArray3 Descriptor ˆ Dummy ˆˆfwrite(fid,0,’long’);

%Instrument Name Descriptor ˆˆ Dummy ˆˆfwrite(fid,’WAVEDESCWAVEDESC’,’char’);

% Instrument Numner Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’long’);

%Trace Label ˆ Dummy ˆfwrite(fid,’WAVEDESCWAVEDESC’,’char’);

% Reserved1 Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’short’);

% Reserved2 Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’short’);

% Write the length of the array. Again check if the array is in the right% format. If so compute ands_str = length(DataInfo.DataArray);


if (DataInfo.CommType)size_trig = s_str*2; % Size of the array in bytes.

elsesize_trig = s_str;

end

fwrite(fid, size_trig, ’long’);

%pntsPerScreen Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’long’);

%firstValidPnt Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’long’);

%lastValidPnt Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’long’);

%firstPoint Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’long’);

%sparsingFactor Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’long’);

%segmentIndex Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’long’);

% Write subarrayCountfwrite(fid, DataInfo.SubArrayCount,’long’);

%sweepsPerAcq Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’long’);

%obsolete1 Descriptor ˆ Dummy ˆˆfwrite(fid,0,’long’);

% Write verticalGainfwrite(fid, DataInfo.VerticalGain,’float’);

% Write verticalOffsetfwrite(fid, DataInfo.VerticalOffset, ’float’);

%maxValue Descriptor ˆ Dummy ˆˆ


fwrite(fid,0,’float’);

%minValue Descriptor ˆ Dummy ˆˆfwrite(fid,0,’float’);

%nominalBits Descriptor ˆ Dummy ˆˆfwrite(fid,0,’short’);

%nomSubarrayCount Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’short’);

% Write horizInterval sampling interval in secondfwrite(fid, DataInfo.HorizInterval, ’float’);

%horizOffset Descriptor ˆ Dummy ˆˆfwrite(fid,0,’double’);

%pixelOffset Descriptor ˆ Dummy ˆˆfwrite(fid,0,’double’);

%vertunit Descriptor ˆ Dummy ˆˆfwrite(fid,’WAVEDESCWAVEDESCWAVEDESCWAVEDESCWAVEDESCWAVEDESC’,’char’);

%horunit Descriptor ˆ Dummy ˆˆfwrite(fid,’WAVEDESCWAVEDESCWAVEDESCWAVEDESCWAVEDESCWAVEDESC’,’char’);

%reserved3 Descriptor ˆ Dummy ˆˆfwrite(fid,0,’short’);

%reserved4 Descriptor ˆ Dummy ˆˆfwrite(fid,0,’short’);

% Enter the time stamp. It is assumed that this will use afwrite(fid,DataInfo.TrigTimeSeconds, ’double’);fwrite(fid,DataInfo.TrigTimeMinutes, ’char’);fwrite(fid,DataInfo.TrigTimeHours, ’char’);fwrite(fid,DataInfo.TrigTimeDay, ’char’);fwrite(fid,DataInfo.TrigTimeMonth, ’char’);fwrite(fid,DataInfo.TrigTimeYear, ’short’);

%trigTimeUnused Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’short’);


%acqDuration Descriptor ˆ Dummy ˆˆfwrite(fid,0,’float’);

%recordType Descriptor ˆ Dummy ˆˆfwrite(fid,0,’ushort’);

%processingDone Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’ushort’);

%processingDone Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’short’);

%risSweeps Descriptor ˆ Dummy ˆˆfwrite(fid,0,’short’);

%timebase Descriptor ˆ Dummy ˆˆfwrite(fid,0,’ushort’);

%vertCoupling Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’ushort’);

%probeAtt Descriptor ˆ Dummy ˆˆfwrite(fid,0,’float’);

%fixedVertGain Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’ushort’);

%bandwidthLimit Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’ushort’);

%verticalVerniert Descriptor ˆˆ Dummy ˆˆfwrite(fid,0,’float’);

%AcqVertOff Descriptor ˆ Dummy ˆˆfwrite(fid,0,’float’);

%waveSource Descriptor ˆ Dummy ˆˆfwrite(fid,0,’ushort’);

% Write the original trigger array


fwrite(fid,DataInfo.TriggerTime,’double’);%fwrite(fid,DataInfo.TriggerOffset,’double’);

% Write the data to the file.

if (DataInfo.CommType == 1)fwrite(fid,DataInfo.DataArray,’short’);

elsefwrite(fid,DataInfo.DataArray,’int8’);

end

if (DataInfo.UserText)

temp_str = [];if ( strcpm(DataInfo.USERTEXT(1:8),’USERTEXT’))

temp_str = strcat(temp_str , ’USERTEXT’);strcat(temp_str,DataInfo.USERTEXT);

elsetemp_str=[];temp_str = strcat(temp_str , DataInfo.USERTEXT);

endfwrite(fid, temp_str(1:user_text_length),’char’);

end

fclose(fid);flag = 1;

end


MATLAB File: compute delay.mThis function computes the number of samples that need to be delayed based on the targetlocation from the radar assuming a maximum round trip delay. τ = 2∗R/Cp. This will bediscretized based on the sampling frequency.

Input ArgumentsR : Range in metersCp : Speed of lightFs : Sampling frequency

Max : Maximum buffer lengthOutput Arguments

Delay : Estimated delay

function [delay] = compute_delay(R, Cp, Fs, Max)delay = R/Cp;delay = round(Fs*delay);delay = max(delay,1); delay = min(delay,Max);

end


MATLAB File: ccompute gain.mThis function computes gain for the multiple reflection path. The ground reflection coeffi-cients as well as the radar cross section are used for this computation. Radar gain in direct-direct path is assumed as maximum.

Input Argumentsantenna gain : Gain of the receiver antennaR : RangeRi : Sampling frequencyRCS : Radar cross sectionlambda : Wavelengthangle : Grazing angleP : Transmission power

Output Argumentsd d gain : Gain of direct direct pathd r gain : Gain of direct reflected pathr r gain : Gain of reflected of reflected reflected pathr d gain : Gain of reflected direct path

function [d_d_gain, d_r_gain, r_r_gain, r_d_gain]= compute_gain(antenna_gain,R,Ri,RCS,lambda,r1,r2,angle,P)

N = length(antenna_gain);% Compute the radar gain for direct direct path.in = round( (pi)/2/pi*N );in = max(in,1); in = min(in,N);d_gain = antenna_gain(in);d_d_gain = P*d_gain 2*lambdaˆ2*RCS/((4*pi)ˆ3*Rˆ4);% Compute reflected reflected gainin = round((angle+pi)*2/pi*N );in = max(in,1); in = min(in,N);dr_gain = antenna_gain(in);r_r_gain = P*r1*r2*dr_gainˆ2*lambdaˆ2*RCS/((4*pi)ˆ3*Riˆ4);% Compute direct reflected Gaind_r_gain = r2*P*d_gain*4*pi*RCS/((4*pi*R)ˆ2)*dr_gain*lambdaˆ2/((4*pi*Ri)ˆ2);% Compute reflected direct pathr_d_gain = r1*P*dr_gain*4*pi*RCS/((4*pi*Ri)ˆ2)*d_gain*lambdaˆ2/((4*pi*R)ˆ2);

end


MATLAB File: compute phase.mThis function computes phase values of different reflection paths. It is assumed that therewill be four different rays arriving at the radar after the reflection from the target: direct-direct, direct - reflected, reflected - reflected, and reflected- direct. Surface scattering basedon specular reflection is assumed.

Input ArgumentsR : RangeRi : Distance traveled by the reflected componentslambda : Wavelengthphi1 : Phase of the complex reflection

coefficient at the sea surfacephi2 : Phase of the complex reflection

coefficient at the sea surfaceOutput Arguments

Phi1 : Phase of path1Phi2 : Phase of path2Phi3 : Phase of path3Phi4 : Phase of path4

function [Phi1, Phi2, Phi3, Phi4] = compute_phase(lambda,R,Ri,phi1,phi2)Phi1 = exp(mod(2*pi*2*R/lambda,2*pi)*i);Phi2 = exp(mod(2*pi*(R+Ri)/lambda,2*pi)*i)* exp(phi1*i);Phi3 = exp(mod(2*pi*(R+Ri)/lambda,2*pi)*i)* exp(phi2*i);Phi4 = exp(mod(2*pi*2*Ri/lambda,2*pi)*i)* exp(phi2*i)*exp(phi1*i);

end


MATLAB File: compute reflection par.mThis function computes all the parameters associated with specular reflection model. Someof the parameters need to be modified.

Input Argumentsht : Height of Electronic warfarehr : Height of radarRd : Wavelengthhrms : Surface roughnessfreq : Operating frequencypol : Vertical or horizontal polarization

Output ArgumentsR1 : Phase of path1R2 : Phase of path2sact fact : Phase of path3theta : Phase of path4

function [R1, R2, scat_fact,theta] =compute_reflection_par(ht,hr, Rd, hrms, freq,pol)

eps = 0.0015;ro = 6375e3; % Earth radius.re = ro* 4/3; % 4/3 earth modellambda = 3e8/freq; % Wavelengthepsp = 13.7;epspp = 0.01;

% Compute the reflection coefficient and change in the angle oval1 = (re+ hr) 2 + (re+ht)ˆ2 - Rd.ˆ2;val2 = 2.*(re+hr).*(re+ht);phi = acos(val1./val2);r = re.*phi;p = sqrt(re.* (ht+hr) + (r.ˆ2./4)).*2./sqrt(3);exci = asin((2.* re.*r.*(ht-hr)./p.ˆ3));r1 = (r./2) - p.*sin(exci./3);phi1 = r1./re;

r2 = r - r1;phi2 = r2./re;R1 = sqrt(hr. 2 + 4.*re.*(re+hr).* (sin(phi1./2)).ˆ2);R2 = sqrt(ht. 2 + 4.*re.*(re+ht).* (sin(phi2./2)).ˆ2);


psi = asin((2.*re.*hr+hr.ˆ2-R1.ˆ2)./(2.*re.*R1));theta = acot((R1+R2*cos(2*psi))/(R2*sin(2*psi)));deltaR = (4.*R1.*R2.*(sin(psi)).ˆ2)./(R1+R2+Rd);

psi = psi.*180/pi;

Sr = surf_rough(hrms,freq,psi);D = divergence(r1,r2,ht,hr,psi);[rh rv] = ref_coef(psi,epsp,epspp);

Ri = R1 + R2;

% We will assume that the polarization% is either vertical or horizontal% which is indicated by variable pol;if (pol)

scat_fact = rv*Sr*D;else

scat_fact = rh*Sr*D;end

end


MATLAB File: convert data type.mThis function converts raw data into char or int format for writing it in Lecroy format. Ifthe type is 1, 16 bit is assumed else 8 bit. If ’b’ is the number of bits, the data needs to bewritten as −2b to 2b −1.

Input Argumentsdata : Input data to converttype : Type of data

Output Argumentsdata : Data in ’char format’vertical gain : Vertical gainvertical offset : Vertical offset

function [data, vertical_gain, vertical_offset]= convert_data_type(data,type);

max_data = max(abs(data));vertical_offset = 0;data = data./max_data;

if typedata = floor(255*data);vertical_gain = max_data/256;

elsedata = floor(127*data);vertical_gain = max_data/256;

end


MATLAB File: divergence.mThis function computes the divergence of the radar reflection.



function [D] = divergence(r1, r2, ht,hr, psi)psi = psi.*pi./180;re = 4/3 * 6375e3;r = r1 + r2;arg1 = re.*r.*sin(psi);arg2 = ((2.*r1.*r2./cos(psi))+ re.*r.*sin(psi)).*(1+ hr./re).*(1+ht./re);D = sqrt(arg1./arg2);


MATLAB File:targetsReturn.mThis function computes the radar target return.



function [t a phi] =targetsReturn(targets, antenaGain,Amp,t,w,targetsTime,IF_Freq)

% returns the time the pulse returns and the amplitudedt = targetsTime-t;radarAngle = t*w;N = length(antenaGain);M = length(targets);a = zeros(M,1);t = zeros(M,1);phi = zeros(M,1);for n = 1:M

cor = targets(n).XY;v = targets(n).v;acc = targets(n).a;cor = cor + v*dt + acc/2*dtˆ2;dist2 = sum((cor. 2)); % this is the distance squered !t(n) = 2*sqrt(dist2)/3e8;targetsAngle = atan2(cor(2) , cor(1));targetsAngle = mod(targetsAngle-radarAngle+pi,2*pi);%finding the antena gain in this anglein = round( (targetsAngle)/2/pi*N );in = max(in,1); in = min(in,N);a(n) = targets(n).RCS / dist2ˆ2*antenaGain(in).ˆ2*Amp;phi(n) = mod(IF_Freq*2*pi*t(n),2*pi);

endend


MATLAB File: surf rough.mThis function calculates the surface roughness, given the frequency and rms roughnessparameters.



function [Sr] = surf_rough(hrms, freq, psi)

clight = 3e8;psi = psi.*pi /180;lambda = clight/freq;g = (2.*pi.*hrms.*sin(psi)./lambda).ˆ2;Sr = exp(-2.*g);

end


MATLAB File: ref coef.mThis function calculates the surface reflection coefficients.



function [rh, rv] = ref_coef(psi,epsp,epspp)

eps = epsp - i* epspp;psirad = psi.*(pi/180);arg1 = eps - (cos(psirad).ˆ2);arg2 = sqrt(arg1);arg3 = sin(psirad);arg4 = eps.*arg3;rv = (arg4-arg2)./(arg4+arg2);rh = (arg3-arg2)./(arg3+arg2);

end


MATLAB File: normalize.mNormalize the data for the Lecroy write program



function [d] = normalize(data)

m = mean(data);data = data-m;m1 = max(data);m2 = min(data);a = m1-m2;d = (2/a)*data;m = mean(d);d = d -m;

end


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DOCUMENT CONTROL DATA (Security classification of title, body of abstract and indexing annotation must be entered when the overall document is classified)

1. ORIGINATOR (The name and address of the organization preparing the document. Organizations for whom the document was prepared, e.g. Centre sponsoring a contractor's report, or tasking agency, are entered in section 8.)

Defence R&D Canada – Ottawa 3701 Carling Avenue Ottawa, Ontario K1A 0Z4

2. SECURITY CLASSIFICATION (Overall security classification of the document including special warning terms if applicable.)

UNCLASSIFIED

3. TITLE (The complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S, C or U) in parentheses after the title.)


4. AUTHORS (last name, followed by initials – ranks, titles, etc. not to be used)

Kurian, A.P., Leung, H., Lee, J.P.Y.

5. DATE OF PUBLICATION (Month and year of publication of document.)

February 2010

6a. NO. OF PAGES (Total containing information, including Annexes, Appendices, etc.)

64

6b. NO. OF REFS (Total cited in document.)

14 7. DESCRIPTIVE NOTES (The category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of report,

e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered.)

Technical Memorandum

8. SPONSORING ACTIVITY (The name of the department project office or laboratory sponsoring the research and development – include address.)

Defence R&D Canada – Ottawa 3701 Carling Avenue Ottawa, Ontario K1A 0Z4

9a. PROJECT OR GRANT NO. (If appropriate, the applicable research and development project or grant number under which the document was written. Please specify whether project or grant.)

15df02

9b. CONTRACT NO. (If appropriate, the applicable number under which the document was written.)

10a. ORIGINATOR'S DOCUMENT NUMBER (The official document number by which the document is identified by the originating activity. This number must be unique to this document.)

DRDC Ottawa TM 2010-014

10b. OTHER DOCUMENT NO(s). (Any other numbers which may be assigned this document either by the originator or by the sponsor.)

11. DOCUMENT AVAILABILITY (Any limitations on further dissemination of the document, other than those imposed by security classification.)

Unlimited

12. DOCUMENT ANNOUNCEMENT (Any limitation to the bibliographic announcement of this document. This will normally correspond to the Document Availability (11). However, where further distribution (beyond the audience specified in (11) is possible, a wider announcement audience may be selected.))

Unlimited

13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual.)

In electronic warfare (EW), signals emitted from different radars are collected and analyzed for various purposes. The radar signals received often contain multiple returns due to ground or sea surface reflections. These reflections are received at different delays and can result in waveform distortions. This situation can be treated as a convolution of transmitted signal with an unknown channel. Coefficients of such channels are time varying and dependant on the surface reflection characteristics. Waveform distortions can be compensated using appropriate equalization methods at the receiver. Since the amplitude and phase of the unwanted multipath component are unknown to the receiver, blind equalization schemes are suitable. We consider blind equalization schemes, such as, constant modulus algorithm (CMA), higher order statistics (HOS) based algorithms, subspace algorithms and maximization of negative entropies (MNE), in our study. We first simulate a phase coded radar waveform with multipath properties using the sea surface reflection characteristics and 4/3 earth model. Real data collected by DRDC were used to analyze each of the blind equalization schemes. We conclude that the best performance is obtained when training sequence based least mean square algorithm is used for the equalization. In blind equalization, HOS algorithm based on the 4th order cumulant gives the best performance such that it reconstructs the radar waveform more faithfully. We further conclude that the best philosophy for the equalization of radar waveforms is the semiblind identification method, where some prior information about the incoming waveform is used.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus, e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.)

Multipath Effects; Blind Equalization; Radar Signal Interception and simulation; constant modulus algorithm; higher order statistics based algorithms, subspace algorithms and maximization of negative entropies.

Equalization of multipath effects in radar...

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