The paper presents a simple method for estimating nonlinear frequency distortions of linear frequency modulated (LFM) signals used in FMCW radars. This method, derived from the polynomial model of the nonlinear FM signal phase, is based on finding the maximum of two-dimensional chirp-like transform of the IF video signal. The IF signal is obtained by mixing transmitted FM signal with its delayed copy. Using suggested transform we show that the presented method is able to detect and classify signal distortions

Kulpa, Krzysztof S.

Misiurewicz, Jacek

Nałecz, Marek

Wojtkiewicz, Andrzej

Biblioteka Cyfrowa Instytutu Łączności / National Institute of Telecomunications: Digital Library

Paper The simple analysis methodof nonlinear frequency distortionsin FMCW radarKrzysztof S. Kulpa, Andrzej Wojtkiewicz, Marek Nałęcz, and Jacek MisiurewiczAbstract — The paper presents a simple method for estimat-ing nonlinear frequency distortions of linear frequency modu-lated (LFM) signals used in FMCW radars. This method, de-rived from the polynomial model of the nonlinear FM signalphase, is based on finding the maximum of two-dimensionalchirp-like transform of the IF video signal. The IF signal isobtained by mixing transmitted FM signal with its delayedcopy. Using suggested transform we show that the presentedmethod is able to detect and classify signal distortions.Keywords — radar, nonlinear frequency distortions, linear fre-quency modulation.1. IntroductionIn many applications such as radars, sonars, biomedicalengineering etc. the constant amplitude complex harmonicsignalss(t) = S0 exp( j f M t) (1)with varying frequency are used. The signal phase f M(t)can be modeled by the M-th order polynomialf M(t) =Måm=0am tm (2)with coefficients am. For example the chirp signal cor-responds to the second order phase polynomial (M = 2)and so-called quadratic frequency modulated (FM) signalcorresponds to the third order phase polynomial (M = 3).To estimate unknown parameters of frequency modulated(FM) signal, well-known time-frequency analysis, both lin-ear (spectrogram, scalogram) and bilinear (such us Wigner-Ville distribution) are commonly used.These tools are, however, inefficient for nonlinear frequencymodulation. The recent works [6 8] on generalization ofWigner-Ville distribution are very useful for analyzing thesignal (1) for M > 2 and for estimating instantaneous signalfrequency f (t) = 12 pd f M(t)dt .One of the most important practical problems is to estimatephase coefficients aM of unknown signal contaminated bywhite Gaussian noise. Such estimation allows identifica-tion and classification of polynomial phase signals [2 4].These methods, however, are not well suited to such prob-lem.The paper presents a simple method for analyzing nonlin-ear distortions of LFM signals used in FMCW radars [1].In theory, the radar transmitter generates continuouswave s(t) with sawtooth frequency modulation (M = 2) ofperiod T . The return echo reflected from a stationary tar-get at distance R from the radar can be considered as thedelayed and attenuated copy of the transmitted signal withtime delay equal to t = 2R=c, where c is the light velocity.This received signal is mixed with transmitted signal and atthe output of a homodyne receiver the pure harmonic videosignal is obtained. The video signal frequency fR is equalto a t , where a = D f=T is the slope of frequency modula-tion and D f is signal frequency deviation. In FMCW radarthe unknown frequency fR is estimated by finding the maxi-mum of signal y(t)= s(t)s(t  t ) spectrum ( denotes com-plex conjugation). In practice, the transmitted signal s(t)has nonlinear frequency modulation. Its phase can be mod-eled by the polynomial (2) of higher order (M > 2) withnon-zero coefficient aM . In the paper we restrict analysis tothe third order (M = 3), however our method can be easilyextended for higher order (M > 3) effects.2. Statement of the problemLet us assume that the signal transmitted by the FMCWradar is given by:s(t) = S0 exp  j f 3(t); (3)where f 3(t) is periodical function with the period T , andfor time interval 0 t < Tf 3(t) = a0+a1 t+a2 t2+a3t T23(4)is the third order polynomial (for convenience, we useslightly different notation than (2)). The instantaneous an-gular frequency w (t) = d f 3(t)=dt of the signal (3) (for0 < t < T ) is equal tow (t) = a1+2a2 t+3a3t T22; (5)where a1=(2 p ) is the carrier frequency f0, 2a2=(2 p ) is theslope of frequency modulation a , and the third term inEq. (5) is responsible for quadratic modulation distortion.After mixing the signal s(t) with the return echo s(t  t )we obtain (for t < t < T )y(t) = s(t)s(t  t ) = Y0 exp  j f (t): (6)26The simple analysis method of nonlinear frequency distortions in FMCW radarThe differential phase f (t) = f 3(t)  f 3(t   t ) is givenby:f (t) = b0+b1 t+b2t T22; (7)whereb0 = a1 t  a2 t2+a3 t3+32 a3 t2 T ; (8)b1 = 2a2 t  3a3 t2; (9)b2 = 3a3 t : (10)To simplify further considerations we have assumed unitamplitude of the signal (S0 = 1) and we do not take intoconsideration initial phase b0. To measure nonlinear effectsof modulation, the unknown parameters b1 and b2 of thesignalx(t) = exp jb1t+b2t T22!(11)have to be estimated. For small nonlinear distortions thesecond term in Eq. (9) is negligible and we can assumeb1 = 2a2 t . The parameter b1 is the unknown angular fre-quency of the signal, proportional to time delay t (range tothe target), and b2 = 3a3 t is the unknown quadratic mod-ulation distortion. It must be noted that due to distortionsthe signal (11) is an LFM signal instead of a pure harmonicsignal.3. Estimation of unknown parametersIn the modern digital FMCW radar receivers the video sig-nal (11) is sampled with sampling period D T = T=N, whereN is a natural number (typically N = 1024; 2048; : : : ).A discrete-time version of Eq. (11) has the formx(n) = expjhn q R+ g q R(n N=2)2i; (12)where q R = w R D T is the unknown range angular frequencynormalized with respect to sampling frequency fs = 1D T andthe parameterg =3a3T2a2N(13)is an unknown coefficient of nonlinear distortions.Let us apply the idea of the matched filter to the estimationof unknown parameters q R and g of a deterministic sig-nal (12) contaminated by additive white Gaussian noise. Ifthe impulse response h(n) of the deterministic N 1 orderFIR filter fulfills the conditionh(n) = x(N 1 n) (14)then the filter is matched to the signal x(n) and its re-sponsez(n) =N 1åk=0x(n k)h(k) =N 1åk=0x(n k)x(N 1 k) (15)for n = N 1 is equal to the signal energy Ez(N 1) =N 1åk=0x(k)2= E: (16)The energy E is obtained only when estimated values ˆq Rand ˆg of matched filter impulse response are equal to actualvalues q R and g of the analyzed signal. In the estimationprocess it is necessary to search for values ˆq R and ˆg thatmaximize (15) for n = N 1, or equivalently to maximizeabsolute value of the two-dimensional transformX( q R; g ) ==N 1åk=0x(N 1 k) exp j(N 1 k)q R ++ g q RN2 1 k2: (17)For g = 0 Eq. (17) becomes the classical discrete Fouriertransform (DFT) of the signal x(n), often used for esti-mating range frequency ˆq R in FMCW radars with linearfrequency modulation. The second order term in Eq. (17)(for g 6= 0) can be interpreted as the correction of non-linear effects. In general, Eq. (17) can be treated as theextended, generalized chirp transform, which allows detec-tion and estimation of modulation distortions. It is worthto notice that transform (17) is a special case of the so-called generalized chirp transform (GCT) and the estimatedvalues ˆq R and ˆg are the maximum likelihood estimatesof q R and g parameters [5].4. Measurement results and conclusionsTo verify the ability of the transform (17) to detect andestimate nonlinear distortions in FMCW radar signal, therecorded and simulated nonlinear modulated FMCW sig-nals were tested. In Fig. 1 the recorded signal transformis presented. It is easy to notice that the transform (17) ofrecorded signal reaches its maximum for g approximatelyequal to  1%.For comparison purposes in Fig. 2 the transform of simu-lated signal with  1% third order component is presented.These two figures are very similar, but it can be easily seenthat recorded signal transform is less symmetrical than thesimulated one. This is due to the presence of higher or-der nonlinear distortions in the recorded signal. Furtherinvestigations show that the fourth order component causesvertical asymmetry, and the fifth order term causes hori-zontal asymmetry. This example proves that the presentedmethod may be used successfully to estimate third ordernonlinear coefficients and to detect higher order effects.27Krzysztof S. Kulpa, Andrzej Wojtkiewicz, Marek Nałęcz, and Jacek MisiurewiczFig. 1. Recorded signal transform (17) magnitude; f   g plane( f = q R =2p ).Fig. 2. Simulated signal transform (17) magnitude.References[1] A. Wojtkiewicz, J. Misiurewicz, M. Nałęcz, K. Jędrzejewski, andK. Kulpa, “Two-dimensional signal processing in FMCW radars”, inProc. XX KKTOiUE, Kołobrzeg, Poland, 1997, pp. 475–480.[2] S. Peleg and B. Porat, “Estimation and classification of polynomial-phase signals”, IEEE Trans. Inform. Technol., vol. 37, no. 2, 1991.[3] S. Peleg and B. Porat, “The Cramer-Rao lower bond for signals withconstant amplitude and polynomial phase”, IEEE Trans. Signal Proc.,vol. 39, no. 3, pp. 749–752, 1991.[4] S. Peleg, B. Porat, and B. Friedlander, “The achievable accuracyin estimating the instantaneous phase and frequency of a con-stant amplitude signals”, IEEE Trans. Signal Proc., vol. 41, no. 6,pp. 2216–2224, 1993.[5] B. Boshash, “Estimating and interpreting the instantaneous frequencyof a signal” (Part 2), Proc. IEEE, vol. 80, no. 4, pp. 540–568,1992.[6] M. Benidir and A. Oudali, “Polynomial phase signal analysis based onthe polynomial derivaties decomposition”, IEEE Trans. Signal Proc.,vol. 49, no. 7, pp. 1954–1965, 1999.[7] B. Barkat and B. Boashash, “Instantaneous frequency estimation ofpolynomial FM signals using the peak of PWVD: statistical perfor-mance in the presence of additive Gaussian noise”, IEEE Trans. SignalProc., vol. 47, no. 9, pp. 2480–2490, 1999.[8] B. Barkat and B. Boashash, “Design of higher order polynomialWigner-Vilde distributions”, IEEE Trans. Signal Proc., vol. 47, no. 9,pp. 2608–2611, 1999.Krzysztof S. Kulpa was born in Warsaw, Poland onApril 13, 1958. He received the M.Sc. and Ph.D. degrees inelectronic engineering from Warsaw University of Technol-ogy in 1982 and 1987, respectively. Since 1982 he has beenin the Institute of Electronics Fundamentals at Warsaw Uni-versity of Technology on the posts of a Teaching Assistantand Assistant Professor. His teaching activities included theareas of measurements and digital signal processing. In theyears 1988–1990 he was also at the Technical Universityof Bialystok, Bialystok, Poland. His professional experienceincludes training visits in IBM Spain or Elmer-Perkins inEngland, as well as being a technical consultant in CN-PEP RADWAR in Poland. He was involved with severalresearch and development projects, granted from the Pol-ish radar industry, in the area of digital processing of radarsignals. His interests include Kalman filtering and objecttracking problems, adaptive MTI filtering, and continuouswave radars.e-mail: k.kulpa@ise.pw.edu.plInstitute of Electronic SystemsWarsaw University of TechnologyNowowiejska st 15/1900-665 Warsaw, PolandAndrzej Wojtkiewicz was born in Nowogrodek (Poland)in 1938. He received the M.Sc. and Ph.D. degrees fromthe Gdansk University of Technology (Poland) in 1961and 1971, respectively, both in electronic engineering.Since 1974 he has been with the Warsaw University ofTechnology, where he currently holds the position of Pro-fessor. His fields of interest concentrate on analysis anddesign of digital filters, adaptive signal processing and pa-rameter estimation, applied primarily in digital processingof radar signals. He has authored and co-authored morethan 100 research papers and technical reports in his areasof interest. He is also author of the book, Introduction doDigital Filter Synthesis (1984). He has been a member ofthe IEEE.e-mail: a.wojtkiewicz@ise.pw.edu.plInstitute of Electronic SystemsWarsaw University of TechnologyNowowiejska st 15/1900-665 Warsaw, PolandMarek Nałęcz was born in Warsaw (Poland) in 1961. Hereceived the M.Sc. degree in 1984 and the Ph.D. degreein 1992 from the Warsaw University of Technology, bothin electronic engineering and both conferred with honors.Since 1984 he has been with the Warsaw University of28The simple analysis method of nonlinear frequency distortions in FMCW radarTechnology, where he currently holds the position of As-sistant Professor. His early works were devoted to the analy-sis of switched-capacitor circuits. Now his fields of interestconcentrate on digital processing of radar signals, applica-tions of digital signal processors, adaptive filters, modernspectral estimation methods and robust algorithm of systemidentification. He has authored and co-authored more than50 research papers, conference papers and technical reportsin his areas of interest. He has been a member of the IEEE.e-mail: m.nalecz@ise.pw.edu.plInstitute of Electronic SystemsWarsaw University of TechnologyNowowiejska st 15/1900-665 Warsaw, PolandJacek Misiurewicz was born in Warsaw, Poland onMarch 26, 1965. He received the M.Sc. and Ph.D. de-grees in electronic engineering from Warsaw University ofTechnology in 1988 and 1996, respectively. Since 1988 hehas been in the Institute of Electronics Fundamentals atWarsaw University of Technology on the posts of Teach-ing Assistant and Assistant Professor in the area of digitalsignal processing. He was involved with several researchand development projects, granted from the Polish radarindustry, in the area of digital processing of radar signals.His interests include non-uniform sampling problems, MTIfiltering and filter design, as well as programmable logicapplications and real-time programming. He was a mem-ber of an Organizing Committee of National Circuit The-ory conferences in 1991 and 1992. He is a member of theIEEE since 1993, and IEEE Poland Section Newsletter Ed-itor since 1998.e-mail: j.misiurewicz@ise.pw.edu.plInstitute of Electronic SystemsWarsaw University of TechnologyNowowiejska st 15/1900-665 Warsaw, Poland29

The simple analysis method of nonlinear frequency distortions in FMCW radar, Journal of Telecommunications and Information Technology, 2001, nr 4

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The simple analysis method of nonlinear frequency distortions in FMCW radar, Journal of Telecommunications and Information Technology, 2001, nr 4

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