Cancellation of Doppler Distortion in PulseCompression for Targets Moving in an Arbitrary

DirectionJie Yang and Tapan K. Sarkar, Fellow, IEEE

resolution of a radar system does not necessarily depend on theAbstract It can be shown that the hyperbolic frequency duration ofthe transmitted pulse, but depends on the bandwidthmodulated waveform is Doppler-invariant only when the Doppler of the pulse [1]. In the modulation stage, the transmitted signalfactor is a constant number, i.e., the target has a constant velocity is modulated by a linear chirp to increase the bandwidth of theand along the direction from the radar to the target. If the target pulse.Ith opesostgth eevdsga1Spsdmoves in an arbitrary direction, the degradation of the pulse

tI thedompressionsae mtch ed sn ispansed

compression caused by the mismatch between the reflected signal through a matched filter which is matched to the transmittedand the matched filter may still exist. In this paper we waveform, which results in an extremely narrow pulse with ademonstrate that the Doppler effect caused by the moving target large peak value, thus the transmitted pulse is compressed inin an arbitrary direction can be approximated by a target with an the time domain [2]. However the received signal is distortedinitial velocity and a constant acceleration along the direction and does not match with the matched filter due to the Dopplerfrom the radar to the target, which results in a frequency shift inthe~~~~~ ~relce.aeom hrfr abn fmthdfleswt effect if the relative velocity between the radar and the target isthe reflected waveform. Therefore a bank of matched filters withpre-selected values of frequency shifts can be utilized to relatively large compared with the velocity of signalcompensate for the Doppler effect. Numerical examples with propagation. The performance degradation of a lineartarget moving in an arbitrary direction are presented to illustrate frequency modulated (LFM) pulse compression system causedthis effect, which have been successfully compensated by shifting by this mismatch has been studied by many researchers in [3-5].the frequency response of the matched filter. Kelly and Wishner [6] parameterize the returned signal and

modify the matched filters and ambiguity functions tocompensate for the Doppler effect. Rihaczer [7] extended their

Arbindexry s-Doppler-Invariant, Pus Copein, work by applying a complex notation to take into account theArbitrary DirectionDoppler-caused amplitude changes in the returned signal.Kramer [8] uses different reference signals (lines with different

I. INTRODUCTION slope in f-t plane) for the matched filter to eliminate the

ulse compression (also known as pulse coding) technique is mismatch. Another way of solving this problem is utilizing117 lied in many pulsed radar systems in order to maximize some other frequency modulation schemes instead ofLFM. It is

shown that the hyperbolic frequency modulated (HFM)the sensitivity and the resolution of radar systems. The shw tathe yproifeqnc mdutd HM

thesensitivityof andtheresolutradar systems.thusthemaximun Tge, waveform is insensitive to the target velocity [9-10]. It is alsosensitivity of a radar system, thus the maximum detection range,depends on the energy transmitted in the radar pulses. This can interesting to know that some kinds of bats use similar signals

in terms otafor echolocation purpose [11-12]. The design andbe expressed intermsoftheaveragetransmittedpower,tha implementation of a matched filter for a HFM waveform arethe peak power multiplied by the transmitter duty cycle. Sincethe peak power mutispalwalimited forthe transmitterdut le. the discussed by Rhodes [13] and Altes and Skinner [14]. Howeverrheliean sfetis sues aoiated with a h - tae the Doppler-invariant property of aHFM waveform is based on

reliability andsafety issue associated witha very important assumption: the Doppler factor should besystem, transmitting longer pulses improves the radar's y pppsystem, transmitting l ter aver transmtte power. constant, i.e., the target should have a constant velocity andsensitivity by increasing the average transmitted power. along the direction from the radar to the target. If the target hasHowever a narrow pulse provides a number of advantages overa changing velocity or moves in an arbitrary direction, similara wide pulse, including better range resolution and range to the Doppler effect on a LFM waveform, the received

accuracy and better performance under cluttered environment.Puls comresion ombnes he igh nery ofa lng plse waveform will have a mismatch with the matched filter and thePulse with the highrsui of s ort lse withb output is again degraded. In this paper we explore the Dopplerwidth with the high resolution of a short pulse width by

increasing the bandwidth of radar pulses, because the range effect on the HFM waveforms when the target has a relativeconstant velocity with respect to the radar but the direction is

Manuscript rceived Januay 11, 2007.arbitrary. A constant velocity in an arbitrary direction can beJ. Yang and T. K. Sarkar are with the Department of Electrical Engineering approximated by a target with an initial velocity and a constant

andComputer Science, Syracuse University, Syracuse,NY 13244USA (phone: acceleration in the direction from the radar to the target, which315-443-1406; e-mail: jyangl4@syr.edu).

1-4244-0284-0/07/$20.OO ©2007 IEEE 297

leads to frequency shifts in the HFM waveform [15]. The scale caused by the relative velocity between the radar and thedegradation of the output caused by the frequency shift, which target. Assuming both the transmitted and the returned signalsdepends on the angle of the direction vector between the radar have approximately the same shape with a delay, the returnedand the target locations, can be eliminated by shifting the signal can be express as [6]:frequency response of the matched filter.

In Section 2, we derive the Dopper effect on the HFM r(t) = 'xE[s (t - r)] (3)waveform when the target has a constant velocity and moves inan arbitrary direction. Some numerical examples are presented where x(t) is the transmitted waveform, r is the propagationin Section 3, followed by the conclusion in Section 4.

time delay and s is the Doppler factor:

II. EFFECT OF CONSTANT VELOCITY IN AN ARBITRARY = 2D / c (4)DIRECTION ON A HFM WAVEFORM 5(t) c -v (t)1 I 2v(t) (5)

Assume that a target has a relative constant velocity with c+ v'(1) crespect to the radar but the direction is arbitrary, which isillustrated in Figure 1. Substituting Equation (2) into Equation (5) and omitting the

terms higher than the second order, the Doppler factorbecomes:

v s(0)l_ 26'ts(t) - 2v cos(00 2V2 sin2 (0) (6)c DcV / So the Doppler factor can be approximated by adding a

constant acceleration term along the direction of the velocity,where the magnitude of the acceleration is:

Radar /a=2v2sin2(00)/D (7)

Fig. 1. Target moving in an Arbitrary Direction. The HFM hyperbolic frequency modulated pulse can bewritten as:

At the initial time, the target velocity has an angle 00 withthe direction between the radar and the target. At time t, which 1is the duration of the transmitted pulse, the target has a x(t) = A(t) exp jL k log (I + kfot) (8)displacement vt and the angle is changed to Ot . If the target

velocity is large enough and the duration of the pulse is long, where k is a constant factor and fo is the starting carrierthe value of vt is not negligible, so the angle 0 will not be a frequency. To simplify the derivation, Equation (8) is rewrittenconstant, but will be a function of time t, which can be written as:as:

0, = 00 - O00 -arcsin(vtsin(OO)/D) (1) x(t-t0) = A(t-to)cosL klog(kfot)] (9)So - vt sin(O0) ID

where t = 1/ kf0. Substituting Equation (9) into Equationwhere D is the distance between the radar and the target. (3), the reflected signal is:Applying a Taylor Expansion to cos(Ot) , the velocitycomponent along the direction between the radar and the target r(t to = /A[s(t -t, r)]expFj 2k2°(z S(tr)can be expressed as:

v'(t)= vcos(0 ) =sA[s(t -to r)]expLJ klog(kt (t r)) + j I log(s) (10)V'(=O)+ v2 Sin2 (09)1v cos(00,)sin2(0) (2 + (2) - r)]expLJ 2 log(kfk(t r))JexPLj k log(s)This velocity component results in a Doppler-distortion on

the etuned ignl,wichcanbe vewe as fuctio oftwo Comparing with the carrier chirp signal, the dilation on thevariables: a time delay caused by the distance and a Doppler envelope A1 (f) can be assumed to be negligible. Thus Equation

298

(10) can be rewritten as: velocity 0. = 600. The output of the matched filter when the

velocity of the target is v=O.Olc is plotted in Figure 2. Here c isr(t -t) =s/x(t-10 --r) expj-2T log(s)0 (11) the velocity of the signal propagation and v is the initial

Lk i velocity of the target. From this figure we can see that theoutput of the matched filter degrades dramatically due to the

Substituting Equation (6) into Equation (11), we get: angle between the velocity of the target and the line from theradar to the target. Figures 3-5 show the output of the matched

r(t - to) = vX (t - to- ) filter when Af is equal to 0.005MHz, 0.013MHz and 0.02MHz

21wr log 2v cos(OO) 2V2 sin 2 (00) (12) respectively. Here Afis the frequency shift ofthe matched filterexp lo 1- ) 2v2Sin2(0t) from the frequency response of the transmitted waveform.

k c Dc j Figures 4 and 5 indicate that when we shift the frequencyresponse ofthe matched filter, the performance resulting in the

Also we assume: pulse compression becomes better and better. At a certainoptimum Af(in this example the value is around 0.013MHz) the

log(1 + x) x for x << 1 (13) system achieves the best performance. This optimum frequencyshift is proportional to the approximated linear acceleration of

Substituting Equation (13) into Equation (12), we have: the target. From Figure 5 we can see that the normalized peakvalue approaching 1, i.e., the distortion caused by the Doppler

r(t-to) effects are almost all cancelled. Figure 6 shows that if we

i-F.2~~2vcos(O0) 22 sin2 7continue increasing the Af the result starts getting worse.

= fsx(t - to - -r) exp Lj.2T ( 2vcos( e') _2V2 sin2 (0) t);vsx(t~to~r)exPLJk(C Dc t)1 Corrpressed I-FM Pulse (v=0.01 c, 0°, 6f=0)

(s)x(t to r) exp 2T i (t 2vcos(OO) 0]( 1Dkc kc ]0.

= [x(t - to - O)exp[2;Tj (Af (t - r)+ A9)] 0.8

0.7-

where Af = -2v2 sin2 (00) / Dkc is the frequency shift and Ay , 0.6-2v cos(00) / kc is the phase shift. Equation (13) indicates that Ub-

the reflected signal can be approximated by a delayed, 04frequency-shifted and phase-shifted version of the transmitted 0.3signal. The phase shift will be ignored by the matched filter but 0.2the frequency shift yields the mismatch between the reflected 0.1signal and the matched filter. Therefore we can construct amatched filter by shifting the frequency response of the 15 20 25 30 35

transmitted signal to match the reflected waveform. Ifthe angle t (,us)00 is unknown, we may also select a number of Af s to Fig. 2. Compressed Pulse withv=O.OOlc, 00 =600 and AfrO.

construct a set of filter bank which covers the exact frequencyresponse ofthe reflected signal. This guarantees the elimination 1Copessed FFMRs (.01c, ° f.O5MU)of the Doppler distortion. 0

0.8-

III. NUMERICAL EXAMPLE 0.7

In this section we use a rectangular envelope modulated , 0.6pulse to illustrate the Doppler effect caused by a point target .05moving with a constant velocity in an arbitrary direction for the O0HFM waveform, and how the output of the matched filter isimproved by shifting the frequency response of the matched 0.3filter. The outputs of the matched filters for different Doppler 0.2factors are normalized to the output without the Doppler effect 0.1 0to keep the same scale. In the first example, a rectangular pulse O y,,,,, ,7ptTl1l l <ois transmitted with the duration 100 microseconds. The carrier 15 20 25 30 35frequency for the hyperbolic chirp is from 2MHz to 20MHz, t(Ps)the propagation time delay r = 20,us . The initial angle of the Fig. 3. Compressed Pulse with v=O.OOlc, 00 =60° and A,ftO.O5MHz.

299

Con-pressed F-FM Pulse (v.01 c, 0=6e, 6f=0.01 3M-z)1

Con-pressed HFM PlIse (v=0.01 c, o-9bO, 6f=0)0.9 ( 1

0.8 0.9 P

0.7 -0.8-

0.6 0.7

= 1 5 0.6

1.5 -02 03

0.4- 0.4

0.3 -04

0.2- 0.3-

0.1 0.2-

0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~0.1(S) 15 20 25 30 35

15202303

Fig. 4. Compressed Pulse with v=O.OOlc, 00 =600 and Af0013MHz. t (Ps)Fig. 6. Compressed Pulse with v=O.OOlc, 00 =90 ° and A4fO.

Con-pressed F-FM PRIse (v=0.01 c, 0=6be 6f=0.02MU-iz)oftCorpressed F-FM Pulse (v=0.01 c, 0=O9bO, ifo=0.01M-6z)

0.9 1

0.8- 0.9

0.7- 0.8-

0.6- ~~~~~~~~~~~~~~~~~~~~~~~0.7-0.60.0. -)0.

0.4- 0.4

0.3- 04

0.2 w 0.3

0.1 0.2-

C 0.115 20 25 30 35

t(Ps) 2015 2025 30 35

Fig. 5. Compressed Pulse with v=0.001c, 00 =600 and AftO0.02MHz t (PS)Fig. 7. Compressed Pulse with v=0.001c, 00 =9u and AflO.OeMHz.

In the second example, the initial angle is changedto0 =o900. The output of the matched filter when the velocity C Fc5of the target is v=..3Mc is plotted in Figure 6. Comparing with 09Figure 2 we can see that the degradation increases when the 08initial angle increases. This is reasonable because the largerinitial angle means the larger acceleration component along thedirection from the radar to the target, i.e. a higher Doppler °06distortion. The output of the matched filter when Af is equal to 250.0oMHz, 0.002MHz and 0.03MHz are plotted in Figures 7-9 0.4erespectively. Similar to the previous example, these results Fig 8indicate that when the frequency response of the matched filteris shifted, the performance of the pulse compression improves 02

gradually and approaches the best result for a certain Af.After othat, the result gets worse and worse. It is also as expected thatthe optimum frequency shift (in this example the value is 15 20 25 30 35

arondi( 0n0MHz7) is larger than that in the- previousv example t Gi5S)

Corrpressed F-FM PRise (v=0.01 c, 0=9Ob, 6f=0.03M-z) [11] R. A. Altes and E.L. Titlebaum, "Bat signals as optimally Doppler tolerant1 waveforms". J Acoust. Soc.48, pp. 1014-1020, 1970.

[12] R. A. Johnson and E. L. Titlebaum, "Range-Doppler uncoupling in theDoppler Tolerant Bat Signal", Proceedings of IEEE Ultrasonic

0.8 Symposium, pp 64-67, New York, 1972.[13] J. D. Rhodes, "Matched-Filter Theory for Doppler-Invariant Pulse

0.7- Compression", IEEE Trans. on. Circuit Theory, vol CT-19, no. 1,).60 pp53-59, January 1972.

-r [14] R. A. Alters and D. P. Skinner, "Sonar Velocity Resolution with a0.5 Linear-Period-Modulation Pulse", J. Acoust. Soc. Am., vol 61 no. 4, pp

0.4- 1019-1030, April 1977.[15] J Yang and T. K Sarkar, "Acceleration-Invariant Pulse Compression using

0.3- Hyperbolic Frequency Modulated Waveforms", Proc. 2006 WaveformDiversity and Design Conf, Kauai, HI, January 2006.

O.101

15 20 25 30 35t (,ts)

Fig. 9. Compressed Pulse with v=0.001c, 00 =90 and Aff0.03MHz.

IV. CONCLUSION

In this paper we demonstrate that the Doppler effect causedby the moving target along an arbitrary direction can beapproximated by a target with an initial velocity and a constantacceleration in the direction from the radar to the target.Therefore we can apply a HFM waveform and frequency shiftof the matched filter to obtain better Doppler tolerant pulsecompression. The numerical example indicates that the outputof the matched filter is significantly improved by applying thisfrequency shift technique. The distortion caused by the Dopplereffect is greatly minimized. We may conclude that the filterbank method used in combination with the HFM pulse is veryuseful in a pulse compression radar system when the movingobject has a constant velocity along an arbitrary direction.

REFERENCES[1] J. R. Klauder et al. "The Theory and Design of Chirp Radars," Bell Syst.

Tech. J, Vol. 39, pp. 745-808, July 1960.[2] Ian Faulconbridge, Radar Fundamentals, Argos Press, Australia, 2002.[3] H. 0. Ramp and E. R. Wingrove, Jr., "Performance degradation of linear

FM-pulse-compression systems due to the Doppler effect." Proc. IRE, vol.49, pp 1693, Nov 1961.

[4] C. E. Cook et al, "Linear FM pulse compression Doppler distortioneffects." Proc. IRE, vol 50, ppl535-1536, June 1962.

[5] W. L. Rubin and J. V. Difranco, "The effects of Doppler dispersion onmatched filter performance." Proc. IRE, vol 50, pp 2127-2128, October1962.

[6] E. J. Kelly and R. P. Wishner, "Matched-filter theory for high-velocity,accelerating targets", IEEE Trans. Mil. Electron. MIL-9, pp. 56-65,January 1965

[7] A. W. Rihaczer, "Delay-Doppler Ambiguity Function for WidebandSignals", IEEE. Trans. on Aero. and Electron. Sys., vol AES-3, no. 4, pp705-71 1, July 1967.

[8] S. A. Kramer, "Doppler and Acceleration tolerances of High-Gain,Wideband Linear FM correlation Sonar", Proceedings ofIEEE, vol 33, no.3, pp 627-636, May 1967.

[9] J. J. Kroszczynski, "Pulse Compression by means of linear-periodmodulation", Proc. IEEE., pp. 1260-1266, July 1969.

[10] J. Yang and T. K. Sarkar, "Doppler-Invariant Property of HyperbolicFrequency Modulated Waveforms". Microwave and Optical TechnologyLetters, Vol. 48, June 2006, pp.1 174-1 179.

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