Pulse Compression Sidelobe Reduction by Minimization of L...

[10] Robey, F. C., Coutts, S., Weikle, D. D., McHarg, J. C., andCuomo, K.MIMO radar theory and exprimental results.In Proceedings of the 38th Asilomar Conference on Signals,Systems and Computers, vol. 1, Nov.2004, 300—304.

[11] White ,L. B., and Ray, P. S.Receiver design for MIMO tracking radars.Presented at the 2004 Waveform Diversity and DesignConference, Edinburgh, UK, Nov. 2004.

[12] Fuhrmann, D. R., and Antonio, G. S.Transmit beam-forming for MIMO radar systems usingpartial signal correlation.In Proceedings of the 38th Asilomar Conference on Signals,Systems and Computers, vol. 1, Nov. 2004, 295—299.

[13] Fuhrmann, D., and San Antonio, G.Transmit beamforming for MIMO radar systems usingsignal cross-correlation.In Proceedings of the 38th Asilomar Conference on Signals,Systems and Computers, Pacific Grove, CA, Nov. 7—10,2004.

[14] Mecca, V., Ramakrishnan, D., and Krolik, J.MIMO radar space-time adaptive processing for multipathclutter mitigation.In IEEE Sensor Array and Multichannel Signal ProcessingWorkshop, July 2006, 249—253.

[15] Xu, L., Li, J., and Stoica, P.Adaptive techniques for MIMO radar.In IEEE Sensor Array and Multichannel Signal ProcessingWorkshop, July 2006, 258—262.

[16] Fishler, E., Haimovich, A., Blum, R., Chizhik, D., Cimini,L., and Valenzuela, R.MIMO radar: An idea whose time has come.In Proceedings of IEEE Radar Conference, Apr. 2004.

[17] Fishler, E., Haimovich, A., Blum, R., Cimini, L., Chizhik,D., and Valenzuela, R.Statistical MIMO radar.In The 12th Conference on Adaptive Sensors ArrayProcessing, Mar. 2004.

[18] Fishler, E., Haimovich, A., Blum, R., Cimini, L., Chizhik,D., and Valenzuela, R.Performance of MIMO radar systems: Advantages ofangular diversity.In Proceedings of the 38th Asilomar Conference on Signals,Systems and Computers, 2004, 305—309.

[19] Klemm, R.Principles of Space-Time Adaptive Processing.London: Institution of Electrical Engineers, 2002.

[20] Scharf, L. L.Statistical Signal Processing.Reading, MA: Addison-Wesley, 1991.

[21] Jin, Y., and Friedlander, B.Detection of distributed sources using sensor arrays.IEEE Transactions on Signal Processing, 52, 6 (June2004), 1537—1548.

[22] Haykin, S. (Ed.)Array Signal Processing.Englewood Cliffs, NJ: Prentice-Hall, 1985.

[23] Ljung, L., and Söderström, T.Theory and Practice of Recursive Identification.Cambridge, MA: MIT Press, 1983.

[24] Haykin, S.Adaptive Filter Theory (4th ed.).Upper Saddle River, NJ: Prentice-Hall, 2004.

[25] Moon, T. K., and Stirling, W. C.Mathematical Methods and Algorithms for SignalProcessing.Upper Saddle River, NJ: Prentice-Hall, 1999.

Pulse Compression Sidelobe Reduction byMinimization of Lp-Norms

Most modern radar systems make extensive use of pulse

compression techniques. This paper presents a technique for the

design of mismatched receive finite impulse response (FIR) filters

based on the minimization of Lp-norms of the sidelobes. The

goal of the minimization process is to reduce the range sidelobe

levels of the convolution of the transmit pulse and the receive

filter. A closed-form solution is derived for the least-squares

case (which is equivalent to the L2-norm) and an expression

for the optimization of the higher order norms is developed.

The solutions for the higher order norms have to be obtained

by means of iterative numerical methods. The effect of using

receive filters which are longer than the transmit pulses is

also investigated. Results are presented for linear FM transmit

waveforms having time-bandwidth products ranging from 10

to 100 in combination with selected values of the norm order

ranging from 2 to 200. Receive filter lengths up to three times

the transmit pulse lengths are investigated. Results are presented

which highlight the tradeoffs between sidelobe level, mismatch

loss and mainlobe width. The effect of Doppler shift on the

sidelobe response of these receive filters is also investigated.

I. INTRODUCTION

Since the development of pulse compression inthe mid-1950s [1, 2] the concept has become anindispensable feature of modern radar systems. Pulsecompression gives radar designers the ability to obtainsufficient energy on a target for detection withoutdecreasing the range resolution of the system orresorting to the use of very high transmitter powerlevels. Pulse compression thus allows for the useof lower power transmitters but with longer pulselengths to maintain the energy content of a pulse. Amatched filter is used on reception to maximize thesignal-to-noise ratio (SNR) of the received signal. Theactual waveforms that are transmitted are chosen soas to have an autocorrelation function (ACF) with anarrow peak at zero time shift and values (referredto as sidelobes) as low as possible at all other times.

Manuscript received May 13, 2006; revised January 2, 2007;released for publication July 27, 2007.

IEEE Log No. T-AES/43/3/908447.

Refereeing of this contribution was handled by E. S. Chornoboy.

This work was supported by the South African Department ofScience and Technology.

0018-9251/07/$25.00 c° 2007 IEEE

1238 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 3 JULY 2007

The sidelobes have the undesirable effect of maskingsmaller targets which are in close proximity to largetargets, such as clutter returns.The quest for good pulse compression waveforms

and techniques for the reduction of sidelobe levelshas been an active area of research since the inceptionof pulse compression. Several approaches to theproblem have been suggested since the 1960s whichrange from analytical techniques [3, 4] throughexhaustive searches to noise radar concepts [5]. Nodirect techniques for the construction of codes withoptimally low sidelobe levels are known, so severalalternative approaches have been proposed for thedesign of such codes. Some of these techniquesfocus on the design of transmit waveforms with goodsidelobe properties, while others focus on the designof the receiving filter.The most common design techniques for pulse

codes with good sidelobe properties rely on searchingfor codes with low sidelobe levels. These proceduresare based on exhaustive searches thus limiting themaximum code length that can be found. Binarycodes having low peak sidelobe levels have beenreported by Lindner [6] for lengths up to 40, Cohen[7] and Mertens [8] for lengths up to 48, and Coxson[9] for lengths up to 70. The searches are usuallylimited to binary phase codes [10, 7] or quadriphasecodes [3] to reduce the size of the search spaceor to adhere to implementation limitations in theradar or both [11]. Gartz [12], and more recentlyNunn [13], have addressed the problem of designingpolyphase codes. The computational complexityof these search techniques limits the length of thecodes that can be found in a reasonable amountof computing time. To alleviate the time constraintsome researchers have published techniques forthe construction of codes which have “good,” butnot optimal performance. The Frank codes [4] andP(n,k) codes proposed by Felhauer [14] are someof the well-known codes in this category. De Witteand Griffiths [15] generated chirp type codes withultralow sidelobe levels (more than 70 dB below thepeak for a time-bandwidth product (TBWP) of 270)by direct design of the time-frequency characteristicof the transmit waveform. Alternative techniquesfor producing low range sidelobes rely on specialfilter techniques in the receiver, which in most casesrepresent some form of mismatched filtering. Sidelobecancellation [16] and sidelobe smoothing [17]schemes have also been used to reduce the sidelobelevels.This paper focuses on the design of the receiving

filter for a linear frequency chirp transmit waveform.The goal is to achieve sidelobe levels that are not onlylow, but approach a constant value over all non-zerotime shifts.In Section II the formulation of the sidelobe

minimization problem is developed. Simulation results

obtained for codes with various TBWP, norm orders,and receive filter lengths are presented in Section III.Section IV contains a discussion of the results andproposals for extensions of the Lp-norm sidelobeminimization concept. The derivation of the Lp-normsidelobe minimization equations is included in theAppendix.

II. PROBLEM FORMULATION

A. Matched and Mismatched Filters

The transmit pulse of a radar can be representedin the discrete time domain by a sequence of complextransmit coefficients fang. Digital pulse compressionis performed by convolving the received signal, whichis assumed to be a time-delayed and scaled versionof the transmitted pulse, with the complex receivefilter coefficients fxng. For the purpose of analyzingthe sidelobe response of the pulse compressor, a zerotime-delay and unity scaling factor can be assumedwithout loss of generality. In this paper a P4 code [18](i.e., sampled linear frequency chirp) is used for thetransmit coefficients.The receive filter can be chosen to be either the

matched filter for the transmit pulse or some form ofmismatched filter. It is known that the matched filtermaximizes the SNR at the output of the receive filter[19]. For the transmit pulse fang, the matched filteris given by hn = a

¤N¡n where

¤ denotes the complexconjugate and N is the number of transmit pulsecoefficients. If the matched filter is used, the output ofthe pulse compressor will be the ACF of fang whichis equivalent to the discrete convolution

bi =Xk

ai+1¡khk: (1)

The convolution sequence fbig for the matched filtercase has the maximum attainable SNR at zero timeshift. The SNR at zero time shift of any mismatchedfilter will be lower than that of the matched filter.This loss in SNR is known as mismatch loss Lm, andis given by the ratio of the peak output value of themismatched filter bpeak,mismatched, relative to the peakoutput value of the matched filter bpeak,matched, i.e.,

Lm =¡20log10Ãbpeak,mismatchedbpeak,matched

!(2)

where the coefficients of both filters have beennormalized for unity noise gain, such thatsX

n

khnk2 = 1: (3)

The sidelobe values of the convolution result canbe minimized by introducing cost functions whichmap the set of sidelobes to a single real value. By

CORRESPONDENCE 1239

minimizing the cost functions, mismatched receivefilters with reduced sidelobe responses can be found.In this paper Lp-norms are applied to the complexsidelobe values and used as the cost functions tobe minimized. Minimization of the L2-norm of thesidelobes will minimize the integrated sidelobe level,whereas minimization of the L1-norm will minimizethe peak sidelobe level. Given that from an analyticalpoint of view, the L1-norm is not a well-behavedfunction, Lp-norms will be used instead, with p= 2P,where p is an even integer larger than or equal to 2.The mismatched filter solutions for the

least-squares sidelobe measure, which is equivalentto the L2-norm solution, and the generalized L2P-normare developed in Sections B and C, respectively.

B. Least-Squares Sidelobe Minimization

The output of the pulse compression filter, whichis the convolution sequence of the transmit pulse andthe receive filter, can be written in matrix form as

b=AFx (4)where

b= [b1 b2 ¢ ¢ ¢b2N¡1]T (5)x= [x1 x2 ¢ ¢ ¢xN]T (6)

and

AF =

266666664

a1 a2 ¢ ¢ ¢ aN 0 ¢ ¢ ¢ 00 a1 ¢ ¢ ¢ aN¡1 aN ¢ ¢ ¢ 0...

......

0 ¢ ¢ ¢ a1 a2 ¢ ¢ ¢ aN 00 ¢ ¢ ¢ 0 a1 a2 ¢ ¢ ¢ aN

377777775

T

: (7)

Note that T denotes the transpose of a vector ormatrix and that AF is the full convolution matrix. Theabove formulation leads to the following convenientexpression for the sum-of-squares of the convolutionsequence

bHb= kb1k2 + kb2k2 + ¢ ¢ ¢+ kb2N¡1k2 (8)where H denotes the complex conjugate transpose.The sidelobe measure function for a compressed pulsecan now be formulated by defining a new matrix, A,which is similar to AF, except that the rows in AFwhich produce the compression peak are removed.The sidelobe measure cost function to be minimizedcan therefore be written as

f(x) = bHb= xHAHAx

= xHCx (9)

whereC=AHA: (10)

Using the method of Lagrange multipliers [20], asolution for x can be found that will minimize thesidelobe measure cost function while satisfyingthe constraint that a pulse compression peak withamplitude bpeak must be produced. This constraint canbe written as

ax= bpeak (11)where

a= [aN aN¡1 ¢ ¢ ¢a1]: (12)This leads to the constraint function

g(x) = ax¡ bpeak = 0: (13)Note that no constraint is placed on the samplesadjacent to the peak sample and no symmetryconstraints are placed on the filter response. It is theauthors’ opinion that removing the constraint on thesamples adjacent to the peak allows the optimizationprocesses to widen the pulse compression peakand force energy from the sidelobes into these twosamples.A system of simultaneous equations arise from the

complex Lagrangian

d

dx¤(f(x)) +

d

dx¤(Ref¸¤g(x)g) = 0 (14)

and the constraint given in (13). This extended systemof simultaneous equations can now be solved to obtainthe value of x that minimizes the sidelobe measure.The closed-form solution for x is of a similar form tothat given by Frost [21]:

x=bpeakC

¡1aH

aC¡1aH: (15)

The above approach is similar to that presented in[22].This solution for x produces a mismatched receive

filter for the transmit pulse fang that minimizes thepulse compression sidelobes in the least-squaressense, i.e., it is equivalent to minimizing the L2-normof the sidelobes. Following a similar approach, thegeneralized expression for the L2P-norm case can nowbe derived.

C. Generalized L2P-Norm Sidelobe Minimization

Noting that any element in the convolutionsequence fbig can be expressed as the product of ai(the ith row of A) and x, the following expression forthe cost function of a single sidelobe sample can bederived:

[kbik2]P = [bHi bi]P

= [(aix)Haix]

P

= [xHCix]P (16)

whereCi = a

Hi ai: (17)


Using (16) the generalized sidelobe measure costfunction (i.e., the L2P-norm of the sidelobes) can beexpressed as

f(x) =

Ã2N¡1Xi=1

[xHCix]P

!1=2P: (18)

By once again applying the method of Lagrangemultipliers, a formulation for x can be found that willminimize the sidelobes (in the sum-of-2Pth-powerssense) while satisfying the constraint for acompression peak as given by (13).From (14) the following set of simultaneous

equations arises for the generalized L2P-norm sidelobemeasure cost function:P2N¡1

i=1 ([xHCix]

P¡1Cix)

(P2N¡1i=1 ([xHCix]P))1¡1=2P

+¸aH = 0 (19)

with the constraint repeated here for clarity:

g(x) = ax¡ bpeak = 0: (20)The derivation of (19) can be found in the Appendix.Note that (19) reduces to the L2-norm form whenP = 1 is substituted.The solution of the above set of simultaneous

equations (together with the constraint given by (20))can be found using iterative numerical methods.

D. Additional Non-Zero Coefficients

The pulse compression sidelobe response canfurther be improved by increasing the length ofthe receive filter by adding additional non-zerocoefficients (ANZCs) to the receive filter. Themethod presented above provides a suitable means todetermine values for these ANZCs that will minimizethe sidelobe measure of the compressed pulse inthe L2-norm or generalized L2P-norm sense. This isachieved by increasing the receive filter length Nby the (even) number of ANZCs to be added andincreasing the number of columns in A to match thenew length of x. For the constraint it is required tozero pad a symmetrically to match the new receivefilter length.

E. Doppler Shifted Return Signals

In most radar applications the target is movingrelative to the radar and the return signal will beshifted in frequency due to the Doppler effect. Theresulting Doppler frequency shift is given by

fd = fc

·c¡ vc+ v

¡ 1¸»= 2vfc

c(21)

where v is the radar relative radial speed of the target(defined as being positive for outbound targets), fc isthe carrier frequency, and c is the speed of light. Toevaluate the effect of Doppler shift on the sideloberesponse of the optimized received filters, Dopplershift was added to the received signal by modulatingthe received samples as follows:

dn = ane¡j2¼fdnTs (22)

where Ts is the sampling period. A set of Dopplershifts were applied to the received pulse and each ofthese was then passed through the pulse compressionfilter to generate the delay-Doppler response of thefilter.

III. RESULTS

A suite of Matlab functions were implementedto generate the various transmit pulses, performthe minimization process and collect the resultsthereof. These routines were used to find optimizedmismatched receive filters for linear chirp transmitpulses with TBWPs from 10 to 100 using the methoddescribed above. The values for P were variedbetween 1 and 100 and the number of ANZCswere varied from zero to twice the length of thetransmit pulse, i.e., 0% to 200% of the length ofthe transmit pulse. Some specific examples of thematched and mismatched filter coefficients and filteroutputs are presented, followed by graphs showingthe performance trends for the zero Doppler caseas the waveform and optimization parameters arevaried. This section is concluded by the presentationof some examples of the delay-Doppler response andperformance trends for a fixed Doppler frequency.

A. Results for the Zero Doppler Case

In Fig. 1 the complex coefficient values andenvelope of the matched filter and two mismatchedreceive filters for a linear chirp pulse with a TBWP of50 are shown. Both mismatched receive filters weregenerated for P = 10. No ANZCs were added to thefirst mismatched receive filter and 50 ANZCs (i.e.,100% of the length of the transmit pulse) were addedto the second.The resulting pulse compressor response for a

linear chirp pulse with a TBWP of 50 is shown inFig. 2. The matched filter response, mismatched leastsquares and two L2P-norm filter responses (P = 2,P = 40) are shown. No ANZCs were added to thereceive filters.Fig. 3 shows the detail of the compression peak

of the pulse compression responses in Fig. 2. Themismatch loss of the mismatched filters is clearlyvisible and it is clear that the compression peaksare wider than that of the matched filter. A slight

CORRESPONDENCE 1241

Fig. 1. Complex coefficient values and envelope of three receivefilters for linear chirp transmit pulse with TBWP of 50.

(a) Matched filter. (b) Mismatched receive filter with no ANZCs.(c) Mismatched receive filter with 100% ANZCs.

Fig. 2. Pulse compression response for linear chirp transmitpulse with TBWP of 50. Pulse compressor output for matched

filter and several mismatched filters are shown.

narrowing of the peak, as well as a slight reduction inmismatch loss, can be observed for increasing valuesof P. The wider compression peaks are due to thefact that the samples on either side of the peak arenot constrained in the formulation of the optimization.This allows the optimization algorithm the freedomto widen the peak in order to reduce the sidelobelevel and to force the sidelobe response to be more“flat.”Fig. 4 shows the sidelobe response for a linear

chirp pulse with a TBWP of 50, now with 100ANZCs (i.e., 200% of the length of the transmit pulse)added to the mismatched receive filters. A significantreduction in the achievable peak-to-highest-sideloberatio (from 35.7 dB to 60.2 dB for P = 40) has been

Fig. 3. Detail of compression peaks in Fig. 2 showing slightnarrowing of mainlobe and reduction in mismatch loss with

increasing values of P.

Fig. 4. Pulse compression response for the same transmit pulseas in Fig. 2, with 100 ANZCs added to mismatched receivefilters. Addition of ANZCs reduces achievable sidelobe level

significantly.

achieved through the addition of ANZCs to thereceive filter.Fig. 5 is a zoom plot of the pulse compression

peak in Fig. 4. The mismatch loss is still visible butthere is no noticeable change in the peak widths ormismatch loss of the mismatched filters for differentvalues of P.Fig. 6 depicts the mismatch loss as a function of P

for various values of ANZC. The same linear chirptransmit pulse with TBWP of 50 was used for thegeneration of these results. The number of ANZCswas varied from 0% to 100% of the length of thetransmit pulse. It is noteworthy that for values ofANZCs higher than about 60%, increasing values ofP lead to increased mismatch loss, whereas for lowervalues of ANZCs, it leads to reduced mismatch loss.Fig. 7 shows the relationship between the

mismatch loss and the number of ANZCs used forvalues of P from 1 to 10. It can be seen that the effect


Fig. 5. Detail of compression peaks in Fig. 4 showing almost nodifference in mainlobe widths for different values of P.

Fig. 6. Mismatch loss for linear chirp transmit pulse with aTBWP of 50 and mismatched receive filters with increasing valuesof P. Number of ANZCs was varied from 0% to 100% of length

of transmit pulses.

Fig. 7. Mismatch loss for linear chirp transmit pulse with TBWPof 50 and mismatched receive filters with values of P from 1 to10 and ANZCs from 0% to 200% of length of transmit pulses.

of P on the mismatch loss diminishes with high valuesof ANZCs and that the mismatch loss tends towards aconstant value of approximately 1.9 dB.

Fig. 8. Mainlobe widths for linear chirp transmit pulses withTBWPs ranging from 10 to 100 and mismatched receive filterswith P = 10. Number of ANZCs was varied from 0% to 200% of

length of transmit pulses.

Fig. 9. Highest-sidelobe-to-peak ratios for linear chirp transmitpulse with a TBWP of 50 and mismatched receive filters for

increasing values of P. Number of ANZCs was varied from 0% to100% of length of transmit pulses.

Fig. 8 shows the 3 dB mainlobe width for linearchirp pulses with TBWPs ranging from 10 to 100 andmismatched receive filters with P = 10. It should benoted that the mainlobe width was measured aftersinc(¢) reconstruction of the mainlobe. The numberof ANZCs for the receive filters were varied from 0%to 200% of the length of the transmit pulses.Fig. 9 shows the highest-sidelobe-to-peak ratios for

a linear chirp transmit pulse with a TBWP of 50 andmismatched receive filters for increasing values of P.The number of ANZCs was varied from 0 to 50, i.e.,0% to 100% of the length of the transmit pulse.Fig. 10 is a graph of highest-sidelobe-to-peak

ratio versus ANZC length for linear chirp pulses withTBWPs of 10, 20, 40, 70, and 100. The mismatchedreceive filters were designed with P = 10. The numberof ANZCs added to the receive filters was varied from0% to 200% of the length of the transmit pulses.Fig. 11 is a contour plot of the highest-sidelobe-to-

peak ratios for linear chirp pulses with TBWPs

CORRESPONDENCE 1243

Fig. 10. Highest-sidelobe-to-peak ratios for linear chirp transmitpulses with TBWPs ranging from 10 to 100. Mismatched filterswith P = 10 were used and the number of ANZCs ranged from

0% to 200% of length of transmit pulses.

Fig. 11. Contour plots of the highest-sidelobe-to-peak ratios forlinear chirp transmit pulses with TBWPs ranging from 10 to 100and mismatched receive filters for increasing values of P. The

number of ANZCs was varied from 0% to 200% of the length ofthe transmit pulses.

ranging from 10 to 100 and mismatched receive filterswith P values of 1, 3, and 10. The number of ANZCsfor the receive filters were varied from 0% to 200% ofthe length of the transmit pulses.

B. Response of Optimized Receive Filters for DopplerShifted Return Pulses

The ambiguity function describes thedelay-Doppler response of a radar receiver for thematched filter case [23]. For the mismatched receivefilters presented here it is therefore appropriateto refer to the delay-Doppler response of thefilter. Fig. 12 and Fig. 13 show examples of thedelay-Doppler response for two of the optimizedreceive filters. In these figures only the positiveDoppler shifts are plotted as the delay-Dopplerresponses were found to be symmetric around zero

Fig. 12. Positive Doppler portion of the delay-Doppler responsefor linear chirp transmit pulse with TBWP of 50 and mismatched

receive filter with P = 2 and 0% ANZCs.

Fig. 13. Positive Doppler portion of the delay-Doppler responsefor linear chirp transmit pulse with TBWP of 50 and mismatched

receive filter with P = 40 and 200% ANZCs.

Doppler. The maximum Doppler shift of 100 kHzwas chosen to correspond to a high speed aircrafttarget (1000 m/s) at a transmit frequency of 15 GHz.In both figures the peak sidelobe level increasessubstantially but still stays below the original matchedfilter sidelobe level for zero Doppler (approximately¡23 dB in Fig. 2). It can also be seen that thesidelobe level is a smooth function of the Dopplershift which means that the degradation due to Doppleris gradual and the optimized receive filters do notfail catastrophically. To evaluate the sensitivity of themismatched receive filters to the parameters of theoptimization, namely the value of P and the numberof ANZCs, graphs of the highest-sidelobe-to-peakratio as a function of P for ANZCs ranging from 0%to 100% was generated (Fig. 14) for a fixed Dopplershift of 10 kHz. This Doppler shift corresponds toa 750 m/s target at a carrier frequency of 2 GHz.Fig. 14 should be compared with Fig. 9 which is thezero Doppler version of this result. It can be seenfrom Fig. 14 that the peak sidelobe level decreasesinitially for low values of P, but then transitions to


Fig. 14. Highest-sidelobe-to-peak ratios for linear chirp transmitpulse with TBWP of 50 versus P for ANZCs varied from 0% to

100% and for Doppler shift of 10 kHz.

an increasing tendency with increasing P. This is dueto a decrease in the level of the peak sidelobe untila different sidelobe becomes the new peak sidelobe.This sidelobe level then increases with increasingP. The increase does however seem to tend to anupper bound for the values of ANZC that wereevaluated. The 100% ANZC case is the most sensitiveto Doppler shift, relative to the zero Doppler case, asthe peak sidelobe level increases by approximately3.5 dB. It is interesting to note that the peak sidelobelevel for the 20% ANZC case rises and remains abovethat of the 0% ANZC case.

IV. CONCLUSION

This correspondence presents a new method fordesigning mismatched pulse compression receivefilters which is based on the minimization of aLp-norm sidelobe measure cost function. The resultsin Section III show that it is possible to achieve verylow sidelobes using linear frequency chirp transmitwaveforms together with the proposed mismatchedreceive filter design technique.From Fig. 3, Fig. 5, and Fig. 6 it is clear that the

mismatched receive filters for linear chirp transmitpulses have moderate mismatch loss. The mismatchloss does not change drastically over the range ofparameter values used and could be considereda worthwhile tradeoff for the low sidelobe levelsachieved. For the parameter value combinationsconsidered, the mismatch loss did not exceed 2.4 dB.It is also noteworthy that for ANZC values of lessthan 60%, there is a decrease in mismatch loss withincreasing P.In all cases considered, the 3 dB mainlobe width

of the mismatched receive filter output is slightlywider than for the corresponding matched filter. Themainlobe width for the matched filter of a linearfrequency chirp is 0.89 samples wide [24], whereasthe mainlobe widths obtained with the mismatched

receive filters ranged between 1.2 and 1.8 samples.As can be seen in Fig. 8, the mainlobe width initiallybroadens and then becomes slightly narrower asmore ANZCs are added. The mainlobe broadeningis due to the fact that the samples adjacent to the peaksample are unconstrained in the optimization process.Much lower sidelobe levels can be achieved using thisapproach.Despite the above disadvantages, there is a

significant gain in the peak-to-highest-sidelobe ratiosof the mismatched receive filters compared with thatof the matched filter. In Fig. 9 it can be seen that thesidelobe levels can be reduced by increasing the valueof P. The effect of P becomes less pronounced withincreasing P and values greater than 10 do not seemto add significant improvement.From Fig. 2 it should be noted that as P is

increased, the energy in the peak sidelobes is forcedinto regions of the sidelobe response with lowerenergy levels, thus lowering the peak sidelobes and“flattening” the entire sidelobe response function.The achievable sidelobe levels can be further

reduced by the addition of ANZCs to the mismatchedreceive filter. From Fig. 10 and Fig. 11 it is clearthat very low sidelobe levels can be achieved by theaddition of sufficient ANZCs. From Fig. 11 it can alsobe seen that the TBWP of the transmit pulse limits theachievable sidelobe level except for a range of ANZCsbetween 80% and 160%. In this region the achievablesidelobe level is limited by the number of ANZCs.The peak-to-highest-sidelobe ratio therefore

improves with increasing TBWP, P and ANZCs.Given the parameter ranges explored, the bestpeak-to-highest-sidelobe ratio of just more than 68 dBwas achieved for a TBWP of 100, P = 10, and 200%ANZCs.It is expected that the trends regarding decreasing

sidelobe levels will continue beyond the parameterlimits of the current investigation. For greater TBWPsit should therefore be possible to obtain even lowersidelobe levels than those reported here.The effect of Doppler shifted return pulses on the

sidelobe level was also investigated. It was shownin Fig. 14 that high values of P combined withhigh values of ANZC results in receive filters withincreased sensitivity to Doppler shifts. The currentclass of mismatched pulse compression filters shouldtherefore be suitable for applications where lowDoppler shifts are encountered or in systems whereDoppler compensation is performed.An important result of this research is the tradeoff

between achievable sidelobe level and mainlobe width.A secondary result is that for this application, valuesof P larger than approximately 20 provide very littleimprovement in the sidelobe level.Future work will extend the parameter space of

this investigation by increasing filter lengths, TBWPs,and values of P. The performance of the sidelobe

CORRESPONDENCE 1245

reduction technique for other transmit waveformswill also be analyzed as well as the sensitivity toquantization of the filter coefficients. The effect ofDoppler shift on the resulting sidelobe response willbe analyzed in more detail and the technique couldbe extended to optimize the receive filter sideloberesponse in range and Doppler space. Additionalconstraints can also be added to the formulation of thetechnique to control the mainlobe width and/or controlsidelobe levels in specific regions of the sideloberesponse function.In general the technique presented here should also

be applicable to other fields within the digital signalprocessing domain such as the design of windowingfunctions and array beam patterns.

APPENDIX

In order to find the value of x that minimizes thegeneralized L2P-norm sidelobe function

f(x) =

Ã2N¡1Xi=1

[xHCix]P

!1=2P(23)

it is necessary to solve the complex valued Lagrangianequation

d

dx¤(f(x)) +

d

dx¤(Ref¸¤g(x)g) = 0: (24)

By substituting the gradients

d

dx¤(f(x)) =

d

dx¤

0@Ã2N¡1Xi=1

[xHCix]P

!1=2P1A=12P

Ã2N¡1Xi=1

[xHCix]P

!1=2P¡1d

dx¤

Ã2N¡1Xi=1

[xHCix]P

!

=12

P2N¡1i=1 [x

HCix]P¡1Cix

(P2N¡1

i=1 [xHCix]P)1¡1=2P

(25)

andd

dx¤(Ref¸¤g(x)g) = d

dx¤(Ref¸¤(ax¡ bpeak)g)

=d

dx¤

μ12(¸¤ax+(¸¤ax)¤)¡Ref¸¤bpeakag

¶=12(0+¸aH)¡ 0

=12¸aH (26)

into (24) we obtain the system of equationsP2N¡1i=1 ([x

HCix]P¡1Cix)

(P2N¡1i=1 ([xHCix]P))1¡1=2P

+¸aH = 0 (27)

which, when solved together with the constraint

g(x) = ax¡bpeak = 0 (28)

gives the value of x that minimizes the generalizedL2P-norm sidelobe function. The solution can not beexpressed in closed form for P > 1, but can be foundusing numerical methods.

ACKNOWLEDGMENT

The authors would like to thank Professor M. A.Van Wyk at the Technical University of Tshwane forvaluable inputs during the preparation of this paper.

JACQUES E. CILLIERSJOHAN C. SMITCouncil for Scientific and Industrial ResearchDefence, Peace, Safety and SecurityPO Box 395Pretoria, 001South AfricaE-mail: ([email protected])

REFERENCES

[1] Cook, C. E.The early history of pulse compression radar–Thehistory of pulse compression at Sperry GyroscopeCompany.IEEE Transactions on Aerospace and Electronic Systems,24, 6 (Nov. 1988), 825—833.

[2] Siebert, W. M.The early history of pulse compression radar–Thedevelopment of AN/FPS-17 coded-pulse radar at LincolnLaboratory.IEEE Transactions on Aerospace and Electronic Systems,24, 6 (Nov. 1988), 833—837.

[3] Mow, W. H.Best quadriphase codes up to length 24.IEE Electronic Letters, 29, 10 (May 1993), 923—925.

[4] Frank, R.Polyphase complementary codes.IEEE Transactions on Information Theory, 26, 6 (Nov.1980), 641—647.

[5] Axelsson, S. R. J.Noise radar using random phase and frequencymodulation.IEEE Transactions on Geoscience and Remote Sensing, 42,11 (Nov. 2004), 2370—2384.

[6] Lindner, J.Binary sequences up to length 40 with best possibleautocorrelation function.IEEE Proceedings, 11, 21 (Oct. 1975).

[7] Cohen, M. N., Fox, M. R., and Baden, J. M.Minimum peak sidelobe pulse compression codes.Record of IEEE 1990 International Radar Conference,633—638.

[8] Mertens, S.Exhuastive search for low autocorrelation binarysequences.Journal of Physics A: Math. Gen., 29 (1996), L473—L481.

[9] Coxson, G., and Russo, J.Efficient exhaustive search for optimal-peak-sidelobebinary codes.IEEE Transactions on Aerospace and Electronic Systems,41, 1 (Jan. 2005), 302—308.

[10] Boehmer, A.Binary pulse compression codes.IEEE Transactions on Information Theory, 13, 2 (Apr.1967), 156—167.


[11] Luke, H. D., Schotten, H. D., and Hadinejad-Mahram, H.Binary and quadriphase sequences with optimalautocorrelation properties: A survey.IEEE Transactions on Information Theory, 49, 12 (Dec.2003), 3271—3282.

[12] Gartz, K. J.Generation of uniform amplitude complex code sets withlow correlation sidelobes.IEEE Transactions on Signal Processing, 40, 2 (Feb.1992), 343—351.

[13] Nunn, C.Constrained optimization applied to pulse compressioncodes, and filters.Record of IEEE 2005 International Radar Conference,190—194.

[14] Felhauer, T.Design and analysis of new p(n,k) polyphase pulsecompression codes.IEEE Transactions on Aerospace and Electronic Systems,30, 3 (July 1994), 865—874.

[15] De Witte, E., and Griffiths, H. D.Improved ultra-low range sidelobe pulse compressionwaveform design.IEE Electronic Letters, 40, 22 (Oct. 2004), 1448—1450.

[16] Lee, W. K., Griffiths, H. D., and Benjamin, R.Integrated sidelobe energy reduction technique usingoptimal polyphase codes.IEE Letters, 35, 4 (Nov. 1999), 2090—2091.

[17] Sato, R., and Shinriki, M.Simple mismatched filter for binary pulse compressioncode with small PSL and small S/N loss.IEEE Transactions on Aerospace and Electronic Systems,39, 2 (Apr. 2003), 711—718.

[18] Lewis, B. L., Kretschmer, F. F., and Shelton, W. W.Aspects of Radar Signal Processing.Norwood, MA: Artech House, 1986.

[19] North, D. O.An analysis of the factors which determine signal-noisediscrimination in pulsed carrier systems.RCA Laboratory, Report PTR-6C; reprinted inProceedings of IEEE, 51 (July 1963), 1016—1027.

[20] Moon, T. K., and Stirling, W. C.Mathematical Methods and Algorithms for SignalProcessing.Upper Saddle River, NJ: Prentice-Hall, 2000.

[21] Frost, O. L.An algorithm for linearly constrained adaptive arrayprocessing.IEEE Proceedings, 60 (Aug. 1972), 926—935.

[22] Griep, K. R., Ritcey, J. A., and Burlingame, J. J.Poly-phase codes and optimal filters for multiple userranging.IEEE Transactions on Aerospace and Electronic Systems,31, 2 (Apr. 1995), 752—767.

[23] IEEE Standard Radar Definitions.IEEE Standard 686, 1997.

[24] Harris, F. J.On the use of windows for harmonic analysis with thediscrete Fourier transform.IEEE Proceedings, 66, 1 (Jan. 1978), 51—83.

Addenda and Errata: Optimal AcknowledgmentFrequency over Asymmetric Space-Internet Links1

On page 1312, right column, 3 lines from thebottom, VJ should read Van Jacobson (VJ).

The name of the second author in Reference[10] is Floreani, D., and the work is available inProceedings of the First Workshop on Wireless MobileInternet, Rome, Italy, 2001.

In Mr. Wang’s biography, the second sentence ofthe last paragraph should read:

He is serving as an editorial board member of JohnWiley InterScience’s Wireless Communicationsand Mobile Computing (WCMC) Journal. He isalso serving as a guest editor for other internationaljournals. Dr. Wang is currently serving as technicalprogram committee cochair for the IEEE ICC’07Wireless Communications Symposium.

1Wang, R., Horan, S., Tian, B., and Bonasu, S., OptimalAcknowledgment Frequency over Asymmetric Space-InternetLinks, IEEE Transactions on Aerospace and Electronic Systems,42, 4 (Oct. 2006), 1311—1322.

Addenda and errata received January 12, 2007.

IEEE Log No. T-AES/43/3/908448.

0018-9251/07/$25.00 c° 2007 IEEE

CORRESPONDENCE 1247

本文献由“学霸图书馆-文献云下载”收集自网络，仅供学习交流使用。

学霸图书馆（www.xuebalib.com）是一个“整合众多图书馆数据库资源，

提供一站式文献检索和下载服务”的24 小时在线不限IP

图书馆。

图书馆致力于便利、促进学习与科研，提供最强文献下载服务。

图书馆导航：

图书馆首页文献云下载图书馆入口外文数据库大全疑难文献辅助工具

http://www.xuebalib.com/cloud/http://www.xuebalib.com/http://www.xuebalib.com/cloud/http://www.xuebalib.com/http://www.xuebalib.com/vip.htmlhttp://www.xuebalib.com/db.phphttp://www.xuebalib.com/zixun/2014-08-15/44.htmlhttp://www.xuebalib.com/

Pulse Compression Sidelobe Reduction by Minimization of L/sub p/-Norms学霸图书馆link:学霸图书馆

Pulse Compression Sidelobe Reduction by Minimization of L...

Documents

Transcript of Pulse Compression Sidelobe Reduction by Minimization of L...