g P3 and P4 Codes

8/3/2019 g P3 and P4 Codes

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1. INTRODUCTION

An Extended Frank Codeand New Technique forImplementing P3 and P4 Codes

CHI-CHANG WANGHSUEN-CHYUNSHYUChung Cheng Institute of TechnologyBiwan

A method for generating a class of polyphase pulsecompression codes of length other than N 2 i th an efficient digitalhqlem entatio n technique is presented. The new structure can beapplied to linear frequency modulation wave form for compressionratios ofN x M . Th e P3 an d P4 codes [4, 51 can be easily realizedby adding only phase shifters to this structure.

Manuscript received November 12, 1987; revised August 9, 1988.IEEE Log No. 28653.Authors' address: Department of Electronic Engineering, ChungCheng Institute of Rchnology, 'Ih-Hsi, Bo-Yuan, Thiwan 33509,Republic of China.

0018-9251/89/0700-0442$1.00 @ 1989 IEEE

Frank [2] described a polyphase code, by a matrixas follows:0. .0 0

21 ; 4 .. . 2 ( N -N j)0 (N-1) 2 ( N - 1 ) . . . ( N - 1 ) 2

where the numbers represent multiplying coefficientsof a basic phase angle, 2s p /N , where p and Nare relatively prime integers. It is noted that whenp = 1, N can be an arbi t rary in teger and such acode is known as a Frank code. In this paper, th eN x N matrix for the Frank code is extended to anN x M matrix where N and M ar e arbitra ry integers.Hence a n extended Frank (EF) polyphase code ca nbe obtained. The main advantage of a n EF code isthat when N = I C M , here IC is an integer, the codestructure can be easily realized by a small M-point fastFourier transform (FFT) ather than a large N-pointFFTwhich is needed for a Fr ank code. However, theautocorrelation of the extended Frank co de has highand periodic sidelobes.radar application [l-51. The P3 and P4 polyphase pulsecompression codes [4] ar e conceptually derived froma linear frequency modulation waveform (LFh4W).Such codes and compressors can be employed toobtain much larger time-bandwidth products, i.e., pulsecompression ratios, R, and analog dispersive delaylines. The significant advantages of P3 and P4 codesare low peak sidelobes, which are approximately 1/4Rto mainlobe peak, and that they ar e more Do pplertolerant than other phase cod es derived from a ste papproximation t o a n LFh4W. Also, the P3 and P4codes can b e realized by using Frank polypha se code[2] digital processing circuits with additive pha se shiftsin filter tim e samp les and r phase shifts in every otherfilter output (frequency) port [4].The main limitation for imp lementing the P3 a ndP4 codes by a structure that implem ents a F rank co deis that the com pression ratio must be a squa re ofintegers. W hen the P 3 and P4 codes ar e derived by anE F code, this limitation is removed. As a consequence,both the good properties of low sidelobe for the P3and P4 cod es and the simple structure for the EFcode a re preserved. H ence, a new and very efficienttechnique for implementing the P 3 and P4 c odes withcompression ratios other than N 2 s obtained.

Many pulse compre ssion waveforms ar e for search

II. EF POLYPHASE CODESimilar to Frank's structure [2], a periodicpolyphase code can be described by a matrix shown in

442 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 25, NO. 4 JULY 1989


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T4BLE IMatrix For E F Code

0 0 0 ... 00 1 2 ... (N 1)0 2 4 .. . 2(N - 1)0 (M-1) 2(M-1) ... (N - )(M - 1)

0 0 0 0 0 0 0 00 t / 2 t 3 t / 20 7 r O 7 r O t O 7 ri 3 t / 2 t ~ / 2

' lhble I. In B b l e I, the num bers represent multiplyingcoefficients of 2 t / M , the b asic phase angle, withN an d M arbitrary integers. It is noted that whenN = M, this phase matrix represents the Frank code.Hence, Bb le I can b e obta ined by extending the phasematrix with specified pro perties of the Frank code ineither row or column direction, depending on whichone is longer than the o ther. He nce this new class ofcode can be called an EF code.Th e EF coded waveform consists of a constantamplitude signal whose carrier frequency is modulatedby the phases discussed above and then sent ou trow by row. By doing so, the phase angle of the i thcode e lement in the j t h row, o r code group, may beexpressed mathematically as@(EF)i,j= ( 2 % / M ) ( i- ) ( j - ),where i = 1,2 ...,N an d j = 1,2 ...,M .

(1)In (l), he tota l number of cod e elements formed isN x M. So the m aximal pulse compression ratio isequal to N x M. In oth er words, the comp ressionratio for the EF cod e is more flexible than that for theFrank code. Som e general properties for the EF codeare now described a s follows.

I *7r/2 t 3 ~ / 20 3 t / 2 t 7r/2

1) When N = M, th e EF c o d e is a Frankpolyphase code. This cod e has b een widely discussedin many papers and hence is not studied again here.2 ) When N > M, @i,jcan b e expressed as

$i,j = @((i))M+aM,j = $((i))M,j (2)where a is an integer and ( ( i ) ) ~ enotes the moduloof M . By ( 2 ) each cod e group can be constructed byduplicating the first M cod e elements again and againuntil the last element is obtained. In a special casewhere N = ICMwith IC an integer, each code groupcontains exactly IC M-element subgroups. In this case,th e N x M phase matrix XN,M f th e EF code can bepartitioned into IC identical, small, M x M submatrices.That is,

X N , M = [XM,M XM,M 1 * IXM,M]* ( 3 )total IC termsIn (3), each submatrix XM,Mquals the phase matrixof the Frank co de with M 2 phases. This property isshown in the following example.WANG & SHYU: NEW TECHNIQUE FOR IMPLEMENTING P 3 AND P4 CODES 443


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COMPRESSORXPANDEhINPUT INPUT- I 1 I -(SIGNALT cI I CONJI 7)

OUTPUlUTPUT

F 2 F 3

Fig. 1. Block diagram for EF code expandercompressor with N = 8, M = 4.W L E 1

Phase Matrix For E F Code With N = 4 and M = 8

0 0 0 00 a 1 4 s J 2 3 x 1 40 i r / 2 A 3 x 1 20 3nJ4 3 n J 2 a 1 4o r 0 A0 5 ~ J 4 ~ / 2 ir/40 3 a J 2 A ~ / 20 7x14 3 irJ2 5ir /4

Now the special case nN = M where, K is aninteger, is discussed. In this case, the se quen ce of th ephase angles within the first N rows of phase matrixno longer fits an N-po int FFT.Henc e the simplestructure in Fig. 1cannot directly apply to this case.However, the phase angles for the ( j + d ) t h r ow i nthe phase matrix can be expressed a s followswhere $i,j denotes the element at i th column and j t hrow in the phase matrix.( j + n)th, ( j + 2n)th,. .., and ( j + (N - 1)d ) t h rowsform a new cod e group. It is noted that in each newcode group, the value for phase angles in (5) consistsof two parts. The first part, $ j , , , acts as a constant inthe identical code group but with distinct values fordistinct groups. The second part, K ( i - 1)(2w/M),contributes identical form t o all code groups butwith distinct values for distinct positions in eac h codegroup. Now let n = i - 1, then

When 15 j 5 IC, he phase angles in jth,

d ( i - 1)27r/M = 2nnn/M= 2 n n / N , f o r O < n < N . (6)

By (6), it is observed that the second part in (5 )coincides the phase angles for the N -point FFTHenc e the special case for the EF code with N x Melements where IC = M can now b e realized by anN-point FFT tructure with additive phase shifters ofvalues 4 i . j . Since $ 0 , ~= 9i .o = 0, it is obvious that onlyN - 1 phase shifters ar e required for each code group.And the number of code groups which need additivephase shif ters is IC- 1.Therefore, the total numberof additive phase shifters required to realize the EFcode in this case is (K- 1)(N - 1).Th e block diagramwhich realizes the EF code with N = 4, M = 8 is nowillustrated in Fig. 2. By comparing Fig. 1and Fig. 2,it is found that the hardware realization for the codeexpandercompressor of case 3) is more complex thanthat of case 2).The c ompression characteristics for the c odecompressor is usually evaluated by the autocorrelationfunction. By (3) an d (6), it is found that a periodicproperty exists for both c ase N = ICM nd nN = M.Henc e the autocorrelation for the EF code will havenot only a main peak with amplitude N x M but alsoperiodic sidelobes with period M an d N fo r N = nMand KN= M, respectively. The diagram for theautocorrelation functions for N = 8, M = 4 and N = 4,M = 8 are shown in Figs. 3(a) and (b), respectively. Inthis figure, it is found that the properties of these twodiagrams ar e very close to each other. Howe ver, themain drawback is that the amplitude of the sidelobesappear much higher than that for the Frank co dein [4].Finally, it should b e emphasized that IC acts asan index for the EF code. When N = R M an d n isnot restricted to be an integer, the value of n can beused to describe many properties for EF code. Forexample, the structure which realizes the EF codefo r N = 8,M = 4 and N = 16, M = 8 is identical.To realize these two EF codes, only the magnitude ofphase shifters and number of points for the FFT a r e

444 IEEE TRANSACTIONS ON AEROSPACE A N D ELECTRONIC SYSTEMS VOL. 25, NO. 4 JULY 1989


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EXPANDER COMPRESSOR

SED

Fig. 2. Block diagram for EF code expandercompressor with N = 4, M = 8.0

TABLE 11P3 Code Modulo 2r, R = 32

0 *I32 U f 8 9x132 RI2 25x132 9x 18 49x1320 337~132 iT/8 4 1x 13 2 x f 2 5 7 ~ / 3 2 9K /8 1 7 ~ 1 3 20 1 1r /3 2 9 r f 8 5 7 ~ 1 3 2 a 1 2 4 1 ~ 1 3 2

r / 8 3 3 ~ 1 3 20 4 9 x 1 3 2 9 x 1 8 2 5 ~ 1 3 2 x/ 2 9x132 r / 8 x i 3 2

different. Please note that both of them have index IC =2. Meanwhile, the sam e results can b e easily obtainedfor other values of IC. Hence the index IC can be usedto determ ine the realization structure for EF code.Two special cases for the EF code are well known.When IC = 1, this is the Frank code. When IC = N,i.e., M = 1, th e EF code becomes the f in i tedurat ionimpulse response (FIR) matched filter which is widelyused for the LFMW applications.

Ill. MOR E EFFICIENT IMPLEMENTATION FO R P3AN D P4 CODEST he P3 and P4 codes [4] are conceptually derivedby converting a LFMW to baseband using a localoscillator, with frequency fo an d fo + k T / 2 respectively,on one end of the frequency sweep and sampling theinphase Z and quadrature Q video at the Nyquist rate.The phase of th e P3 and P4 codes can be expressedmathematically as

$?) = ~ ( i1) 2 /B T = ~ ( i )2/Rand $jp4) = [T(i- 1) 2 /R ]- ( i- ) (7)where B = kT is the bandwidth, T is the pulse length,an d R is the pulse compression ratio.The matrices for the phase angles of th e P3 and P4codes modulo 27r with R = 32 are now shown in ThbleI11 a nd n b l e IV, respectively.

(4 S A M P L E N U M B E R0

(b) S A M P L E N U M 9 E RFig. 3. Autocorrelation function of EF codes. (a) N = 8, M = 4.

(b) N = 4, M = 8.

TABLE IVP3 Code Modulo 2x , R = 320 33x132 rf8 41x132 r / 2 57x132 9x18 11x / 320 4 9x 13 2 9 R/8 2 5 ~ 1 3 2 9 ~ 1 3 2 TI 8 r / 3 20 x / 3 2 x / 8 9x132 n / 2 25x132 9x18 49x1320 11r f32 9x18 57x132 a12 41x132 x / 8 33x132

The advantage of th e P3 and P4 polyphase pulsecompression codes is that they do not produce largetime sidelobes with large Doppler shifters. Thisadvantage, the better Doppler tolerance, of th e P3and P4 codes allows large time-bandwidths whichWANG & SH W : NEW TECH NI Q UE F O R I M P LEM ENTI NG P 3 AND P4 CODES 445


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EXPANDER COMPRESSORINPUTI I (SIGNALINPUT

EXPANDEDOUTPUT1

COMPRESSEDOUTPUT___)

Fig. 4. P3 expandercompressor using E F code for R = 32.EXPANDER COMPRESSOR

INPUTCONJU?ATEIJ)

EXPANDED4OUTPUT

Fig. 5. P4 expandercompressor using EF code for R = 32.

TABLE VISubtracting P4 Code By EF Code With R = 32TABLE VSubtracting P3 Code By EF Code With R = 320 x / 3 2 n/ 8 9x132 a12 25x132 9x18 49~132 0 33~132 n /8 41x132 x/2 51x/32 9x18 11x1320 a132 x / 8 9x132 n/2 25x132 9x18 49x132 0 33x132 x / 8 41x132 n /2 57~132 9x18 17~1320 ~132 n/ 8 9x132 x / 2 25x132 9x18 49x132 0 33x132 r / 8 41x132 x / 2 51x132 9x18 17x/320 x i 3 2 x / 8 9x132 n/2 25x132 9n/8 49x132 0 33x132 x / 8 41x132 x / 2 57x/32 9x18 11x132

are effective in the presence of large Doppler shiftson radar echo pulses [4]. By comparing the phasematrices of th e P3 code in Thble I11 and the EFcode in Example 1, it is found that the differencesbetween phase angles repeat by every N samples.These differences between the P3, P4, and EF codesare illustrated in lhble V and Table VI, respectively.Note that the added phase shifts for the P3 and P4codes are caused by the linear frequency shift duringthe time the equivalent EF code frequency is constant.By the fact that the phase differences for the phasematrices are identical within each column in Table V

an d VI, both P3 and P4 codes can be implementedin a pulse expander-compressor employing digitalFourier transform, adders, and delay circuits whichare similar to those for the EF codes in Fig. 1orFig. 2with identical compression ratio. Extra N - 1phase shifters are also required to compen satethe Qhase differences between the P3, P4, an d EFcodes. T he block diagram for implementing the P3an d P4 codes by using th e EF code s t ructure areshown in Fig. 4and Fig. 5, respectively. M eanwhile,Fig. 6illustrates the autocorrelation functions orcompresse d pulse waveforms for the digital pulse



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0

0N-WW OE:a 7E?

VYW O

0n

' 0 . 0 0 2 0 . 0 0 4 0 . 0 0 8 0 . 0 0S A M P L E NUMBER

Fig. 6. Output result of Fig. 4 and 5 with R = 32, zero Dopplershift, no bandwidth limitation.

I ' '0 . 0 0 4 b . 0 0 ' s b . 0 0 ' l20.00 160 . 0 0 2 0 0 . 0 0S A M P L E N U M 9 E R

Fig. 7. Autocorrelation function for P3 or P4 code with R = 96 ,zem Doppler shift, no bandwidth limitation.

expander-compressors shown in Fig. 4and Fig. 5,respectively, with no D oppler shift and no bandw idthlimitation. In this figure, the pulse compression ratiois R = N x M = 8 x 4 = 32. The autocorrelationfunction for the exp ande rcom pres sor for compressionratio R = N x M = 24 x 4 = 96with no Dopplershift and no bandwidth limitation is also shownin Fig. 7.By these figures, it is observed that thehighest range-time sidelobe is 4R below the peakresponse. The perform ance of the compressor inFig. 7is very close to that of [4, Fig. 31, of which thecompression ratio is 100. The m ain advantages ofthis new expander-compressor are that the hardwareimplementation is much simpler than that in [4], also,the compression ratio is more flexible than that usingFrank code.IV. CONCLUSION

In this paper a new class of polyphase codes, calledth e EF code, is introduced. Th e phase matrix

of th e EF code has the sa me properties as that of th eFrank code, but the restriction of identical numbersof columns and rows for the Frank c ode is removed.That is, the phase matrix can now have M rows andN columns where M and N are distinct integers.A special case N = KM, here K is an integer, isdiscussed. In this case, the hardware implementationfor the EF code needs one M- point FFT, GM adders,an d KM elay lines only. In other words, the hardwarecomplexity is largely decreased from O(N log2N) fo rthe Frank code to O ( K ) = O ( N / M ) or the EF code.Unfortunately, the autocorrelation function shows highsidelobes for the EF code.The disadvantage of high sidelobes for the EFcode can b e recovered when P3 an d P4 codes ar econsidered. By comparing the phase matrices of th eP3 an d P4 codes with the EF code, it is found thatonly N - 1 phase shifters are required to transferth e EF code to P3 o r P4 code. Henc e a n effectiveimplementing technique for the P3 o r P4 polyphasecode ca n now b e obtained by using the simple EFcode structure. As a consequence, a new, flexible, andefficient technique for implementing an LFh4W withcompression ratio oth er than N 2 is obtained.REFERENCES(11 Cook, C ., and Bernfeld, E. (1%7)Radar Signals, An Introduction to Theory and Application.Ne w York Academic Press, 1967.

Polyphase d e s with good nonperiodic correlationproperties.IEEE Transactions on Information Iheo ry, IT-9 Jan.1%3), 43-45.A new class of polyphase pulse compression cod es andtechniques.IEEE Transactkm on Aerospace and Electronic Systems,AES-17 (May 1981), 364-371.Linear frequency modulation derived polyphase pulsecompression codes.IEEE Transactions on Aerospace and Electronic Systems,

[2] Frank, R. L. (1%3)

[3] Lewis, B. L., and Kretschmer, E E, Jr. (1981)

[4] Lewis, B. L., and Kretschmer, E E, Jr. (1982)

A B - 1 8 (Sept. 1982), 637-641.[ 5 ] Lewis, B. L. , Kretschmer, E E, Jr., and Shelton, W. W. (1986)Aspects of Radar Signal Processing.

Dedham, MA: Artech House, 986.

WANG & SHW: NEW TECH NI Q UE FOR IMPLEMENTING P3 AN D P4 CODES


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Chi-Chang Wang was born in Chiai, 'hiwan, Republic of China on Nov. 19,1957. He received the B.S. degree in electrical engineering from the Chun gCheng Institute of Rchnology, 'h-Hsi, Tao-Yuan, 'hiwan, and the M.S. degree inelectronical engineering from th e National Taiwan University, B ip ei , in 1979 and1984, respectively.Engineering, Chung Cheng Institute of Technology. His re search interests includerada r signal processing, digital signal processing in VLSI, co mm unica tion system,and com puter architecture design.

Since 1984 he has been a L ecturer in the Depa rtment of Electronic/---

Hsuen-Chyun Shyu was born in W p e i on Nov. 15, 1951. He received the B.S. andM.S. degrees in electrical engineering from Chung Cheng Institute of Technology,'h-Hsi, B o-Y uan, 'hiwan , and the National 'hiwan University, 'hipei, in 1974and 1978, respectively, and the Ph.D . degree in com puter engineering from theUniversity of Southern California, Los Angeles, in 1986.From 1978 to 1982 he was a Lecturer in the D epartm ent of ElectronicEngineering, Chung Cheng Institute of Technology, whe re he is presently a nAssociate Professor. His current research interests include the VLSI algorithm,residue number system, computer architecture design, radar signal processing, anddigi tal signa l proce ssing in -1.


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