Technical Report No. 216 036040- 9-T MODULATION BY LINEAR-MAXIMAL SHIFT REGISTER SEQUENCES: AMPLITUDE, BIPHASE AND COMPLEMENT- PHASE MODULATION by a Theodore G. Rirdsall Richard M. Heitmeyer Kurt Metzger COOLEY ELECTRONICS LABORATORY Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan for Contract No. N00014- 67-A- 0181- 0032 Office of Naval Research Department of the Navy Arlington, Virginia 22217 December 1971 Approved for public release; distribution unlimited. TW2 WPQGNEERITV OF LRp'nA ENGINEERING LIBRARY

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ABSTRACT This report describes the use of linear- maximal shiftregister sequences either to amplitude modulate (AM) or to angle modulate a sinusoidal carrier. Two cases of angle modulation are considered: biphase modulation (BM) and complement-phase modulation (CM). In biphase modulation the carrier angle switches between +7T/2 and -,7/2, whereas in complement-phase modulation the carrier angle switches between +v/4 and -7/4. It is shown that the ratio of the average power to the peak power for AM is only one-half as large as for either BM or CM. On the other hand, the ratio of the carrier power to the average power for either AM or CM is one-half, but for BM this ratio is approximately zero. Finally, a replica-correlation technique for use with any of the three modulations is described. iii

AC KNOWLEDGMENT The techniques described in this report have been developed for use in the project MIMI (University of Michigan - University of Miami) underwater-sound-propagation experiments sponsored by Code 468 of the Office of Naval Research.

TABLE OF CONTENTS Page ABSTRACT iii ACKNOWLEDGMENT iv LIST OF ILLUSTRATIONS AND TABLES vi 1. INTRODUCTION 1 2. A REVIEW OF LINEAR-MAXIMAL SHIFTREGISTER SEQUENCES 3 3. AMPLITUDE MODULATION, BIPHASE MODULATION AND COMPLEMENT- PHASE MODULATION 8 4. A REPLICA-CORRELATION TECHNIQUE 17 APPENDIX 23 REFERENCES 27 DISTRIBUTION LIST 28

LIST OF ILLUSTRATIONS Figure Title Page 1 One period of a LMSR binary waveform 5 2 (a) Autocorrelation function R (7) (b) RMS power-spectrum S (f)- 7 3 The low-pass waveform s(t) 10 4 The modulated signal s(t) 11 5 The output of the digital modulator after being filtered by a Q = 4 bandpass filter 16 LIST OF TABLES Table Title Page I The function A(f), for AM, BM and CM 12 II The ratios ca and 3 for AM, BM and CM 14 III The coefficients A and B 22 vi

1. INTRODUCTION A common technique for sounding the multipath structure of an acoustical or an electromagnetic channel is to transmit a periodic signal consisting of a sequence of rectangular pulses, modulating a carrier. The implementation of this technique, however, often leads to the following problem. If d is the duration of each of the rectangular pulses, then d must be kept small in order to resolve the different arrivals from each transmission path. On the other hand, if T denotes the period of the transmission, then T must be kept large so that all of the arrivals from one transmitted pulse can be distinguished from those of the succeeding pulse. Thus, the nature of the propagation mechanism places an upper bound on the duty cycle d/T. Furthermore, the equipment used to transmit the signal may also place an upper bound on the signal's peak power P so that the quantity P(d/T) may be bounded from above. However, P(d/T) is the average power in the signal, and if this quantity is not large enough to compensate for both the attenuation in the channel, and the channel and system noise, then this technique cannot be successfully implemented. Over the years, however, various pulse-compression techniques suitable for periodic transmissions have been developed to circumvent this basic problem. The idea behind these techniques is

-2to construct a signal that has the same peak power, and effectively the same duty cycle, as the repeated pulse signal but with a larger average power. One such technique, which has gained in popularity since the advent of digital circuitry, involves the use of linearmaximal shift-register (LMSR) sequences to modulate the carrier. The transmitted signal is constructed to have a sufficiently large average-power to peak-power ratio while, at the receiver, the reception is correlated with a replica in order to achieve the effect of a small duty cycle. This report describes three different LMSR modulation techniques that have been used in the MIMI (University of Michigan - University of Miami) underwater sound propagation experiments. Section 2 briefly reviews the linear-maximal, shift-register sequences. Section 3 describes the modulation techniques themselves and derives the power spectra for each case. In Section 4, a replicacorrelation technique for achieving the effect of a small duty cycle is described. In large part the material contained in this report is taken from "MIMI Processing Techniques" (Ref. 1).

2. A REVIEW OF LINEAR- MAXIMAL, SHIFT-REGISTER SEQUENCES A LMSR sequence {mk} is a periodic, binary sequence mk e {0, 1}, which can be generated by certain shift-register generators using mod-2 arithmetic. Methods of generating these sequences and their algebraic properties have been studied extensively (Refs. 2 and 3). Here only the following three properties are presented: 1. Periodic Property The period L of a LMSR sequence is of the form L = 2n 1 n= 1, 2,.. The number n can be associated with the number of stages in the shift-register generator. 2. Balance Property The total number of l's in one period of an LMSR sequence is (L+ 1)/2 =2 and the total number of O's is (L-1)/2 = 2n-1- 1 -3

-43. Pseudo-Random Property The occurrence of l's and O's in an LMSR sequence has the appearance of having been generated by successive trials of a fair- coin tossing experiment. (A more precise statement of the pseudo-random property can be found in Ref. 2.) The above three properties state (i) that LMSR sequences can only have periods of the form L = 2n-1, (ii) that there are approximately an equal number of l's and -l's in an LMSR sequence, and (iii) that the order in which the I's and O's occur does not follow a simple pattern. A specific example of an LMSR sequence from a 4-stage generator, n = 4, with period, L = 24-1 = 15, is... 1111 0 0 000 1 0 0 1 1 0 1 0... In order to describe the use of LMSR sequences in modulation schemes, it. is convenient to introduce the notion of the associated binary waveform. Basically, the idea is to construct a periodic, binary-valued function of time, m(t), by associating logical I's in Imk} with +1 values of m(t) and logical O's in {mk} with -1 values of m(t). Specifically, we define +1 for t e [kd, (k+l) d) if mk = +1 m(t) = -1 for t e [kd, (k+l) d) if mk = -1

- 5The number d is known as the digit duration and the period of m(t) is easily seen to be equal to T = Ld The binary waveform associated with the sequence {mk} illustrated above is shown in Fig. 1. m(t)~ T = Ld l! t Fig. 1. One period of a LMSR binary waveform The utility of LMSR waveforms as modulation waveforms is due to the form of the autocorrelation function T R () = _f m(t) m*(t+ T)dt (1) mand Tpower spectra and the corresponding power spectra

-63C -j27Tf Sm(f) = S Rm(T) ej27 dT (2) m m -3c In Ref. 2 it is shown that R (.) is periodic with period T and for m T [-T/2, T/2], 1 (1+L) 11 T1I d L d otherwise Moreover, the power spectra for m(t) is a line spectra given by, Sm(f)= A(f) P(f) 6(f - n/T)] (3) n where sin wdf 2 P(f) [X df and 1/L2 f = 0 A(f) = (1 + L)/L2 f 0 Illustrations of R (T) and the root-mean-squared (RMS) power spectrum, [Sm(f)], appear in Fig. 2.

Rm o HV-d [MI2 1f 0f (b) Fig. 2. (a) Autocorrelation function Rm (1T) (b) RMS power spectrum [S m (f)J 2

3. AMPLITUDE MODULATION, BIPHASE MODULATION AND COMPLEMENT- PHASE MODULATION The basic idea behind LMSR modulation is to cause either the amplitude or the phase of a carrier to switch between two values depending on the value of the modulating waveform m(t). In the case of amplitude modulation (AM), the amplitude switches between +1 and 0, in biphase modulation (BM), the phase switches between +7T/2 and -ii/2 and in complement-phase modulation (CM), the phase switches between +ir/4 and -I/4. More precisely, if s(t) is the low-pass waveform associated with the modulated signal s(t), s(t) = Re [(t) e (4) then for AM s (t) = {[1 + m(t)]/2} V1- sin (2rf t) (5) Sa(t) = [1 + m(t)] j/2 and for BM sb(t) = m(t) Jv sin 2VfCt = t cos [27fCt - m(t) m/2] (6) sb(t) = m(t) j and for CM -8

-9sc(t) = V cos [2tfct - m(t) i/4] (7) c(t) = [1+ m(t) jl/|In each of the above cases, it is assumed that there is an integral number of cycles of carrier for each digit in m(t). If this number is denoted by D, then s(t) is periodic with period T = dL where d = D/fc The complex low-pass signals for each of the three cases are illustrated in Fig. 3 and portions of the modulated waveforms appear in Fig. 4. In order to obtain the power spectra for the three different modulation schemes, the autocorrelation functions for the low-pass signals must first be determined. These functions may be determined from the definition of the autocorrelation of a complex signal (Eq. 1) by a straightforward computation. The results are, for AM Ra() = [(1 + 2/L)+ Rm(T)]/4 (8) for BM Rb(7) = Rm (7) (9)

s(t) for m(t) = +1 s(t) for m(t) = +1 s(t) for m(t) = 1 s(t) for m(t) =-1l s(t) for m(t) = -1 s(t) for m(t) =-1 Fig. 3. The low-pass waveform s(t); (a) AM, (b) BM, (c) CM and for CM R (7) [1 +R (T)]/2 (10) C m Next, the spectra for the low-pass signals are obtained as the Fourier transforms of the corresponding autocorrelation functions. We have, for AM Sa(f)= [(1 + 2/L) 6(f) + Sm(f)]/4 (11) for BM Sb(f) - Sm(f) (12)

~~- -~~~~~ -i11 d D/f (a) E I- l/fC (b) (c) (d) Fig. 4. The modulated signal s(t); (a) a portion of m(t), (b) CM, (c) BM, (d) AM

and for CM Sc(f) = [6(f)+ Sm(f)]/2 (13) or alternatively, if S(f) denotes either Sa(f), Sb(f) or S (f), then S(f) = A(f) P(f) L (f- n/T) (14) n where again P(f) = [sin ldf/lTdf]2 and A(f) is given in Table I for the three cases. Table I. The function A(f) for AM, BM and CM AM A(f) = 1 (1 + L)2 f = O A(f) = 1L fO BM A(f) = 1/L2 f = O i ++L A(f) = +L f #O L2 CM A(f) f=0. 2L2

To find the power spectrum for the modulated waveform, S(f), we note that if the number of cycles per digit, D, is larger than one, then the first zero of P(f), d' = f /D, is less than f C C Thus, s(t) is essentially bandlimited so that, S(f) = [s (f -f ) + S (f+ )] (15) Furthermore, the total average power in the modulated signal, Ps, can be obtained from the autocorrelation function of the low-pass signal by noting that, P= -f S(f) df -cc 2 [S(f- f )+ S(f+ fc)] df 2, f S(f) df or P = R(O) (16) Moreover, it is easy to show that the power in the carrier frequency line, Pc, is given by, Pc = A(O) (17) and that the peak RMS power, P, is

P = maxi[(t)] = 1 (18) Finally, the above equations (Eqs. 16, 17 and 18) may be used to determine the ratio of the total average power to the peak power, -a = Ps/P = A(0) (19) and the ratio of the carrier power to the total average power, P = Pc/P = A(0)/R (0) (20) for the three different cases. The results are summarized in Table II. Table II. The ratios a and I for AM, BM and CM = Ps/P c = Pc/Ps AM +- 1 1../, 1.,. 1, ~2 L2 C l l1 ( 1 i, 1 BM 11( 1 Ill the above p1aragraphs it has been seen that the power spectra for each of the three modulation schemes has the basic [(sinx)/x] z2

- 1 5form except at the carrier frequency line. The difference in the three modulation schemes appears in the ratios a and j. For a fixed peak power, the average power in the AM signal is only about half of the average power in either the BM or CM signal. On the other hand, both the AM and CM signal have approximately half of the total power contained in the carrier line, whereas the BPM signal has essentially no power contained in the carrier line. The choice of which modulation scheme to use in a particular setting depends, of course, not only on the quantities a! and A, but on the ease with which the modulation is implemented. Of the three types, AM modulation is the easiest to implement since it only involves gating the carrier on or off depending on the bits in the LMSR sequence. If more signal power is needed, however, the choice between BM and CM can be made on the basis of whether or not a large carrier power is desired, since either signal can be generated with about the same amount of difficulty. Reference 4 describes a method used to generate all three types of modulations using digital circuitry. The principle behind this method is to generate a square wave version of the modulated signal and then to filter out that portion of the spectrum centered about the harmonics of the carrier frequency. Figure 5 illustrates the output of the modulator, after being filtered by a Q = 4 bandpass filter.

(a) (b) (c) Fig. 5. The output of the digital modulator after being filtered by a Q = 4 bandpass filter (a) AM, (b) BM and (c) CM

4. A REPLICA-CORRELATION TECHNIQUE As mentioned in the introduction, the high resolution property of repeated pulse modulation can be achieved for LMSR sequence modulation by using replica-correlation at the receiver. To be more specific, assume that the transmitted signal can be considered as being propagated over a finite number of distinct transmission paths with transient times 1,..., TN and attenuations aYl,..., a'N. We may then write the demodulated reception, r(t), as the sum N r(t) = t. s(t-.) (21) where s(' ) is the low-pass signal associated with the transmission. Now a standard technique for processing a repeated pulse transmission is to cross-correlate r(t) with a replica of the pulse envelope to produce the function N C(T) = a. R(T- i) (22) i=' where R(.) is a triangularly shaped function with base width 2d From Eq. 22, it is seen that the ith transmission path appears in C(T) as the triangular peak R(T- Ti) centered at the transient time T. (mod T) with height proportional to the attenuation ai.. Thus, C(T) may be used as a measure of the multipath structure of the -17

channel. In the following paragraphs, a replica- correlation technique is described for obtaining C(7) when s(t) is the low-pass waveform associated with LMSR sequence modulation. To begin, assume that a biphase modulated signal is transmitted, [ (t) = sb(t)], and consider the cross-correlation of r(t) with the low-pass signal s (t). From Eq. 21, we have a T R (T) = 1 f s *(t) r(t+ T) dt a, r T 0 a N T a1 Sa i (t - i 0' -1 T I *(t+ T- T.)dt sa 1 or Ra?() a 2. R (7 — Ti) (23) where Ra r(T) = *(t) sb(t+ T) dt (24) But by substituting for sa(t) and sb(t) from Eq. 8 into Eq. 24 it is easily seen that for I 1 < T/2

-19(1+ /L) (1- d Ra b(T) = j (25) 0 otherwise Thus, Ra b(T) is a triangularly shaped function with base width 2d and height 1 + 1/L. [Note that the graph of Ra b(T) can be obtained by adding 1/L to the graph of Rb(T) in Fig. 2(a).] It then follows from Eq. 23 that if a biphase modulated signal is transmitted, we may determine C(z) as C(T) = Ra (T) (26) Next, consider the result of transmitting either an amplitudemodulated signal or a complement-phase modulated signal and again cross- correlating the demodulated reception with a replica of s (t). If the low-pass transmitted signal is denoted by s(t), then by proceeding as before, we have, N R a,(7) O Ra'(T- T.i) (27) ar i=1 a,'s where T R () = *(t) s(t + T) dt (28) a, s T a In the Appendix it is shown that R A(?) can be written as a, s

- 20Ra,(T) = C1Ra, b(7)+ C2 where C1 and C2 are constants depending on whether s(t) is an AM signal or a CM signal. Thus, we may substitute for R.(T) a, s from Eq. 29 into Eq. 27 to obtain N RaR r(T) ( I ci[CRa, b(- i) + C2] i-1 or Next note that the first bracketed term on the right-hand side of Eq. 30 is equal to C(T) so that Eq. 30 may be rearranged to give, C(T) = C1 Rar(T)- C2 [i (Y i] (31) It remains to find an expression for N a H i=l in terms of a measurable quantity. To this end, consider the mean of the received signal

-21R (T) =4T r(t + ) dt = T c[iai os(t + T- -i) dt 0 i=1 N T - i s(t + 7 -.)dt I T 1 i= 01 or, since s(t) is periodic with period T, R, r(T) =Rr [= a [ f (t) dt (32) 1, r(T) =R, r i 0 Finally, substituting for [I from Eq. 32 into Eq. 31 we obtain C(T) = A r (T)- BR, r (33) where A C 1 B [C2/c1] [ T s(t) dt] From Eqs. 33 and 26 it is seen that C(T) can be determined as a linear combination of Ra, r(T) and R1, r for each of the three modulations considered in this report. The advantage of this technique is that only the constants A and B depend on which modulation is used and thus the basic form of the receiver does not have to

- 22be changed when the modulation is changed. The constants A and B for the three modulations are derived in the Appendix and summarized in Table III below. Table III. The coefficients A and B A B AM 2 -j BM 1 0 [CM + j/L]

APPENDIX In this appendix the coefficients in Table III are derived. To this end it is first necessary to derive the coefficients in Eq. 29. We begin by noting from Eqs. 5 and 7 that sa(t)= [j + b(t)]/2 (34) and sc(t) [1 + sb(t)]/v (35) For the AM case, we set s(t) = sa(t) in Eq. 28 and then substitute a for s (t) from Eq. 34 to give T Ra, a(T) T s sa *(t)[ s(t + 7) dt 0 a T j + Sb(t + T) f T _ S *2(t) _ dt or R (,) = ( T s a (t) dt + b() (36) Thus, by comparing Eq. 36 with Eq. 29 it is seen that for the AM case C1 = (37) -23

-24and C2 = (j/2) T S *(t) dt] (38) For the CM case, we set s(t) = sc(t) in Eq. 28 and then substitute for s (t) from Eq. 35 yielding T R (T) T= f *(t) (t + -T)dt ) [1 + b(t + 7)] s *(t) S dt T 0 a2 or R (T) =- T sa (t) dt + R b(T 0a,c a, Thus, for the CM case, C 1 (39) 1,2 and C =2 d [I T a*(t) dt (40) The next step in determining the constants A and B is to colmlute thle nleans of the low —).Iss signals s (t) and s (t). From Eq. 5,

-25-Tf J a(t) dt = jij f I 2] dtl (41) aT 0 a 0 L or 1 T L+I T a(t dt 2 L (42) where we have used Property 2 of Chapter 2 to evaluate the integral on the right-hand side of Eq. 41. In a similar manner it can be shown that T T' 1 I + t ut 1 - s (t) dt T S 1+ m(t) dt or T f sc(t) dt = [1 + (j/L)] - (43) 0 2 Finally, the constant A1 is determined by substituting for C1 from Eqs. 37 and 39 into A = 1/C1. The constant B, in the AM case, is given by C2 T -S(t) dt1 = 1 -

-26and, in the CM case, by C -1 CB = T c(t) dt 1T s (t) dt T S (t) dt (-j/2) (1 + 1/L) (1/f2)(1 + j/L) = — ~) [I + I/L]

REFERENCES 1. T. G. Birdsall, "MIMI Processing Techniques, " (Internal Memorandum), Cooley Electronics Laboratory, The University of Michigan, Ann Arbor, Michigan, December 1968. 2. S. W. Golomb, Shift Register Sequences, Chapters III and IV, Holden-Day, Inc. 3. C. C. Hoopes and R. Randall, Study of Linear Sequence Generators, Cooley Electronics Laboratory Technical Report No. 165 (6576-4-T), The University of Michigan, Ann Arbor, Michigan, June 1966. 4. J. Stewart, "Operating Principles of the ELMSG, " (Internal Memorandum), Cooley Electronics Laboratory, The University of Michigan, Ann Arbor, Michigan, July 1971. -27

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Unclassified S' K t(IIr (.,lssit'ii nltitotl DOCUMENT CONTROL DATA - R & D.',,"- trlrxt,. l;I..tili, iti Iflt~ ef f it l. hty ef tr. I-tr e l u't o utis u i rn.,i~R maiIlt i uuti it, titt e iw 1t dtvr It i t h I,t,,r(1 i i'rat, i ie'lrt II If.'.fie'I) k IIt. 1N A N h A IA VI I Y (('orpotrite ititther) 2a. N 1'OI I I I CUL II (I A S5 1II A ION Cooley Electronics Laboratory Unclassified University of Michigan 2b. GROUP Ann Arbor, Michigan 48105 3. REPORT TITLE MODULATION BY LINEAR-MAXIMAL SHIFT-REGISTER SEQUENCES: AMPLITUDE, BIPHASE AND COMPLEMENT-PHASE MODULATION 4. DESCRIPTIVE NOTES (Type of report and.inclusive dates) C.E.L. Technical Report No. 216 -- December 1971 5. AU THOR(S) (First name, middle initial, last name) Theodore G. Birdsall Richard M. Heitmeyer Kurt Metzger 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS December 1971 40 4 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) N00014- 67-A- 0181- 0032 036040-9- T b. PROJECT NO. c. 19b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) d. TR216 10. DISTRIBUTION STATEMENT Approved for public release; distribution unlimited. 1 1. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Office of Naval Research Department of the Navy Arlington, Va. 22217 13. ABSTRACT This report describes the use of linear-maximal shift-register sequences either to amplitude modulate (AM) or to angle modulate a sinusoidal carrier. Two cases of angle modulation are considered: biphase modulation (BM) and complement-phase modulation (CM). In biphase modulation the carrier angle switches between +?r/2 and -?r/2, whereas in complement-phase modulation the carrier angle switches between +7r/4 and -7r/4. It is shown that the ratio of the average power to the peak power for AM is only one-half as large as for either BM or CM. On the other hand, the ratio of the carrier power to the average power for either AM or CM is one-half, but for BM this ratio is approximately zero. Finally, a replica-correlation technique for use with any of the three modulations is described. DD1 NOV 65473 Unclassified S/N 0101-807-6811 Security Classification. A-.q aL~

Security Classilication KEY''4 LINK A L.INK B LINli C KY W0DI... ii_.ROLE, WT ROLt WT wo L - WT Linear-maximal shift register sequences Amplitude modulation Biphase modulation Complement-phase modulation Replica- correlation technique

UNIVERSITY OF MICHIGAN 1111111111 90I1111111111111111111111111114 1 790411111 I I91L 02514 7904