Technical Report No. 206 036040-3-T COHERENT DEMODULATION OF THE IMPULSE RESPONSE OF BANDPASS FILTERS by D. Jaarsma Approved by: ______ Theodore G. Birdsall for COOLEY ELECTRONICS LABORATORY Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan Contract No. N00014-67-A-0181-0032 Office of Naval Research Department of the Navy Arlington, Virginia 22217 August 1971 Approved for public release; distribution unlimited

ABSTRACT This report contains the analysis appropriate for the complex demodulation of the impulse response of realizable lumped-element filters. Plots of the time behavior of the log of the envelope, the instantaneous phase, and the instantaneous frequency, together with the conventional phase and amplitude of the transfer characteristic, are presented for a number of bandpass filters. iii

FOREWORD In the signal processing of MIMI underwater acoustic propagation receptions, complex demodulation is used to determine the amplitude and phase of the complicated multipath arrivals. The interpretation of the amplitude as the envelope of a conventional pulse reception, and hence "path strength, " is straightforward. The interpretation of the instantaneous phase as an indication of physical path structure is not so obvious. As an aid to the interpretation of phase, a number of examples of the complex demodulation of common classes of bandpass filters were analyzed, and the results plotted for comparison with physical measurements. iv

TABLE OF CONTENTS Page ABSTRACT iii FOREWORD iv LIST OF ILLUSTRATIONS vi 1. INTRODUCTION 1 2. THE COHERENT DEMODULATION OF A BANDPASS IMPULSE RESPONSE 7 3. DISPLAY OF THE COMPLEX DEMODULATION FOR VARIOUS TYPICAL BANDPASS FILTERS 10 APPENDIX A: CLASSICAL LOW-PASS FILTERS 18 APPENDIX B: CONVERSION FROM LOW-PASS TO BANDPASS 21 DISTRIBUTION LIST 25 v

LIST OF ILLUSTRATIONS Figure Title Page 1 Cartesian vs. polar coordinates 3 2 AMSEQ 11 August 1966 1445-1457 hours 5 3 Time and frequency domain representation of the impulse response of a Butterworth bandpass filter, n = 2 11 4 Time and frequency domain representation of the impulse response of a Butterworth bandpass filter, n = 3 12 5 Time and frequency domain representation of the impulse response of a Butterworth bandpass filter, n = 5 13 6 Time and frequency domain representation of the impulse response of a Chebyshev bandpass filter, n = 2, e =.5 14 7 Time and frequency domain representation of the impulse response of a Chebyshev bandpass filter, n = 3, e =.5 15 8 Time and frequency domain representation of the impulse response of a Chebyshev bandpass filter, n = 5, e =.5 16 9 Demodulated time response of the impulse response of a Chebyshev bandpass filter, n = 2, e =.5 17 vi

1. INTRODUCTION In the study of underwater acoustic propagation, the multipath structure of the propagation is often studied using sharp impulsive sources, that is, explosions. The signal processing on reception consists normally of high-speed graphic recording of the reception. An alternative technique is to use high-power short pulses. This concentrates the transmission energy in the band of frequencies which one is most interested in. This latter is most appropriate in studying the difficulties that will arise in using a sonar system at this chosen carrier frequency. The usual signal processing upon reception is to envelope demodulate the reception; in conjunction with this high-speed graphic recording of the pulse, reception may be utilized to look at the fine detailed structure in the carrier. Perhaps additional processing in terms of spectrographs will be used to study dispersion of the pulse caused by the propagation. In the propagation studies of Project MIMI the medium has been shown to be very complicated, but yet phase stable. This means that there is a possibility of extremely high gains in signal processing possible if suitable transmissions are used. The actual transmissions used in Project MIMI are pseudo-random coded transmissions. The reception technique converts these back to the equivalent pulse -1

-2at much higher power, utilizes extremely narrow comb filters to reject surface reverberation, and then coherently demodulates the result so that the fine detail structure of the reception, as well as the envelope, is available. Both the envelope function and the phase function are displayed. This has two basic advantages. First, the stability in phase allows the comparison of repeated receptions, and the determination of very weak receptions by noting the stability in received phase; in the absence of received energy the indicated phase is uniformly distributed over all possible angles, and quite recognizable from a weak arrival which yields a concentrated or nearly repeated phase. Secondly, the instantaneous frequency of the arrival can be measured as a function of delay from the main arrival, by differentiating the phase function. The equations for the processed pulse reception are usually written in the form of Eq. 1. h(7) = R(T) cos [W0T + 0 (r)] (1) fi(T) = w0 + d/dT where R(T) is the envelope of the reception, co0 is the carrier frequency, and 0 (t) is the relative phase shift of the received waveform compared to a stable reference at the carrier frequency, fi(T) is the instantaneous frequency of the reception.

-3The actual demodulation process used in Project MIMI is referred to as "phase coherent demodulation," and is generally known in recent years as "complex demodulation. " For the actual processing it is more appropriate to write the received waveform of Eq. 1 in Cartesian coordinates. h(T) = x(T) cos WCOT- y(T) sin cO0T (2) The relationship between the envelope and phase functions, and these processing coordinates is shown in Fig. 1. R Ix(T) = R(T) cos 0(T) y() = R(T) sin 0(T) x Fig. 1. Cartesian vs. polar coordinates This relationship is the usual triangular relationship between magnitude and phase on the one hand and orthogonal components on the other. This type of demodulation has earned the nickname "complex demodulation" because one may write the reception in terms of a complex quantity by imagining that x(T) is the real part and y(T) the imaginary part, or if one chooses, that R(T) is the magnitude and 0 (T) is the phase.

-4z(T) = x(T) + j y(T) = R(T) ej (T) (3) This is merely a compact form of writing the coordinate systems shown in Fig. 1. Using the complex form shown in Eq. 3 the received and demodulated pulse response shown in Eq. 1 can be written as the real part of a complex number h(T) = Re (T) e (4) Although this may seem a little artificial, it displays to the analyst that the actual reception is simply formed by complex demodulation and then shifted in frequency back around the carrier frequency wo. Complex demodulation and display of the envelope and phase functions has proved to be an extremely useful signal processing technique. Figure 2 shows a typical overlay of ten such demodulations taken by Project MIMI on a 43-mile transmission. The upper graph is the envelope, the lower graph displays the demodulated phase as a function of time. From the repeatability of the phase diagram one can see immediately that the arrivals are quite phase coherent, that the instantaneous frequency shifts throughout the duration of the reception, and that one may determine the presence of very weak paths in the tail or later arrivals of the reception by noting the repeatability of the phase, even though the magnitude is quite small.

-5B, MI' -'j — h3'-H-+ -- H -H- -H - -H —-''' —3-12 t 1-10 13-21 t 11-20 R -I - -A.o.e.-t- _ 1 —, _ 22-30 t 21-30.....,..... 19 I -+' 1.......'' r':j.. - __-____- _ 31-39 t 31-40 SEQ, R = 0.5V/div CW, R = 0.2 V/div t = 120 msec/div t = 120 msec/div Fig. 2. AMSEQ 11 August 1966 1445 - 1457 hours

-6Very little theory is available to guide one in using such a detailed picture to infer the properties of the medium. The present study was undertaken in order to gain some insight into the meaning of the analysis, and in particular in the behavior of the phase function when the amplitude is very low. Specifically the medium was modeled as a bandpass filter of any one of many classical types familiar to electrical engineers. The impulse response of the chosen filter was then coherently demodulated and the envelope and phase functions of this impulse response displayed similar to the displays used in the Project MIMI propagation studies. This report contains the derivation of the technique for coherently demodulating the impulse response of a realizable filter, and displays many of these graphs so that they may be used as a basis for comparing actual receptions of underwater acoustic signals.

2. THE COHERENT DEMODULATION OF A BANDPASS IMPULSE RESPONSE Some classical low-pass filters are reviewed in Appendix A. A technique for converting from bandpass to low-pass is discussed in Appendix B, and it is shown that the transfer function of a bandpass filter which has no repeated poles can be written in its partial fraction expansion as n r. r. H(s) = S- s+ 1 (5) i=1 1 s- S. 1 The summation has been written in two parts where the poles, s., that occur in the upper left quadrant have been singled out. The poles in the lower left quadrant are the complex conjugates of those in the upper left quadrant, and it is assumed that all of the poles are in the left half plane; that is, we are studying a stable realizable bandpass filter. The * denotes complex conjugate, and the real and imaginary parts of the pole are denoted by Si i + ji, <i < 0, Wi> 0 (6) 1 i i 1 For simplicity of complex demodulation, the complex impulse response due to just the upper half plane poles will be utilized. -7

-8n s T u 1 hu(T) = ri e U(i ) (7) i=l where u(T) is the unit step function. The total impulse response of the filter is the sum of the contributions from both upper and lower half planes and is h() =h( hu(T)+ h(T) = 2Re[h ()] (8) Comparing Eq. 8 and Eq. 4 one determines immediately jW0T z(r) e = 2 h () (9) This is why the complex response from the upper half plane poles only was singled out. Shifting the exponential term to the other side of the equation, we have -jaoT n Pi. z(T) = 2 () 2 2 1 ri e = u(z) (10) where Pi = Si- jo = a + j(- w) (11) This shows that z(T) is the response due to poles at new positions shifted from their bandpass location down to near the origin with the

-9residues r. the same as for the bandpass function. Complex demodulation can be obtained by writing Eq. 10 in Cartesian coordinates n piT z(7) = x(T) + j y() = 2 J r. e u(r) (12) i=1 and using the computer to evaluate the real and imaginary parts of this complex summation. The instantaneous frequency as defined in Eq. 1 can be determined from the Cartesian coordinates, and the use of the coordinate change shown in Fig. 1. f ( 7) d/ (7) f.(=) = d + ( 1 0 dr (13) X+ (T) y(T) - x(T) y(T), T > = a).+', r> o x(7) + y2(7) The derivatives of the complex demodulation are determined by differentiating Eq. 12 n pi. z(=) = (T)+ j y(T) =2 riPie, T> 0 (14) i= 1 and again using the computer to evaluate the real and imaginary parts of this complex summation.

3. DISPLAY OF THE COMPLEX DEMODULATION FOR VARIOUS TYPICAL BANDPASS FILTERS Figures 3- 5 show the results of complex demodulation for various Butterworth filters. Figures 6-8 show the same results for various Chebyshev filters. The figures on the left side show the time domain representation of the impulse response while the figures on the right side show the frequency domain representation. Figure 9 shows the time domain representation of the magnitude function R on a linear scale. The bandwidth of each of the filters of Figs. 3-8 is 100 Hz with a center frequency of 420 Hz. The parameter n refers to the order of the filter while the parameter e is the ripple factor for the Chebyshev filter. The dominant feature that shows up on all of the figures is the sweeping phase for fairly low amplitude levels. The same sweeping phase appears in MIMI data of Fig. 2. At these low amplitude levels the nth order filter behaves as if it were a first order filter with the dominant effect due to the pole nearest the imaginary axis. The conclusion to be drawn from this comparison is that the MIMI channel appears to behave as a first order bandpass filter for low level signals. -10

-11~0 1.0 -40 - -80,, 0___ 0 20 40 320 420 520 t (msec) f (Hz) 180 - 180 bf bn i -'\-0 0 -180 -8 0 II I, 0 20 40 320 420 520 t (msec) f (Hz) W 400 200 I 0 20 40 t (msec) Fig. 3. Time and frequency domain representation of the impulse response of a Butterworth bandpass filter, n = 2

-120 1.0 -40 -80 O I 0 30 60 320 420 520 t (msec) f (Hz) 180 - \ 180 0_0 -180, -180 0 30 60 320 420 520 t (msec) f (Hz) 400 200 |, i_ 0 30 60 t (msec) Fig. 4. Time and frequency domain representation of the impulse response of a Butterworth bandpass filter, n = 3

-130 1.0 -40 -80 0 0 40 80 320 420 520 t (msec) f (Hz) 180 -3 180 -180 \ -1 801 0 40 80 320 420 520 t (msec) f (Hz). 40.0 200 - 0 40 80 t (msec) Fig. 5. Time and frequency domain representation of the impulse response of a Butterworth bandpass filter, n = 5

-14O~~~ iX ~~~~~~1.0 -40 -80 0 I I I 0 40 320 420 520 t (msec) f (Hz) 180 - 180 0 0 -180 -180L I I I_ 0 40 320 420 520 t (msec) f (Hz) 400 f 200 I I 0 40 t (msec) Fig. 6. Time and frequency domain representation of the impulse response of a Chebyshev bandpass filter, n = 2, e =. 5

-150 1.0 ^40 - I I I I I 0 60 120 320 420 520 t (msec) f (Hz) 180 - \ 1 80 0 -180 I I I - 80 0 60 120 320 420 520 t (msec) f (Hz) 400 200 0 60 120 t (msec) Fig. 7. Time and frequency domain representation of the impulse response of a Chebyshev bandpass filter, n = 3, e =. 5

0 1.0 -40 -80, I I I I 1 0 ~ - I 0.I 0 150 300 320 420 520 t (msec) f (Hz) 180 1800 0 - 180 i I _ I -180. 0 1 50 300 320 420 520 t (msec) f (Hz),400 200 I l 0 1 50 300 t (msec) Fig. 8. Time and frequency domain representation of the impulse response of a Chebyshev bandpass filter, n = 5, e =. 5

-17-.04 -02 0OL-4 0 40 t (msec) 180 -180 0 40 t (msec) Fig. 9. Demodulated time response of the impulse response of a Chebyshev bandpass filter, n = 2, e =. 5

APPENDIX A CLASSICAL LOW-PASS FILTERS The Butterworth and the Chebyshev filters were considered as appropriate representations of low-pass filters. For convenience, the center frequency was normalized to w0 = 1. 1) Butterworth filter of nth order —maximally flat magnitude function IH(jw)l (1+ w2n) i) and H(s) = ( n n (s- Si) i=l where Si = ai + jW and -18

-19jw (2i- 1 a Poles are on j ao = sin unit circle i=1,2,..., n -c ).= o(V2i- I V 1i cos( n 2 i =1,2,..., n 2) Chebyshev filter of nth order —equal-ripple magnitude function I H(jco) I IH(j()I 1 1 l l l+e2 C2(w) 0 and H (s) = 1 n n n (s-s.) i=1 1 where Si = ai + jwi i=1,2,... n and i. = sin (2 1 *2 tanh(v) 2i- I1 i wi = cos.2) v =1 - 1l v (- sinh() n

-20jw j Poles are on ellipse -j C (w) is the nth order Chebyshev polynomial cos [n cosl(w)] 0 < w < 1 Cn() cosh[n cosh (w)] w > 1 with Co(w) = 1 C1(W) = Ck+l () = 2w Ck(w) - Ck- 1()

APPENDIX B CONVERSION FROM LOW-PASS TO BANDPASS Let L(z) be a low-pass transfer function of nth order. Assume only single order poles are present. Denote the poles by z. and the residues by p. Hence n p. L(z) = i=l z z. If L(z) is positive real (the network is stable), then complex poles occur in conjugate pairs and their residues are conjugates. If the poles are in the vicinity of the origin with Izil < 1, i = 1, 2,..., n, then the normalized transformation z= S+S is used to convert from the low-pass transfer function L(z) to the bandpass transfer function A 1 H(s) L(s +) n t. t.' i=1S-S S-S where -21

-22S. = 5z +.25z.2- 1 Pi Si s.' = s.5z. - t.25z.2t -- iS i s. - s. 1 1 Some Observations 1) Each pole z. (in the vicinity of the origin) is mapped into an UHP pole s. (close to j) and a LHP pole si' (close to -j) with Is.-j I is + jl - 21zil, i=1,2,..., n. six Z. X z*x z plane s plane s5 x S' X -- s* -~ -j Pole Locations In terms of Cartesian coordinates ^, ^v, +ya +x aya x+jyW 2 + j sgn(y) where ( y>o sgn(y) = - (- y<O

-232) Symmetry considerations enable one to write H(s) as a sum of UHP poles and LHP poles, i.e., n 3. * H(s) = +' - oz. S-O/. i=l 1 1 where s. Im s. > 1 1 1 s.* Im s. < 0 1 1 t. Im s.> 0 P ti. < 1 Im s < 0 1 t 1

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Security Classification DOCUMENT CONTROL DATA R & D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION Cooley Electronics Laboratory Unclassified University of Michigan 2b. GROUP- -'. Ann Arbor, Michigan 48105...... REPORT TITLE Coherent Demodulation of the Impulse Response of Bandpass Filters DESCRIPTIVE NOTES (Type of report and.inclusive dates) Technical Report No. 206 - 036040-3-T AU THOR(S) (First name, middle initial, last name) David Jaarsma REPORT DATE 7a. TOTAL I4Q. OF PAGES 7b. NO. OF REFS August 1971 40 a. CONTRACT OR GRANT NO. 9a. ORIGINATOtrS REPORT NUMBER(S) N0001 4- 67-A- 0181- 003 2 036 0-3b. PROJECT NO. 3 -3C. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) A _d. X TR 206 0. DISTRIBUTION STATEMENT Approved for public release; distribution unlimited. 1. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Office of Naval Research Department of the Navy Arlington, Virginia 22217:. ABSTRACT This report contains the analysis appropriate for the complex demodulation of the impulse response of realizable lumped element filters. Plots of the time ehavior of the log of the envelope, the instantaneous phase, and the instantaneous frequency, together with the conventional phase and amplitude of the transfer characteristic, are presented for a number of bandpass filters. ID FOR 1473 (PGE 1 I NOV 651 /N 0101-807-6811'Security Classification....... A-314(08

Security Classification 14 LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Signal Processing Acoustic Propagation Complex Demodulation Phase Behavior DD FORMv a1473 (BACK) _

UNIVERSITY OF MICHIGAN 1111111111111 II 3 9015 03025 1642