Contains reports on three research projects.Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E

Bruce-Lockhart, M. P.

Johnson, D. H.

Kuleshov, V. N.

Nelsen, D. E.

Oppenheim, Alan V.

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XVI. SIGNAL PROCESSINGAcademic and Research StaffProf. A. G. Bose Prof. A. V. Oppenheim Dr. M. V. CerrilloProf. J. D. Bruce Prof. C. L. Searle Dr. V. N. KuleshovProf. D. E. Nelsen Prof. H. J. ZimmermannGraduate StudentsS. A. Hendryx R. M. Mersereau R. K. StockwellM. P. Bruce-Lockhart S. G. Peltan J. L. VeranthM. R. SamburA. FIRST-PASSAGE TIME STATISTICS OF A FIRST-ORDERMARKOV PROCESS WITH A LINEAR RAMP1. Statement of the ProblemSome problems concerning time jitter in regenerative switches can be reduced tothe calculation of the statistical characteristics of the first zero passage time (FPT)of the sum y(t) of a linear ramp x(t) = Et and a first-order Markov process x(t) definedby the differential equationdx + x = n(t). (1)dtWe assume that n(t) is stationary white Gaussian noise with the autocorrelation functionR (T) = n(t) n(t+T) = 26(T), (2)nwhere 6(T) is the unit impulse function. If the coefficients of Eq. 1 and the noise spec-tral density were arbitrary rather than as specified above, the problem could easily bereduced to that described with suitable scaling of x and t.The problem is illustrated in Fig. XVI-1. Solution for the first zero of y(t) = Et +x(t) is equivalent to determining the first time instant that x(t) = -Et. As the slope,E, decreases, we would expect the mean FPT to become more negative, and thestandard deviation of the FPT to increase. In this report, we have obtained asymp-totic expressions for the probability distribution w(O), the mean value 0, and thestandard deviation cr6 0- 2 of the FPT. The results are valid when E << 1, whichturns out to be the region where computer simulation of the process is most diffi-cult.This work is supported by the Joint Services Electronics Programs (U. S. Army,U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E).QPR No. 97 181(XVI. SIGNAL PROCESSING)FIRST-PASSAGE TIME (FPT)y(t)X= EtFig. XVI-1. First-passage time for the sum y(t) of the linear rampx = Et and noise x(t).2. Method of SolutionThe results were obtained by a Fokker-Planck equation approach. We utilized theseries solution of the problem for R(t) = const. (obtained by Stumpers ) and proceededto the case of slowly varying x(t). Asymptotic formulas for the parabolic cylinder func-2tions were used to present the results in a very simple form.3. Statistical Characteristics of the FPTThe asymptotic distribution, w(6), of the FPT is1 (E) e - ( )2 / Z exp 1 e -( 6)/0 0w(O) = p(E6) = (3)S0 > 0In terms of the integrated distribution function, W(6) = . w(r) dr a simpler formexp [e -E6 < / 01 - W(O) = (4)0 0 >0is obtained. Examples of the distribution in Eq. 3 are shown in Fig. XVI-2.QPR No. 97 182(XVI. SIGNAL PROCESSING)= 10-6  10-5  10-4  10-3  10-22.0- 1.51.0 p(E9)- 0.50-6 -5 -4 -3 -2 -1Fig. XVI-2. First-passage time probability distributions p(EO) forseveral values of the ramp slope E.a. Average Value 0 of the FPTWe were unable to get an analytical expression for 0 from Eq. 3. A numerical eval-uation of 0 was made and is presented in Fig. XVI-3. Fortunately, for E << 1, 0 isAasymptotically close to the value 0 of the maximum of the distribution w(0), which caneasily be approximated. An expression for the mean EO(E) is thus found to be-10 -8 -6 -4 -2 og 0 E08060.i- AFig. XVI-3. Plots of the mean value 0, mode 0, and standard deviation06 of the FPT distribution as a function of ramp slope E.The parameters are multiplied by E to emphasize deviationof the laws from 1/E.QPR No. 97 183(XVI. SIGNAL PROCESSING)6 E6 - -2 In Er- - 1 - (5)-Z In (E -Tr)which is also shown in Fig. XVI-3.b. Standard Deviation (6The variation of the FPT standard deviation cr6 with slope E was calculated numeri-cally and is also shown in Fig. XVI-3. An approximation for cr- can be obtained byAusing analytic results for the height of w(0) at the maximum, 0, and approximating theshape of w(6) by a unit-area Gaussian distribution. We obtain the approximate formula,-e 1. 083ZK Ed EOc. Range of Validity of the Asymptotic ResultsThe formulas above are certainly valid for E << 0. 005 when the assumptionw(0) = 0 for 0 > 0 gives negligible error. Under this condition, most of the first passagesoccur far back in time where the threshold, -Et, is still several standard devia-tions (of x(t)) high.4. Discussion of ResultsThere are two points in these results that deserve to be mentioned.The first is connected with applications. If we apply this FPT model to a calculationof jitter statistics in a regenerative switch where noise is caused by wideband shot and-2thermal noise, the region where E < 10-2 is the range of practical interest.The second point is concerned with the generalization of the results. Replacing(t) = Et with any function with dt < 0. 01 and using the same approach, we can getapproximate analytical expressions for w(O). It is also possible to adapt this analysisto the case in which the spectral density of n(t) is multiplied by a slowly changingcoefficient.V. N. Kuleshov, D. E. NelsenReferences1. F. L. H. M. Stumpers, "On a First-Passage-Time Problem," Philips Res.Rep. 5, 270-281 (August 1950).2. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover Publi-cations, Inc., New York, 1965).QPR No. 97 184(XVI. SIGNAL PROCESSING)B. DISCRETE REPRESENTATIONS OF ANALOG SIGNALS1. IntroductionIn many applications we are concerned with implementing or simulating analog signalprocessing on a digital computer. In carrying out such an implementation, an analog(continuous) signal is represented by a sequence that is processed digitally, and the out-put sequence is then converted to an analog signal. The most common example of a dis-crete representation of continuous signals is periodic sampling. More generally,Steiglitz1 has discussed the equivalence of digital and analog signal processing throughan isomorphic mapping between the s-plane and the z-plane, and he suggests the bilineartransformation as a specific mapping that permits linear, time-invariant continuous fil-ters to be represented by linear, shift-invariant discrete filters. Masry, Steiglitz, andLiu discuss other isomorphisms between continuous and discrete signals. Piovosoand Bolgiano 3 have proposed the Poisson transform as a means for associatingsequences with continuous functions to implement linear continuous time-invariantfilters digitally.A linear representation of a continuous function f(t) with a sequence fn is of theform+of(t) = f n n(t), (1)n= -where the functions cn(t) should, for convenience, be linearly independent so that therepresentation is unique, but need not be orthogonal. The requirement that a linear,continuous time-invariant filter be represented by a linear discrete shift-invariant fil-ter corresponds to requiring that if f , hn, and gn are the discrete representations off(t), h(t), and g(t), respectively, and if g(t) is the continuous convolution of f(t) and h(t),then gn will be the discrete convolution of the sequences fn and h n. It is straightforwardto show that this requires the Fourier transform of cn(t), denoted (n(jw), to have theproperty4n(j) m (j )= n+m(j )" (2)When fn is derived by periodic sampling of f(t) so that fn = f(nT), the functions (t)are given bysin wr(t-nT)n( = (t ) (3)v(t-nT)QPR No. 97 185(XVI. SIGNAL PROCESSING)withe-jwnT 0 < @ IS 0 otherwiseso that (2) is satisfied.In general, any mapping from the s-plane to the z-plane corresponding to expressingz as a function of s will lead to a representation of a time function f(t) in the formof Eq. 1 with the n(j) satisfying Eq. 2. This can be seen in several ways. Forexample, from (1), the Laplace transform F(s) of f(t) can be expressed as+ooF(s) = fn n ( s ) . (4)n=-ooAThe z-transform of the sequence fn, denoted F(z), is given by+oA fZ-nF(z) = f -nn=-ooand since the s-plane is mapped to the z-plane, we have z = G(s), and consequently+oF(s) = F[G(s)] = fn[G(s)]n. (5)n= -ooComparing Eqs. 4 and 5 yields(n(s) = [G(s)] - n, (6)so that Eq. 2 is clearly satisfied. In general there is no guarantee that the set of func-tions 4n(t) corresponding to (6) is complete. The conditions under which they are havebeen discussed by Masry et al. 2Any mapping from the s-plane to the z-plane of a form corresponding to Eq. 6 willpreserve convolution whether or not it maps the jw axis in the s-plane to the unit circlein the z-plane. For the case of the bilinear transformation, z = G(s) =a-s so thatn() = s. With z = e corresponding to the unit circle and s = jc,ejR +1j = a . = ja tan -ej- 1QPR No. 97 186(XVI. SIGNAL PROCESSING)orQ(w)= 2 tan - . (7)In particular, then, the bilinear transformation maps the jw axis in the s-plane to the unitcircle in the z-plane. The functions cn(t) corresponding to this transformation are2a(-1) n - 1 -at e L (2 at) u (t) + (-1) u (t), n > 0n-1 1 0n(t ) = uo(t) n = 02a(n-1 at LM (-Zat) u_ (-t) + (-1 )n u(t), n < 0 (8)-n-1 1 owhere L(1) (x) is the first-order Laguerre polynomial of degree (n-1) defined asn-1L ( ) (x) = d [L (x)(n-l1) dx n(9)ex d n -xL(x) = n (x n e .dxnIn addition to simulation, it is often desirable to choose a discrete representation ofanalog signals which permits a determination of the analog spectrum digitally. This ismotivated in part by the fact that an efficient algorithm exists for computing samples ofthe z transform on the unit circle. In order to utilize this algorithm for measuring ananalog spectrum, we would want to choose a discrete representation that maps thejw axis to the unit circle. A common procedure is periodic sampling correspondingto a representation in terms of Eqs. 1 and 3. If the sampling rate is sufficientlyhigh so that aliasing is avoided, then a spectral analysis of the discrete sequenceresulting in samples at equally spaced angles on the unit circle will correspond to ameasurement of the analog spectrum at equally spaced frequencies on the jw axis. Fur-thermore, since the computation of the spectrum of the discrete sequence must becarried out on a finite-length sequence, a finite-duration window is put on thesequence. This then corresponds to a "smearing" of the spectrum with a spectral win-dow that is the same at all frequencies. Consequently, measurement of the analog spec-trum by generating a sequence through periodic sampling, applying a window, and thencomputing the discrete Fourier transform is similar to measurement of the analog spec-trum with equal-bandwidth filters. More generally, if we choose a discrete repre-sentation of the form of Eq. 1 with D n(s) given by (6), then the jw axis in the s-planewill be mapped onto the unit circle in the z-plane if G() = , that jn(w)will be mapped onto the unit circle in the z-plane if IG(jw) = , so that n (jw) = e -QPR No. 97 187(XVI. SIGNAL PROCESSING)The function Q(w) then represents a warping between analog frequencies W and digitalfrequencies 2. In many instances, it is desirable to measure an analog spectrum withunequal resolution across the band, as obtained for example, with a constant-Q mea-surement. Such a spectral measurement can be obtained digitally by using a discreterepresentation corresponding to the bilinear transformation between the s- andz-planes. For Eq. 7 we see that frequencies equally spaced in 2 correspond to fre-quencies unequally spaced in c with spectral samples closer at low frequencies thanat high frequencies. Just as in periodic sampling, the discrete sequence must be trun-cated before computing the transform. If fn corresponds tothe discrete representationof f(t), and wn is a finite-duration window, then the sequence to be transformed, gn,would be given bygn = W nnAIf F(Q), W(Q), and G(Q) represent the transform of the sequences fn' wn, and gn'respectively, then1 ^G(2) = -L W( A-) F( ) d.  (10)-1Or if F(jw) corresponds to the analog spectrum of f(t), then from (7)G() = W(Q-) F(ajtan 2 d. (11)From Eq. 11 we see that the digital spectrum, G(Q), is obtained by smearing the analogspectrum in such a way that spectral resolution decreases as the analog frequencyincreases. Samples of G(Q2) equally spaced in Q2, would then correspond to a measure-ment of the analog spectrum with a filter bank for which the filter bandwidth wasproportional to the tangent of half the center frequency. Over a range of frequencies,this can be made to look similar to a constant-Q filter bank, with the specificshaping of the frequency resolution governed by the parameter a.One of the major advantages of the discrete representation of analog signalsthrough periodic sampling is the ease with which samples can be obtained. To obtain thecoefficients in an expansion in terms of the set of functions given in Eq. 8, there areseveral alternatives. One possibility is to utilize the fact that for these functions,St(t) m(t) dt = 0 n m (12)= In n=mQPR No. 97 188188(XVI. SIGNAL PROCESSING)so thatf n tf(t) cn(t) dt. (13)n n -ooIf we consider, for convenience, f(t) to be zero for t < 0, then f for n > 0 can benobtained by exciting the linear time-invariant network shown in Fig. XVI-4 withf(-t) 2a a-s a-s(s+a)2 a+s a+s-0'f1 (t) f2 (t)Fig. XVI-4. Analog network for obtaining the discrete representationcorresponding to the bilinear transformation.f(-t). The coefficients fn are then equal to the outputs at each of the taps at t = 0,that is,f = f (0).n nAn alternative procedure is periodic sampling of f(t) at a sufficiently high sampling rateto avoid aliasing, and then converting these samples to samples corresponding to a rep-resentation in terms of the functions in Eq. 8. If f(nT) correspond to the periodicsamples, and fn correspond to the coefficients for an expansion in terms of the functionsin Eq. 8, then0ofn f(kT) Cnk (14)k=000 sin L (t-kT)1 TC 1 t n(t) dtnk n 0 n r (t-kT)Tcorresponding to a matrix multiplication.The transformation from periodic sampling to the samples fn in Eq. 14 can be viewedin the z-plane as a transformation from the unit circle to the unit circle that preservesconvolution and also provides a frequency warping. In future work, other means foraccomplishing this frequency warping will be investigated.A. V. Oppenheim, D. H. JohnsonQPR No. 97 189(XVI. SIGNAL PROCESSINGReferences1. K. Steiglitz, "The Equivalence of Digital and Analog Signal Processing," Inform.Contr. 8, 455-467 (October 1965).2. E. Masry, K. Steiglitz, and B. Liu, "Bases in Hilbert Space Related to the Repre-sentation of Stationary Operators," SIAM J. Appl. Math., Vol. 16, No. 3, May 1968.3. M. Piovoso and L. Bolgiano, Jr., "Digital Simulation Using Poisson TransformSequences," 1969 Polytechnic Institute of Brooklyn Symposium (to appear in theProceedings of the Symposium).C. NONLINEAR PROCESSING OF SIGNAL ENVELOPESIn recent months our group has been considering some problems relating to the ques-tion of style in music. Some of these ideas were suggested by M. V. Cerrillo, in1957.1,2 At that time, he pointed out that the gross essence of style may be characterizedPREAMPLIFIER X TO POWER AMPLIFIER(a)PREAMPLIFIER X TO POWERAMPLIFIERFULL-AV ENVELOPE TRANSFERRECTIFIER DETECTOR CIRCUITDC(b)Fig. XVI-5. (a) Circuit for altering level contrast.(b) Circuit for altering note-to-note contrast.by a small number of parameters, such as contrast, attack, tempo, and so forth. Thesein turn may be related to more physical quantities, such as the time between twoaccented notes for tempo, or the magnitude of note envelopes for contrast. Whiletempo cannot be altered in real time, contrast or attack can be.Thus far, we have concentrated mainly on contrast, which, qualitatively at least,is the difference between the loud and the soft passages of music. We have definedQPR No. 97 190(XVI. SIGNAL PROCESSING)two different kinds of contrast. The first is level contrast, that is, the differencein level between two successive passages of music, while the second is note-to-notecontrast, or the difference between the level of a note and the background. Savefor the noticeable lag when a fast transition between two passages of different levelsoccurs, level contrast is easily accomplished (see Fig. XVI-5a). On the other hand,note-to-note contrast is somewhat more complicated (see Fig. XVI-5b). In orderto avoid unnecessary delay, the envelope detector must be mainly a peak detector,rather than an averaging detector. The problem is to set the time constant. If itis too short, the detector will pick up carriers, as well as envelopes, while if itis too long, it will tend to skip notes, particularly any preceded by a much loudernote or a very short one. Furthermore, while the music is playing, the peak detec-tor seldom gets a chance to decay completely before the next note, so that a super-fluous DC level is injected into the signal.While there is no optimum solution to the question of a time constant, it is pos-sible to find a working compromise so that only very few notes are skipped. Elim-inating the excess DC, however, requires some kind of nonlinear transfer curve,and several such curves have been found that are effective to one degree or another.Both kinds of contrast can now be altered in a manner that sounds quite natural.M. P. Bruce-LockhartReferences1. M. V. Cerrillo, Quarterly Progress Report, Research Laboratory of Electronics,M. I. T. , April 15, 1957, p. 100.2. M. V. Cerrillo, Quarterly Progress Report, Research Laboratory of Electronics,M.I.T., July 15, 1957, p. 89.QPR No. 97 191

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