Contains reports on two 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)National Aeronautics and Space Administration (Grant NsG-496

Oppenheim, Alan V.

Schafer, R. W.

Schindall, J. E.

DSpace@MIT

COMMUNICATION SCIENCESANDENGINEERINGXVII. STATISTICAL COMMUNICATION THEORYAcademic and Research StaffProf. Y. W. Lee Prof. J. D. Bruce Prof. D. E. NelsenProf. A. G. Bose Prof. A. V. Oppenheim G. GambardellaGraduate StudentsT. Huang E. M. Portner, Jr. F. P. TuhyD. J. Matthiessen L. R. Poulo J. L. WalkerM. F. Medress A. E. Roland J. J. WawzonekV. Nedzelnitsky S. K. Samson C. J. WeinsteinA. D. Pitegoff R. W. Schafer D. H. WolaverJ. E. SchindallA. DISCRETE ANALYSIS OF HOMOMORPHIC DECONVOLUTIONA method for separating convolved signals based on the canonic system of Fig. XVII-1has been previously discussed. Some preliminary analytical results for continuous sig-nals and a brief discussion of the problems involved in implementing such a system haveA L Ax W x (t) (t) y(t)Fig. XVII-1. Canonic form for deconvolution filter.also been given. Since such a system must be implemented by using a digital computer,it is of importance to consider the characterization of the system of Fig. XVII-1 whenall of the signals are discrete time functions.In this report we shall discuss the discrete characterization of the system AO. Thisis a nonlinear system that is characterized by the property that if x = x l 0 x 2 , thenS= 1 + ~' with & denoting discrete convolution. If X(z) and X(z) denote the two-sidedz-transforms of 4xn) and x(n), respectively, this property can be realized through therequirement thatX(z) = log X(z),provided that when z = e j o , arg[X(z)] is continuous in the interval -rT < w < Tr. Since thez-transform is periodic in w, arg[X(z)] must also be periodic in w. It is not necessaryto associate the region of convergence of X(z) with X(z).We shall make use of just two choices for the region of convergence of X(z). One*This work was supported principally 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),and in part by the National Aeronautics and Space Administration (Grant NsG-496).QPR No. 85 231(XVII. STATISTICAL COMMUNICATION THEORY)choice is the region outside all poles and zeros of X(z). This results in a sequence, x(n),which is zero for n < 0. The other choice is an annular region that includes the unitcircle. This choice results in a sequence, x(n), which always approaches 0 as napproaches oo00.It will be assumed that X(z) is of the form11 (1 -a z-1X(z) = Azr k=l(1)pP (i)S(1-biz - l)i=lwhich can be rewritten in the formm mr-mo +p m -p o o m ( -1)z A(-i) IE ak l-ak-1z I -akzk=l k=1 k=m +1X(z) = oPo Po P11 b i . ( 1-bi) 1 ( 1-bizi= 1 i=l i=po+ 1where the first m0 zeros and po poles are assumed to be outside the closed contour thatis used for determining x(n), and the rest inside. For X(z) of the form of Eq. 1 and forwhich the angle is computed as discussed above, it can be shown by integrating by partsthatmoA r akk=1(O) = log n + c(r-mo+po) (Za)i=1andcrn cos r nx (n) = Ke + c(n) n = ±1, 2, .... (2b)nThe function c(n) is defined by1 C X' (z) n-lc (n) = r z- z dz, (3)C X(z)where C is a circular contour defined by z = e , which lies in the region of conver-gence of X(z). The constant K in Eq. 2b is given byK= r+Po -m .QPR No. 85 232(XVII. STATISTICAL COMMUNICATION THEORY)It can also be shown thatK = r + m i - p. + p -m,where m i and p.i are the number of nonzero zeros and poles inside the contour C.By applying the properties of z-transforms, it can be shown that c(n) also satisfiesthe equation1 x(n-k)c (n) = L kc(k) n 0.e n, - x(0)k=-ooIn the special case for which x(n) = 0 for n < 0, x(0) # 0, and the contour C in Eq. 3encloses all of the poles and zeros of X(z), so that x(n)= 0, n <0, the constant K in Eq.2is zero and therefore ^(n) = c(n). Under these conditions, x(n) satisfies the recursionformulan-1x (n) 1 (n-k)(n) = -- 1 kx(k) n > 0 (4a)x(O) n x(O)k=O'(0) = log A = log x(0). (4b)It should also be noted that the inverse operation, corresponding to the charac-terization of the system A-, can easily be obtained simply by solving Eqs. 4for x(n).If x(n) = 0, n < 0 and x(0) * 0, Eq. 4 can be used to compute x^(n). The computationis practical only in the case in which X(z) has no poles or zeros outside the unit circlenbecause if this is not true, x(n) will grow at least as fast as-, where a is some num-ber greater than 1. It should be mentioned, however, that multiplication of x(n) bypn (<1) will sometimes result in a new function whose transform has no poles or zerosoutside the unit circle. It can also be shown that if X(z) is a rational function, the effectof multiplying x(n) by 3n is simply to multiply the output x(n) by 3n.In many cases of interest it may not be possible or desirable to use Eq. 4 in thecalculation of ^(n). In such cases, the inverse discrete Fourier transform of the loga-rithm of the discrete Fourier transform of x(n) must be computed. If N denotes thenumber of points in the discrete Fourier transform, then the resulting sequence, x(n),is related to x(n) by the equation(n) X (n+aN) n = 0, i,..., N- 1,a=-oocorresponding to aliasing of the sequence X^(n). This aliasing can be made lessQPR No. 85 233(XVII. STATISTICAL COMMUNICATION THEORY)noticeable by exponentially weighting x(n) or by augmenting the sequence x(n) byterminating in zeros.R. W. Schafer, A. V. OppenheimReferences1. A. V. Oppenheim, "Nonlinear Filtering of Convolved Signals," Quarterly ProgressReport No. 80, Research Laboratory of Electronics, M. I. T. , January 15, 1966,pp. 168-175.2. R. W. Schafer, "A New Approach to Echo Removal," Quarterly ProgressReport No. 84, Research Laboratory of Electronics, M. I. T., January 15, 1967,pp. 194-201.B. UNITY FEEDBACK TWO-STATE MODULATOR CONFIGURATIONS1. IntroductionThe transfer function for the two-state modulation configuration described by Bose 1has been derived in detail. 2 It has been shown that the dynamic ramp response, withinan error term, is of the formT(t) = -x(t) - *rx (t). (1)The frequency response of the modulator, then, as inferred from the ramp response,goes asY(s)X(s) - -(I+Ts) (2)which, as in the case of a continuous amplifier with feedback, is the inverse of the feed-back network response.The system frequency response can be made flat by placing an RC filter identical tothe feedback network in front of the modulator loop. The same effect is achieved in thecircuit of Fig. XVII-2, in which the RCfeedback filter has been located in the placewhere it also acts to filter the inputTHRESHOLDx e(t) R g(t DEVICE () waveform.INPUT C WITHN __ C  HYSTERESIS OUTPUTIn Fig. XVII-2 the hysteresis switchwith delay, previously used as the forwardFig. XVII-Z. Unity feedback two-state element of the modulator loop, has beenmodulation system. replaced by an arbitrary threshold devicewith hysteresis. This has been done topermit application of the analysis to more general effects that are present in a physicalrealization, such as switching transients and variations in the switching levels. TheQPR No. 85 234(XVII. STATISTICAL COMMUNICATION THEORY)threshold device has two states or modes of operation: the charge state, entered whenits input g(t) drops below wc, during which the output will be designated yc(t); and thedischarge state, entered when g(t) rises above wd, during which the output will be des-ignated yd(t). The only contraints on the device behavior to insure recurrent alternationof states or cyclic operation of the modulator are that x(t) plus yc (t)be sufficiently posi-tive that g(t) tends to increase or "charge," eventually reaching wd, and that x(t) plusYd(t) be sufficiently negative that g(t) tends to decrease or "discharge," eventuallyreaching w c . A switch with hysteresis and delay, for example, meets this requirement,provided the input is bounded by hd - wc < x < hc - wd.A general transfer expression for this system configuration will be derived. For2static input, the system is shown to be identical to the RC feedback configuration ifideal hysteresis with delay is assumed. For dynamic inputs - again, under the assump-tion of ideal hysteresis with delay - it can be shown that a flat frequency response isobtained with error equivalent to the static error. Finally, an improved system will beintroduced for which the transfer function predicts zero error, even for the arbitrarythreshold device described above.2. AnalysisThe transfer function for the circuit of Fig. XVII-2 will now be derived. For theRC circuit,e(t) = g(t) + Tg'(t). (3)The input to the nonlinear element has the form sketched in Fig. XVII-3. As long asthis waveform is cyclic, the desired modulation will obtain.g(t)CHARGE _ DISCHARGESTATE STATEWd Fig. XVII-3. Typical input to the thresh-old device.WTC -wc / ----------' ------ T --...Define a cycle of operation to be from one w state transition to the next. Then,choosing t = 0 at the start of a cycle, we haveg(T) = g(0). (4)From (3), since e(t) = x(t) + y(t),QPR No. 85 235(XVII. STATISTICAL COMMUNICATION THEORY)[x(t)+y(t)] dt = g (t) dt + g(t) T (5)Rearranging terms, using (4), and dividing by the period T yieldsT T T- . y(t) dt = x(t) dt + g(t) dt. (6)The output mean over the cycle is equal to the negative of the input mean over the cycleto an error term in g(t), independently of the form of x(t) and y(t), as long as cyclicbehavior is maintained. For efficiency, y(t) should be a switching waveform, and, toenable accurate reconstruction of the input waveform, the modulator parameters shouldbe such that the cycle time is short compared with the rate of change of x(t); however,neither of these considerations is involved in the derivation of Eq. 6.As indicated by Eq. 6, the modulator output in the unity feedback configuration isreferenced to the mean of the input during a cycle, rather than depending only on theinput values at the switching instants as was the case with the RC feedback configura-tion. The RC network continuously looks at the input waveform and effectively storesits behavior over the entire cycle in g(t).3. Investigation of the Error TermIn the preceding development arbitrary parameters that were appropriate for cyclicoperation were assumed. It is intuitively clear that a narrow hysteresis window, boundedinput and output, and small delays will limit the excursion of g(t) to a restricted rangeand permit a bound on the error term. For typical operation this bound is exceedinglyloose because the approximate symmetry of g(t) reduces the value of its integral over acycle. Expressions for the modulator error term have been developed for the case inwhich the forward path is an ideal switch with hysteresis and delay, and the input isstatic or characterized by its midpoint and slope over the cycle.A typical errror waveform g(t) over a cycle with static input is sketched inFig. XVII-4. Notice that g(t) is pre-cisely the static error waveform eval-'g , (uated in a previous report, 2 in whichTdc the RC feedback modulator configura-,to hcWd- tion was discussed. The average ofo / g(t) was shown to be equal to its mid-M -- to point, which can be evaluated in termsTdd of the modulator parameters, plus aterm expressing the discrepancyFig. XVII-4. Error waveform, static input. between mean and midpoint. ThisQPR No. 85 236(XVII. STATISTICAL COMMUNICATION THEORY)result, for the symmetric case, is reproduced in Eq. 7.TdTy = -x e 1 + (7)x e1-The error function for ramp input can be derived by characterizing the ramp asaltering the effective midpoint and half-width of the exponential window function g(t),which is the same procedure previously used to discuss the ramp response of the RCmodulator configuration.2 The detailed mathematics will be omitted here; the resultsupports our intuitive claim that the device should exhibit flat frequency response bypredicting the same transfer expression as for the static case. The "static" input valueof Eq. 7 is simply replaced by the mean of the ramp during the modulator cycle.4. Integral ModulatorIn the derivation presented in this report, the error term was found to be the integralof g(t) over a cycle. If the RC network is replaced by an integrator, as drawn inFig. XVII-5,e(t) = Tg'(t), (8)and the error term in the derivation vanishes! Equation 6 becomes simplyS y(t) dt = - I x(t) dt. (9)Independent of the form of y(t) -as long as it is appropriate to produce cyclicoperation - and independent of the parameters of the hysteresis element and the natureFig. XVII-5. Integral two-state modu-+ e (t) THRESHOLD lation system.x(t): INTEGRATOR - WITHCE y (t)INPUT HYSTERESIS OUTPUTof the input, the average of the output over a cycle is precisely the negative of the meanof the input over the cycle.A qualitative explanation of this type of modulator operation is easy to provide. Theoutput of the integrator represents the input mean plus the output mean, and the hyster-esis element guarantees that this output is cyclic. Then, over a cycle, the sum of theQPR No. 85 237(XVII. STATISTICAL COMMUNICATION THEORY)means, the output error, is brought to zero.In practice, the realization of the integrator may not be ideal. The transfer functionof the integrator may then be represented in the form.e(t) = Ag(t) + Tg'(t), (10)where A is zero ideally, approximately .001 for a typical operational amplifier real-ization of an integrator, and equal to one for the simple RC approximation to an integra-tor. We have shown that the modulator error resulted from the integral of g(t) over acycle. In the analysis of this error, A was taken as unity (an RC approximation) anderror expressions were derived. These same error expressions can be applied to thecase of incomplete realization of the integrator in an integral modulation scheme simplyby multiplying the error expressions by the factor A. Of course, for an ideal integratorA is zero and there will be no error, and for most practical realizations A is sufficientlysmall that the error may be neglected.J. E. SchindallReferences1. A. G. Bose, Quarterly Progress Reports No. 66 and No. 67, Research Laboratoryof Electronics, M. I. T., July 15 and October 15, 1962, pp. 187-189 and 115-119.2. J. E. Schindall, Quarterly Progress Report No. 84, Research Laboratory of Elec-tronics, M. I. T. , January 15, 1966, pp. 184-194.QPR No. 85 238

Statistical Communication Theory

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Statistical Communication Theory

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