Contains research objectives and reports on two research projects.National Institutes of Health (Grant MP-4737

Lee, Y. W.

Schetzen, M.

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XVIII. STATISTICAL COMMUNICATION THEORY*Prof. Y. W. Lee Prof. H. L. Van Trees, Jr. D. F. DeLong, Jr.Prof. A. G. Bose V. R. Algazi H. HemamiProf. I. M. Jacobs M. E. Austin W. S. WidnallProf. D. J. Sakrison A. M. Bush R. B. ParenteProf. M. Schetzen C. E. Wernlein, Jr.RESEARCH OBJECTIVESThis group is interested in a variety of problems in statistical communicationtheory. Our current research is concerned primarily with: phase-locked oscillators,two-terminal nonlinear systems, location of noise sources in space by correlationmethods, iterative methods in nonlinear theory, measurement of the kernels of a non-linear system, and factors that influence the recording and reproduction of sound.1. The threshold behavior of phase-locked oscillators in high noise-signal environ-ments is being studied. A suitable model for the actual system has been shown to be anonlinear time-variant system. This model is now being simulated on an analog com-puter to try to find out how the stability of the loop depends on the various systemparameters.2. Two-terminal nonlinear passive systems are being studied from the point of viewof statistical theory. The study of these systems is producing some interesting results.3. Noise sources in space can be located by means of higher-order correlationfunctions. A study of the errors in locating sources by this method is continuing.4. The mean of a convex weighting function of the error of a system can be mini-mized by an iterative adjustment of certain parameters of the system. A study of someiterative methods is being made.5. In the Wiener theory of nonlinear systems, a nonlinear system is characterizedby a set of kernels. A method for the determination of these kernels was reported inQuarterly Progress Report No. 60 (pages 118-130). Some experimental results haveindicated that the method is practical in many problems. Work on the method, boththeoretical and experimental, is continuing.6. We are also studying the factors that influence the accurate recording and repro-duction of sound. In this study the tools of statistical communication theory are appliedto spectral analysis under different methods of recording. In addition to the spectralstudies, the transient behavior of the various links in the reproduction process will beinvestigated. Associated with this project, a filter of the Wiener-Lee type, which hascontrollable amplitude with a fixed phase over part of the audio spectrum, will be con-structed as a tool for studying the effects of magnitude and phase perturbations onsound signals.Y. W. LeeA. POWER AMPLIFICATION WITH TWO-TERMINAL NONLINEAR SYSTEMSSome results relating to the power absorbed by a two-terminal linear or nonlinearnetwork, as depicted in Fig. XVIII-1, have been previously presented.1 It was shownthat the average power, P, absorbed can be expressed asP = i(t) e(t) = ie(0) = 2 Re {ie (w)} dw (1)in which Re {ie(w)} is the real part of the Fourier transform of the crosscorrelation*This work was supported in part by the National Institutes of Health (Grant MP-4737).165(XVIII. STATISTICAL COMMUNICATION THEORY)function, e. (T) = i(t) e(t+T). In this report, we shall present some results for passivesystems for which Re {@ie(w)} < 0 over some range of frequencies.i(t) (t)Fig. XVIII-1. A two-terminallinear or non-linear network.Fig. XVIII-2. A nonlinear resistivecircuit.If, for some excitation, Re {(ie( )} < 0 over some range of frequencies, thenthe passive system can not be linear and it can be used as a power amplifier.For example, consider the nonlinear resistive circuit depicted in Fig. XVIII-2. Thevalue of the resistor can be expressed ash(cr) i(t-c-) do-h(oa) h(-r 2 ) i(t-o-1 ) i(t-c 2 ) do- 1 do-which is a quadratic function of the current flowing through the resistor. An-'i(t)Fig. XVIII-3. An illustrative method ofsynthesizing the system ofFig. XVIII-Z.illustrative method by which such a system can be synthesized is depicted inFig. XVIII-3. The voltage across the resistor ise(t) = R i(t)h(c-l) h(-r2 ) i(t--r 1 ) i(t-o-2) i(t) do-ld- 2 'Thus, the crosscorrelation function, ie (T), isle16600 00--00 0000 00-oo -oo[ 00R =e(1R Sh(t) SUAE(XVIII. STATISTICAL COMMUNICATION THEORY)ie (T) = e(t) i(t-T)leh(- l) h(o-2 ) i(t-- 1) i(t-o 2 ) i(t) i(t-T) do-ld 2'For simplicity, we shall consider the case for which i(t) is a white Gaussian processwhose autocorrelation function isii(T) =i(t) i(t+T)= Fo(T)Then 2i(t--l) i(t-o-2) i(t) i(t-T) = po(T) 1o(o-l- 2 )By substituting Eq. 6 in Eq. 4, we obtaini4. (T) = 0 (T) h2 (c) do- + 2h(0) h(T)-00The cross-power density spectrum is, then,1 °Sie( ) - 0-00+ L(o-) (T-- 2 ) + 2(o-Z ) 1(T-oI )h2 (a) do- +1 h(O) H(w)in which the transfer function, H(w), isH(w) =-0oh(t) e - jwt dtThus, from Eq. 8, we obtainRe ie(w)} = h2 (a) do- + h(O) Re H(H))}.ie Z 00ooTWe can make Re {ie(w)} < 0 over a range of frequencies(10)by choosing the impulseresponse, h(t), to be the first Laguerre function. That is, we leth(t) = [2pt-1] e-Pt t >0t<0(11)The network realization of this impulse response has been given by Lee. 3function, H(w), is, then,p -jH(w) = N2p pj)2(p+j)Substituting Eq. 12 in Eq. 10, we obtainThe transfer(12)167(9)-o0 -o00(XVIII. STATISTICAL COMMUNICATION THEORY)Re ie()} 1 4 p(p-32)= 2 (p2+ 2) 2We note from this equation that Re { ie()} < 0(13)for w < 0. 45 p. Such a nonlinearsystem, N, can be used as a power amplifier, as illustrated;( tin Fig. XVIII-4.Fig. XVIII-4. A power amplifier having the circuit of Fig. XVIII-2.this figure, the current, i(t), is obtained by summing the outputscurrent sources, il(t) and i 2 (t), so thati(t) = i l (t ) + i2(t )in whichil(t) = X-00of two separate(14)k l (a-) i(t-a-) do-. (15)(16)The total average power supplied to the nonlinear system, N, is, then,P 1 + 2in whichP, = e(t) il(t)andP2 = e(t) i 2 (t).By use of (15), we can express P1 in terms of ie (w) as1aeP 1 = 2 Oc Re K 1(w)ie (w)} dw. (17)168(XVIII. STATISTICAL COMMUNICATION THEORY)We thus observe that we can make P1 < 0 by making kl(t) a lowpass filter so that K1(0)Iis small for w > 0. 45 p. That is, P 1 < 0 if il(t) is a Gaussian process for which littleof its power is in the band of frequencies higher than 0. 45 p. For such a current source,the average power dissipated in the load resistor, R L , is2il(t) RL = Ps - 1in which Ps is the average power delivered by the current source, il(t). But, sinceP 1 < 0, we have Ps < i(t) RL. That is, more power is delivered to the load resistorthan is supplied by the source, il(t). Of course, the additional power is being suppliedby the other current source, i 2 (t). In this sense, we can consider the current source,i2(t ) , as the power supply and the nonlinear system, N, as a power amplifier.In this manner, any two-terminal passive nonlinear system can be used as a poweramplifier with any excitation for which Re {fie(w)} < 0 over a range of frequencies. Wealso note that if, for some excitation, the passive nonlinear system can be used as apower amplifier, then Re { ie(w)} < 0 over a range of frequencies. Thus the require-ment that for some excitation Re {(ie()} < 0 over a range of frequencies is a necessaryand sufficient condition for the passive nonlinear system to be used as a power amplifier.M. SchetzenReferences1. M. Schetzen, Power absorbed by a nonlinear two-terminal network with a whiteGaussian input, Quarterly Progress Report No. 62, Research Laboratory of Electronics,M.I.T., July 15, 1961, pp. 152-154.2. M. Schetzen, Average of the product of Gaussian variables, Quarterly ProgressReport No. 60, Research Laboratory of Electronics, M.I.T., January 15, 1961,pp. 137-141.3. Y. W. Lee, Statistical Theory of Communication (John Wiley and Sons, Inc.,New York, 1960).B. ERRATAIn my report entitled "Measurement of the Kernels of a Nonlinear System by Cross-correlation with Gaussian Non-White Inputs," published in Quarterly Progress ReportNo. 63 (pages 113-117), each lower-case x in Eqs. 12-14 should be replaced by alower-case z.M. Schetzen169

Statistical Communication Theory

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Statistical Communication Theory

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