Schetzen, M.

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VIII. STATISTICAL COMMUNICATION THEORYProf. Y. W. Lee Prof. M. Schetzen D. F. De Long, Jr.Prof. A. G. Bose Prof. H. L. Van Trees, Jr. C. L. FeldmanProf. I. M. Jacobs V. R. Algazi H. HemamiProf. D. J. Sakrison W. S. WidnallA. MEASUREMENT OF THE KERNELS OF A NONLINEAR SYSTEM BYCROSSCORRELATION WITH GAUSSIAN NON-WHITE INPUTSIn the Wiener theory a nonlinear system is characterized by the kernels, hn , of thefunctionals, G .1 For a Gaussian white-noise input, the functionals, G , are orthogonaland their kernels, hn, can be determined by crosscorrelating the output with a multi-dimensional delay of the input.2 In this report a procedure is developed for determiningthe kernels of a set of functionals which are orthogonal for a Gaussian non-white inputprocess.z(t) y(t) Fig. VIII-1. Nonlinear system with Gaussian non-whitenoise input.Consider that the system N shown in Fig. VIII-1 is to be characterized with aninput, z(t), which is a Gaussian non-white process. We shall assume that the powerdensity spectrum, zz(w), of the input, z(t), is factorizable. 3 It then can be written+-: z) + (W)  ( (W)1)zz zz zzin which +zz(w) is the complex conjugate of zz(); also, all of the poles and zeros of+ ) are zz++Szz(w) are in the upper half of the complex X-plane. Thus zz ( ) and I/,zz(w) are eachAr---'-~ -- -- 1-z(t) x(t) z(t) Y(t)L- ------- jFig. VIII-2. An equivalent form of the nonlinear system N.realizable as the transfer function of a linear system. We can then consider the systemof Fig. VIII-1 in the equivalent form shown in Fig. VIII-2, in which the transfer functionsof the two linear systems, kl(t) and k2 (t), are:113(VIII. STATISTICAL COMMUNICATION THEORY)K1  +I (WS( ) (2)Kz() D+z(Wo)2 zzAlso, as shown, the system A is the system formed by the tandem connection of thelinear system k2 (t) and the system N. We observe that the input to the system A is aGaussian white process whose power density spectrum is 1 watt per radian per second.Thus, as previously described, 2 the kernels, hn , of the orthogonal functionals, G , forthe system A are(zw)nhn(T '... Tn) n! y(t) x(t--T1 ) ... x(t-T) (3)except when, for n > 2, two or more T' s are equal.We therefore need to know the crosscorrelation functionS n(T 1' ' Tn ) = y(t) x(t-T ) ... x(t-Tn) (4)yxin order to determine the kernels, hn , of the system A. Since only z(t) is available tous, we shall express the desired correlation function in terms of the corsscorrelationbetween the output, y(t), and a multidimensional delay of the input, z(t). By substitutingthe relation0ox(t) = 0 k l ( r ) z(t-c) do" (5)in Eq. 4, the desired correlation function can be expressed aso00yxn (T1 T.. n) =  kl 1) d l  k 1 (n) d n  n ( T  1 n-n) (6)in whichn (T1' .. Tn ) = y(t) z(t+T1) ... z(t+Tn) (7)yzis the crosscorrelation between the output, y(t), and a multidimensional delay of theinput, z(t). In the frequency domain, Eq. 6 can be expressed asn(1o . 'wn) = K 1(w1 ) K 1(w2 ) ... Kl(wn) " 5 n( 1" . ' n) (8)in which114(VIII. STATISTICAL COMMUNICATION THEORY)1 _o-jc1 T1 noo-jon-nn1 (W... n ) e dT ... e d n 4 n( T... T n )  (9)yz (2r)n  oo -00 yzand the transfer function Kl(w) is given byKl(w) = kl(t) e-jt dt (10)-00Substituting Eq. 2 in Eq. 8, we have the desired relation in the frequency domain.n(1'...'''.. nyzSyxn(it . On) = )(wi +() (11)yx :n. +(zz 1 zz nEither Eq. 6 or Eq. 11 can be used to determine the kernels, hn , as given by Eq. 3 interms of the measured crosscorrelation function between the output and a multidimen-sional delay of the input, z(t).Once the kernels hn have been determined, a representation of the system N is asgiven in Fig. VIII-3a which can be redrawn as shown in Fig. VIII-3b. We note fromFig. VIII-3b that the outputs of the parallel branches are orthogonal for the input z(t).Thus, we have expanded the nonlinear system N in a set of functionals that are orthog-onal for Gaussian inputs with a power density spectrum of 2zz(w). Note that for thisprocedure, we never need construct either of the linear systems kl(t) or k2 (t). Oncethe representation in the form of Fig. VIII-3b is known, the functionals that are orthog-onal for the input z(t) can be calculated. We shall call these functionals L . The firstnfew functionals are:Ll[h,,z(t)] = 5 hl(rl)kI(T1 -To1 )X(t-T 1 ) d-IdTI  (12)00 00 00LZ[h 2 , z(t)] = hZ(CI , ,r 2 ) k 1(T 1 -r 1) k 1 (-) x(t-T 1 ) x(t-T2 )dr Idr 2d dTd20000 -00 00- 21 h 2 ( 2 , 2 ) d, 2  (13)-00L3[h3zlt). = "'" ~h3("l' " ' "3 ) kl (Tl-l) kl(T- 2) kl(3-3 ) x(t-TI) x(t--T2) xlt-T3 )-00 00d 1dcT2dC 3 d dTzdT3- 3(2w) 5 5 h3 ( 1 ' 2' 2) kl(T -1 ) x(t-T1 ) d IdT2dT 1  (14)00 -00 -00115r-I G , o- h, - -)IG, [h ,1]0 I y(t)I -( oay(t)z (t)0 00o0 0(b)Fig. VIII-3. Representation of the expansion for system N.116z (t)(VIII. STATISTICAL COMMUNICATION THEORY)in whichkl(t) =-- jwt dw. (15)1 (too (u)zzIn terms of these functionals, the output of the system N for an input z(t) can be writtenooy(t) = I Ln[h n , z(t)]. (16)n= 1In this manner, we have characterized the nonlinear system N with a Gaussiannon-white input in terms of a set of functionals, Ln , that are orthogonal for the giveninput process.M. SchetzenReferences1. N. Wiener, Nonlinear Problems in Random Theory (The Technology Press of theMassachusetts Institute of Technology, Cambridge, Mass., and John Wiley and Sons,Inc., New York, 1958).2. Y. W. Lee and M. Schetzen, Measurement of the kernels of a nonlinear systemby crosscorrelation, Quarterly Progress Report No. 60, Research Laboratory of Elec-tronics, M.I.T., January 15, 1961, pp. 118-130.3. Y. W. Lee, Statistical Theory of Communication (John Wiley and Sons, Inc.,New York, 1960).117

Statistical Communication Theory

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Statistical Communication Theory

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