# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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The chapter is organized as follows. Section 9.2 gives some preliminaries regarding multi-port Volterra series where the symmetry properties of the transfer functions are analyzed, and a modification of the probing method to allow commensurate frequencies is presented. In section. 9.3 the theoretical formulation for the determination of multi-port Volterra transfer functions is presented. Section 9.4 discusses the implementation of the method in a symbolic programming language on a digital computer. Section 9.5 presents some types of time domain multi-port Volterra kernels and the corresponding frequency domain transfer functions. In section 9.6 three examples are presented to illustrate the method. Finally, section 9.7 presents some concluding remarks.

9.2 Preliminaries

The frequency domain response v(f) at a frequency / from a non-linear multi-port Volterra system (a system which can be described by a convergent Volterra series) with Ii input ports and input signals {-з-ц/)...., ■злЧ/)} is given by

'In Chua and Ng [7] a set of frequencies {П...,П„} is called incommensurate (and a fte-nuencv base ', if there does not exist a set of rational numbers {ri, , rm (not ail zero) such that

r, O, 4- ... 4- r„ 0_ — q, Th|s >s indeed a sufficient but actually not a necessarv condition for the conventional probing method to be vaiid. it can be shown that a sufficient anc! necessary condition for the conventional probing method to be valid is that there does not exist a set ol integer numbers , у— where -fl -*-16 Zo + (nnt all zero) such that -/iП; -f* • • • -f- qmV-rn — 0- t'or

example, the Chua and Ng condition says that the probing method can not be used for the set of frequencies {f2i = 4,(^2 = 7} since rjfii + Г2П2 = 0 for {ri = —1/4, Г2 = 1/7}. According to the sufficient and necessary condition, the probing method can be used for the set of frequencies {fii = 4, fh = 7} sincc gifti 4- ф 0 for all integer numbers <ji, q? where q\ 4- 1, q? 4- 1 -Zo + except for (71 — qi — 0. This is also confirmed from Equations f9.9) and (3.10) m section 0.2.

9.2. Preliminaries

217

oo

Ci.oo(!|m|])

mi=Q тд'=0 °°

^ тп i ,..., m rv-1 /1,1 ? • • • i J 1, rn i i.* f К, 1 > • • • • /к, m i. - )

•*l(/l,l) ' ' ••sl(/l,m1)...................$ К ( fК ,1 ) ' ■ ‘$1\(/к.тк)

/1,1 ' * ‘ /l,771.1 . /am ■ ' ' /л',пя.

^/*1,1 * ’ ‘ 1,^M....................'i-fК' ' '^/к,тр-

(9.1)

where

m

[ть...,тА-]7' 6 2(f+xl

(9.2)

_ J 1 for 7 fc {а, а + 1, а + 2,. . . ,i3 - 1. ii}

t 0 otherwise

and j| • j[ denotes the sum of all elements in the given vector, i.e.

(9.3)

l T • • ■ T ТПК

(9.4)

In Equation (9.1) the quantity 'Hmi..mAr(-) is a (possibly) unsymmetrical multi-

port frequency domain Volterra transfer function of order |lmj| and £(■) is the Dirac ci-function [8]. In Equation (9.1) the zeroth order contribution corresponding to ||m[| = 0 has been excluded due to the factor /^.^(ЦтЦ). This contribution, which is usually not of interest, is identical to the response v(f) when no input signals are applied (the contribution may be called a mathematical offset). As pointed out in the previous section, this is no restriction in the noise analysis since the internal sources in the non-linear systems are applied at external ports.

9.2.1 Symmetry properties

For one-port Volterra systems ( К = 1) it is usually assumed, without loss of generality, that the one-port Volterra transfer function fU, f/i,i, ■ • ■, fi.mi) is symmetrical. If the transfer function is not symmetrical it, can be replaced by times the sum of all permutations of the unsymmetrical transfer function. In this wav a symmetrical one-port transfer function can be obtained. This transfer function can directly be substituted with the given (and possibly unsymmetrical) transfer function in Equation (9.1).

For multi-port Volterra systems this approach for obtaining a symmetrical multiport Volterra transfer function is not directly possible. As seen from Equation (9.1) the response v(f) remains the same for any arbitrary permutation of the transfer function ..../l.m,:......:/км-----With respect to

218

9. Multi-port Volterra transfer functions

variables from the same set { fk l...., fk,mk } where к {1, 2,..., К]. This means, for example, that two signals may generally be interchanged to yield the same response г>(/) unless they enter the system through two different input ports. Thus a partly symmetrical Volterra transfer function, indicated by Sym, can be obtained as

where Vkjk {fk.i. ■ ■ ■, fk,mk) indicates permutation number 4 fc {1, 2, . .mi..!} of {Л.1, •••>/*.!»*} for к E {1,2,..K}. All permutations Vk,i {•},•■■, 'Pk, mt!{'l are required to be different for all A: {1,2, ...,A'} to assure (partial) symmetry of ЯШ1....„,..(■). The symmetrical transfer function Hmi..тк(') can directly be substituted for the corresponding (possibly) unsymmetrical transfer function Hmi...mj,.( •)

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