# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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in Equation (9.1).

It should be noted that the (possibly) unsymmetrical Volterra transfer function 'Hmi...mA-(0 is not unique in the sense that there may be several distinct trans-

fer functions which give the same response v(f) in Equation (9.1) for the same input. However, the (partly) symmetrical multi-port Volterra transfer function Hmi....mA.(-) is unique since

for all permutations 'Pk,!k{fk,i, • ■ •, fk,mk} where к e {1, 2,. .., A'}, and Ik 6 {1,2,...,m(!}. The use of partly symmetrical multi-port Volterra transfer functions, and not (possibly) unsymmetrical Volterra transfer functions, implies

soidal or complex exponential inputs may be substantially reduced by exploiting the symmetry properties to avoid recalculating contributions which are just permutations of variables from the same set.

9.2.2 Determination of transfer functions

The time domain signal applied at input port к fc {1,2,....Л'} is given by a sain of I-K complex exponentials:

12 mi , • - • , fl ,m; ■......> IK, 1, • • • , J К,тп^ )

H,

тп 1

v'(/l.l, • • • , /l,mi ,........i /а', • • • , /к',тд- )

that the amount of computations needed to determine the response v{j) to sinu-

9.2. Preliminaries

219

h

Sk(t) = Y, ехр[7'(2-г'л4,,*г + (9.7)

»A = I

Thus sk(f) = = f™Msk(t) exp{-j?~ft} dt where 'F {} denotes the (Inte-

gral) Fourier transform is given by

h

sk(f) = £ «Ptf &.«'*! *(/ - ФкЛь) (9.3)

— i

Insertion of Equation (9.8) into (9.1) leads to

со со l\ 11 Ik ik

-(f) = E ••• E E - E ...............E • E

"4=0 mK-0 ,1,1 = 1 !i.m, =1 4,-.i=l >Ктк=1

A.ocUMI)

Иггц...‘ ■ ■ • > i........• , • • - »‘ФкЛгс,тК ^

It should be noted that in Equation (9.9) it is generally not possible to reduce the amount of computations required by permutation of the variables, e.g. 2l jb .. ., il mi, since the multi-port Volterra transfer function Hmi...mK(') is generally not part-

ly symmetrical in the sense described earlier. Choosing Ik — mk for all k 6

{1,2,..., A’} a partly symmetrical multi-port Volterra transfer function ...

can be determined as

1----- Фк.ТПгА

nil'. ■ • • mK-

X | coefficient of exp

К тпь

j E E °ь.

«(/

К 7/г t. .

^ >i’u ) in v{ f)\

1,9.10)

for ||m|! 6 {1,2,..., oo} and provided that the set of phases ____<31>m......

• • • ? rK,i5 • • •) is chosen as а phase base up to order j]mjj as defined below.

Definition 9.1 A set of phases -(Oi.i...... ........ PAM- ■ ■ ■ • is de-

fined as a phase base up to order и •£+ if for positive (zero included) integers

{?1,1, * ■ • , 91,774 .Як, 1, - ■ ■ > 4K,mK}

К mk К 771(5

£qkj = ^k'1 k=[ 1=] ,<=[ 1=1

(0.11)

220

9. Multi-port Volterra transfer functions

only for qk,i = 1 for all к 6 {1,2,...,A'} and I £ {1,2,..., mt,} with the

restriction that Чк,: £ {0,1, 2,. . .,,«}.

When Equation (9.10) is used to determine the multi-port frequency domain Volter-

Г 2, t Г H.riS fer ^nCtlOR it V/’Ч l-\ о могассоп/ fn ^rtnollv rlinneo cnonfir vi] hoc

the phase base. Only the properties which are associated with {<?i,i, ■ •.. <?i ,m[, • • • ..., фк,\• ■ ■ ■, Фк,тК} being a phase base will be used in the following. Note that

if {01,1 > ■ • ■ i Л.Л11 >.i Фк,1, ■ ■ ■, Фк,тк} is a phase base then <pkj ф 0 for all

k £ {1, 2,. .., ft’} and I 6 {1,2,..., mj.

Note that Ik = mi for all k £ {1,2,..., A'} is used in the derivation of Equation

(9.10). If Ik < for one or more к £ {1,2, ...,A'} then at least one frequency

in {Фк,11 ■ ■ - ,Фк,тк} appears twice in the argument to 'Hmi<...,mK{')- In this case Equation (9.10) does not follow since frequencies in the set { Фкл, ■ ■ ■, Фк.тк} f°r the symmetrical are n°t different. This is necessary to obtain a gen-

erally correct Jfmil...,mK(') independent of the argument frequencies. If

for к £ {1,2,..., A'} then Equation (9.10) can be derived. However, if Д. > mк for к £ {1,2,..., A'} then the determination of v(f) in Equation (9.9) requires more calculations than are actually necessary. Thus, choosing = m-K for al-l к £ {1,2, ...,K} is the proper choice for the analysis which leads to Equation

(9.10).

The conventional probing method to determine one-port Volterra transfer functions uses only a sum of (5-functions and no phases in the probing signal [3,6]. This situation is a special case of Equation (9.3) when фк,гк = 0 for all к £ {1, 2,..., A'} and ih £ {1, 2,..., Д}, and precludes the situation of commensurate frequencies as noted by Chua and Ng [6].

9.3 Theory

In Figure 9.1 the overall non-linear network under consideration is shown. The non-linear network is separated into a purely linear network and Q pureiv nonlinear subsystems.2 Non-linear subsystem number q £ {1,2,... .Q} is controlled by a number of P(q) variables. There are R controlling variables for the overall non-linear network represented by {x\{xMf)} ■ Thus P(q) 6 {1,2, ...,-R} for all q £ {1, 2,. ... Q \. There are Q controlled variables, identical to the number of non-linear subsystems, for the overall non-linear network represented by {Vil}),. .., yoif) }■ Thus there are not necessarily equally many controlling as controlled variables. For example, one controlling variable may control several controlled variables, and one controlled variable may be controlled by several control-

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