# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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7.3.1 Basic theory

Noise sources are generally included as in Figure 7.3 in which the possibly modulated (dependent) noise source nq(f) is identical with the response from a non-linear noise free subsystem not containing dc sources with inputs {пчЛ{ f).. . . ,uqj4yf)}. which are referred to as controlling variables applied at ports {(>i7, 1)...., (u,q. [a)}. and a fundamental (unmodulated, independent) noise source wq(f) where q £ {1,2,..., Q}. The non-linear noise free multi-port system in Figure T.3 must not contain dc sources. However, the controlling variables {u,,i(/),..u7./,(/)} may have a dc value, and also the fundamental noise source wq(t) where q t {1, 2,..., Q} may

have a non-zero mean value. The controUing variable where q £ {1.2.....Q j

and iq £ {1,2,.may be either an open circuit voltage or a short circuit current (or actually any system variable) at any node or branch respectively in the underlying network of the non-linear system. The non-linear system in Figure 7.3 is described by a multi-port frequency domain Volterra series j'j with inputs ubi(f),_____________%/,(/)} and output nq(f) where q t {1,2,...,Q} as

152

7. Noise in non-linear systems: Theory

Figure 7.3: Representation of modulated (dependent) and unmodulated (independent) noise sources. The input signals {u9ji(/), • • •, where q £ {1,2, are con-

trolling variables at ports {(uq,l),... ,{uq, Iq)} — deterministic currents and voltages at any place in the non-linear noisy network. The signal wq{f) where q 6 {1,2,...,Q} is a fundamental (unmodulated) noise voltage or current source, and nq(f) is the (possibly) modulated noise source.

7.3. Noise sources

153

м м м

*</> = EE-E

mo=Omi=0 0

г со r oo лсо roc л со f CO

У—CO j—OO J—OO J— 'УЭ j— 'У0 J—CO

A ,;V/(mG + miT-" + m/J (G,) Я1П.ТП1 m/^

f”0.ii • i ^o.mn; Г2] j,..., Г21 ]m,; ■ ■ ■

' ' ' I ^/,,1 ! ' • • I Qlq,m,q)

Mq(Qorl ) С ^ 1,1 ^ ) ' ‘ '

' ’ 'Uq,Iq{^Iq,l) ' ' ' UQ,Iq(^lvml4)

Ш ~ Под - ■ ■ ■ — Ooi77LO - nlfi — • ■ ■ — •

----------П/„т/,)

dQo i • • • d£lq[771c dilij • • ■ (Шl m] • • •

■■■dQ.iq,l---dSlIq<rni (7.19j

where

, . f 1 for '/£ {e.a + l,...,3 - 1,3}

£-a.p{l) = 1 n ■ 7.20}

[ 0 otherwise

and (G,)m0imi....mi (•) is the partly symmetrical multi-port frequency domain Volter-

ra transfer function relating the inputs {w.JJ), uql(f),..., u,,/ (/)} to the output nq(f) of order mo + тг + ■ ■ • + mjq. In Equation (7.19) the function £i,.v/(-) limits the order rao + mi + ■ • ■ + mj to be from 1 to M. Thus, M is the order of truncation of the multi-port Volterra series.

A partly symmetrical multi-port frequency domain Volterra transfer function is defined as follows. Let Vkjik {S*,i, ■ • • } denote permutation lk £ {1,2,.... Of;l}

of the o.i! total number of permutations of the frequency variables {Н^л ,. .. where к 6 {1, 2,..., A'}. Then, a multi-port frequency domain Volterra transfer function (Fq: )0l...o-v(') is said to be partly symmetrica! if

.,ил'(,-'1.1' ■ ■ ■ * —•..................................■ —^.i’ ■ ■ ■ ?

_ ip л i-n . f— — >I •.........................-n

Г 7.21)

for all the ojI-'-oa'! possible permutations of the variables. An imsymmetrical Volterra transfer function lrJ]0l.r i ■ I can be made partlv symmetrical аь

154

7. Noise in non-linear systems: Theory

(7.22)

All permutations {Pk,i {'}>•••, 'Pk,mfc!{-}} are required to be different for all к £ {1,2,.. ., ft'} to assure (partial) symmetry of (F,,;

Note that the assumption of partly symmetrical transfer functions is not a loss of generality, since the response n,.(f) where q £ {1, 2,..., Q} in Equation (7.19) remains the same if a possibly unsymmetrical frequency domain Volterra transfer function is substituted with the corresponding partly symmetrical transfer function. Note also from Equation (7.19) that there may be several different (<37)m0imii..,im/ (-) multi-port frequency domain Volterra transfer functions of the same order m0 + "ii + • ■ ■ + rn/ . Observe from Equation (7.19) that even if (G7)m0imi,...,m, (•) where q £ {1,2, ...,(2} is unsymmetrical then any variables {fi,,,!,..., } where

к t {1,2,..., A'} may be permuted and one still obtains the same result for nq(f). The reason for using partly symmetrical Volterra transfer functions is that the amount of computations required to determine the response may be significantly reduced when the transfer functions are symmetrical compared to when the transfer functions are unsymmetrical.

Full symmetry of a multi-port Volterra transfer function can generally not be utilized. A fully symmetrical Volterra transfer function is defined by

(F.........Or,;( —1,1 ' ■ ‘ • , —1 ,.' , “Я .1 , ■ • ■ , “Л ,0^- )

= (f»,>,)oi,...,oK(7:H-i,b • ■ ■, -t.3,;....; , • ■ •, ~k,o;,-}) (, ■ -2.3)

where q 6 {1,2,. . - ,Q}, and Vi{-} denotes permutation number I e {l,2,...,(oj + + 0fr)!} of the frequency variables. The full symmetry in Equation (7.23) is generally not fulfilled since the signals are applied at A' possibly different input excitation ports. Thus, two signals from two different input ports may generally not be inter changed Vo give the same output response. Tb.ereicne, avuUi-poi't Volterra. transfer functions are generally not fully symmetrica!, and a fully symmetrical multiport Volterra transfer function can not be substituted for the possibly unsymmetrical transfer function in Equation (1 19)

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