# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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to the output Ti(j). The response r;(f) where I £ {1,2,..., L} may conveniently be separated into two contributions: (ij a purely deterministic contribution rjj(f), and (ii) a noise contribution rnj(f) such that

n(/) = rd.iif) + rnAf) (7.81)

where I £ {1,2,...,X}. The deterministic response I'ijif) consists of the linear transfer of input signals to the output port, (r.l) where / £ {1/2,.... L] and of all intermodulation contributions of the applied deterministic input signals up to order M. The noise response |\Д/) consists of the linear transfer ot the noise sources

to the output port (r.l) where I - {1,2...........[.) and of all non-linear transfers due

to inter modulation between deterministic signals and noise. (_ ontnbutious lee ‘o mixing of noise with noise are not included due to the low-level noise assumption.

7.4.1 JJeterministic response

The deterministic response rjj(f) at port (r, /) where I £ {1.2.....A,} can be determined from Equation i i .30) With ■>’ — ■ - - oq — 1) <is

172

7. Noise in non-linear systems: Theory

м и

гчлл = £ ••■ E

mj =0 m/v-=0

с СО Г ОО ЛОО f со

-со у — ос J — оо J — оо

Г- -L___________L т.Л

"—11 ,jw V, "ь I 1 1 ' t\ )

(Hl)m1 к,0,...,0(^1,11 • ■ • ? ^ 1 ,mi 1............, ^ К , 1 > • ■ ■ ?

*’l (^1,1) ' ' ' -^1 (^1 ,mi ).....А’ЛЧ^АМ) ' ' ' ^ А'{ ‘,шл- )

S (/ ^1,1 '*■ ^1,771! ....... Пк,1 ,m^- )

(Ш,д ■ ■ .......dQh',1 ■ ■■d&K,mK (7.82)

The deterministic response r,;/(/) where / £ {1,2, ...,£} is identical with the response r;(/) when all noise generators are quiet, i.e. nx(}) = ■■■ = nq(f) = 0. Thus, using Equation (7.8) for Sk(f) in Equation (7.82) leads to the determination of the contribution rdj(f) where / £ {1,2, as

JV/ \! J i J [ ^/v

rd,i(i) = £ •■■ £ E ••• E .......£ E

7711 —0 mj(=0 jl,l=l i7\,mA-=l

£l,<vr(ml H---------1- mA')

(-^/)mi 1...,m^-,0,...1o(Vl,ji i , • ■ ■ , ^l.ji.in, i...’ ^XJk.l ’ ‘ ' ' ’ )

«i(V'iji.i) • ••«i(^iji,mi).............sK(i>K,JKA)- ■ -ЫФк^к)

KS - V-'i.ji,,------------.....................- ------------ш,ж,тк)

(7.83)

Using the symmetry properties of the partly symmetrical multi-port frequency domain Volterra transfer function leads to

rdj

M M J\ Ji ,/A- Jrv-

E - E E -.£ E •• E

mi =0 m/4-=0 _7l, 1 — 1 J1,mi — A JAT,1—1 JK.mj,-

-Сг.лД771! + • • • 1 + mK)

'4l(jl,b • • m7i ,ml ) ’ ’ " К ,1 7 • • • ? JK,miK-)

П-

.. m г.- I) oi 'Ф]. м ^ ’i ; • • •

,si ( vi ,,,}■•• >h ! ,n ,! ■ • •

■ •' sk(vk,]Ki1 )■■■ зк(Фк.]К,тк)

^(/ — Wl .J1 ,1 — • • • “ ■

(7.84)

7.4. Responses

173

where Ak(jk,i, ■ ■ ■, jk,mk) is defined by Equation (7.28) and jVJk(jkil______________,jk.mk) is

defined by Equation (7.29). Equation (7.84) can be rewritten as

с

r.-i it f) = rjj( v_.) Л’(/ - vO (7.85)

C=1

where

* = {r(Po)| Po £ {0,1, 2-■ ■ • • +'"+Ja'x 1

Л llPoll £ {1.2,.... A/} J (7.86)

= {.Yi,---,Xc} (7.87)

and tl’(p0) and pa are given by Equations (7.32) and (7.33) respectively. The frequencies in the set X are organized such that \\ Ф • r Xc ■ The Fourier series coefficients РлДхс) where / £ {1, 2... ., 1} and с £ {1, 2,..., С} may be determined from Equation (7.84).

Note from Equations (7.30), (7.60) and (7.86) that Q X. The only

case where = X is if and only if фк^к = 0 for all к £ {1,2.. .., A'} and

jk £ {1,2,..., .7^}. This corresponds to both frequency sets consisting of dc and harmonics of dc (which is also located at dc).

In some cases it is convenient to write the deterministic response at port (r, /) where I £ {1,2,..., L] as

OO

n.i(J) = £ nAS,)bU-Zr) (7.88)

p=-co

In this case the deterministic signal is represented at the same frequencies as the

noise signals. However, for Equation ( 7.88) to be correct it must be that

3 p £ 2: H-----1- +......+- ФЫкл + "•+ I ”89)

for all mi,..., ток £ {0,1...., A/}, mi+- • -Fm;s- £ {1, 2,. .., A/}, jkj £ {1,2,..., for A- £ {1,2,..., /t'} and / £ {1,2,. .., mt}. Equivalently, the requirement is

с ! ' ' т ■ < VI'-Jka > ■ 'rh.JK,„K\ t “ V"'u/

Since £ = l/(2r) can be chosen arbitrarily small it is always possible to choose a ? such that the requirements in Equations ;7.Ss)) and (7.90) are fulfilled. Tlie formulation in Equation (7.88) means that

rd,t(Zr) = 0 for sPf^ (7.91)

and 7ц[^р) may be different from 0 if qP £ .V. where X is given by Equation ! 7.86).

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7. Noise in поп-linear systems: Theory

7.4.2 Noise response

The contribution to r;(/) where / {l,2.....£} due to the noise sources, rn>;(/), can be determined from Equation (7.80) as

Q 1 |V/ 1 /"OO r-yrj roo ГСО

v;(7) - E £ ■■■ E i: L......................................./.■■■/./„

<7 — 1 rni=0 тд-=0 ^ '4'

^o,,v/-i('ni + • • ■ + тк)

(Hi )m .Tn/v',U....,0,Oq = l,0.0 (^ 1,1 > ■ ■ • i ,

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