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Theory of Linear and nonlinear circuits - Engberg J.

Engberg J. Theory of Linear and nonlinear circuits - Wiley & sons , 1995. - 154 p.
ISBN 0-47-94825
Download (direct link): noisetheoryoflinearandnonlinear1995.pdf
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7.3.3 Modulated noise source
The possibly modulated noise source n,.(f) where q £ {1,2, ...,Q} is given as the response from a non-linear noise free system not containing internal sources with the deterministic modulating input signals {u?iI(/),. - ., u,,/ (/)} and a fundamental (unmodulated) noise source wq(f). Insertion of Equation (7.40) into Equation (7.24) leads to
Л/ — 1 M — 1 oc
»,(/) = E E E
77ll=0 ТП/ =0p = — OC
M-l E»I.l M-1 Б°1,т, М-1 Е”ЦЛ Xf— I
E E E E ........... E E ••• E E
"1.1 = 1 4,1=1 oi.m, =1 =1 »/,,i=l'/,.i=l j/,,m; =l =1
А.Л/—l(°l,l T • ’ • + 01,mi +.............+ 0/,,l + ■ • ■ + )
(C',))l,m1,.-,m,1 (^p'i ®oi j ,= i,i i ■ ■ • , mi ' ' '
" ' > Ф%,I’ = /„!’ ' ■ Ф =
”1 .ei.i ) " ^ uq.\( )
■ ■ • Ufjj„{^or ,.c., ) • • йПш[„{'Эг _ )
• ‘V-' 'V 1 -i.-, -Ч’--'! j
$( f С — \lf — ... — ш ......
L \J sp k 01,1 .<!; : ''i.mj 1<?1
-----Ф -------Ф ,, r ) (7.16)
Note from Equation (7.46) that there are infinitely many sum terms in the expression for fiqi f)• This is due to the fact that the fundamental (unmodulated) noise source w.j(f) where q e {1.2,.. ..Q} is represented is a Fourier series at an infinite number
7.3. Noise sources
16.3
of frequencies. The effects of the modulating signals {uql(f),..., u,h / (/)} where q £ {1.2are (among other things) to frequency shift the noise component corresponding to a given frequency, and to change (modulate) the amplitude of the fundamental noise source component. The computational cost of determining nq(J) where q £ {I, '2,. . .. Q} in Equation (7.46) can be reduced significantly by using the symmetry properties of the partly symmetrical multi-port frequency domain Volterra transfer function (Gg)iimii...im;.4(-) which leads to
Л/— 1 Л/ — 1 oo
niU) = £ ••• £ E
rn i=0 m/(J=0p = ~'X
M-1 E°l,i A/-1
E E - E E -
01.1=1 «1,1 — 1 Jj.rnj —1 «1.ГП!
Л/-1 £°V 1 Л/-1
- E E - E E
^l,A/-l(°l,l + "' + <>1,711! +.........+ %.! + ' ■ • + )
•4l(0l,l, . . .,0i,mt) • ■ ' Л/,(0/„1, • • .,0/„m,4)
•4i(ei.i>- ■ •. ei,mi) • • -Af4{eiq
П mu'
;,=1
I; Л/-1 E°'<t _
П П П {л^.,.e,, (Oi',1, • . • , ; e;,д , . . . , e,)! J
*'t=l = l e.^ = l
(^-7)1,7711.('fpi ^Oi.bS!,’. ) " ’ " J ^Jl.m] .fl.mj 1 ‘ ’
......^o.”V7-Vm,J
W,(£p) <’,>1 «7,l(«o,«7.1(Фо,.т, )”•
5.Vi 77 »Т/ \ Z . 11Г1
,S ^ A' — £ — \[/ _ . . . _ 1[Г _ . . .
~ ''У - '-'1.11=1,1 .ГГ1 [ >“1 .mj
.....Ф,. . . ■ • ■ ■ Ф,. .. 'i
where
'V>1f7,eIf7(°^a , • • ■ , Oi4.:ntq> 6s'7.b---)
= Number of {(o,,.i. с;,|Д)..................which
164
7. Noise in non-linear systems: Theory
and and ■ ■ ■, ) are defined by Equation (7.28).
In Equation (7.46), £o,m-i{') used with a different argument compared to Equation (7.24). This is because otherwise there would be contributions of a total order greater than M. This is avoided by using Equation (7.46). Thus, from Equation (7.46) it is seen that the (possibly) modulated noise source nt{f) where <7 6 {1,2, can be written as the Fourier series in Equation (7.9) with some
rather complicated Fourier series coefficients n,(fp) where p t 2 is an integer. However, this requires that
(7.49)
for any integer p £ Z, o; / £ {1,2— 1}, and e; / G {1,2,---------------------E0. ,} where
г, £ {1,2,...,/,} and I G {1,2,. .ттг,- }. Equivalently, the requirement in Equation
(7.49) can be expressed as
7.50)
for any integer p G Z, o,tjj £ {1.2,...,M - 1}, £ {1,2,. .., E, ,} where
i„ £ {1,2,...,/,} and I P {1,2,--------rn, }. Tlie fulfilment of this requirement is
of no concern since the frequency resolution in the Fourier series representation f = l/(2r) can be made arbitrarily small by choosing r sufficiently large (2r is the time interval of observation). For example, if ail applied frequencies are integers then 2r = 1 fulfils the equivalent requirements in Equations (7.49) and (7.50). Assuming that the requirements in Equations (7.49) and (7.50) are fulfilled, then the coefficient ng(£p) where q 6 {1, 2,. .., Q} in the Fourier series representation of «,(/} in Equation (7.47) can be determined as
7.3. Noise sources
165
M-1 Л/-1
Mtp) = E E
mi sO m j j —0
Л/-1 Л/-1
E E E E'
131,1—161,1=1 ^1,ггг, =1 ег.лц =1
,Vf-l ■Б°1,д ,Vf_l E"!4-mlq
- E E E E
=/,.1=1
£i,a/_ifoij -i------------------------------------------r Oi.m, +•.+ 0/„i +-h £>/„,„, )
A(»l.b--------Ol.m, ! • ' •Al'ioi.'.l,--)
A(ei,i,..., fii,mi) ■ ■ -Д/т(е/,.1, ■. •, efT,m, )
^4
Д "S! tq —1
A, Af-1
II П П ----; «i„l, • • •, е,-„т,? )! |
;,=i o,,=i e,?=i
I-1
^1,171] .^1 ,1711
■- Ф0
Ф:,
°/fl,l-e/q.l -'1q>7nIq'''tq*mIq
ф Ф
X 01,1 r«l,l ’ • • • » *01
“4^ ~ Ф»1,«М
Jl ,mi .ei ,m
«,д(Ф
1^1 , !
'«••И
Г.51)
For the noise response it is not of interest directly to operate on the exDtession for n„(f) but rather on the Fourier series coefficient oi 0(0; in the expression for !77(/). This is because (i) the noise .epieseulaiion has a. [practically — due to a. very small f) continuos frequency spectrum, and (ii) only iow level noise is included in the analysis, in which case it is not necessary to consider contributions caused by non-linear mixing of noise with noise. In Equation (7.51) a part of the argument frequency to i?7(-) is
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