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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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i ,1! ■ ■ ■ 7 Q I\ , m д-, — q. 1)
■Sl(^l,l) ‘ ' '’5l(fK1m1 ) ' ' '
'5Л'(0ЛМ) ‘ ‘ 4/v 1 *■- К T'\j '< ; ■ . 1 )
^'(/ — ^1,1 - ■ ' ■ - fil.jTij - • • '
• • ■ — Г£/\‘д — ■ ■ • — &К,тк ~ *-q, 1 )
<Ш1Д • • • d£li mi...................................dp.л-д ■ • -d9.K,mK dEqji (7.92)
In Equation (7.92) the sum over the g-variable means that the contributions to ?„,,(/) are determined for each noise source successively by adding all the contributions from the individual noise sources. Insertion of Equations (7.8) and (7.19) into Equation (7.92) leads to
Q M-1 Л/-1 Ji Ji Jk Jk «>
r»Af) V V . . V V v ............................................... V V v
<(=1^=0 m/c —0 j](1—1 JAM-1 kmrc=l ? —CO
£o,Af-l(ml + • • ' + mK)
I, h ;)mi ,...,771 j7—1 ,(J,...,0v ^'1 ji,i ’ ■ ■ ■ ; гц '
' ’ ’ ’ 1 - ■ ■ ’ ^ 1 )
■ ■ ' " ,\ i ) ' ' ' -'n r\\ JA'.mA- /
£(/ - f/’i.i:,:-------------
• • • — t!\,J 4|1 — • • - /v ,j/4 .rti ,, '->P/ ' *
Using the symmetry properties of the partly symmetrical frequency domain multi-port Volterra transfer function ( Hi) m , 771,-0 'jo,j=!0 o(‘) to
7.4. Responses
175
Q Л/-1 Л/-1 J, J i JK jK ao
= ££■■■£ E ■■■ E ................................E E E
(?=г1тщ=0 ТП =0 J i, j — 1 Jl.mj =1 JK, 1=1 J;:,:^y = 1 p = -'»
£о.Л/-1("Ч + •• *4*
-^l f jl.b • ■ • . il.-n.) ' ■ 'Л-K (jK.l.jK.rn;;)
ri"**! ri ri
k-l k=l jk = l
( Я,)т1 a',0,...,U,j4=1 ^1 ,j\.i - ■ ■ ■ 5 ^;1 ,j\ ,rriI ‘ '
' • ' ■ ■ - ’ 1 Vp)
) ’ ‘ ) ' ' '
■ • • sk(wk\jKi)••■ Ja(&K.jK,mK) щ(£р)
fiif-v 1.Л.1--------------------------
• — — ■ ■ • — yA'.j,ViJn;. ~ fp) i 7.94)
where / 6 {1,2,..., Л}. It is seen from Equation (7.94) that rnti[f) where I 6
{1,2,. .. yL} can be written as a Fourier series as
Tn.iij) = £ ?пЛ) Kf - ip) (7-95^
p = -eо
where
Q M M Ji J i J к J’<
,,(fP) = EE-E E ■■■ E ...................................................E E
pi ”4=0 «Ift-O лл=1 Jl.mj=l JAM=1 Л0,тл-=1
£q,a/-i(™i + ■••-!- mK)
(j i.i I ■ ■ •i\ .rr, i j ■ ■ ■ к i jk,i j x.-rn*.-)
X A JK _ ^
П •' П П {-vJtO'w...................
fc=l k-lJk=l
( I )mj tv,0,...,0.c'.j = 1.0,...10* • • • •• « ‘ - ‘
-----'-/'.JAM----------------tK.jK.mK) !7'!)6)
TIiik r i where I f 1 2, . Z-) fan •v written as
176
7. Noise in non-linear systems: Theory
Q
uAZp) = E'U«p) (7.97)
7=1
w})PfP
TiMp) = [ч,(?р,'Е,1),---,т711Др>Фе)|Г e CExl (7.98)
= [n,(^-tPi),...,n,(fp-$£:)]2' 6 CExl (7.99)
In Equation (7.97) the vector T|i?(£p) where I £ {1, 2,..., L} and q £ {1,2,.... Q} describes the non-linear conversion from the possibly modulated noise source ng((v) to the noise response r„,i((p) at port (r,l). Thus, the cross-correlation between two Fourier series coefficients r^i^p,) and рл,12(цР2) where £ {1,2, ...,£} at the two response ports (r./j) and (r,/>) can be determined as
Q Q __
Ы(01)К,12(^)) = (7.100)
where
-^71.7з(£р1 • Zp2) = {^91 (spi) nlziZp?)) (7.101)
In Equations (7.100) and (7.101), JV<jbq2(£pY,£p2) is a noise cross-correlation matrix describing the correlation between the possibly modulated noise sources {raqi(fp, -Ф1),.. .,n71(fPl - ФЕ)} and {пЯ2((Р2 - Ф1), • ■й,2(£Р2 - Фя)} applied to ports (n.qi) and (n, 52) respectively where 51, 52 6 {1, 2, • . ■, Q} at the various frequencies given. Using Equation (7.57) it can be shown that
_ EE
n41 ,«(fp.. £») = E E < (£» - ф=.)
e1 = l e2 = L
= -'5ej^(sfP2 -
(7.102)
where
d, = [di =0,.. =0,rfe= 1,4+1 = 0,...,<i£ = o]J £ {0,l}Exl
(7.103)
Thug, using Equations (7,100) a.Tid (7,101) gives
7.4. Responses
177
Q Q E E
(’WW = E E E E r^,5.(?pJ d2, ^(?P, -
91 = 1 72 — 1 ei —1 e2 = l
- Фе,,ел - *„)
x^2(sp2 - ф«) df2 T',,72(^2) (7.104)
where /j.L 6 {l,2,...,i}. Equation (7.104) can be rewritten as
Q Q E E
^J2(sCpJ) = J2YlY.Ylah,
71 — 1 72 = 1 -1=1 ej —1
- Фе,.?р2 - Ф=3)аГ21,7г.е2(?Р2:Ф,2)
(7.105)
where /(, /2 £ {1, 2,..., L]. and where
a,.qMP,Ve) = t,(£p-$e)df r,,,(£„) (7.106)
where / 6 {l,2,...,i}.gt {1,2,.... 0} and e 6 {1,2,..., E}. In Equation (7.106), al,q,e{£p, Фе) is a vector describing the transfer of the fundamental noise source {®7(£p-®e —Ф1), ■ • ■. w,(fp-$, — Фд)} where q 6 {1,2.. .., Q} at the given specified frequencies to the noise response r„,;(sp) at port (r,l) where I £ {1,2,. .., L].
7.4.3 Total response
The Fourier series coefficient for the response at port- (i\l) where / £ {1, 2,. .., £} at frequency £p where p в Z is given by
n( ip) = + 'n.K'sp) (7.107)
where rjj((p) is given by Equations (7.88)-(7.91), and rn i(£p) is given by Equation
(7.97). Thus, the cross-correlation between two arbitrary Fourier series coefficients at arbitrary ports and frequencies can be determined as
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