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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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2, there are uou-zero noise contributions to the response caused by a total order of
4. The two noise sources wi and w2 are zero mean white noise Gaussian processes with standard deviations a\ and a2. The continuous time autocorrelations for the two fundamental noise sources u5i(fp) and «^(fp) are given by
where q £ {1,2}. The cross-correlation between two Fourier series coefficients for the two fundamental noise sources can be shown to be (wi(fPl) щЦР2)) = 0 for all Pi>?2 с 2. The frequency sets So, Si, S2 and S3 can be determined as
amplitude modulated by the controlling variable ui,i(})- The non-linear element is a volt age/current (non-linear resistance) element given by
v(i) = c(i(t),t)
(.8.103)
So = {«0,1} , E0 - 1 = {0}
(8.104)
(8.105)
(8.106)
(8.107)
S2 — {«2,1, «'2.2) «2,3} ? E2 — 3
= {2Ju, —2^ii1; 0}
(8.108)
(8.109)
S3 = {«3,1, «3,2, «3,3, «3,1} - E3 — 4
= {3^1,1, -3^1д, -tfi.i}
(8.110)
(8.111)
and thus t,— l, and
(1,, . ., Ф7} ~ ) 0. i»i д, -’Jiд. 2 д. —2i^i д. 3 д, — Зг>1 д }
(8.112)
ti(sP) = [Cb 0, 0, 0. 0, 0, 0]T To determine t2(?p) observe that
(8.113)
8.3. Example 3
207
1 ^2,i('Pi.1) — (#i " R) (8.114)
(R\ -f R) (— д) i S. 115)
a2 ■Sl(^l,l)'Sl(’-)l,l) (8.116)
a2 (— ^1,1)s\( — 0i.i) (8.117)
2 а-i si(0i,i)si( —^l.i) (8.118)
аз ?ii’^i,i)'?i(i^i.i)5i(0i,i) (8.119)
“2,1 ( Ф3,2J — “3 .Sl(_^l,l)Jl(— ^l,l)Jl(— 1,1) (8.120)
3 «3 .?1 ( 01,1 )5l( 01,l )si ( —-0; 1 ) (8.121)
3a3^i(0i,i)5i(-0i,i)Ji(-0I,i) (8.122)
1^2,l(4,l,2)
‘“2.l(®2,l)
^Й2л(Ф2,2)
"и2,1(Ф2,з)
3S2,l(^3.l)
322д('Рз,з)
3и2д(Фз,4)
using the fact that
Ri a 2 а з 0
R for Oi = 1
for oi = 2
for Oi = 3
otherwise
(8.123)
This means that the controlling variable ii2,i(/) includes up to third-order contributions due to si{f). Using the above controlled variable Ui,i(/) and Equations (8.100) and (8.101) leads to
^2(£p) = [2 C2a2si(i)ij)si( — 0i,1),
Ci(R 1 4- R) ^ 0i,i) i- 3 C'2a3s 1 (01,1)j 11. 01,1)^ 1 ( — 70i,i),
C-2(Rl R) ■> 1 ( — 01,1) -b 3 C2(13-b’i (01Д).?i( —01,1 ).Si ( — 7^1 д ),
CT2<3;2^l(01.l)^l(01.lj-
C2<l2-?l( —^1,1 )?l( —’?l,l). C^Ojii (011 ).Si( 01,1 ).■>1 ( 01Д ),
iT
С 2a 351( — 01,1,) -41 (— 0i .1 i-’i( — 0i .1) (8.124)
Next, the conversion vectors and tu(£p) cap. be determined using Equations
(7.98) and (7.96) as
Ti..,(f?> = ----->1 ^.125)
П Atp-Vl) = №)o.'2-?.?-l(fp)
+ 2(ffi)2,2-.,J-i!’)i.b-0i,bfP) ?i(^i.i}?i(-01.1
208
8■ Noise in non-linear systems: Examples and Conclusion
ri= (#i)j,2_w_i(tfj,i;£p - tfi.i) 5i(t>i,i)
+ 3 '\i, — ^ 1,1 i tp ~
'?i(-^i.i) (8.127)
П.,(^,Фз) = + *д) Ji(—*1Д)
+ 3 (ffi)3,2-?.?-i(^i,b ip + ^i,i)
5”iС ^i,i) 5i(—^1,1) 5i (— t?iti) (8.12S)
rl,?(sp; ^-l) = (-ffl)2,2-3,?-l(^l,l > ^l,b ip ~ 2$1 ,l) ($l,t) -51 ( I? 1,1) (8.129)
T\,g(ip, Ф5) = - ^1,1 i ip + 2l?l,l) Si( - i*u ) -31 ( — ^1,1 )
(8.130)
г1,7(?р,Фб) — ( 1 )з.2 —7,^— I ( rJ 1,1! $1,1- ’^1,1 • ip - 3^i,l) 5l(tfl,l) -5l (^1,1) 3"l (^l,l)
(8.1.31)
n,7(?p, Ф7) — (^?l)s,2 —1( — $1,1) ~ ^I,l> — ^l.li ip + 3l?l,l)
3"i(—^1,1) si(—A,i) ?i( —i?i,i) (8.1-32)
for q £ {1,2}. Thus the autocorrelation {[5\i,i(fp)|2) where p £ Z can be determined using Equation (7.104). The multi-port frequency domain Volterra transfer functions in Equations (8.126)—(8.132) can be determined as
(-ffi)o,i,o(—1,1! - Ri + R
(8.133)
(-ffl)o,0,l(-2,l) - R
(-^Оид^и; —i,i) = 2n2
(8.134)
(8.135)
(-ffi)i,o,i(^i.b-2,i) — 2a2
(^i)2,i,o('Ti,i, fii.2-2i,i) — 3 a3
[Hi h.O.H^l.l, ^1,2, — 2,iJ — 3 «3
(8.136)
(8.137)
(8.138)
( /fi)3,i,o(ftl,lr ^1,2, ,3' -1,1)
Iя nai
(-ffi)3,u,i№,i> ^i. 2, fii,,i; —2,i) - 0
('8.140)
8.3. Example 3
209
Thus the vectors TU(£p) and TU2(iP) are given by
ri,i(fp) = j^i + R + 6«3Si(^1,1 i( — i)\ д), 2<It J\), 2a.?31 ( — <)•, i),
3«33i(tfi,i)»i(^i,i)> 3a35i(— — 0. o] (8.141)
and
rl,2(fp) = [л + 6a3?i(l?i,i)s,( -1?1Л ), 2rt2?i(l)j,i), 2e2.?l(-!?t,i),
3аз^i(i31 !)^i (г)i,i), ЗС13.Н1 (— д) j5 ( — 0\д ), 0, oj (8.142)
The autocorrelation (|?„,i(fp)l2) ‘s given by
(i^i(^p)i2) = E E E E W^pMei<r1ucP-'M
7i=l 92 = 1 ej =1 £2 = 1
xW„^(fp - Фе1,{р - фч) г„((р - 9tJ) d£ TlM{S„)
(8.143)
This leads to
2 2 7 7 7 7
(|rn,l(fp)| ) — ^ ^ 7"i(9l ffp. ФС1 ) ~ii42 ($p, Фе2 )
<?i=l ?2 = 1 =1 ег = 1 = l “2”1
Х*«иР-Фе1.Ф-у,)^(6-Ф«-Ф^)
X{5,,(fp - Ф?1 - Ф-„) ш:Др - Ф=2 - Ф.,2 )> (8.144)
where the autocorrelation of the fundamental noise sources is given by Equation (8.103).
Numerical Experiment: To investigate the correctness of the theoretical results presented in this example, numerical simulations are nerformpH in '•'•'-'■'--i-.r.C" Section 7.3. In the simulation it is first necessary to determine the coiHro'ilittj» variable «2.1 (^)• This can be det.crm.ined as
«гДО - (Й1 + R)*i{t) 4- a2s{[t) j- aj.i'ilt) (8.1451
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