# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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■n(^i,i) A'i( $1,1) -Sl($l,l) 'l($l,l) ?l($l,l)

si( $1,1) 3i(t?i,i) (8.72)

Г1,1(^.Ф15) = (Я1)7л(—t)j,i, -tfl.J, - tfj.i, — 01,1, —^1.1, ~^1,1. - l^l,i;

ip + 70i,1) $i(_0i,i) 5i(~^i,i) 5i(-^i,i)

?i(~$1,1) ^i( — $i,i) Si(-$i,i) Ji( —$i.1) (8.73)

Thus the autocorrelation (гпД(£Р! )г*л(£Р2)) where P1.V2 S Z can be determined using Equation (7.104). The multi-port, frequency domain Volterra transfer functions in Equations (S.59)—(8.73) can be determined as

(#iki(Hi.i) = Ri (8-74)

(#1)1,1(^1,ii —1,1) = —2 Щд-2 (8.75)

= eR\gj (8.76)

(#1)3,1(^1,ь fit,2, ^1,3; —1.1) = —20 Л[з2 (8-77)

= 70 R\g\ (8.78)

(Я1)5,1(Й1,1,П1,2,П1,з,П1,4,П1,5;Н1,1) = -252 R^gi (8.79)

(Я,)6,i(ft,,,, Qi,3, 1JW. fii,5: fii,rf Hi,,) = 924 rY'/1 (8.80)

(Я,)7,!(П1,1, П.,2, «,,3. Пм, nit5, П,.*, Е,л) = -?A32R)bgl (8.81)

Thus. (r„.ii5„, ! r; , f f„„ )) can be determined аз

(r„.i(f,,)r~ ,(£,,)) = } , Y. rf.i(^,)dsi - Фс.)

et=l c2=l

t! j f.;r.) i 8.S‘2)

8.2. Example 2

199

by use of Equations (7.105) and (7.106). It can be shown that the autocorrelation (|»4i(£?)i*) is given by

(l?»,i(^)!7> = C\ — 1 'a ,? !1г1д(чР)||2 (S.S3)

^ i Ispi

Note tл.01 the output ciutocorrel<itioii (j’’has the same frequency domain shape as the autocorrelation for the fundamental noise source (j ffii(fp)|2). The only difference is the multiplication factor C\ ||Ti,i(fp)]|2 which depends oil the amplitude of the input sinusoidal signal, and oil the linear and non-linear network parameters.

Numerical Experiment: To investigate the correctness of the theoretical results presented in this example, numerical simulations are performed in accordance with Section 7,3. To obtain the desired correlation properties for the fundamental noise source Wi(t) as given by Equation (3.39), Wi(t) is given by a linear filtering of a noise signal £i(() which is a zero mean white Gaussian noise process with standard deviation <Ti. Choosing the frequency domain transfer function of the linear filter

Hiin(f) as

= — i/k.C2A + Г) ——— Г S.S4)

а l v ' + J }

gives the desired correlation properties for itq(£?). Thus the fundamental noise source is given by

5;(f?1) = Hiin(tPX)h{{Pl) (S.S5)

which leads to the derived correlation properties for wi(f?) as given by Equation

(8.39). It is necessary to determine fundamental noise samples uq(t-л), ■ • шд’л) from the noise samples e;(i_л)••■•,£i(iл) taking into account the ’linear filtering. One way to do this is first to generate ■ • •, £i(fл) as a zero mean white

Gaussian noise process with standard deviation tX], then determine all the Fourier series coefficients Si(£-.\), ■ ■ -, ^i(чЛ), then multiply Я«л(|_л), • •Я{,ч(?л) on

the corresponding £-Fourier series coefficients to obtain uq(£_;\).......and

then finally translate the ir-Fourier series coefficients to the time domain samples

Wi(t_.v),..., u>i(t.\) bv inverse Fourier series transformations. Thus where

A G . ..,-1.0,1...., Л} can be determined from .....as

ЛЛ,

- — \jTm T^tT- \, 1 oxpI" о~

A

x Y1 -if,A2)exp

а2 = -л

/\i/\2

2Л + 1

(8.86)

io increase ьие speeu oi

jiii,a.iion ooservf; i.nai

200

8. Noise in non-linear systems: Examples and Conclusion

(8.87)

where

S^Ui) = fi U.\J exp -j'2-уг-^т

(8.88)

— £i(*o)

(8.89)

with 5з(—Aj) = ^(Aj) assuming that £\(t) is real valued for all t 6 [— r; rj. Note from Equation (8.89) that

It is also important to note that for arbitrary finite pi 6 Z and r 6 Лл. then

In this case wi(£Pl) = Si(£Pl) for all pt 6 {—A...., -1, 0,1,..., Л} which means

that wi(t\) = £t(t.\) for all A £ {-A. ..., — 1,0,1.........A}. This gives a simple (but

of course not complete) way to test the correctness of the implemented numerical simulation.

The network equation for the non-linear noisy system in Figure 8.7 is given by

л

.wo) = y; fi(<Aj

(8.90)

Л2 — ~"Л

lim HunUm) = 1 for <7! = v'2A + 1 (8.91)

Si =K^ —«oo

(8.92)

1-1

Thus the svstem is described bv a secon.d-d,;,gree equation as

O ’l It ( г [ {1) -г г i(

и

The correct solution to this equation is given by

■ri(£) = -—— I — 1 T- \/l + 4 g<2 Rf[s\(t) -f- Tii(Oil (8.94)

2 g2 Ki I * *' ' J

8.3. Example 3

201

by which the solution to the linearized system may be found as r^t) = ЛLi) + ni(0i as 92 approaches 0 (this is not the case for the other possible solution to Equation (8.92)).

_• .Ши j. j. ^ uu^uLunciduuu hji iiic suuiue \ jCp )| ”/39901 ls ^imu-

lated versus the frequency point p 6 {-Л...........-1,0,1,_____A}. The data

and results of the simulation are shown in Figure 8.8. All simulated values (|t5i(fp)|2)399o? where \p\ 6 {1,3,5,7,9,11,13,13} agree with the theoretical results (|’3i(fp)|") with deviations less than ±1.3 %. To avoid making assumptions on the simulation it is only possible to make direct noise comparisons for |p| £ {1,3,5,7,9,11,13,15} since the simulated results for |p| 6 {0,2,4,6,8,10,12,14,16} contain both a noise contribution and a deterministic distribution which can not be separated without making some assumptions.

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