Books
in black and white
Main menu
Home About us Share a book
Books
Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics
Ads

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
Previous << 1 .. 45 46 47 48 49 50 < 51 > 52 53 54 55 56 57 .. 85 >> Next

t e {-«Л-With t.\ - t,>_i = 2t/(2A + 1) for all A {-Л 4-
1, -A + 2,..., A}, and
■' - <1Л> where A {-A,. ..,-1, 0,1,..., A}. Note from Equation (7.69) that /_д =—t\. In this case it can be shown that the Fourier series coefficient is given by
A
-1~^Р .7
2 A i i
70)
where p t {—A, ...,—1,0, 1, • ■ ■, A}. Thus the one time domain stochastic variable is represented as 2Л + 1 frequency domain stochastic variables. Assuming that wq(t) where q 6 {1,2,... ,Q} is a white noise Gaussian stochastic process with mean pq and standard deviation <j„ it can be shown that
7.3. Noise sourccs
169
= £ (7.71)
where Ai,A2 6 { —Л,. . ., —1, 0, 1,..., Л}. Then the cross-correlation between two arbitrary frequency domain Fourier coefficients is
S2/(2A + !) + flq f°r Pi = P2 - 0
(“'«(fpi) «£(£»*)> = { <^/(2Л + 1) for Pl - p., / 0 (7.72)
0 for pi ф p2
Note that {)£5g(^p)|2) where q 6 {1, 2...., Q} and p £ {-Л,. .., -1, 0,1...., A} is
the average signal power at the frequency This means that
л
I] (i'Vfp)!') = °q+l4 (7.73)
p=-.\
is the totai average noise signal power.
7.3.5 Some special cases
The following are special cases related to Equation (7.62):
• If two zero mean noise sources wqi(t) and w,,(i) where qi,q2 £ {1,2.... ,Q} are uncorrelated then
- °for
(7.74)
for all ei, c2 £ (1,2,..., E}, where 0 6 {Q}ExE' is the zero matrix.
• If the fundamental (unmodulated) noise source ?c3(/) where q £ {1, 2, .... Q]
generates Gaussian zero mean white noise with standard deviation a then
= о1 I (7.75)
where I £ {0, j_JSxb ;s the identity matrix. Note that ^;,1(^()., <f„.,; where q £
(1,2.....Q) and pi,Pj £ Z are integers can be the zero matrix of dimension
E x E, but it can also be a non-zero and 11011-identity matrix of dimension
E x E depending on (Ф[..........'$f) . and when the fundamental noise
source where q £ 1 L2T. .., (J } generates white noise.
• If the noise source nq(t) where q £ {1.2,...,Q} is unmodulated then lq = 0
and
170
7. Noise in non-linear systems: Theory
Tims, in this case n,(fp) = tq(£p, 0) tD,,(fp) where q £ {1,2...., Q} as seen from Equation (7.57).
As an example consider the following situation:
• Determine (nq(£p + Ф) nq(£p)) when {Ф1.,..., Фд} = {0, - Ф, Ф} where Ф ф 0 and '«.’?(i) is a zero mean white noise source with standard deviation oq and autocorrelation function
„ f а;/(2т) for pi = P2
WsVMMW/ = |0- othervvise I '■<<!
where q £ (1,2,.... Q}. In this case
(H,(fp + *)nJ(fp)) = tq(^p+ Ф, 0) ^(fp, —Ф) Cj/(2r)
+ t^£p + 9,9) <J(^p, 0) crj*/(2r) (7.78)
which is generally different from zero. Note that the modulated noise source iiq(t) is correlated at two different frequencies £p + Ф and (,p even when, as in this case, the fundamental (unmodulated) noise source wq(t) is a zero mean white noise source.
7.3.6 Algorithm
To determine the cross-correlation (nqi (fpi) ~i'q2((P2)) where q\,q2 £ {1,2,.... Q} the following algorithm can be used:
1. Determine the frequency sets So, S\, S2, ■ ■ ■, Sm-i from Equation (7.30).
2. Determine the controlling variables
{«,1Д(Фе),...,^1Л1(Фе)} U {5.)3.1(Фе)..........«,_„/„(*<)} (7.79)
for alt Фг £ 5i U S2 U • • • U 5vf-1 and 91.92 с {1,2,.. .,0} using Equations (7.40), (7.41) and (7.27).
3. Specify the multi-port frequency domain Volterra transfer functions
(G- _ — I Л where m 1 + • • • + .71 г £ (1.2.....M — ll. 4i £ 11.2.....О К
Ч1 /1 ,..«£ • - y] - «. , , ■
and (G^h.,»,.....where mt + ••• + m/4 £ {1.2,..., A/- 1} and q2 E
{1.2....Q}.
4. Determine the transfer vectors tq. (£p. 1 and S,;, () where q\,q2 £ {I, 2,.... Q) niici ’?i f 2 uslnc Equations iT.51 i—(7.59).
5. Specify the noise cross-correlation matrix vVqi^2 (, ip2) </1,-72 G
{1,‘2....,Q} given by Equations (7.63)—(7.64).
гл T)otprminn /л (t I ii* (f„ )\ from Fmiatinn Г7.62).
.v......................................-i..... '
7.4. Responses
171
7.4 Responses
The response r/(/j where I £ {1,2....,£} from port (r,/) Figure 7.2 can be Jeter-mined as the response from a multi-port Volterra system as
м л f \i ,\f
n(/) = E - E E ■■ E
£l.\Ami +---------i- тк -j- oi + • • • + oQ)
.....oq ( ^1,1 ' - • • ; ^1 ,mi •...................' ; ■ ■ ■ • ^/\ , rn ^ •
—i,i......-i,v..........: -q.i- ■ ■ •, HQ.»,)
sl(^l,l) ' ■ ■ S1 (J...........sr<[Cl/x,i) ■ ' ■ *к№к,тк ?
»»l(5i,i) ■ ■ •n1(5i«.l.......nQ(Eдд) • ■ -nq(Hq.c.,)
«(/ - fii.i----------...............- Пл-д------------Пк.тк
——l.i - ■ ■ ■ - -i.jj -........- г.дд - • • ■ - -q:0q )
<2^1,1 ‘ ' .............• ■ • d.\lj^,m ^
^“i,i ■ ■ 'd—\.0l....................................dz.Q'i ■ ■ -d^QiOQ (7.80)
where (Я|)П|.......Я1.д....oq(') is a partly symmetrical multi-port frequency domain
Volterra transfer function relating the inputs {si(/)...............s/\(/)-ni(})• ■ ■ ■ ■ 71q(/)}
Previous << 1 .. 45 46 47 48 49 50 < 51 > 52 53 54 55 56 57 .. 85 >> Next