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 .. 41 42 43 44 45 46 < 47 > 48 49 50 51 52 53 .. 85 >> Next

+ ■ ■ ■ + °k)
...0.v(—'1,1- ■ • • : -*1,01 >.: — A',l> ■ ■
-Si (—1,1) ■ ’ •5l(-l,oI).Зк{~к,\) • • ■ SK(~K.JK)
Kf ~ -1,1 - ■ • • - -......- -A',1 - ■ • • - Ha-,ok)
-* К, ;
1,1 • • • dz. i
• dz A',,
(7.25)
where (Fqja)c,....oh-(') *s a partly symmetrical multi-port frequency domain Volterra
transfer function relating the input signals {.5\( f),..., sr,;(f)} to the controlling variable и„;_(/). The maximum order of Ihe signals {i /)..... дд-(/)} is M — I since the noise signal w„(j) where q £ {1,2,...,Q\ in itself accounts for order 1 in the expression for nq(f) in Equation (7.24). In Equation (7.25) the signal Sk{f) where k £ {1,2,..., A'} is given by the Fourier series in Equation (7.8). Insertion of Equation (7.S) into Equation (7.25) leads to
7.3. Noise sources
157
nq(t) mi'(' ' ’)
СqWq(t^ 0 Cq
CqXUq,l(t)Wq(t) 1 J Cq<i for my ~ I ] 0 otherwise
C4,l Uq(t) Til4,1<0 1 f j'2TrC.j,i(l1}1 for irn - I ] 0 otherwise
Cq,lUq,l(t) 1 f j2xC„,for mi = I t 0 otherwise
СчЛтЛичА^ шз(г)] 1 ( _/2л-Сд11(П0,1 + fijj) for mi = 1 0 otherwise
Cq,lWq(t) J-oo^Ar)^ 1 f Cqti/'(j2-xQ.iti) for mi = 1 t 0 otherwise
1 f Cq д for m i = 2 | 0 otherwise
Cq,l Uq,l [t) Uq}2(t) ^q(^) 2 J C,,i for rjli = I7!i = 1 t 0 otherwise
Table 7.1: Examples of partly symmetrical multi-port, frequency domain Volterra transfer functions (modulating functions) versus the time domain modulated noise function n{,(t) and the number of controlling variables where q 6 {1. 2,..., Q}. A partly symmetrical Г.)i,mi,(•) multi-port frequency domain Volterra transfer function i.s derived from the expression for the time domain modulated r.oise function n\ t; where q £ {1.2, .. ..Q) by (integral) Fourier transforming then making a dircct parameter identification from
Equation (7.U4).
158
7. Noise in non-linear systems: Theory
Л/ - i Л/-1 J\ Ji Jk J/v-
“»..,(/) = E-E E - E ...........................................E E
31=0 O/x=0 JlA =1 JA’,or4-=1
-f--------h «Я')
I )oj ,....0/.; I. ' ■ ■ ■ ’ V'l jj(0l ’.5 VKJx i ■ j oj- )
«i(*i4i)’ ’-?1(^Л,01)...................*k(^ava-.,)- ■■SK(i'K,]Ki0I.)
S(f - --------^’l.jl.o, -..........- ^KJA-.,----------------
(7.26)
The computational cost of determining the controlling variable (f) where q G {1,2, and i, £ {1,2,...,/,} can be reduced significantly by using the sym-
metry properties of the partly symmetrical multi-port frequency domain Volterra transfer function (F,,; )01l...,OA.(-) as
M-1 Д/-1 Ji J] -/л- ■<'/(■
«?..,(/) = E E E ■■ E ..........E E
01=0 Ox=0 Л,1 =1- >1,0! =1 jK,l=1 i/\,oA- = l
jV/—-f* • • • ~f- о к)
•^l(jl,l. ■ ■ - I.7l.ai) • ■ ■^лЧ^’я'. ■ ■ -Jk.0K)
IT-.: П П{4(д.......................л.,):}4
k~ 1 A:=l jk =1
......Ofr{ > - ■ ■ 1 Л.Л.О, t...’ 1 , - . * ? )
^lCV’i.ji.i) ■ ■ ■51(01,л|О1).......^к(,Фк.л<л)'' ■ sK(^K.jKrJK)
<5(/ — ii’L.ji j — ■ ■ ■ — ?4.j,,n —......... ФкЛк,1 — ■ • ■ - Фк.]Ко1.)
(7.27)
where
and
j / . 4 { 1 for jk, l <
A(;M,-..,.n-,ot) = j 0 otherw.se
: 3k,ok (7.281
-'Vjjjk,i-■ ■ ■,) = number of {дд,.. } which
are equal to j* £ {1,2........./*} (7.29)
The Л*(-) functions in Equation (7.27) may be used to significantly reduce the number of sum-terms in the expression for the controlling variable u,.; (/) where q £ {1,2,..., Q} and iq £ {1,2,...,/,,}. Note from Equation (7.27) that the only difference between determining «„.;, (/) and (/) where fli.oi P {1,2,...,0}
7.3. Noise sources
159
with ?! ф q2 is that )„,...0A.(-) mast be replaced with (/',2.,uL,......0K{-) re-
spectively. This may also be used to reduce the number of computations required to determine uq^ [j) for all q £ {1,2..... Q} and iq £ {1, 2..... /,}.
To avoid noise contributions to the output response r;(/) where I £ {1, 2,. .., L} at port (r,/) due to frequencies caused by orders higher than M - 1 it is necessary to keep track of the orders of the frequencies of the individual contributions to u„ ,• (/)
where q £ {1,2______,Q} and i, £ {1,2,. in Equation (7.27). This is done by-
defining frequency sets Si,S-i,. . . as
So = {ф(Ра)\ P0 £ {0, 1.2....0}!Л+-+^)*1 л j|pJ| = oj (7.30)
= {Ф^ь-.^Ф^Л (7.31)
where о £ {1, 2,. .., M — 1} ls the order of the given frequencies, and
and
yip.,) = ibT p0 (1.32)
Pa = [Pl,l , ■ ■ --Pl.J,.........РЛМ»
{0,1.2,...,o}!J‘+"'+Ja')x! (7.33)
Ф = ---Фи,...........><£70>--4^kVk]*
£ (7.34)
In Equation (7.30) the quantity i| • Ц denotes the sum of ail elements in the vector,
i.e.
!|p:il = /->1,1 + • ■ • + P\JX +......-r PAM H-------4- PA-.J,; (7.35)
Equation (7.35) follows since Pk,]k for all к £ {1,2,..., K},jk £ {1 > 2,..., Д \. Note from Equation (7.30) that there may be one or more frequencies which belongs to more than one of the sets S\, • • •. vSv/— i- For example, n the applied deter-
Previous << 1 .. 41 42 43 44 45 46 < 47 > 48 49 50 51 52 53 .. 85 >> Next