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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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Thus
(9.20)
(9.21)
"(/) = £ M/)M7) + ^ h(f)sk(f) (9.22)
r=l
The controlling variable xr(f) where г t {l,2,...,i?} in Equation (9.22) is related to the known input signals {si(f ).. .., sk(I)} as a multi-port Volterra series. The controlling variable xT(f) where r 6 {1,2, ...,R} is given as the response from a non-linear multi-port Volterra system with frequency domain input signals {.sj(/),..-, sr'(/)} where s*(/) with к 6 {1,2,..., A'} is given by Equation (9.8) with Ik = Thus
£ CO ••• £ £ • mi . V m r; £ mjs ••• £
oi =0 cA-=0 i,l=i ‘дм — 1 ‘K.o/f-1
A,oo(°l + • • + Ok)
(IIr)oi,...,jK(i!l,ii i ■ ■ . .. j > ’ • ■ ■ ’ J^A'.i,\-,o ,)
Г A' ok / I\ Jk
exp j££ L k=l1=1 Фкм,, if- ££ 'A t=i (9.23)
(•) is the mil; ti-port frequency domain Volterra transfer function
between input signals {«i(/),...,«*•(/)} and the controlling variable zAf\. From Equation (9.23) it can be seen that
к—1 (—i
for ikj G {1,2,..., nik} and where
' }_ 4k.i !
k=l1=1
qj. i = number of {г&д,. .., } which arc equal to
9.3. Theory
225
for к £ {1, 2,..., К } and i £ {1, 2,.. ., m*}. This means that the qkj variable where к 6 {1,2,..., К} and I 6 {1,2,... , 77ij..} fulfils the following two properties
Thus
qkj 6 {0,1,..., ок} (9.26)
E <!k,l = Ok (9.27)
l=i
7П! E mi ... E ■••• m к ■■ E • m К К E EE**.*,
4.1=1 1 1 , O-J — 1 •AM = i i.v,oA- = W-*=l i=l
01 0\ 0 v О А' Д' ^-k
E ... E ... V ■ ■ ■ E EE «•'^
о It 4l ,rri} —0 4K,l~Q 7j:,mA-=0fc = l 1=1
•+91,771! =<^1 4h\l+" •+'7/С,тк=^К'
Then the response xr(/) where r t {1,2....,i?} can be written as
(9.28)
CO OO Oj 0\ O’' О!'
-(/) = E-E E E .............E - E
01-0 0 70= 0 91,1=0 ?1,т1=й ?к,;=0 Чк,тК-
where
91,1-!-hoi.mj =°1 -\-ЯК, mIv-=07v
£l,oo(<>l + ■ • ■ + OK )
(хг)„1+...+ол.(ч/’7У) exp[j<£rg] <5(/ - Фтд) (9.29)
Я = .......,?A\l,---,<7A\m:,-]J 6 (9.3,0)
■Ф = .........,Фкл,---Л’к.тХ]т e (9.31)
Ф = ............feb-.-.fe./ S ^lrn||xl (9.32)
and (гг )a:-j_го:<\~фТ*l) has been introduced as a coefficient instead of (Hr)m,.rar..(•}
'which is just a coefficient to exp[7 0' q\ n(f — ф1 q!. Then, ~r(/) can, after .some careful considerations, be rewritten into tim following то;*:1 convonipnr form:
xAf) = E E Л»(1Ы1) (ir),(/?) exp [7 фГq] S{f- ipTq) (9.33)
0=1 </
where
226
9. Multi-port Volterra. transfer functions
. f 1 for 9 = в
Ла(0) = n . (9.34)
( 0 otherwise
0 0 0 0
E = E E ..........E E (9-35)
Я 71.1=0 1Ц,т,=0 чк, 1=0 QX,mK= 0
The variables a and fi in Equation (9.34) may generally be scalars or vectors. In Equation (9.33), (xr)cl(tpTq) is the (presently unknown) oth order coefficient of expL7</>‘<7j 6(f — ■фл q) in xr(f). Substituting Equations (9.33) and (9.8) with h = mk into Equation (9.22) gives v( f) as
R oc
v(f) = Yl E E Mikll) ат(фТч) (хт)0(-фТч) exp [j 6T q] S(f-ipTq)
Г-1 0=1 q К mk
+ E E ехрУФкДЦ/- Фк,1) (9.36)
A-l 1=1
Thus two expressions for the output response v(f) are obtained. In Equation (9.9) the response i:(J) is given by the multi-port Volterra. transfer function which is to be determined (with the substitutions mentioned earlier in this section), and in Equation (9.36) the response v(f) is given by network specific vectors and the non-linear controlling variables.
9.3.3 Transfer functions
From Equations (9.9), (9.10) and (9.36) the expression for an arbitrary first order (j|m|| = 1) frequency domain Volterra transfer function can be determined as
R
..o(Фк,д = аг(^и) (xr)l(i'k,l) + bLc(lbk,l) (9.37)
r = l
for given к £ {1,2,..., A'} and l а {1,2,..., Using vector notation the expression for a first order frequency domain Volterra transfer function is given bv
IIq Q,rr. i£ = l — al {‘I'k.l) xl(’I'kj) '°<i ) (9-3-4)
and where
Xo(i>Tq) - l(2ri)o(V,"i<7)> - • - j [хн)о{'Ф1 <7)]J e CHX1 (9.39)
is the oth order vector of the controlling variables at frequency ij>Tq. Thus to determine the first order transfer function from a given input port /сё {1.2...., Л'}
9.3. Theory
227
and a given I e {1,2------,m*}, the quantities «(>£%,/), and bk( фк:) must be
determined.
The multi-port Volterra transfer functions of second and higher order (11m.11 £ {2, 3,. .., со}) can be determined from Equations (9.9), (9.10) and (9.36) as
^ ■ .-I. '-1 . I..’-I.-".;'..............; 1 - . • ■ , 71' J\ , til ;l- )
= mi! ... nl[<\ £ MlMI) (*rW||(iiV>l|) (3.40)
Equation (9.40) can also be written in vector notation as
/V (?r 1.1 i - ■ - ; ^-’l ,171! t."j Фкл ; ■ ■ ■ i ^К,771 д- )
= mi~!. ,пк\ ж||тп||(11^!1) (9,11)
for |jm|| G {2, 3,..., со}. Thus to determine the multi-port frequency domain Volterra transfer function of order 2 and higher, the system vector a( ||V*II) and the ||m||th order coefficients of the controlling variables at frequency |jxi|| must be determined.
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