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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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...r.p!?) (wT<Ji,i...’..................•- i>T4pu),i,. ■ ■ - Фтчр{,),яр(,))
PM «,
П П 1.Хи.г)°Г,р{ФТЧг.р') (9-70'
r=l p = l
for 5 £{1,2,where
S(9r,i......9r,nr)
for ?■ £ {l,2,...,P(g)} and
с = [2U, 21,....2ilmli_1jr £ г|т||х1 (9.72)
Equation (9.70) has used the fact that the multi-port frequency domain Volterra
transfer function (G,)ni....nP,,)(') >s partly symmetrical for all q £ {1,2. ...,Q}.
The condition in Equation (9.71) is used to avoid contributions which are simply permutations of {qr 1, - ,qr Яг} where r £ {1, 2...., P(q)}. As seen from Equations
(9.68) and (9.70) there must be summation over many 17-variables. From Equation (9.70) it is seen that there are ||mi| x ||nj| q-variables. According to the properties, only j j m. j] of these (/-variables are different from 0 (those (/-variables that are not 0
are 1). Thus it would be more convenient to use a pointer wt,i to indicate which
«jr-variable in the set {71,1,w,..., <7i,ni,fc.:...?P(7).l.W------^P{q).np,„,.k.i} is equal
to 1. In this case only ||mij u;-pointers are needed instead of j|m!i x |jnj| (/-variables.
Now the requirements jiven ‘or the g-vaiiables 111 Equation (9.70) — and in turn in properties 9.2, 9.1, 9.3 and 9.4 — must be transferred to the rr-pomters. As seen from the properties
№*.; £ {1.2,.... ||n||} (9.73)
for к £ {1, 2,.... A’} and / £ {1,2, ...,mj.}.
Next the correct qr vectors where г £ {1. 2, . . .. P(q)} and p £ {1.2......nT}
inusi be determined. To do this define vectors
1 fo‘ cVi<-..<cVnr ,()7
0 otherwise
236
9. Multi-port Volterra transfer functions
~r.p(^0 ^r.p.l.lV1-^); ■ - • j "r.p,l,Tni('^)> ■ ■ •
6 4m|ixl
(9.74)
for r £ {1,2,..., P(q)} and p 6 {1,2,..., nT}, where
to = [uj1;1----,wi,m„........wKil,..., WK,mK]T 2|[’m!|xl (9.75)
Г 1 for wkJ = m +---------h nT-1 +p . ,
= ( 0 otherwise (9'‘6)
for к £ {1, 2,. .., A'} and /6 {1,2,..., m*}. Thus
(9.77)
Ят,р — z7 Aw)
.e above, a modified version of Equation (0.70) is obtained as
KWi!^!!)
iVni) Oi.i
E E ■ •• E ■ E E
nj =0 r •P(q)=0 01,1—1 J1 ,n^ —1 “fllt.i-1 34«),»pr,)-
ini INI INI INI
E - • E ■ £ ■ E
^1.1=i «"l.mj =1 “A',l=1 1LJK,rnj(^ i
PM Р{я)
П n- у 1 1- —l v (■“>))
PM 71.
A.codMD П П лог.Р(11гг.Р(^)1!)
r = l p=1
I ,"T5 / HI ^ Zl,l'\W }l ■ - ■ 5 ”V*’ ' Z 1 .71 Л I
■ • •; ^р^дСш), • ■ • >^TzP(,),npf?)(™))
-P(?) Пг
II Ii (^.г)огЛФ1 ZM>(«)) I9'78)
r = l ? = 1
In Equation (9.78) it is not necessary to include Ai(-) from Equation (9.70) because the use of w pointers ensures that Aj(-) = 1.
9.3. Theory
237
9.3.6 Algorithm
To determine the non-linear multi-port frequency domain Volterra transfer function Hmi.....mA.(-) the following algorithm can be used.
1. Specify the number of controlling variables for all Q non-linear subsystems.
{/’(1),.. ,,P(Q)}. Specify j q for all q 6 {1,2,...,Q}
where jq contains all subscript numbers for the relevant controlling variables x(J) = [ii(/),...,ih(/)]t. For example, if j:j = [l,2,4]r for non-linear subsystem q 6 {1,2, ...,Q} then the corresponding controlling variables are {xi(f),xi(f),x±(f)}.
2. Determine expressions for the system matrices and vectors of the linear system:
A~\f), and b(/).
3. Specify the multi-port frequency domain Volterra transfer function (G,j)„lr..i„p(,|)(j for all the non-linear subsystems q £ {1,2,..., Q}.
4. Determine the controlling vector Zjjmi|(j|Y>j|). This is done by a recursive method where highex order variables are determined from lower order variables as follows:
• Determine xi(0t,i) for all к £ {1. 2,..., A'} and I £ {1,2,..., m.t} using Equation (9.53).
• Determine X2(i>Te) for all e where ||e|| = 2 using Equations (9.68) and
(9.78). The vector e is given by
e = ......, ед-,ь .. ., eK.m..}1 £ {0, l}ilT*»)!xl
(9.79)
where the element £ {0,1} for all к £ {1,2, ...,A'} and
/6 {1,2,..., от*}.
• Determine cc3('tiTe) for all e where [|el] = 3 using Equations (9.68) and (9.73).
• Determine cc js фГе) and so on.
• • •
• Determine хщги\(Ф' e) ‘or e where ilejl — using Equations (9.6?) and (9.78). The only e which fulfils this is e = 1. Thus the desired controlling variables ii|m||(!i'0ii) are determined.
5. Determine ffm, тк(Фi,i, • , .I 4’K,i, ■ ■ ,i’y 4K) using Equation
(9.38) if ]|m|| = 1 and Equation (9.41) if ||m|| £ {2,3. :*}.
238
9. Multi-port Volterra. transfer functions
9.4 Computer implementation
This section briefly discusses the computer implementation of the theory in Section
9.3 to determine non-linear multi-port frequency domain Volterra transfer functions. The implementation is made in the MAPLE V Release 3 symbolic programming language which, for example, makes it possible to determine algebraic expressions for the multi-port Volterra transfer functions. The source code for the program is included in appendix E.
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