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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 Previous << 1 .. 64 65 66 67 68 69 < 70 > 71 72 73 74 75 76 .. 85 >> Next ...r.p!?) (wT<Ji,i...‚Äô..................‚Ä¢- i>T4pu),i,. ‚ñÝ ‚ñÝ - –§—Ç—á—Ä{,),—è—Ä(,))
PM ¬´,
–ü –ü 1.–•–∏.–≥)¬∞–ì,—Ä{–§–¢–ß–≥.—Ä') (9-70'
r=l p = l
for 5 ¬£{1,2,where
S(9r,i......9r,nr)
for ?‚ñÝ ¬£ {l,2,...,P(g)} and
—Å = [2U, 21,....2ilmli_1jr ¬£ –≥|—Ç||—Ö1 (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 –Ø–≥} 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,‚Äû,.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 ‚Äòor the g-vaiiables 111 Equation (9.70) ‚Äî and in turn in properties 9.2, 9.1, 9.3 and 9.4 ‚Äî must be transferred to the rr-pomters. As seen from the properties
‚Ññ*.; ¬£ {1.2,.... ||n||} (9.73)
for –∫ ¬£ {1, 2,.... A‚Äô} and / ¬£ {1,2, ...,mj.}.
Next the correct qr vectors where –≥ ¬£ {1. 2, . . .. P(q)} and p ¬£ {1.2......nT}
inusi be determined. To do this define vectors
1 fo‚Äò cVi<-..<cVnr ,()7
0 otherwise
236
9. Multi-port Volterra transfer functions
~r.p(^0 ^r.p.l.lV1-^); ‚ñÝ - ‚Ä¢ j "r.p,l,Tni('^)> ‚ñÝ ‚ñÝ ‚Ä¢
6 4m|ixl
(9.74)
for r ¬£ {1,2,..., P(q)} and p 6 {1,2,..., nT}, where
to = [uj1;1----,wi,m‚Äû........wKil,..., WK,mK]T ‚Ç 2|[‚Äôm!|xl (9.75)
–ì 1 for wkJ = m +---------h nT-1 +p . ,
= ( 0 otherwise (9'‚Äò6)
for –∫ ¬£ {1, 2,. .., A'} and /6 {1,2,..., m*}. Thus
(9.77)
–Ø—Ç,—Ä ‚Äî z7 Aw)
.e above, a modified version of Equation (0.70) is obtained as
KWi!^!!)
iVni) Oi.i
E E ‚ñÝ ‚Ä¢‚Ä¢ E ‚ñÝ E E
nj =0 r ‚Ä¢P(q)=0 01,1‚Äî1 J1 ,n^ ‚Äî1 ‚Äúfllt.i-1 34¬´),¬ªpr,)-
ini INI INI INI
E - ‚Ä¢ E ‚ñÝ ¬£ ‚ñÝ E
^1.1=i ¬´"l.mj =1 ‚ÄúA',l=1 1LJK,rnj(^ i
PM –Ý{—è)
–ü n- —É 1 1- ‚Äîl v (‚ñÝ‚Äú>))
PM 71.
A.codMD –ü –ü –ª–æ–≥.–Ý(11–≥–≥.–Ý(^)1!)
r = l p=1
I ,"T5 / HI ^ Zl,l'\W }l ‚ñÝ - ‚ñÝ 5 ‚ÄùV*‚Äô ' Z 1 .71 –õ I
‚ñÝ ‚Ä¢ ‚Ä¢; ^—Ä^–¥–°—à), ‚Ä¢ ‚ñÝ ‚Ä¢ >^TzP(,),npf?)(‚Ñ¢))
-P(?) –ü–≥
II Ii (^.–≥)–æ–≥–õ–§1 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.
{/‚Äô(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)‚Äûlr..i‚Äûp(,|)(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:
‚Ä¢ Determine xi(0t,i) for all –∫ ¬£ {1. 2,..., A'} and I ¬£ {1,2,..., m.t} using Equation (9.53).
‚Ä¢ Determine X2(i>Te) for all e where ||e|| = 2 using Equations (9.68) and
(9.78). The vector e is given by
e = ......, –µ–¥-,—å .. ., eK.m..}1 ¬£ {0, l}ilT*¬ª)!xl
(9.79)
where the element ¬£ {0,1} for all –∫ ¬£ {1,2, ...,A'} and
/6 {1,2,..., –æ—Ç*}.
‚Ä¢ Determine cc3('tiTe) for all e where [|el] = 3 using Equations (9.68) and (9.73).
‚Ä¢ Determine cc js —Ñ–ì–µ) and so on.
‚Ä¢ ‚Ä¢ ‚Ä¢
‚Ä¢ Determine —Ö—â–≥–∏\(–§' e) ‚Äòor e where ilejl ‚Äî 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, —Ç–∫(–§i,i, ‚Ä¢ , .I 4‚ÄôK,i, ‚ñÝ ‚ñÝ ,i‚Äôy 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. Previous << 1 .. 64 65 66 67 68 69 < 70 > 71 72 73 74 75 76 .. 85 >> Next 