# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

**Download**(direct link)

**:**

**71**> 72 73 74 75 76 77 .. 85 >> Next

9.4.1 Some introductory considerations

It can be derived from the method described in section 9.3 that the number of contributions of x0(ibTe) for a given о = j|ej| 6 {l,2,...,|!m||} is given by

j I 777,11 f

r0(!M!) = —J и (9-80)

o\ (!|m(| - o)l

Thus the total number of evaluations required to determine XjjTnj|(J|V’!!) is given by

T(IHI) = £ r<(!Hi) (9-81)

0=1

= 2limll - 1 (9.82)

In Table 9.1, 7^(||m||) and X(jjm||) are listed for some values of о and ||m||. When the multi-port frequency domain. Volterra transfer function of a given order ||mj| is to be determined, all lower order Volterra transfer functions at any permutation of {mi,.. are determined at a very low computational cost. This is because all

the required controlling x vectors have already been calculated. Thus Я0,,...10л-(•) where 0\ + • • ■ + ox' E {1,2,..., ||m|| — 1} is easily determined using Equation (9.38) if 0! + ■ ■ ■ + Ofc = 1 and Equation (9.41) if Oi + • ■ • + ok £ {2, 3,..., со}.

It can also be shown that the number of Я................тл-(‘) with different frequency

arguments that must be determined to evaluate the response o({) is given by

1

Dinii, .. .. rrtR-: /i,. ... Ih') = TT ---------J J, 11. + \ ) - • ■ (Ik Ir,-k ~ 1) i 9.83)

mi;!

In Table 9.2 examples are given of the number of different transfer functions (‘different' in the sense that the frequency argument are different: that must be determined versus tile number of input ports and applied I incommensurate ) frequencies.

When К > 1 there have to be determined several Hmi..............mK(’) transfer functions

of the same order j|m||. The number of ЯГ„1|_..1Г„А,(-) of order ]|m|| that must be determined for given К is shown in Table 9..3.

9.4. Computer implementation

239

Order IMI jmil, T(;|mi|)

0=1 o = 2 о = 3 о = 4 II О И 1 ai i , о : II —1

1 1 I

2 2 1 3

3 3 о 1 7

4 4 6 4 1 15

5 5 10 10 5 i 31

6 6 15 20 15 с i 63

7 7 21 35 35 21 7 1 127

Table 9.1: Number of controlling vectors which must be determined. 7^(j|m||), for a given order ||m|| versus the orders of the individual controlling variables o. If the frequencies form a frequency base then the number of controlling variables listed is the same as the number of different frequencies at which the controlling variables must be determined. The rows indicate that a parallel computer could be efficient in computing the Volterra transfer functions as the calculation of different contributions of the same order о can run in parallel.

9.4.2 Precalculated tables

To make the program run relatively fast, tables are precalculated to determine valid n„ о and w vectors from Equation (9.78) where

Р(я) nr P(g)

£2,co(|MI) П П -V,,(IK?(™)ii) Па(гм(и)....,гг.Пр(и)) = 1(9.84)

r=l p=l r = l

In the computer implementation the valid o- and -ш-values are combined in one pointer to give the location of each which controlling jr-variable involved in the

К Ji = • • • = Ir- i*r d 2 #Я’з ■••• /7. #Ih

1 2 2 3 4 5 6 -

1 4 4 10 20 35 56 84

1 6 (j 21 oij 126 252 462

2 4 10 ■)f) 35 . j6 SI

2 4 8 30 L20 330 702 1716

r, 12 1 J •i(i4 !365 4368 i 2376

Table 9.2: #tf0 is the number of different Hmu where mi +------mK — о that

must be evaluated at different frequencies to determine vf f) when the symmetry properties of the Volterra transfer functions are utilized.

240

9. Multi-port Volterra transfer functions

Order Number of of order \\m !l

ll™!l К = 1 К = 2 К = 3 К = 4 К = 5 К = 6

l 1 2 3 4 5 6

2 i 3 6 10 15 21

3 1 4 10 20 35 ■56

4 1 5 15 35 70 126

5 1 6 21 .56 126 252

6 1 7 28 84 210 462

Table 9,3: Number of ffmii ,mK(') ^at must be determined for a given order m and number of input ports K.

calculation of (^?)||m|[(||'0!]) where q G {1, 2,. .., Q}. These table entries are of the form

{{ni,..{!|n|l pointers to i-variables}} (9.85)

for a given P(q) and ]]m[|. In (9.85) the coefficient is derived from the fact that there are tij! • • -iipj,)! identical contributions to (ur;)i|m||(ll1/,!l) due to permutations of the partly symmetrical (G, transfer function as seen from Equation

(9.78). The number of entries in this type of table versus the number of controlling variables P{q) for the given non-linear subsystem and the order ]|m)| are shown in Table 9.4.

Order IH! Number of table entries

P(q) = 1 P(q) = 2 P(q) = 3

2 1 4 9

3 4 20 54

4 14 92 306

■3 51 452 1863

6 202 2428 12348

7 876 142! 2 88560

Table 9.4: Number of table entries for the determination of the number

of coutrolUuK, vdiia.b!es F(q) and order ;rrv

Also tables are precalculated for the locations of the - 1 evaluations re-

quired to determine i||m||(ll,/;li) 35 seen from Equation (9.82). To illustrate why this type of table is necessary consider the following example.

9.4. Computer implementation

241

For a given order, o, compare х0(П[) and x0(fl2) where ф fi2. As has been described previously, the oth order contribution is calculated from contributions of order 1,2,. ■. ,o — 1. Thus the frequencies at which x must be known for lower order contributions to determine x„(Q.,) are not the same as these used to delennine x0(fi'>). Thus it is necessary to know where to find the lower order contributions dependent on the given frequency (actually a pointer to the given frequency).

**71**> 72 73 74 75 76 77 .. 85 >> Next