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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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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).
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