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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 .. 65 66 67 68 69 70 < 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 ąŠ = j|ej| 6 {l,2,...,|!m||} is given by
j I 777,11 f
r0(!M!) = ŌĆöJ ąĖ (9-80)
o\ (!|m(| - o)l
Thus the total number of evaluations required to determine XjjTnj|(J|VŌĆÖ!!) is given by
T(IHI) = ┬Ż 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 ąŠ 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 ą»0,,...10ą╗-(ŌĆó) where 0\ + ŌĆó ŌĆó Ō¢Ā + ox' E {1,2,..., ||m|| ŌĆö 1} is easily determined using Equation (9.38) if 0! + Ō¢Ā Ō¢Ā Ō¢Ā + Ofc = 1 and Equation (9.41) if Oi + ŌĆó Ō¢Ā ŌĆó + ok ┬Ż {2, 3,..., čüąŠ}.
It can also be shown that the number of ą»................čéą╗-(ŌĆś) 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. + \ ) - ŌĆó Ō¢Ā (Ik Ir,-k ~ 1) i 9.83)
mi;!
In Table 9.2 examples are given of the number of different transfer functions (ŌĆśdifferent' 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 ąÜ > 1 there have to be determined several Hmi..............mK(ŌĆÖ) transfer functions
of the same order j|m||. The number of ą»ąōŌĆ×1|_..1ąōŌĆ×ąÉ,(-) of order ]|m|| that must be determined for given ąÜ is shown in Table 9..3.
9.4. Computer implementation
239
Order IMI jmil, T(;|mi|)
0=1 o = 2 ąŠ = 3 ąŠ = 4 II ą× ąś 1 ai i , ąŠ : II ŌĆö1
1 1 I
2 2 1 3
3 3 ąŠ 1 7
4 4 6 4 1 15
5 5 10 10 5 i 31
6 6 15 20 15 čü 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 ąŠ can run in parallel.
9.4.2 Precalculated tables
To make the program run relatively fast, tables are precalculated to determine valid nŌĆ× ąŠ and w vectors from Equation (9.78) where
ąĀ(čÅ) nr P(g)
┬Ż2,co(|MI) ą¤ ą¤ -V,,(IK?(Ōäó)ii) ą¤ą░(ą│ą╝(ąĖ)....,ą│ą│.ą¤čĆ(ąĖ)) = 1(9.84)
r=l p=l r = l
In the computer implementation the valid o- and -čł-values are combined in one pointer to give the location of each which controlling jr-variable involved in the
ąÜ Ji = ŌĆó ŌĆó ŌĆó = Ir- i*r d 2 #ą»ŌĆÖąĘ Ō¢ĀŌĆóŌĆóŌĆó /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 Ō¢Ā)f) 35 . j6 SI
2 4 8 30 L20 330 702 1716
r, 12 1 J ŌĆói(i4 !365 4368 i 2376
Table 9.2: #tf0 is the number of different Hmu where mi +------mK ŌĆö ąŠ 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
llŌäó!l ąÜ = 1 ąÜ = 2 ąÜ = 3 ąÜ = 4 ąÜ = 5 ąÜ = 6
l 1 2 3 4 5 6
2 i 3 6 10 15 21
3 1 4 10 20 35 Ō¢Ā56
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! ŌĆó ŌĆó -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
Ō¢Ā3 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 čģ0(ą¤[) and x0(fl2) where čä fi2. As has been described previously, the oth order contribution is calculated from contributions of order 1,2,. Ō¢Ā. ,o ŌĆö 1. Thus the frequencies at which x must be known for lower order contributions to determine xŌĆ×(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). Previous << 1 .. 65 66 67 68 69 70 < 71 > 72 73 74 75 76 77 .. 85 >> Next 