CN103164621A - Partial fraction decomposition method suitable for system function with multiple high-order poles - Google Patents

Partial fraction decomposition method suitable for system function with multiple high-order poles Download PDF

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CN103164621A
CN103164621A CN2013100824651A CN201310082465A CN103164621A CN 103164621 A CN103164621 A CN 103164621A CN 2013100824651 A CN2013100824651 A CN 2013100824651A CN 201310082465 A CN201310082465 A CN 201310082465A CN 103164621 A CN103164621 A CN 103164621A
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partial fraction
decomposition
system function
multiple high
decomposition method
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CN103164621B (en
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余锦华
马友能
汪源源
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Fudan University
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Abstract

The invention belongs to the technical field of signal processing and circuit analysis, and particularly relates to a partial fraction decomposition method suitable for a system function with multiple high-order poles. The method comprises the following steps: firstly decomposing parts of general system functions according to a simple-to-complex principle, wherein the numerators of the parts of the system functions are constants; and decomposing partial fraction expression of the system functions by using a recursive relation. The partial fraction decomposition method suitable for the system function with the multiple high-order poles has the advantages that a differential operation, polynomial division and solving and operation of an equation set are avoided, a general function can be decomposed as along as an algebraic operation is required, computer implementation and hand computation are convenient, and the partial fraction decomposition can be effectively conducted on the system equation with the multiple high-order poles.

Description

A kind of partial fraction decomposition method that is applicable to the system function of a plurality of higher order pole
Technical field
The invention belongs to signal processing, circuit analysis technical field, be specifically related to a kind of partial fraction decomposition method that is applicable to the system function of a plurality of higher order pole.
Background technology
Method of Laplace transformation and transform method are widely used in fields such as signal processing, circuit analysis and numerical evaluation.Adopt these two kinds of conversion very big simplified operation of energy on the one hand, be convenient on the other hand frequency domain characteristic and the Systems balanth of analytic signal in transform domain.Especially when the research linear time invariant system, they are very powerful instruments.When utilizing Induction Solved by Laplace Transformation solution system responses, it can be converted into algebraic equation with differentio-integral equation on the one hand, makes its computing simple; It can automatically be included in the starting condition of system in transform on the other hand, and the zero input response of system and zero state response can be obtained simultaneously.The zero-pole analysis of Help of System function, can judge rapidly that the cause and effect of system is stable in addition, expresses intuitively the complex frequency domain characteristic that system function has.Transform plays a part particular importance in discrete linear time-invariant system, it can convert difference equation to algebraic equation.Want easy too much than finding the solution difference equation in time domain, and its utilization scope is wider than discrete time Fourier transform.After system function is carried out Laplace transform and Z, for trying to achieve corresponding time-domain expression, need carry out anti-Laplace transform and inverse Z-transform.It is in practice that partial fraction is decomposed (PFE), finds the solution anti-Laplace transform and inverse Z-transform, the easiest, uses a kind of maximum methods.PFE be with rational function resolve into the low rational function of many number of times and form.After system function was carried out PFE, the time-domain expression of each rational expression can directly obtain by tabling look-up.Classical PFE method has method of residues and the method for undetermined coefficients.Yet these two kinds when processing contains the system function of higher order pole, when especially containing the system function of a plurality of higher order pole, and inapplicable.The method of undetermined coefficients requires to set up and find the solution complicated Algebraic Equation set, and the foundation of equation is very inconvenient, and finding the solution of equation is also very complicated.Method of residues is due to the continuous differentiate of needs, and calculated amount is large, and during computer realization, easily produces larger error.In addition, also have the scholar to propose algebraic method and be used for PFE[1-6], however these methods are only suitable for decomposing specific system function, mostly are difficult to realize with computer programming.The present invention proposes a kind of new PFE algorithm, this algorithm need not differential calculation and solving equation group, only relates to simple algebraic operation in computation process, is easy to programming and realizes, and can keep high precision.Applicable equally to the improper fraction algebraic equation.
Summary of the invention
The objective of the invention is to propose a kind of PFE method that effectively is applicable to contain the system function of higher order pole.
For general system function R(s), remember that the result after its partial fraction is decomposed is:
Figure 505478DEST_PATH_IMAGE001
（1）
Wherein, c ij Be residual, e k For R( s) multinomial coefficient after decomposition, NFor R( s) the degree of molecule, MFor R(s) number of limit, KFor R( s) the degree of denominator; d nBe the multinomial coefficient of molecule, sBe independent variable, s 0Be a constant, s iFor R(s) limit, k, nWith m i Be power exponent.The PFE decomposition concrete steps that the present invention proposes are as follows:
(a) decompose
Figure 867320DEST_PATH_IMAGE002
Wherein
Figure 955362DEST_PATH_IMAGE003
For
Figure 222395DEST_PATH_IMAGE004
Residual;
(b) based on the decomposition result in step (a), decompose
Figure 483613DEST_PATH_IMAGE005
(2)
Wherein
Figure 886912DEST_PATH_IMAGE006
With
Figure 958904DEST_PATH_IMAGE007
For
Figure 713234DEST_PATH_IMAGE008
Multinomial coefficient after decomposition and residual;
(c) based on the decomposition result in step (b), obtain R(s) decomposition result.
1, Decomposition
According to Heaviside differentiate formula, easily know:
Figure 35948DEST_PATH_IMAGE009
（3）
Wherein ,
Figure 254888DEST_PATH_IMAGE011
（4）
List of references [3] utilizes Leibniz differentiate rule to ask G k KInferior leading,
Figure 857907DEST_PATH_IMAGE012
, have
Figure 501378DEST_PATH_IMAGE013
（5）
Wherein
Figure 649594DEST_PATH_IMAGE014
,
Figure 378516DEST_PATH_IMAGE015
, ,
Figure 830674DEST_PATH_IMAGE017
（6）
Can be got by (3), (5) and (6):
Figure 336741DEST_PATH_IMAGE018
（7）
Wherein,
Figure 615276DEST_PATH_IMAGE019
（8）
2, Decomposition
After method in document [2] was expanded, we derived and have obtained according to differentiate rule and mathematical induction
Figure 493393DEST_PATH_IMAGE021
Residual,
Figure 435941DEST_PATH_IMAGE022
With
Figure 749242DEST_PATH_IMAGE023
Residual,
Figure 169859DEST_PATH_IMAGE024
(m is an integer, and there is following relation in 0＜m＜k):
Figure 704745DEST_PATH_IMAGE025
（9）
Especially, when m=1, have
Figure 552616DEST_PATH_IMAGE026
（10）
(10) formula of utilization and the 2nd step obtain c 0 ij Can recursion obtain
Figure 290896DEST_PATH_IMAGE020
Residual.When n 〉=K,
Figure 577520DEST_PATH_IMAGE020
After decomposing, coefficient will comprise multinomial coefficient e nk Yi Zhi:
Figure 904597DEST_PATH_IMAGE027
（11）
（12）
Figure 414523DEST_PATH_IMAGE029
（13）
(2) make in formula n= n+ 1, have:
Figure 239260DEST_PATH_IMAGE030
（14）
(2) the formula both sides are with multiply by
Figure 420843DEST_PATH_IMAGE031
, have:
Figure 689144DEST_PATH_IMAGE032
（15）
Contrast (14) and (15) can get e nk Iterative relation be:
Figure 588967DEST_PATH_IMAGE033
（16）
Therefore e nk Calculating formula can be summarized as:
Figure 217394DEST_PATH_IMAGE034
（17）
3, R(s) decomposition
Due to
Figure 253483DEST_PATH_IMAGE035
（18）
Can derive in conjunction with (2) and obtain:
Figure 692686DEST_PATH_IMAGE036
（19）
Contrast (1) and (19) can get R(s) PFE coefficient is:
Figure 142122DEST_PATH_IMAGE037
（20a）
（20b）
The calculation process of whole algorithm may be summarized as follows:
Step 1: input R(s) parameter: numerator coefficients d n , the denominator limit s i And tuple m i
Step 2: utilize (7) and (8) formula to calculate
Figure 215568DEST_PATH_IMAGE039
Residual
Figure 12623DEST_PATH_IMAGE003
Step 3: utilize (9) or (10) formula to calculate
Figure 683776DEST_PATH_IMAGE020
Residual
Figure 857268DEST_PATH_IMAGE040
Calculate according to (17) formula
Figure 680999DEST_PATH_IMAGE008
Multinomial coefficient e nk If
Figure 648955DEST_PATH_IMAGE008
Be fraction, do not have multinomial coefficient, (17) formula need not to use.
Step 4: utilize (20) formula to calculate R(s) coefficient of dissociation.
The inventive method only need be carried out simple algebraic operation, is easy to computer programming and realizes, also is practically applicable to hand computation, can keep high precision when processing higher order pole.
Embodiment
Embodiment 1
The invention is further illustrated by the following examples.
Ask the inverse transformation of following one-side Laplace transform
Figure 807404DEST_PATH_IMAGE041
Step 1: calculate according to (7) and (8) formula
Figure 253429DEST_PATH_IMAGE039
Residual
Calculate the residual of first limit (1).
；
Figure 330243DEST_PATH_IMAGE043
；
。
In like manner, can get second and third limit residual is:
Figure 225704DEST_PATH_IMAGE045
；
Figure 24027DEST_PATH_IMAGE046
;
Figure 333785DEST_PATH_IMAGE047
；
Figure 404509DEST_PATH_IMAGE048
；
Figure 520233DEST_PATH_IMAGE049
；
Figure 360013DEST_PATH_IMAGE050
；
Figure 653722DEST_PATH_IMAGE051
。
Step 2: utilize (10) formula, decompose
Figure 946163DEST_PATH_IMAGE052
n c n11 c n12 c n21 c n22 c n23 c n31 c n32 c n33 c n34
0 -5/16 1/16 5 -2 1 -75/16 -39/16 -1 -1/4
1 3/8 -1/16 -12 5 -2 93/8 101/16 11/4 3/4
2 -7/16 1/16 29 -12 4 -457/16 -259/16 -15/2 -9/4
3 1/2 -1/16 -70 28 -8 139/2 657/16 81/4 27/4
Step 3: calculate according to (20b) formula R(s) residual
Calculate the residual of first limit
Figure 865578DEST_PATH_IMAGE053
；
Figure 559864DEST_PATH_IMAGE054
。
In like manner can get:
Figure 24475DEST_PATH_IMAGE055
；
Figure 69791DEST_PATH_IMAGE056
。
Therefore former formula can be decomposed into:
。
By the Laplace transform relational expression
Figure 76110DEST_PATH_IMAGE058
, as can be known R(s) contravariant is changed to
Figure 977201DEST_PATH_IMAGE059
If this example is found the solution very complicated with method of residues or the method for undetermined coefficients, adopt method of the present invention can obtain result by hand computation.
List of references
[1] J.F.Mahoney and B. D. Sivazlian, “Partial fractions expansion: A review of computational methodology and efficiency,” J. Comput. Appl. Math., Vol.9, pp.247-269, 1983.
[2] X.C. Liu, “An Easy Algorithm for Partial Fraction Expansion with Multiple Poles,” Journal of Hunan Educational Institute (in Chinese), vol.13, no.2, pp.25-29.1993.
[3] O. Brugia, “A non-iterative method for the partial fraction expansion of rational functions with high order poles”, SIAM Rev., vol. 7, pp.381 -387, 1965
[4] L. Linner, “The computation of the Kth derivative of polynomials and rational functions in factored form and related matters,” IEEE Trans. Circuits and systems. vol.21, no.2, pp.233-236, 1974.
[5] E. Wehrhahn. “On Partial Fraction Expansion with High-Order Poles,” IEEE Trans. Circuit Theory, vol. 14，no.3, pp. 346-347, 1967.
[6] Fielder, D.C., Karni, S., “Supplementary remarks on Easy partial fraction expansion with multiple poles,” Proceedings of the IEEE, Vol. 57, no.10, pp.1769 – 1771,1969.
[7] T. Tomaru, “An algorithm for the expansion of partial fractions using tables,” IEEE Trans. Automat. Contr., vol. AC-19, pp.443-446, 1974。

Claims (4)

1. partial fraction decomposition method that is applicable to the system function of a plurality of higher order pole is to general system function formula R( s):
Wherein, c ij Be residual, e k For R( s) multinomial coefficient after decomposition, NFor R( s) the degree of molecule, MBe the number of limit, KFor R( s) the degree of denominator; d n Be the multinomial coefficient of molecule, sBe independent variable, s 0Be a constant, s iFor R(s) limit, n, m i , kBe power exponent; It is characterized in that functional expression R( s) carry out the step that partial fraction decomposes and be:
(a) decompose
Figure 304512DEST_PATH_IMAGE002
Wherein
Figure 21932DEST_PATH_IMAGE003
For Residual;
(b) based on the decomposition result in step (a), decompose:
Figure 2013100824651100001DEST_PATH_IMAGE005
Wherein
Figure 106880DEST_PATH_IMAGE006
With For
Figure 134059DEST_PATH_IMAGE008
Multinomial coefficient after decomposition and residual;
(c) based on the decomposition result in step (b), obtain R(s) decomposition result.
2. decomposition method according to claim 1, is characterized in that: calculate in step (a)
Figure 270642DEST_PATH_IMAGE003
Concrete formula be:
Figure 939521DEST_PATH_IMAGE009
（1）
Wherein
（2） 。
3. decomposition method according to claim 1, is characterized in that: calculate in step (b) c nij With e nk Concrete formula be:
Figure 624897DEST_PATH_IMAGE011
（3）
Figure 299592DEST_PATH_IMAGE012
（4） 。
4. decomposition method according to claim 1, is characterized in that: calculate in step (c) R( s) the concrete formula of resolution parameter be:
Figure 822977DEST_PATH_IMAGE013
（5） 。
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