ARMA filter and method for designing the same
First Claim
1. A stable digital filter whose amplitude-frequency response approximates an arbitrarily selected frequency spectrum of amplitude, whose total number of parameters m+n+1 is near-minimum, and whose phase response approximates a substantially linear function of frequency with an arbitrarily selected slope, comprising:
- (a) an input terminal and output terminal;
(b) m+1 gain parameters β
0, β
1, . . . β
m associated with the input terminal;
(c) n gain parameters α
1, α
2, . . . α
n associated with the output terminal;
(d) means connecting the input and output terminals with the gain parameters such that ##EQU6## where D is a delay operator operating on the filter'"'"'s output and input signals y(t) and u(t), respectively, such that Dy(t)=y(t-D), etc.;
(e) the gain parameters satisfying the relationship ##EQU7## where n=0,1,2, . . . ;
m=0,1,2, . . . ;
k is an integer;
v is an integer determining the slope of the phase shift versus frequency response of the filter; and
yk and rk are the kth elements of an output sequence and a random sequence, respectively, the output sequence being the sequence obtained by convolving a truncated sequence of coefficients with the random sequence, the truncated sequence being a sequence obtained by truncating the sequence of coefficients resulting from performing an inverse Fourier transform on said arbitrarily selected frequency spectrum.
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Abstract
A near minimum order ARMA type recursive filter with guaranteed stability and convergence is provided together with a method for obtaining the parameters of such filter. The amplitude/frequency response of the filter approximates an arbitrarily selected frequency spectrum of amplitude, and the phase response approximates a substantially linear function of frequency with an arbitrarily selected slope because the parameters are identified, off-line, using a minimization process that minimizes an integral error norm. The first step involves performing an inverse discrete Fourier transform of the arbitrarily selected frequency spectrum of amplitude to obtain a truncated sequence of coefficients of a stable, pure moving-average filter model, i.e., the parameters of a non-recursive filter model. The truncated sequence of coefficients, which has N+1 terms, is then convolved with a random sequence to obtain an output sequence associated with the random sequence. A time-domain, convergent parameter identification is then performed, in a manner that minimizes an integral error function norm, to obtain the near minimum order parameters αi and βj of the model having the desired amplitude- and phase-frequency responses, the parameters satisfying the relationship: ##EQU1## where: αi is the ith auto-regressive parameter, and βj is the jth moving-average parameter, respectively, of an ARMA-type recursive filter; n and m denote the order of the auto-regressive and the moving-average parts of the ARMA model, respectively; yk and uk are associated elements of the kth element of the output sequence and the random sequence, respectively; k is an integer; and v is a shift integer selected to provide the desired slope of the phase response given by 2π(v-N/2).
65 Citations
18 Claims
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1. A stable digital filter whose amplitude-frequency response approximates an arbitrarily selected frequency spectrum of amplitude, whose total number of parameters m+n+1 is near-minimum, and whose phase response approximates a substantially linear function of frequency with an arbitrarily selected slope, comprising:
-
(a) an input terminal and output terminal; (b) m+1 gain parameters β
0, β
1, . . . β
m associated with the input terminal;(c) n gain parameters α
1, α
2, . . . α
n associated with the output terminal;(d) means connecting the input and output terminals with the gain parameters such that ##EQU6## where D is a delay operator operating on the filter'"'"'s output and input signals y(t) and u(t), respectively, such that Dy(t)=y(t-D), etc.; (e) the gain parameters satisfying the relationship ##EQU7## where n=0,1,2, . . . ;
m=0,1,2, . . . ;
k is an integer;
v is an integer determining the slope of the phase shift versus frequency response of the filter; and
yk and rk are the kth elements of an output sequence and a random sequence, respectively, the output sequence being the sequence obtained by convolving a truncated sequence of coefficients with the random sequence, the truncated sequence being a sequence obtained by truncating the sequence of coefficients resulting from performing an inverse Fourier transform on said arbitrarily selected frequency spectrum. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)
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16. A hearing aid comprising:
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(a) the combination of; (1) a microphone for receiving input sound and converting the sound into electrical signals; (2) an amplifier for amplifying the signals; and (3) a speaker for converting the amplified signals into output sound; (b) a filter coupled to the combination for providing compensation for acoustic feedback; and (c) means for selecting the location of the notch to reduce the effect of acoustic feedback (d) said filter comprising; (1) input and output terminals; and (2) means converting the terminals with m+1 gain parameters having the form β
0, β
1, . . . , β
m, and n gain parameters having the form α
0, α
1, . . . α
n such that ##EQU8## Where D is a delay operator operating in the filters output and input signals y(t) and u(t), respectively, such that Dy(t)=y(t-D), etc.;(3) the gain parameters satisfying the relationship ##EQU9## where n=0,1,2, . . . ;
m=0,1,2, . . . ;
k is an integer;
v is an integer determining the slope of the phase shift versus frequency response of the filter; and
yk and rk are the kth elements of an output sequence and a random sequence, respectively, the output sequence being the sequence obtained by convolving a truncated sequence of coefficients with the random sequence, the truncated sequence being a sequence obtained by truncating the sequence of coefficients resulting from performing an inverse Fourier transform on said arbitrarily selected frequency spectrum. - View Dependent Claims (17, 18)
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Specification
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- Resources
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Current AssigneeNoise Cancellation Technologies, Inc. (NCT Group, Inc.)
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Original AssigneeAloysius A. Beex, Causey G. Donald, Daniel Graupe
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InventorsGraupe, Daniel, Causey, G. Donald, Beex, Aloysius A.
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Primary Examiner(s)Smith, Jerry
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Application NumberUS05/852,917Time in Patent Office816 DaysField of Search364/724, 179/1 F, 179/1 FS, 179/107 R, 179/107 E, 179/107 FDUS Class Current708/320CPC Class CodesH03H 17/0258 ARMA filtersH03H 17/04 Recursive filtersH04R 25/453 electronicallyH04R 25/505 using digital signal proces...
