Efficient digital filter design tool for approximating an FIR filter with a low-order linear-phase IIR filter
First Claim
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1. A method of approximating an FIR filter with low-order linear-phase IIR filters by the rational Arnoldi algorithm with adaptive orders containing the following steps:
- a) initialize the first vector of the Krylov sequence for each expansion point;
b) in the jth iteration of the algorithm, choosing an expansion frequency wherein the heuristics of selecting expansion frequencies in advance for the proposed rational Arnoldi method we given by(a) low-pass filters;
the proposed method with the expansion point ω
1=0;
(b) high-pass filters;
the special structures of state-space matrices used to present the duality between low-pass and light pass filters;
let state matrices become Ā
=−
A, b=b, c=c, and h0=−
h0, the expansion point ω
1=0 chosen to perform the Arnoldi algorithm;
when the corresponding orthonormal matrix Vq is obtained and then the high-pass IIR filter, which satisfies the same specifications as the original FIR filter; and
(c) band-pass/band-stop filters;
the passband edge and the stopband edge frequencies being the appropriate candidate expansion points in meeting the specifications of the design, and other expansion points with uniform spacing recommended to be selectedsuch that the frequency gives the greatest difference between the (j+1)st-order output moment of the original FIR filter H(z) and that of the lower-order IIR filter Ĥ
(z) wherein the expression of output moment errors between the ĵ
ith-order moments H(ĵ
i)(zi) and Ĥ
(ĵ
i)(zi) at each expansion point zi are expressed as follows;
is the normalization coefficient when an expansion frequency zi is selected in the jth iteration;
vector c contains the last n impulse response coefficients of a FIR filter with length n+1; and
r(j−
1)(zi) is the residual vector in the (j−
1)st iteration of the disclosed adaptive rational Arnoldi algorithm at the expansion frequency zi;
c) after the choosing the expansion point in jth iteration being determined, the single-point Arnoldi method applied at the expansion point to generate the new orthnormal vector; and
d) determine a new residual at each expansion point for next iteration;
whereby, after the giving total iteration number of the algorithm, outputting the resulting orthogonal projection matrix.
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Abstract
A method and apparatus for designing low-order linear-phase IIR filters is disclosed. Given an FIR filter, the method utilizes a new Krylov subspace projection method, called the rational Arnoldi method with adaptive orders, to synthesize an approximated IIR filter with small orders. The method is efficient in terms of computational complexity. The synthesized IIR filter can truly reflect essential dynamical features of the original FIR filter and indeed satisfies the design specifications. In particular, the linear-phase property is stilled remained in the passband.
3 Citations
1 Claim
-
1. A method of approximating an FIR filter with low-order linear-phase IIR filters by the rational Arnoldi algorithm with adaptive orders containing the following steps:
-
a) initialize the first vector of the Krylov sequence for each expansion point; b) in the jth iteration of the algorithm, choosing an expansion frequency wherein the heuristics of selecting expansion frequencies in advance for the proposed rational Arnoldi method we given by (a) low-pass filters;
the proposed method with the expansion point ω
1=0;(b) high-pass filters;
the special structures of state-space matrices used to present the duality between low-pass and light pass filters;
let state matrices become Ā
=−
A,b =b,c =c, andh0 =−
h0, the expansion point ω
1=0 chosen to perform the Arnoldi algorithm;
when the corresponding orthonormal matrixV q is obtained and then the high-pass IIR filter, which satisfies the same specifications as the original FIR filter; and(c) band-pass/band-stop filters;
the passband edge and the stopband edge frequencies being the appropriate candidate expansion points in meeting the specifications of the design, and other expansion points with uniform spacing recommended to be selectedsuch that the frequency gives the greatest difference between the (j+1)st-order output moment of the original FIR filter H(z) and that of the lower-order IIR filter Ĥ
(z) wherein the expression of output moment errors between the ĵ
ith-order moments H(ĵi )(zi) and Ĥ
(ĵi )(zi) at each expansion point zi are expressed as follows;is the normalization coefficient when an expansion frequency zi is selected in the jth iteration;
vector c contains the last n impulse response coefficients of a FIR filter with length n+1; and
r(j−
1)(zi) is the residual vector in the (j−
1)st iteration of the disclosed adaptive rational Arnoldi algorithm at the expansion frequency zi;c) after the choosing the expansion point in jth iteration being determined, the single-point Arnoldi method applied at the expansion point to generate the new orthnormal vector; and d) determine a new residual at each expansion point for next iteration; whereby, after the giving total iteration number of the algorithm, outputting the resulting orthogonal projection matrix.
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Current AssigneeChang GUNG University
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Original AssigneeChang GUNG University
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InventorsChu, Chia-Chi, Feng, Wu-Shiung, Lee, Herng-Jer
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Primary Examiner(s)An; Meng-Al T.
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Assistant Examiner(s)YAARY, MICHAEL D
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Application NumberUS10/827,504Publication NumberTime in Patent Office1,485 DaysField of Search708/300, 708/304US Class Current708/300CPC Class CodesH03H 17/04 Recursive filtersH03H 17/06 Non-recursive filtersH03H 2017/0072 Theoretical filter design
