ON THE DESIGN OF IIR DIGITAL FILTERS Mitko Kostov 1), Momcilo Bogdanov 2)

Transcription

1 ON THE DESIGN OF IIR DIGITAL FILTERS Mitko Kostov ), Momcilo Bogdanov ) ) St. Kliment Ohridski University, Faculty for Technical Sciences, Bitola, R. Macedonia, mitko.kostov@uklo.edu.mk ) SS Cyril and Methodius University, Faculty of Electrical Engineering, Skoje, R. Macedonia, ogdanov@cerera.etf.ukim.edu.mk Astract: This aer considers the constraints imosed on the values of the design arameters in the efficient method roosed y Hedge and Shenoi. We have examined the relationshis etween the filter arameters in order to make correct secifications of the ermissile values of the design arameters. Index terms: Analytical design method, IIR digital filters, magnitude and grou delay aroximation.. INTRODUCTION It is well known that infinite-imulse resonse (IIR) filters can e designed y transforming an analog rototye filter to the digital domain using either imulse invariant or ilinear transformations. These transformations guarantee the staility of the filter and reserve, to a certain extent, its magnitude resonse. There are also methods to design IIR filters directly in the digital domain. However, all these methods, analog to digital or strictly digital, consist in aroximating only the magnitude resonse. On the other hand, in certain alications such as ulse transmission and high-seed data transmission, it is desirale the filters have constant delay characteristics. In [], R. Hedge and B. Shenoi resent a new technique to aroximate the magnitude resonse of IIR filters with maximally flat or equirile delay characteristics. The magnitude resonse can e matched to rescried values at a numer of frequencies etween ω = and ω = π and has desired degrees of flatness at ω = and/or ω = π. Motivated y this efficient method, we have examined the relationshis etween the filter arameters in order to make correct secifications of the ermissile values of the design arameters. To otain an IIR filter that has to meet magnitude and hase secifications simultaneously, the design arameters have to satisfy certain constraints. In [] the set of all filter arameters is not equal to the set of design arameters, i.e. there are relationshis etween the filter arameters, so they can not e all secified at the eginning of the design rocess. Our aer considers the constraints imosed on the values of the design arameters in Hedge and Shenoi method. It is organized as follows. Section II gives an outline of the Hedge and Shenoi method. Section III resents the relationshi etween the design arameters. A summary is rovided in section IV.

2 . OUTLINE ОF ТHE HEGDE АND SHENOI METHOD This section gives a review of the Hedge and Shenoi method []. In [], using two theorems, they derive sufficient conditions to otain desired degrees of flatness at any given oint in the frequency domain. From these conditions, which are linear in terms of the coefficients of the numerator olynomial, they otain an analytical solution to the rolem of designing IIR digital filters aroximating constant assand delay characteristics (in either maximally flat or equirile sense) and flat assand and stoand magnitude characteristics. Their method also meets the rescried magnitudes at a numer of frequencies, in addition to the zero frequency and half Nyquist frequencies. First, in [] an all-ole transfer function H(z)=/D(z) is chosen. The denominator olynomial of this function is chosen such that the filter has a maximally flat delay characteristic [], i.e. P P P P! k P τ + i D( z) = ( + ) ( ) ( ) τ i z P! i= P+ k = k i= τ + k + i where τ is the grou delay of /D(z) at the origin and P is the order of D(z). This function is then augmented y a mirror image olynomial as its numerator so that the delay characteristic of the resulting transfer function remains unaltered excet y a ure delay, i.e. where N H ( z) = D () z () z z + z z + z N z = + + m +. N(z) = z N a (z), () a The coefficients of N a (z) denoted y,,, are chosen so that the function H(z) has the desired magnitude characteristic. The desired conditions ) degree of flatness L at ω =, ) degree of flatness K at ω = π, 3) magnitudes of and at ω = and ω = π, resectively, and ) a 3dB attenuation at the secified andwidth ω can e exressed jointly in matrix form as A = d I, where = [ ] T, d I = [D () D ω D () D () D (M) ] T, j D ω =.77 D( e ω ), d D (k) = k jω ( ) D e d ω k ω=, π =, M = L, N = K, = (L+K+)/. k

3 o ( ) ( ) ( ) o cosω cos ω o cos ω ( ) ( ) o ( ) ( ) ( ) o ( ) A = s. () M / M / M M / M ( ) ( ) ( ) o ( ) ( ) ( ) o ( ) ( ) o ( ) s N / N / N N / N ( ) ( ) ( ) o ( ) ( ) The analytical solution to the rolem of designing IIR filters with maximally flat delay characteristics and a magnitude resonse that has desired degrees of flatness at ω = and ω = π is given y = A d I. With this method it is ossile to secify the magnitude of the low-ass filter at multile frequencies in the range [, π]. This simly adds more linear equations in (). 3. CONSTRAINTS To otain an IIR filter that has to meet magnitude and hase secifications simultaneously, its design arameters have to satisfy certain constraints. This section resents the relationshis etween the design arameters in the Hedge and Shenoi method []: the stoand frequency ω, the grou delay τ, the order P of the transfer function denominator and the degrees of flatness L and K. First, in order to otain a magnitude resonse with degrees of flatness L at ω = and K at ω = π, the values of the arameters τ and P must e chosen such that the delay characteristic of the function /D(z) aroximates the constant grou delay τ in the maximally flat sense in the region ω ω. If the arameter P increases, while the arameters τ and ω do not change, the region where the delay characteristic of the function /D(z) aroximates the arameter τ, increases as well. This is shown in Figure where solutions for the grou delay are otained for the following filter arameters ω =.35π, τ=.5, L=3, K=3. Then, if the grou delay τ increases, while the arameters P and ω do not change, the region where the delay characteristic of the function /D(z) aroximates the constant grou delay τ decreases. This is shown in Figure where solutions for the grou delays are otained for the following filter arameters ω =.35π, P=6, L=3 and K=3.

4 Grou delay (in samles) P=5 P= P=5 Grou delay (in samles) τ=.5 τ=.8 τ= Normalized frequency (Nyquist == ) Normalized frequency (Nyquist == ) Figure. Grou delay of maximally flat Figure. Grou delay of maximally flat delay filter with desired degrees of flatness at delay filter with desired degrees of flatness at oth ω = and ω = π oth ω = and ω = π The arameters P, τ and ω, affect the magnitude characteristic of the filter H(z) as follows. Increasing the value of P imroves the assand of the filter, ut worseens its stoand, as shown in Figure 3. The filter arameters in Figure 3 are ω =.π, τ=, L=5 and K=3. On the other hand, decreasing the value of τ imroves the assand of H(z), ut worsens its stoand (Figure ). The filter arameters in Figure are ω =.35π, P=6, L=9 and K= P=, P= τ=.5, τ= Normalized frequency (Nyquist == ) Figure 3. of maximally flat delay filter with desired degrees of flatness at oth ω = and ω = π Normalized frequency (Nyquist == ) Figure. of maximally flat delay filter with desired degrees of flatness at oth ω = and ω = π Then, if the filter designed with [] has a relatively high frequency ω, its magnitude has oor assand, ut good stoand. Therefore, we recommend choosing a small value for the arameter τ or a ig value for the arameter P (Figure5). For filter arameters ω =.5π, τ=.6, P=5, L=5 and K=7, there is a rile in the magnitude resonse (shown in Figure5-a). The rile can e eliminated y choosing τ=. and P=5 (shown in Figure5-).

5 Normalized frequency (Nyquist==) Normalized frequency (Nyquist==) a) ) Figure 5. of maximally flat delay filter with desired degrees of flatness at oth ω = and ω = π for filter arameters If the filter has a relatively low stoand frequency ω, its magnitude has oor stoand, ut good assand. So, if it is imortant the filter to have a good stoand, then we recommend choosing a ig value for the arameter τ or a small value for the arameter P (Figure6). For filter arameters ω =.5π, τ=.5, P=6, L=5 and K=7, there is a rile in the magnitude resonse (shown in Figure6-a). The rile can e eliminated y choosing τ=.9 and P=3 (shown in Figure6-) Normalized frequency (Nyquist==) Normalized frequency (Nyquist==) a) ) Figure 6. of maximally flat delay filter with desired degrees of flatness at oth ω = and ω = π for filter arameters Finally, the degrees of flatness at ω =, L, and ω = π, K, cannot e chosen comletely freely, as they have to otain adjacent values. If the degrees of flatness are very different, there will e distortion in the magnitude characteristic of the filter. In addition, if the arameter L is much greater then the arameter K, there will e distortion in the stoand of the magnitude characteristic. In such a case, ig values of the stoand frequency ω and the arameter τ or a small of the arameter P would imrove the stoand. For instance, for L=7 and K=5 (τ=.5, ω =.35π, P=6), the magnitude characteristic of the filter is quite good (shown in Figure7-a), ut for L=9 and K=5, there is a rile in the stoand (shown in Figure7-), which can e eliminated y choosing τ=.8 with the other arameters unchanged (Figure7-c). On other hand, if the arameter L is much smaller then the arameter K, there will e distortion in the assand of the magnitude characteristic. In such a case, small values of the stoand frequency ω and the arameter τ or a ig value of the arameter P should e

6 used. For instance, for L=3 and K=5 (τ=.5, ω =.π, P=3), the magnitude characteristic of the filter is quite good (shown in Figure8-a), ut for L=3 and K=7, there is a rile in the assand (shown in Figure8-), which can e eliminated y choosing τ=. and P= with the other arameters stay unchanged (Figure8-c) Normalized frequency (Nyquist==) Normalized frequency (Nyquist==) Normalized frequency (Nyquist==) a) ) c) Figure 7. of maximally flat delay filter with desired degrees of flatness at oth ω = and ω = π Normalized frequency (Nyquist==) Normalized frequency (Nyquist==) Normalized frequency (Nyquist==) a) ) c) Figure 8. of maximally flat delay filter with desired degrees of flatness at oth ω = and ω = π. SUMMARY We considered the constraints imosed on the values of the design arameters in Hedge and Shenoi method. We showed how the choice of the values of the stoand frequency ω, the grou delay τ and the order of the denominator olynomial P, affect the magnitude and the delay characteristics. Increasing P and decreasing τ yields an increase in the region where the delay characteristic of the function /D(z) aroximate the grou delay. Increasing P and decreasing ω and τ imroves the assand of the filter, ut worsens its stoand. Finally, the degrees of flatness, L and K, must otain adjacent values. Otherwise, there will e distortion in the magnitude of the filter, which may e eliminated y choosing aroriate values for the other design arameters. REFERENCES [] R. Hedge and B. A. Shenoi, "Magnitude Aroximation of Digital Filters with Secified Degrees of Flatness and Constant Grou Delay Characteristics," IEEE Trans. Circuits Syst. II, vol. 5, , Nov [] J. P. Thiran, Recursive Digital Filters with Maximally Flat Grou Delay, IEEE Trans. Circuit Theory, vol. CT-8, , Nov. 97.