Digital Signal Processing IIR Filter Design via Impulse Invariance

Transcription

1 Digital Signal Processing IIR Filter Design via Impulse Invariance D. Richard Brown III D. Richard Brown III 1 / 11

2 Basic Procedure We assume here that we ve already decided to use an IIR filter. The basic procedure for IIR filter design via impulse invariance is: 1. Determine the CT filter design method: 1.1 Butterworth 1.2 Chebychev Type I or Type II 1.3 Elliptic Transform the DT filter specifications to CT (sampling period T d is arbitrary) 3. Design CT filter based on the magnitude squared response H c (jω) 2 Determine filter order Determine cutoff frequency 4. Determine H c (s) h c (t) corresponding to a stable causal filter 5. Convert to DT filter H(z) h[n] via impulse invariance such that h[n] = h c (nt d ) D. Richard Brown III 2 / 11

3 Determining the Continuous-Time Filter Design Method passband stopband Butterworth monotonic monotonic Chebychev Type I equiripple monotonic Chebychev Type II monotonic equiripple equiripple equiripple equiripple D. Richard Brown III 3 / 11

4 Impulse-Invariant Lowpass Butterworth Filter Design Ex. H(ω) 1 1 δ 1 δ 2 ω p ω s π ω We start with the desired specifications of the DT filter. For this example, we will use ω p = 0.2π, ω s = 0.3π, 1 δ 1 = , and δ 2 = D. Richard Brown III 4 / 11

5 Convert DT Filter Specs to CT Filter Specs We will use the Butterworth filter approach in this example. A CT Butterworth filter has a squared magnitude response given by H c (jω) 2 1 = ( ) 2N (1) 1 + jω jω c where Ω c is the cutoff frequency (radians/second) and N is the filter order. When designing filters via impulse invariance, the sampling period T d is arbitrary. It is often convenient to just set T d = 1 so that Ω = ω. This implies the CT filter specs can be written as Ω p = 0.2π and Ω s = 0.3π. Along with our magnitude specifications 1 δ 1 = and δ 2 = , we can substitute these results directly into (1) to write ( ) 2N ( ) 2 0.2π Ω c ( ) 2N ( 0.3π 1 + Ω c ) 2 D. Richard Brown III 5 / 11

6 Determine the N and Ω c We can take the previous inequalities and write them as equalities as ( ) 0.2π 2N ( ) = (2) Ω c ( ) 0.3π 2N ( ) =. (3) Ω c We have two equations and two unknowns. By taking logarithms, we can isolate N and Ω c to get N = Ω c = Note that N must be an integer, so we can choose N = 6. We now can decide whether to pick Ω c to match the passband spec (and exceed the stopband spec) or match the stopband spec (and exceed the passband spec). To minimize the effect of aliasing, we usually choose the former. D. Richard Brown III 6 / 11

7 Determine H c (s) (part 1 of 3) Given N = 6 and our choice to match the passband spec, we have the equality ( ) 0.2π 12 ( ) = (4) Ω c which gives Ω c = Now, to determine H c (s), we can write H c (s)h c ( s) = 1 ( ) 2N. 1 + s jω c The pole locations of H c (s)h c ( s) follow from the fact that 1 + ( x a) M = 0 has M solutions given by x = {ae jπ/m, ae j3π/m,..., ae j(π+(m 1)2π)/M }. The M 2 poles corresponding to H c(s) are those in the left half plane. D. Richard Brown III 7 / 11

8 Determine H c (s) (part 2 of 3) D. Richard Brown III 8 / 11

9 Determine H c (s) (part 3 of 3) Choosing the poles from the left half plane and doing a little bit of algebra, we can write H c (s) = (s s )(s s )(s s ) Since all of the poles are simple (non-repeated), we can write the partial fraction expansion 6 A k H c (s) = s s k which implies that h c (t) = k=1 { 6 k=1 A ke s kt t 0 0 t < 0. D. Richard Brown III 9 / 11

10 Determine H(z) via Impulse Invariance Impulse invariance requires h[n] = h c (nt d ) where we have previously chosen T d = 1. Hence we have { 6 k=1 h[n] = A ke s kn n 0 0 t < 0. and it follows that H(z) = 6 k=1 A k 1 e s k z 1 with ROC z > largest magnitude pole. A bit of algebra yields the final result z 1 H(z) = z z z z z z z z 2 which can immediately be realized in parallel form or rearranged to be realized in cascade or direct forms. D. Richard Brown III 10 / 11

11 Impulse-Invariant Lowpass Butterworth Filter Design Ex CT Butterworth LPF DT Butterworth LPF via impulse invariance magnitude response normalized frequency (times π) D. Richard Brown III 11 / 11