# Recursive digital filters

This demo illustrates the classic filters: Butterworth, Chebyshev 1 & 2, and Elliptic.

Compared to FIR filters, recursive filters of low-order have improved magnitude response (but they do not have linear phase).

## Butterworth filter

```clear

[b, a] = butter(3, 0.4);
% filter order: 3 (3 poles, 3 zeros)
% cut-off frequency: 0.2 cycles/sample
% Note: Matlab uses non-conventional scaling of
% frequency (0.4 instead of 0.2 here).
```

Note that the magnitude response is flat at both dc and the Nyquist frequency (0.5). There are no ripples. The response is monotonic.

```[H, om] = freqz(b, a);
f = om/(2*pi);

figure(1)
clf
plot(f, abs(H))
xlabel('Frequency (cycles/sample)')
title('Magnitude response')
``` See that the half-power frequency is 0.2

```figure(1)
clf
plot(f, abs(H), 0.2, 1/sqrt(2), 'ro')
xlabel('Frequency (cycles/sample)')
title('Magnitude response')
grid

% The frequency response magnitude is 1/sqrt(2) = 0.707
% at f = 0.2
``` The Butterworth filter has all its zeros at z = -1. That corresponds to the flat behavour of the stop band at the Nyquist frequency.

```figure(2)
clf
zplane(b, a)
``` 6th order: the transition band is sharper.

```[b, a] = butter(6, 0.4);

[H2, om] = freqz(b, a);

figure(1)
plot(f, abs(H), f, abs(H2), 'r')
legend('3rd order','6th order')
grid
``` Pole-zero diagram of 6th order Butterworth filter.

```figure(2)
clf
zplane(b, a)
``` ## Chebyshev I filter

The Butterworth has quite a wide transition. For the same order, the Chebyshev filter has a narrower transition band.

```delp = 0.05;             % delp : pass-band ripple
Rp = -20*log10(1-delp)   % Rp : delp in dB

[b, a] = cheby1(3, Rp, 0.4);
```
```Rp =

0.4455

```

The stop-band of the Chebyshev I filter is monotonic (no ripples). The pass-band has ripples.

```[H, om] = freqz(b, a);

figure(1)
clf
plot(f, abs(H))
grid
``` Detail of pass-band: the pass-band has exactly the specified ripple, the pass-band edge is at exactly f = 0.2.

```figure(1)
ylim([0.8 1.1])
``` The Chebyshev I filter has all its zeros at z = -1 like the Butterworth filter.

```figure(2)
clf
zplane(b, a)

% Each pole corresponds to a positive ripple in the pass-band.
% The pole on the positive real axis correpsonds to the positive
% ripple at dc.
``` 6th order filter:

```[b, a] = cheby1(6, Rp, 0.4);
```

The 6th order filter has exactly the same pass-band ripple and pass-band edge. But it has a narrower transition band.

```[H2, om] = freqz(b, a);

figure(1)
plot(f, abs(H), f, abs(H2), 'r')
legend('3rd order','6th order')
grid
``` Pass-band detail:

```figure(1)
ylim([0.8 1.1])
``` The pass-band is shaped by 6 poles. Each pole corresponds to a positive ripple in the pass-band.

```figure(2)
zplane(b, a)
``` ## Chebyshev II filter

The Chebyshev II filter is like the Chebyshev I filter. But the stop-band has ripples and the pass-band is monotonic.

```dels = 0.02;                % dels : stop-band ripple
Rs = -20*log10(dels)        % Rs : dels in dB

[b, a] = cheby2(3, Rs, 0.4);
```
```Rs =

33.9794

```

The stop-band has nulls not just at f = 0.5.

```[H, om] = freqz(b, a);

figure(1)
clf
plot(f, abs(H))
grid
``` Detail of stop-band: the stop-band has exactly the specified ripple, the stop-band edge is at exactly f = 0.2.

```figure(1)
line([0.2 0.2 0.5], [0 dels dels], 'color', 'r')
ylim([0.0 0.1])
``` . . . So the zeros are not all at z = -1. The angles of the zeros correspond to the nulls of the frequency response.

```figure(2)
clf
zplane(b, a)
``` 6th order filter:

```[b, a] = cheby2(6, Rs, 0.4);
```

The 6th order filter has exactly the same stop-band ripple and stop-band edge. But the transition-band is narrower.

```[H2, om] = freqz(b, a);

figure(1)
plot(f, abs(H), f, abs(H2), 'r')
legend('3rd order','6th order')
grid
``` Detail of stop-band: Chebyshev II filters of odd order have a null at the Nyquist frequency; those have a positive ripple there.

```figure(1)
plot(f, abs(H), f, abs(H2), 'r')
legend('3rd order','6th order')
grid
ylim([0.0 0.1])
``` The 3rd order Chebyshev II filter had a zero at z = -1 (3 is odd). Here, the 6th order Cheyshev II filter has no zeros at z = -1 (6 is even).

```figure(2)
clf
zplane(b, a)
``` ## Elliptic filter

Elliptic filters have ripples in both pass-band and stop-band. For the same order, the elliptic filter has a narrower transition band than Chebyshev filters.

```delp = 0.05;             % delp : pass-band ripple
Rp = -20*log10(1-delp)   % Rp : delp in dB

dels = 0.02;             % dels : stop-band ripple
Rs = -20*log10(dels)     % Rs : dels in dB

[b, a] = ellip(3, Rp, Rs, 0.4);
```
```Rp =

0.4455

Rs =

33.9794

```
```[H, om] = freqz(b, a);

figure(1)
clf
plot(f, abs(H))
grid
``` Pass-band detail: the pass-band has exactly the specified ripple, the pass-band edge is at exactly f = 0.2.

```figure(1)
ylim([0.8 1.1])
``` Stop-band detail: the stop-band has exactly the specified ripple. But the stop-band edge is determined by the order of the filter. The Elliptic filter has the minimim possible transition-bandwidth.

```figure(1)
ylim([0.0 0.1])
``` Like the Chebyshev I filter, the poles correspond to positive ripples in the pass-band. Like the Chebyshev II filter, the zeros are not all z = -1.

```figure(2)
zplane(b, a)
``` 6th order filter:

```[b, a] = ellip(6, Rp, Rs, 0.4);
```

The 6th order filter has a narrower transition band than the 3rd order filter.

```[H2, om] = freqz(b, a);

figure(1)
plot(f, abs(H), f, abs(H2), 'r')
legend('3rd order','6th order')
grid
``` Does the Elliptic filter have unity dc gain? It depends if it is of even or odd order.

Does the Elliptic filter have a null at the Nyquist frequency (f = 0.5)? It depends if it is of even or odd order.

```figure(2)
zplane(b, a)
``` Does the Elliptic filter have a null at z = -1? It depends if it is of even or odd order.

## More stop-band attenuation...

When the stop-band attenuation is high, then a logarithmic scale is needed to see it. Note: 0.1 is 20 dB, 0.01 is 40 dB, 0.001 is 60 dB, etc.

## 20 dB in stop-band

```dels = 0.1;
Rs = -20*log10(dels)

[b, a] = cheby2(5, Rs, 0.4);
[H, om] = freqz(b, a);

dB = @(x) 20*log10(abs(x));

figure(1)
plot(f, abs(H))
grid
```
```Rs =

20

``` ```figure(2)
plot(f, dB(H))
grid
ylim([-70 10])
title('Magnitude response in dB')
``` ## 40 dB in stop-band

```dels = 0.01;
Rs = -20*log10(dels)

[b, a] = cheby2(5, Rs, 0.4);
[H, om] = freqz(b, a);

figure(1)
plot(f, abs(H))
grid
```
```Rs =

40

``` ```figure(2)
plot(f, dB(H))
grid
ylim([-70 10])
title('Magnitude response in dB')
``` ## 60 dB in stop-band

```dels = 0.001;
Rs = -20*log10(dels)

[b, a] = cheby2(5, Rs, 0.4);
[H, om] = freqz(b, a);

figure(1)
plot(f, abs(H))
grid
```
```Rs =

60

``` ```figure(2)
plot(f, dB(H))
grid
ylim([-70 10])
title('Magnitude response in dB')
``` 