cheby2 (Signal Processing Toolbox)

Chebyshev Type II filter design (stopband ripple)

Syntax

[b,a] = cheby2(n,Rs,Wn)
[b,a] = cheby2(n,Rs,Wn,'ftype')
[b,a] = cheby2(n,Rs,Wn,'s')
[b,a] = cheby2(n,Rs,Wn,'ftype','s')
[z,p,k] = cheby2(...)
[A,B,C,D] = cheby2(...)

Description

cheby2 designs lowpass, highpass, bandpass, and bandstop digital and analog Chebyshev Type II filters. Chebyshev Type II filters are monotonic in the passband and equiripple in the stopband. Type II filters do not roll off as fast as type I filters, but are free of passband ripple.

Digital Domain

[b,a] = cheby2(n,Rs,Wn) designs an order n lowpass digital Chebyshev Type II filter with normalized stopband edge frequency Wn and stopband ripple Rs dB down from the peak passband value. It returns the filter coefficients in the length n+1 row vectors b and a, with coefficients in descending powers of z.

Normalized stopband edge frequency is the beginning of the stopband, where the magnitude response of the filter is equal to -Rs dB. For cheby2, the normalized stopband edge frequency Wn is a number between 0 and 1, where 1 corresponds to the Nyquist frequency. Larger values of stopband attenuation Rs lead to wider transition widths (shallower rolloff characteristics).

If Wn is a two-element vector, Wn = [w1 w2], cheby2 returns an order 2*n bandpass filter with passband w1 < < w2.

[b,a] = cheby2(n,Rs,Wn,'ftype') designs a highpass, lowpass, or bandstop filter, where the string 'ftype' is one of the following.

'high' for a highpass digital filter with normalized stopband edge frequency Wn
'low' for a lowpass digital filter with normalized stopband edge frequency Wn
'stop' for an order 2*n bandstop digital filter if Wn is a two-element vector, Wn = [w1 w2]. The stopband is w1 < < w2.

With different numbers of output arguments, cheby2 directly obtains other realizations of the filter. To obtain zero-pole-gain form, use three output arguments as shown below:

[z,p,k] = cheby2(n,Rs,Wn) or

[z,p,k] = cheby2(n,Rs,Wn,'ftype') returns the zeros and poles in length n column vectors z and p and the gain in the scalar k.

To obtain state-space form, use four output arguments as shown below.

[A,B,C,D] = cheby2(n,Rs,Wn) or

[A,B,C,D] = cheby2(n,Rs,Wn,'ftype') where A, B, C, and D are

and u is the input, x is the state vector, and y is the output.

Analog Domain

[b,a] = cheby2(n,Rs,Wn,'s') designs an order n lowpass analog Chebyshev Type II filter with angular stopband edge frequency Wn. It returns the filter coefficients in the length n+1 row vectors b and a, with coefficients in descending powers of s, derived from the transfer function.

Angular stopband edge frequency is the frequency at which the magnitude response of the filter is equal to -Rs dB. For cheby2, the angular stopband edge frequency Wn must be greater than 0 rad/s.

If Wn is a two-element vector Wn = [w1 w2] with w1 < w2, then cheby2(n,Rs,Wn,'s') returns an order 2*n bandpass analog filter with passband w1 < < w2.

[b,a] = cheby2(n,Rs,Wn,'ftype','s') designs a highpass , lowpass, or bandstop filter.

With different numbers of output arguments, cheby2 directly obtains other realizations of the analog filter. To obtain zero-pole-gain form, use three output arguments as shown below:

[z,p,k] = cheby2(n,Rs,Wn,'s') or

[z,p,k] = cheby2(n,Rs,Wn,'ftype','s') returns the zeros and poles in length n or 2*n column vectors z and p and the gain in the scalar k.

To obtain state-space form, use four output arguments as shown below:

[A,B,C,D] = cheby2(n,Rs,Wn,'s') or

[A,B,C,D] = cheby2(n,Rs,Wn,'ftype','s') where A, B, C, and D are

and u is the input, x is the state vector, and y is the output.

Examples

Example 1: Lowpass Filter

For data sampled at 1000 Hz, design a ninth-order lowpass Chebyshev Type II filter with stopband attenuation 20 dB down from the passband and a stopband edge frequency of 300 Hz, which corresponds to a normalized value of 0.6:

```
[b,a] = cheby2(9,20,300/500);
```

The frequency response of the filter is

```
freqz(b,a,512,1000)
```

Example 2: Bandpass Filter

Design a fifth-order bandpass Chebyshev Type II filter with passband from 100 to 200 Hz and plot the impulse response of the filter:

n = 5; r = 20;
Wn = [100 200]/500;
[b,a] = cheby2(n,r,Wn);
[y,t] = impz(b,a,101); stem(t,y)

Limitations

For high order filters, the state-space form is the most numerically accurate, followed by the zero-pole-gain form. The transfer function coefficient form is the least accurate; numerical problems can arise for filter orders as low as 15.

Algorithm

cheby2 uses a five-step algorithm:

It finds the lowpass analog prototype poles, zeros, and gain using the cheb2ap function.
It converts poles, zeros, and gain into state-space form.
It transforms the lowpass filter into a bandpass, highpass, or bandstop filter with desired cutoff frequencies, using a state-space transformation.
For digital filter design, cheby2 uses bilinear to convert the analog filter into a digital filter through a bilinear transformation with frequency prewarping. Careful frequency adjustment guarantees that the analog filters and the digital filters will have the same frequency response magnitude at Wn or w1 and w2.
It converts the state-space filter back to transfer function or zero-pole-gain form, as required.

See Also

besself, butter, cheb2ap, cheb1ord, cheby1, ellip

cheby1 chirp