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cheby2

Chebyshev Type II filter design

Description

example

[b,a] = cheby2(n,Rs,Ws) returns the transfer function coefficients of an nth-order lowpass digital Chebyshev Type II filter with normalized stopband edge frequency Ws and Rs decibels of stopband attenuation down from the peak passband value.

example

[b,a] = cheby2(n,Rs,Ws,ftype) designs a lowpass, highpass, bandpass, or bandstop Chebyshev Type II filter, depending on the value of ftype and the number of elements of Ws. The resulting bandpass and bandstop designs are of order 2n.

 Note:   See Limitations for information about numerical issues that affect forming the transfer function.

example

[z,p,k] = cheby2(___) designs a lowpass, highpass, bandpass, or bandstop digital Chebyshev Type II filter and returns its zeros, poles, and gain. This syntax can include any of the input arguments in previous syntaxes.

example

[A,B,C,D] = cheby2(___) designs a lowpass, highpass, bandpass, or bandstop digital Chebyshev Type II filter and returns the matrices that specify its state-space representation.

example

[___] = cheby2(___,'s') designs a lowpass, highpass, bandpass, or bandstop analog Chebyshev Type II filter with stopband edge angular frequency Ws and Rs decibels of stopband attenuation.

Examples

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Lowpass Chebyshev Type II Transfer Function

Design a 6th-order lowpass Chebyshev Type II filter with 40 dB of stopband attenuation and a stopband edge frequency of 300 Hz, which, for data sampled at 1000 Hz, corresponds to rad/sample. Plot its magnitude and phase responses. Use it to filter a 1000-sample random signal.

[b,a] = cheby2(6,40,0.6);
freqz(b,a)
dataIn = randn(1000,1);
dataOut = filter(b,a,dataIn);


Bandstop Chebyshev Type II Filter

Design a 6th-order Chebyshev Type II bandstop filter with normalized edge frequencies of and rad/sample and 50 dB of stopband attenuation. Plot its magnitude and phase responses. Use it to filter random data.

[b,a] = cheby2(3,50,[0.2 0.6],'stop');
freqz(b,a)
dataIn = randn(1000,1);
dataOut = filter(b,a,dataIn);


Highpass Chebyshev Type II Filter

Design a 9th-order highpass Chebyshev Type II filter with 20 dB of stopband attenuation and a stopband edge frequency of 300 Hz, which, for data sampled at 1000 Hz, corresponds to rad/sample. Plot the magnitude and phase responses. Convert the zeros, poles, and gain to second-order sections for use by fvtool.

[z,p,k] = cheby2(9,20,300/500,'high');
sos = zp2sos(z,p,k);
fvtool(sos,'Analysis','freq')


Bandpass Chebyshev Type II Filter

Design a 20th-order Chebyshev Type II bandpass filter with a lower stopband frequency of 500 Hz and a higher stopband frequency of 560 Hz. Specify a stopband attenuation of 40 dB and a sample rate of 1500 Hz. Use the state-space representation. Design an identical filter using designfilt.

[A,B,C,D] = cheby2(10,40,[500 560]/750);
d = designfilt('bandpassiir','FilterOrder',20, ...
'StopbandFrequency1',500,'StopbandFrequency2',560, ...
'StopbandAttenuation',40,'SampleRate',1500);


Convert the state-space representation to second-order sections. Visualize the frequency responses using fvtool.

sos = ss2sos(A,B,C,D);
fvt = fvtool(sos,d,'Fs',1500);
legend(fvt,'cheby2','designfilt')


Comparison of Analog IIR Lowpass Filters

Design a 5th-order analog Butterworth lowpass filter with a cutoff frequency of 2 GHz. Multiply by to convert the frequency to radians per second. Compute the frequency response of the filter at 4096 points.

n = 5;
f = 2e9;

[zb,pb,kb] = butter(n,2*pi*f,'s');
[bb,ab] = zp2tf(zb,pb,kb);
[hb,wb] = freqs(bb,ab,4096);


Design a 5th-order Chebyshev Type I filter with the same edge frequency and 3 dB of passband ripple. Compute its frequency response.

[z1,p1,k1] = cheby1(n,3,2*pi*f,'s');
[b1,a1] = zp2tf(z1,p1,k1);
[h1,w1] = freqs(b1,a1,4096);


Design a 5th-order Chebyshev Type II filter with the same edge frequency and 30 dB of stopband attenuation. Compute its frequency response.

[z2,p2,k2] = cheby2(n,30,2*pi*f,'s');
[b2,a2] = zp2tf(z2,p2,k2);
[h2,w2] = freqs(b2,a2,4096);


Design a 5th-order elliptic filter with the same edge frequency, 3 dB of passband ripple, and 30 dB of stopband attenuation. Compute its frequency response.

[ze,pe,ke] = ellip(n,3,30,2*pi*f,'s');
[be,ae] = zp2tf(ze,pe,ke);
[he,we] = freqs(be,ae,4096);


Plot the attenuation in decibels. Express the frequency in gigahertz. Compare the filters.

plot(wb/(2e9*pi),mag2db(abs(hb)))
hold on
plot(w1/(2e9*pi),mag2db(abs(h1)))
plot(w2/(2e9*pi),mag2db(abs(h2)))
plot(we/(2e9*pi),mag2db(abs(he)))
axis([0 4 -40 5])
grid
xlabel('Frequency (GHz)')
ylabel('Attenuation (dB)')
legend('butter','cheby1','cheby2','ellip')


The Butterworth and Chebyshev Type II filters have flat passbands and wide transition bands. The Chebyshev Type I and elliptic filters roll off faster but have passband ripple. The frequency input to the Chebyshev Type II design function sets the beginning of the stopband rather than the end of the passband.

Input Arguments

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n — Filter orderinteger scalar

Filter order, specified as an integer scalar.

Data Types: double

Rs — Stopband attenuationpositive scalar

Stopband attenuation down from the peak passband value, specified as a positive scalar expressed in decibels.

If your specification, ℓ, is in linear units, you can convert it to decibels using Rs = –20 log10ℓ.

Data Types: double

Ws — Stopband edge frequencyscalar | two-element vector

Stopband edge frequency, specified as a scalar or a two-element vector. The stopband edge frequency is the frequency at which the magnitude response of the filter is –Rs decibels. Larger values of stopband attenuation, Rs, result in wider transition bands.

• If Ws is a scalar, then cheby2 designs a lowpass or highpass filter with edge frequency Ws.

If Ws is the two-element vector [w1 w2], where w1 < w2, then cheby2 designs a bandpass or bandstop filter with lower edge frequency w1 and higher edge frequency w2.

• For digital filters, the stopband edge frequencies must lie between 0 and 1, where 1 corresponds to the Nyquist rate—half the sample rate or π rad/sample.

For analog filters, the stopband edge frequencies must be expressed in radians per second and can take on any positive value.

Data Types: double

ftype — Filter type'low' | 'bandpass' | 'high' | 'stop'

Filter type, specified as a string.

• 'low' specifies a lowpass filter with stopband edge frequency Ws. 'low' is the default for scalar Ws.

• 'high' specifies a highpass filter with stopband edge frequency Ws.

• 'bandpass' specifies a bandpass filter of order 2n if Ws is a two-element vector. 'bandpass' is the default when Ws has two elements.

• 'stop' specifies a bandstop filter of order 2n if Ws is a two-element vector.

Data Types: char

Output Arguments

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b,a — Transfer function coefficientsrow vectors

Transfer function coefficients of the filter, returned as row vectors of length n + 1 for lowpass and highpass filters and 2n + 1 for bandpass and bandstop filters.

• For digital filters, the transfer function is expressed in terms of b and a as

$H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{\text{b(1)}+\text{b(2)}\text{\hspace{0.17em}}{z}^{-1}+\cdots +\text{b(n+1)}\text{\hspace{0.17em}}{z}^{-n}}{\text{a(1)}+\text{a(2)}\text{\hspace{0.17em}}{z}^{-1}+\cdots +\text{a(n+1)}\text{\hspace{0.17em}}{z}^{-n}}.$

• For analog filters, the transfer function is expressed in terms of b and a as

$H\left(s\right)=\frac{B\left(s\right)}{A\left(s\right)}=\frac{\text{b(1)}\text{\hspace{0.17em}}{s}^{n}+\text{b(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{b(n+1)}}{\text{a(1)}\text{\hspace{0.17em}}{s}^{n}+\text{a(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{a(n+1)}}.$

Data Types: double

z,p,k — Zeros, poles, and gaincolumn vectors, scalar

Zeros, poles, and gain of the filter, returned as two column vectors of length n (2n for bandpass and bandstop designs) and a scalar.

• For digital filters, the transfer function is expressed in terms of z, p, and k as

$H\left(z\right)=\text{k}\frac{\left(1-\text{z(1)}\text{\hspace{0.17em}}{z}^{-1}\right)\text{\hspace{0.17em}}\left(1-\text{z(2)}\text{\hspace{0.17em}}{z}^{-1}\right)\cdots \left(1-\text{z(n)}\text{\hspace{0.17em}}{z}^{-1}\right)}{\left(1-\text{p(1)}\text{\hspace{0.17em}}{z}^{-1}\right)\text{\hspace{0.17em}}\left(1-\text{p(2)}\text{\hspace{0.17em}}{z}^{-1}\right)\cdots \left(1-\text{p(n)}\text{\hspace{0.17em}}{z}^{-1}\right)}.$

• For analog filters, the transfer function is expressed in terms of z, p, and k as

$H\left(s\right)=\text{k}\frac{\left(s-\text{z(1)}\right)\text{\hspace{0.17em}}\left(s-\text{z(2)}\right)\cdots \left(s-\text{z(n)}\right)}{\left(s-\text{p(1)}\right)\text{\hspace{0.17em}}\left(s-\text{p(2)}\right)\cdots \left(s-\text{p(n)}\right)}.$

Data Types: double

A,B,C,D — State-space matricesmatrices

State-space representation of the filter, returned as matrices. If m = n for lowpass and highpass designs and m = 2n for bandpass and bandstop filters, then A is m × m, B is m × 1, C is 1 × m, and D is 1 × 1.

• For digital filters, the state-space matrices relate the state vector x, the input u, and the output y through

$\begin{array}{c}x\left(k+1\right)=\text{A}\text{\hspace{0.17em}}x\left(k\right)+\text{B}\text{\hspace{0.17em}}u\left(k\right)\\ y\left(k\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{C}\text{\hspace{0.17em}}x\left(k\right)+\text{D}\text{\hspace{0.17em}}u\left(k\right).\end{array}$

• For analog filters, the state-space matrices relate the state vector x, the input u, and the output y through

$\begin{array}{l}\stackrel{˙}{x}=\text{A}\text{\hspace{0.17em}}x+\text{B}\text{\hspace{0.17em}}u\\ y=\text{C}\text{\hspace{0.17em}}x+\text{D}\text{\hspace{0.17em}}u.\end{array}$

Data Types: double

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Limitations

Numerical Instability of Transfer Function Syntax

In general, use the [z,p,k] syntax to design IIR filters. To analyze or implement your filter, you can then use the [z,p,k] output with zp2sos. If you design the filter using the [b,a] syntax, you might encounter numerical problems. These problems are due to round-off errors and can occur for n as low as 4. The following example illustrates this limitation.

n = 6;
Rs = 80;
Wn = [2.5e6 29e6]/500e6;
ftype = 'bandpass';

% Transfer function design
[b,a] = cheby2(n,Rs,Wn,ftype);      % This filter is unstable

% Zero-pole-gain design
[z,p,k] = cheby2(n,Rs,Wn,ftype);
sos = zp2sos(z,p,k);

% Plot and compare the results
hfvt = fvtool(b,a,sos,'FrequencyScale','log');
legend(hfvt,'TF Design','ZPK Design')


Algorithms

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.

cheby2 uses a five-step algorithm:

1. It finds the lowpass analog prototype poles, zeros, and gain using the function cheb2ap.

2. It converts poles, zeros, and gain into state-space form.

3. If required, it uses a state-space transformation to convert the lowpass filter into a bandpass, highpass, or bandstop filter with the desired frequency constraints.

4. For digital filter design, it uses bilinear to convert the analog filter into a digital filter through a bilinear transformation with frequency prewarping. Careful frequency adjustment the analog filters and the digital filters to have the same frequency response magnitude at Ws or w1 and w2.

5. It converts the state-space filter back to transfer function or zero-pole-gain form, as required.