The classic IIR filter design technique includes the following steps.
Find an analog lowpass filter with cutoff frequency of 1 and translate this prototype filter to the specified band configuration
Transform the filter to the digital domain.
Discretize the filter.
The toolbox provides functions for each of these steps.
Design Task  Available functions 

Analog lowpass prototype  
Frequency transformation  
Discretization 
Alternatively, the butter
, cheby1
, cheb2ord
, ellip
, and besself
functions perform all steps of the filter design and the
buttord
, cheb1ord
, cheb2ord
, and ellipord
functions provide minimum
order computation for IIR filters. These functions are sufficient for many design
problems, and the lower level functions are generally not needed. But if you do have
an application where you need to transform the band edges of an analog filter, or
discretize a rational transfer function, this section describes the tools with which
to do so.
This toolbox provides a number of functions to create lowpass analog prototype filters with cutoff frequency of 1, the first step in the classical approach to IIR filter design.
The table below summarizes the analog prototype design functions for each supported filter type; plots for each type are shown in IIR Filter Design.
The second step in the analog prototyping design technique is the frequency transformation of a lowpass prototype. The toolbox provides a set of functions to transform analog lowpass prototypes (with cutoff frequency of 1 rad/s) into bandpass, highpass, bandstop, and lowpass filters with the specified cutoff frequency.
Frequency Transformation  Transformation Function 

Lowpass to lowpass $${s}^{\prime}=s/{\omega}_{0}$$ 

Lowpass to highpass $${s}^{\prime}=\frac{{\omega}_{0}}{s}$$ 

Lowpass to bandpass $${s}^{\prime}=\frac{{\omega}_{0}}{{B}_{\omega}}\frac{{(s/{\omega}_{0})}^{2}+1}{s/{\omega}_{0}}$$ 

Lowpass to bandstop $${s}^{\prime}=\frac{{B}_{\omega}}{{\omega}_{0}}\frac{s/{\omega}_{0}}{{(s/{\omega}_{0})}^{2}+1}$$ 

As shown, all of the frequency transformation functions can accept two linear system models: transfer function and statespace form. For the bandpass and bandstop cases
$${\omega}_{0}=\sqrt{{\omega}_{1}{\omega}_{2}}$$
and
$${B}_{\omega}={\omega}_{2}{\omega}_{1}$$
where ω_{1} is the lower band edge and ω_{2} is the upper band edge.
The frequency transformation functions perform frequency variable
substitution. In the case of lp2bp
and lp2bs
, this is a secondorder substitution,
so the output filter is twice the order of the input. For lp2lp
and lp2hp
,
the output filter is the same order as the input.
To begin designing an order 10 bandpass Chebyshev Type I filter with a value of 3 dB for passband ripple, enter
[z,p,k] = cheb1ap(10,3);
Outputs z
, p
, and k
contain
the zeros, poles, and gain of a lowpass analog filter with cutoff
frequency Ω_{c} equal to 1 rad/s. Use the
function to transform this lowpass prototype to a bandpass analog
filter with band edges Ω_{1} = π/5 and Ω_{2} = π.
First, convert the filter to statespace form so the lp2bp
function
can accept it:
[A,B,C,D] = zp2ss(z,p,k); % Convert to statespace form.
Now, find the bandwidth and center frequency, and call lp2bp
:
u1 = 0.1*2*pi;
u2 = 0.5*2*pi; % In radians per second
Bw = u2u1;
Wo = sqrt(u1*u2);
[At,Bt,Ct,Dt] = lp2bp(A,B,C,D,Wo,Bw);
Finally, calculate the frequency response and plot its magnitude:
[b,a] = ss2tf(At,Bt,Ct,Dt); % Convert to TF form w = linspace(0.01,1,500)*2*pi; % Generate frequency vector h = freqs(b,a,w); % Compute frequency response semilogy(w/2/pi,abs(h)) % Plot log magnitude vs. freq xlabel('Frequency (Hz)') grid
The third step in the analog prototyping technique is the transformation
of the filter to the discretetime domain. The toolbox provides two
methods for this: the impulse invariant and bilinear transformations.
The filter design functions butter
, cheby1
, cheby2
,
and ellip
use the bilinear
transformation for discretization in this step.
Analog to Digital Transformation  Transformation Function 

Impulse invariance 

Bilinear transform 

The toolbox function impinvar
creates a digital
filter whose impulse response is the samples of the continuous impulse
response of an analog filter. This function works only on filters
in transfer function form. For best results, the analog filter should
have negligible frequency content above half the sampling frequency,
because such highfrequency content is aliased into lower bands upon
sampling. Impulse invariance works for some lowpass and bandpass filters,
but is not appropriate for highpass and bandstop filters.
Design a Chebyshev Type I filter and plot its frequency and phase response using FVTool:
[bz,az] = impinvar(b,a,2); fvtool(bz,az)
Click the Magnitude and Phase Response toolbar button.
Impulse invariance retains the cutoff frequencies of 0.1 Hz and 0.5 Hz.
The bilinear transformation is a nonlinear mapping of the continuous domain to the discrete domain; it maps the splane into the zplane by
$$H(z)=H(s){}_{s=k\frac{z1}{z+1}}$$
Bilinear transformation maps the jΩaxis of the continuous domain to the unit circle of the discrete domain according to
$$\omega =2{\mathrm{tan}}^{1}\left(\frac{\Omega}{k}\right)$$
The toolbox function bilinear
implements
this operation, where the frequency warping constant k is
equal to twice the sampling frequency (2*fs
) by
default, and equal to $$2\pi {f}_{p}/\mathrm{tan}\left(\pi {f}_{p}/{f}_{s}\right)$$if you give bilinear
a
trailing argument that represents a “match” frequency Fp
.
If a match frequency Fp
(in hertz) is present, bilinear
maps
the frequency Ω = 2πf_{p} (in
rad/s) to the same frequency in the discrete domain, normalized to
the sampling rate: ω = 2πf_{p}/f_{s} (in
rad/sample).
The bilinear
function can perform this transformation
on three different linear system representations: zeropolegain,
transfer function, and statespace form. Try calling bilinear
with
the statespace matrices that describe the Chebyshev Type I filter
from the previous section, using a sampling frequency of 2 Hz, and
retaining the lower band edge of 0.1 Hz:
[Ad,Bd,Cd,Dd] = bilinear(At,Bt,Ct,Dt,2,0.1);
The frequency response of the resulting digital filter is
[bz,az] = ss2tf(Ad,Bd,Cd,Dd); % Convert to TF
fvtool(bz,az)
Click the Magnitude and Phase Response toolbar button.
The lower band edge is at 0.1 Hz as expected. Notice, however,
that the upper band edge is slightly less than 0.5 Hz, although in
the analog domain it was exactly 0.5 Hz. This illustrates the nonlinear
nature of the bilinear transformation. To counteract this nonlinearity,
it is necessary to create analog domain filters with “prewarped”
band edges, which map to the correct locations upon bilinear transformation.
Here the prewarped frequencies u1
and u2
generate Bw
and Wo
for
the lp2bp
function:
fs = 2; % Sampling frequency (hertz) u1 = 2*fs*tan(0.1*(2*pi/fs)/2); % Lower band edge (rad/s) u2 = 2*fs*tan(0.5*(2*pi/fs)/2); % Upper band edge (rad/s) Bw = u2  u1; % Bandwidth Wo = sqrt(u1*u2); % Center frequency [At,Bt,Ct,Dt] = lp2bp(A,B,C,D,Wo,Bw);
A digital bandpass filter with correct band edges 0.1 and 0.5 times the Nyquist frequency is
[Ad,Bd,Cd,Dd] = bilinear(At,Bt,Ct,Dt,fs);
The example bandpass filters from the last two sections could
also be created in one statement using the complete IIR design function cheby1
.
For instance, an analog version of the example Chebyshev filter is
[b,a] = cheby1(5,3,[0.1 0.5]*2*pi,'s');
Note that the band edges are in rad/s for analog filters, whereas for the digital case, frequency is normalized:
[bz,az] = cheby1(5,3,[0.1 0.5]);
All of the complete design functions call bilinear
internally.
They prewarp the band edges as needed to obtain the correct digital
filter.