IIR Filter Design

Open Live Script

This example compares classical Butterworth, Chebyshev, and elliptic designs; and explores Bessel, Yule-Walker, and generalized Butterworth filters.

IIR vs. FIR Filters

The primary advantage of IIR filters over FIR filters is that they typically meet a given set of specifications with a much lower filter order than a corresponding FIR filter. Although IIR filters have nonlinear phase, data processing within MATLAB® software is commonly performed “offline,” that is, the entire data sequence is available prior to filtering. This allows for a noncausal, zero-phase filtering approach (via the filtfilt function), which eliminates the nonlinear phase distortion of an IIR filter.

Classical IIR Filters

The classical IIR filters, Butterworth, Chebyshev Types I and II, elliptic, and Bessel, all approximate the ideal “brick wall” filter in different ways.

Signal Processing Toolbox™ provides functions to create all these types of classical IIR filters in both the analog and digital domains (except Bessel, for which only the analog case is supported), and in lowpass, highpass, bandpass, and bandstop configurations. For most filter types, you can also find the lowest filter order that fits a given filter specification in terms of passband and stopband attenuation, and transition widths.

Other IIR Filters

The direct filter design function yulewalk finds a filter with magnitude response approximating a specified frequency-response function. This is one way to create a multiband bandpass filter.

You can also use the parametric modeling or system identification functions to design IIR filters. These functions are discussed in Parametric Modeling.

The generalized Butterworth design function maxflat is discussed in the section Generalized Butterworth Filter Design.

IIR Filter Method Summary

The following table summarizes the various filter methods using Signal Processing Toolbox™ and lists the functions available to implement these methods.

Toolbox Filters Methods and Available Functions

Filter Method	Description	Filter Functions
Analog Prototyping	Using the poles and zeros of a classical lowpass prototype filter in the continuous (Laplace) domain, obtain a digital filter through frequency transformation and filter discretization.	Filter design: `besself`, `butter`, `cheby1`, `cheby2`, `ellip` Order estimation: `buttord`, `cheb1ord`, `cheb2ord`, `ellipord` Lowpass analog prototyping: `besselap`, `buttap`, `cheb1ap`, `cheb2ap`, `ellipap` Frequency transformation: `lp2bp`, `lp2bs`, `lp2hp`, `lp2lp` Filter discretization: `bilinear`, `impinvar`
Direct Design	Design digital filter directly in the discrete time-domain by approximating a piecewise linear magnitude response.	`yulewalk`
Generalized Butterworth Design	Design lowpass Butterworth filters with more zeros than poles.	`maxflat`
Parametric Modeling	Find a digital filter that approximates a prescribed time or frequency domain response. (See System Identification Toolbox™ documentation for an extensive collection of parametric modeling tools.)	Time-domain modeling: `lpc`, `prony`, `stmcb` Frequency-domain modeling: `invfreqs`, `invfreqz`

Classical IIR Filter Design Using Analog Prototyping

The principal IIR digital filter design technique this toolbox provides is based on the conversion of classical lowpass analog filters to their digital equivalents. The following sections describe how to design filters and summarize the characteristics of the supported filter types. See Special Topics in IIR Filter Design for detailed steps on the filter design process.

Complete Classical IIR Filter Design

You can easily create a filter of any order with a lowpass, highpass, bandpass, or bandstop ftype configuration using the filter design functions.

Filter Design Functions

Filter Type	Design Function
Bessel (`besself`, analog only)	`[b,a] = besself(n,Wn,ftype)` `[z,p,k] = besself(n,Wn,ftype)` `[A,B,C,D] = besself(n,Wn,ftype)`
Butterworth (`butter`)	`[b,a] = butter(n,Wn,ftype)` `[z,p,k] = butter(n,Wn,ftype)` `[A,B,C,D] = butter(n,Wn,ftype)`
Chebyshev Type I (`cheby1`)	`[b,a] = cheby1(n,Rp,Wn,ftype)` `[z,p,k] = cheby1(n,Rp,Wn,ftype)` `[A,B,C,D] = cheby1(n,Rp,Wn,ftype)`
Chebyshev Type II (`cheby2`)	`[b,a] = cheby2(n,Rs,Wn,ftype)` `[z,p,k] = cheby2(n,Rs,Wn,ftype)` `[A,B,C,D] = cheby2(n,Rs,Wn,ftype)`
Elliptic (`ellip`)	`[b,a] = ellip(n,Rp,Rs,Wn,ftype)` `[z,p,k] = ellip(n,Rp,Rs,Wn,ftype)` `[A,B,C,D] = ellip(n,Rp,Rs,Wn,ftype)`

By default, each of these functions returns a lowpass filter; you need to specify only the cutoff frequency that you want, Wn, in normalized units such that the Nyquist frequency is 1 Hz). For a highpass filter, append "high" to the function's parameter list. For a bandpass or bandstop filter, specify Wn as a two-element vector containing the passband edge frequencies. Append "stop" for the bandstop configuration.

Here are some example digital filters:

[bB,aB] = butter(5,0.4);                  % Lowpass Butterworth
[bC1,aC1] = cheby1(4,1,[0.4 0.7]);        % Bandpass Chebyshev Type I
[bC2,aC2] = cheby2(6,60,0.8,"high");      % Highpass Chebyshev Type II
[bE,aE] = ellip(3,1,60,[0.4 0.7],"stop"); % Bandstop elliptic

To design an analog filter, perhaps for simulation, use a trailing "s" and specify cutoff frequencies in rad/s:

[bBA,aBA] = butter(5,0.4,"s");    % Analog Butterworth filter

All filter design functions return a filter in the transfer function, zero-pole-gain, or state-space linear system model representation, depending on how many output arguments are present. In general, you should avoid using the transfer function form because numerical problems caused by round-off errors can occur. Instead, use the zero-pole-gain form which you can convert to a second-order section (SOS) form using zp2sos and then use the SOS form to analyze or implement your filter.

All classical IIR lowpass filters are ill-conditioned for extremely low cutoff frequencies. Therefore, instead of designing a lowpass IIR filter with a very narrow passband, it can be better to design a wider passband and decimate the input signal.

Designing IIR Filters to Frequency Domain Specifications

Signal Processing Toolbox™ provides order selection functions that calculate the minimum filter order that meets a given set of requirements.

Filter Type	Order Estimation Function
Butterworth	`[n,Wn] = buttord(Wp,Ws,Rp,Rs)`
Chebyshev Type I	`[n,Wn] = cheb1ord(Wp,Ws,Rp,Rs)`
Chebyshev Type II	`[n,Wn] = cheb2ord(Wp,Ws,Rp,Rs)`
Elliptic	`[n,Wn] = ellipord(Wp,Ws,Rp,Rs)`

These order estimation functions are useful in conjunction with the filter design functions. Suppose you want a bandpass filter with a passband from 1000 to 2000 Hz, stopbands starting 500 Hz away on either side, a 10 kHz sampling frequency, at most 1 dB of passband ripple, and at least 60 dB of stopband attenuation. You can meet these specifications by using the butter function as follows.

[n,Wn] = buttord([1000 2000]/5000,[500 2500]/5000,1,60)

n = 
12

Wn = 1×2

    0.1951    0.4080

[bB2,aB2] = butter(n,Wn);

An elliptic filter that meets the same requirements is given by

[n,Wn] = ellipord([1000 2000]/5000,[500 2500]/5000,1,60)

n = 
5

Wn = 1×2

    0.2000    0.4000

[bE2,aE2] = ellip(n,1,60,Wn);

These functions also work with the other standard band configurations, as well as for analog filters.

Comparison of Classical IIR Filter Types

Signal Processing Toolbox™ provides five different types of classical IIR filters, each optimal in some way. This section shows the basic analog prototype form for each and summarizes major characteristics.

Butterworth Filter

The Butterworth filter provides the best Taylor series approximation to the ideal lowpass filter response at analog frequencies $Ω = 0$ and $Ω = \infty$ ; for any order $N$ , the magnitude squared response has $2 N - 1$ zero derivatives at these locations (maximally flat at $Ω = 0$ and $Ω = \infty$ ). Response is monotonic overall, decreasing smoothly from $Ω = 0$ to $Ω = \infty$ . $| H (j Ω) | = 1 / \sqrt{2}$ at $Ω = 1$ .

n = 5;
[z,p,k] = buttap(n);
[bBp,aBp] = zp2tf(z,p,k);
[hBp,wBp] = freqs(bBp,aBp,4096);
semilogx(wBp,abs(hBp))
grid on
xlabel("Frequency (rad/s)")
ylabel("Magnitude")