IIR Filter Design
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.
This 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 width(s).
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.
IIR Filter Method Summary
The following table summarizes the various filter methods in the toolbox and lists the functions available to implement these methods.
Toolbox Filters Methods and Available Functions
|Filter Method||Description||Filter Functions|
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.
Design digital filter directly in the discrete time-domain by approximating a piecewise linear magnitude response.
Generalized Butterworth Design
Design lowpass Butterworth filters with more zeros than poles.
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.)
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 configuration using the filter design functions.
Filter Design Functions
Bessel (analog only)
Chebyshev Type I
Chebyshev Type II
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
the bandstop configuration.
Here are some example digital filters:
[b,a] = butter(5,0.4); % Lowpass Butterworth [b,a] = cheby1(4,1,[0.4 0.7]); % Bandpass Chebyshev Type I [b,a] = cheby2(6,60,0.8,'high'); % Highpass Chebyshev Type II [b,a] = ellip(3,1,60,[0.4 0.7],'stop'); % Bandstop elliptic
To design an analog filter, perhaps for simulation, use a trailing
specify cutoff frequencies in rad/s:
[b,a] = 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
This toolbox provides order selection functions that calculate the minimum filter order that meets a given set of requirements.
Order Estimation Function
Chebyshev Type I
Chebyshev Type II
These 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) [b,a] = butter(n,Wn);
n = 12 Wn = 0.1951 0.4080
An elliptic filter that meets the same requirements is given by
[n,Wn] = ellipord([1000 2000]/5000,[500 2500]/5000,1,60) [b,a] = ellip(n,1,60,Wn);
n = 5 Wn = 0.2000 0.4000
These functions also work with the other standard band configurations, as well as for analog filters.
Comparison of Classical IIR Filter Types
The 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.
The Butterworth filter provides the best Taylor series approximation to the ideal lowpass filter response at analog frequencies Ω = 0 and Ω = ∞; for any order N, the magnitude squared response has 2N – 1 zero derivatives at these locations (maximally flat at Ω = 0 and Ω = ∞). Response is monotonic overall, decreasing smoothly from Ω = 0 to Ω = ∞. at Ω = 1.
Chebyshev Type I Filter
The Chebyshev Type I filter minimizes the absolute difference
between the ideal and actual frequency response over the entire passband
by incorporating an equal ripple of
in the passband. Stopband response is maximally flat. The transition
from passband to stopband is more rapid than for the Butterworth filter. at Ω = 1.
Chebyshev Type II Filter
The Chebyshev Type II filter minimizes the
absolute difference between the ideal and actual frequency response
over the entire stopband by incorporating an equal ripple of
Rs dB in the stopband. Passband response is maximally flat.
The stopband does not approach zero as quickly as the type I filter (and does not approach zero at all for even-valued filter order n). The absence of ripple in the passband, however, is often an important advantage. at Ω = 1.
Elliptic filters are equiripple in both the passband and stopband.
They generally meet filter requirements with the lowest order of any
supported filter type. Given a filter order n,
Rp in decibels, and stopband ripple
decibels, elliptic filters minimize transition width. at Ω = 1.
Analog Bessel lowpass filters have maximally flat group delay at zero frequency and retain nearly constant group delay across the entire passband. Filtered signals therefore maintain their waveshapes in the passband frequency range. When an analog Bessel lowpass filter is converted to a digital one through frequency mapping, it no longer has this maximally flat property. Signal Processing Toolbox™ supports only the analog case for the complete Bessel filter design function.
Bessel filters generally require a higher filter order than other filters for satisfactory stopband attenuation. at Ω = 1 and decreases as filter order n increases.
The lowpass filters shown above were created with the analog
These functions find the zeros, poles, and gain of an
analog filter of the appropriate type with a cutoff frequency of 1 rad/s. The complete filter design functions (
ellip) call the prototyping
functions as a first step in the design process. See Special Topics in IIR Filter Design for details.
To create similar plots, use
5 and, as needed,
20. For example,
to create the elliptic filter plot:
[z,p,k] = ellipap(5,0.5,20); w = logspace(-1,1,1000); h = freqs(k*poly(z),poly(p),w); semilogx(w,abs(h)), grid xlabel('Frequency (rad/s)') ylabel('Magnitude')
Direct IIR Filter Design
This toolbox uses the term direct methods to
describe techniques for IIR design that find a filter based on specifications
in the discrete domain. Unlike the analog prototyping method, direct
design methods are not constrained to the standard lowpass, highpass,
bandpass, or bandstop configurations. Rather, these functions design
filters with an arbitrary, perhaps multiband, frequency response.
This section discusses the
which is intended specifically for filter design; Parametric Modeling discusses
other methods that may also be considered direct, such as Prony's
method, Linear Prediction, the Steiglitz-McBride method, and inverse
yulewalk function designs
recursive IIR digital filters by fitting a specified frequency response.
yulewalk's name reflects its
method for finding the filter's denominator coefficients: it finds the inverse
FFT of the ideal specified magnitude-squared response and solves the modified
Yule-Walker equations using the resulting autocorrelation function samples. The
[b,a] = yulewalk(n,f,m)
returns row vectors
a containing the
n+1 numerator and denominator coefficients of the
nth-order IIR filter whose frequency-magnitude
characteristics approximate those given in vectors
f is a vector of frequency points
ranging from 0 to 1, where 1 represents the Nyquist frequency.
m is a vector containing the specified magnitude response
at the points in
m can describe any piecewise linear shape magnitude
response, including a multiband response. The FIR counterpart of this function
fir2, which also designs a filter
based on an arbitrary piecewise linear magnitude response. See FIR Filter Design for details.
not accept phase information, and no statements are made about the
optimality of the resulting filter.
Design a multiband filter with
yulewalk and plot the specified
and actual frequency response:
m = [0 0 1 1 0 0 1 1 0 0]; f = [0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1]; [b,a] = yulewalk(10,f,m); [h,w] = freqz(b,a,128) plot(f,m,w/pi,abs(h))
Generalized Butterworth Filter Design
The toolbox function
you to design generalized Butterworth filters, that is, Butterworth
filters with differing numbers of zeros and poles. This is desirable
in some implementations where poles are more expensive computationally
maxflat is just
butter function, except
that it you can specify two orders (one for the
numerator and one for the denominator) instead of just one. These
filters are maximally flat. This means that the
resulting filter is optimal for any numerator and denominator orders,
with the maximum number of derivatives at 0 and the
Nyquist frequency ω = π both
set to 0.
For example, when the two orders are the same,
the same as
[b,a] = maxflat(3,3,0.25)
b = 0.0317 0.0951 0.0951 0.0317 a = 1.0000 -1.4590 0.9104 -0.1978
[b,a] = butter(3,0.25)
b = 0.0317 0.0951 0.0951 0.0317 a = 1.0000 -1.4590 0.9104 -0.1978
maxflat is more
versatile because it allows you to design a filter with more zeros
[b,a] = maxflat(3,1,0.25)
b = 0.0950 0.2849 0.2849 0.0950 a = 1.0000 -0.2402
The third input to
maxflat is the
half-power frequency, a frequency between
0 and 1 with a magnitude response of .
You can also design linear phase filters that have the maximally
flat property using the
ans = 0.0331 0.2500 0.4337 0.2500 0.0331