FIR Filter Design
FIR vs. IIR Filters
Digital filters with finite-duration impulse response (all-zero, or FIR filters) have both advantages and disadvantages compared to infinite-duration impulse response (IIR) filters.
FIR filters have the following primary advantages:
The primary disadvantage of FIR filters is that they often require a much higher filter order than IIR filters to achieve a given level of performance. Correspondingly, the delay of these filters is often much greater than for an equal performance IIR filter.
FIR Filter Summary
FIR Filters
Filter Design Method | Description | Filter Functions |
---|---|---|
Windowing | Apply window to truncated inverse Fourier transform of specified "brick wall" filter | |
Multiband with Transition Bands | Equiripple or least squares approach over sub-bands of the frequency range | |
Constrained Least Squares | Minimize squared integral error over entire frequency range subject to maximum error constraints | |
Arbitrary Response | Arbitrary responses, including nonlinear phase and complex filters | |
Raised Cosine | Lowpass response with smooth, sinusoidal transition |
Linear Phase Filters
Except for cfirpm
, all
of the FIR filter design functions design linear phase filters only.
The filter coefficients,
or “taps,” of such filters obey either an even or odd
symmetry relation. Depending on this symmetry, and on whether the
order n of the filter is even or odd, a linear
phase filter (stored in length n+1 vector b
)
has certain inherent restrictions on its frequency
response.
Linear Phase Filter Type | Filter Order | Symmetry of Coefficients | Response H(f), f = 0 | Response
H(f), f = 1 (Nyquist) |
---|---|---|---|---|
Type I | Even | even: $$b(k)=b(n+2-k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$ | No restriction | No restriction |
Type II | Odd | even: $$b(k)=b(n+2-k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$ | No restriction | H(1) |
Type III | Even | odd: $$b(k)=-b(n+2-k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$ | H(0) | H(1) |
Type IV | Odd | odd: $$b(k)=-b(n+2-k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$ | H(0) | No restriction |
The phase delay and group delay of linear phase FIR filters are equal and constant over the frequency band. For an order n linear phase FIR filter, the group delay is n/2, and the filtered signal is simply delayed by n/2 time steps (and the magnitude of its Fourier transform is scaled by the filter's magnitude response). This property preserves the wave shape of signals in the passband; that is, there is no phase distortion.
The functions fir1
, fir2
, firls
, firpm
, fircls
,
and fircls1
all design type
I and II linear phase FIR filters by default. rcosdesign
designs
only type I filters. Both firls
and firpm
design
type III and IV linear phase FIR filters given a 'hilbert'
or 'differentiator'
flag. cfirpm
can design any type of linear
phase filter, and nonlinear phase filters as well.
Note
Because the frequency response of a type II filter is zero at
the Nyquist frequency (“high” frequency), fir1
does
not design type II highpass and bandstop filters. For odd-valued n
in
these cases, fir1
adds 1 to the order and returns
a type I filter.
Windowing Method
Consider the ideal, or “brick wall,” digital lowpass filter with a cutoff frequency of ω_{0} rad/s. This filter has magnitude 1 at all frequencies with magnitude less than ω_{0}, and magnitude 0 at frequencies with magnitude between ω_{0} and π. Its impulse response sequence h(n) is
$$h(n)=\frac{1}{2\pi}{\displaystyle {\int}_{-\pi}^{\pi}H}(\omega ){e}^{j\omega n}d\omega =\frac{1}{2\pi}{\displaystyle {\int}_{-{\omega}_{0}}^{{\omega}_{0}}{e}^{j\omega n}}d\omega =\frac{\mathrm{sin}{\omega}_{0}n}{\pi n}$$
This filter is not implementable since its impulse response is infinite and noncausal. To create a finite-duration impulse response, truncate it by applying a window. By retaining the central section of impulse response in this truncation, you obtain a linear phase FIR filter. For example, a length 51 filter with a lowpass cutoff frequency ω_{0} of 0.4 π rad/s is
b = 0.4*sinc(0.4*(-25:25));
The window applied here is a simple rectangular window. By Parseval’s theorem, this is the length 51 filter that best approximates the ideal lowpass filter, in the integrated least squares sense. Display the filter's frequency response. Note that the y-axis shown in the below is in Magnitude Squared.
Ringing and ripples occur in the response, especially near the band edge. This “Gibbs effect” does not vanish as the filter length increases, but a nonrectangular window reduces its magnitude. Multiplication by a window in the time domain causes a convolution or smoothing in the frequency domain. Apply a length 51 Hamming window to the filter and display the result:
b = 0.4*sinc(0.4*(-25:25)); b = b.*hamming(51)';
Note that the y-axis shown in the figure is in Magnitude Squared.
Using a Hamming window greatly reduces the ringing. This improvement is at the expense of transition width (the windowed version takes longer to ramp from passband to stopband) and optimality (the windowed version does not minimize the integrated squared error).
Standard Band FIR Filter Design: fir1
fir1
uses a least-squares approximation to compute filter
coefficients and then smooths the impulse response with a window. For an
overview of windows and their properties, see Windows. fir1
resembles the IIR filter design functions in that it is formulated to design
filters in standard band configurations: lowpass, bandpass, highpass, and
bandstop.
The statements
n = 50; Wn = 0.4; b = fir1(n,Wn);
create row vector b
containing
the coefficients of the order n
Hamming-windowed
filter. This is a lowpass, linear phase FIR filter with cutoff frequency Wn
. Wn
is
a number between 0 and 1, where 1 corresponds to the Nyquist frequency,
half the sampling frequency. (Unlike other methods, here Wn
corresponds
to the 6 dB point.) For a highpass filter, simply 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.
b = fir1(n,Wn,window)
uses
the window specified in column vector window
for
the design. The vector window
must be n+1
elements
long. If you do not specify a window, fir1
applies
a Hamming window.
Kaiser Window Order Estimation. The kaiserord
function
estimates the filter order, cutoff frequency, and Kaiser window beta
parameter needed to meet a given set of specifications. Given a vector
of frequency band edges and a corresponding vector of magnitudes,
as well as maximum allowable ripple, kaiserord
returns
appropriate input parameters for the fir1
function.
Multiband FIR Filter Design: fir2
The fir2
function also
designs windowed FIR filters, but with an arbitrarily shaped piecewise
linear frequency response. This is in contrast to fir1
, which only designs filters in standard
lowpass, highpass, bandpass, and bandstop configurations.
The commands
n = 50; f = [0 .4 .5 1]; m = [1 1 0 0]; b = fir2(n,f,m);
return row vector b
containing the n+1
coefficients of
the order n
FIR filter whose frequency-magnitude
characteristics match those given by vectors f
and
m
. 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 specified in f
. (The IIR counterpart of this
function is yulewalk
, which also designs
filters based on arbitrary piecewise linear magnitude responses. See IIR Filter Design for details.)
Multiband FIR Filter Design with Transition Bands
The firls
and firpm
functions provide a more
general means of specifying the ideal specified filter than the fir1
and fir2
functions. These functions
design Hilbert transformers, differentiators, and other filters with odd symmetric
coefficients (type III and type IV linear phase). They also let you include
transition or “don't care” regions in which the error is not
minimized, and perform band dependent weighting of the minimization.
The firls
function is an extension of the fir1
and
fir2
functions in that it minimizes the integral of the
square of the error between the specified frequency response and the actual
frequency response.
The firpm
function implements the Parks-McClellan algorithm, which uses the
Remez exchange algorithm and Chebyshev approximation theory to design filters with
optimal fits between the specified and actual frequency responses. The filters are
optimal in the sense that they minimize the maximum error between the specified
frequency response and the actual frequency response; they are sometimes called
minimax filters. Filters designed in this way exhibit an
equiripple behavior in their frequency response, and hence are also known as
equiripple filters. The Parks-McClellan FIR filter design
algorithm is perhaps the most popular and widely used FIR filter design
methodology.
The syntax for firls
and firpm
is
the same; the only difference is their minimization schemes. The next
example shows how filters designed with firls
and firpm
reflect
these different schemes.
Basic Configurations
The default mode of operation of firls
and firpm
is to
design type I or type II linear phase filters, depending on whether the order
you want is even or odd, respectively. A lowpass example with approximate
amplitude 1 from 0 to 0.4 Hz, and approximate amplitude 0 from
0.5 to 1.0 Hz is
n = 20; % Filter order f = [0 0.4 0.5 1]; % Frequency band edges a = [1 1 0 0]; % Amplitudes b = firpm(n,f,a);
From 0.4 to 0.5 Hz, firpm
performs no error
minimization; this is a transition band or “don't care”
region. A transition band minimizes the error more in the bands that
you do care about, at the expense of a slower transition rate. In
this way, these types of filters have an inherent trade-off similar
to FIR design by windowing.
To compare least squares to equiripple filter design, use firls
to
create a similar filter. Type
bb = firls(n,f,a);
and compare their frequency responses.
Note that the y-axis shown in the figure below is in Magnitude Squared.
The filter designed with firpm
exhibits equiripple
behavior. Also note that the firls
filter has a
better response over most of the passband and stopband, but at the
band edges (f
= 0.4
and f
= 0.5
), the response
is further away from the ideal than the firpm
filter.
This shows that the firpm
filter's maximum error
over the passband and stopband is smaller and, in fact, it is the
smallest possible for this band edge configuration and filter length.
Think of frequency bands as lines over short frequency intervals. firpm
and
firls
use this scheme to represent any piecewise linear
frequency-response function with any transition bands. firls
and firpm
design lowpass, highpass, bandpass, and bandstop
filters; a bandpass example is
f = [0 0.3 0.4 0.7 0.8 1]; % Band edges in pairs a = [0 0 1 1 0 0]; % Bandpass filter amplitude
Technically, these f
and a
vectors
define five bands:
Two stopbands, from 0.0 to 0.3 and from 0.8 to 1.0
A passband from 0.4 to 0.7
Two transition bands, from 0.3 to 0.4 and from 0.7 to 0.8
Example highpass and bandstop filters are
f = [0 0.7 0.8 1]; % Band edges in pairs a = [0 0 1 1]; % Highpass filter amplitude f = [0 0.3 0.4 0.5 0.8 1]; % Band edges in pairs a = [1 1 0 0 1 1]; % Bandstop filter amplitude
An example multiband bandpass filter is
f = [0 0.1 0.15 0.25 0.3 0.4 0.45 0.55 0.6 0.7 0.75 0.85 0.9 1]; a = [1 1 0 0 1 1 0 0 1 1 0 0 1 1];
Another possibility is a filter that has as a transition region the line connecting the passband with the stopband; this can help control “runaway” magnitude response in wide transition regions:
f = [0 0.4 0.42 0.48 0.5 1]; a = [1 1 0.8 0.2 0 0]; % Passband, linear transition, % stopband
The Weight Vector
Both firls
and firpm
allow
you to place more or less emphasis on minimizing the error in certain
frequency bands relative to others. To do this, specify a weight vector
following the frequency and amplitude vectors. An example lowpass
equiripple filter with 10 times less ripple in the stopband than the
passband is
n = 20; % Filter order f = [0 0.4 0.5 1]; % Frequency band edges a = [1 1 0 0]; % Amplitudes w = [1 10]; % Weight vector b = firpm(n,f,a,w);
A legal weight vector is always half the length of the f
and a
vectors;
there must be exactly one weight per band.
Anti-Symmetric Filters / Hilbert Transformers
When called with a trailing 'h'
or 'Hilbert'
option,
firpm
and firls
design FIR filters
with odd symmetry, that is, type III (for even order) or type IV (for odd order)
linear phase filters. An ideal Hilbert transformer has this anti-symmetry
property and an amplitude of 1 across the entire frequency range. Try the
following approximate Hilbert transformers and plot them.
b = firpm(21,[0.05 1],[1 1],'h'); % Highpass Hilbert bb = firpm(20,[0.05 0.95],[1 1],'h'); % Bandpass Hilbert
You can find the delayed Hilbert transform of a signal x
by
passing it through these filters.
fs = 1000; % Sampling frequency t = (0:1/fs:2)'; % Two second time vector x = sin(2*pi*300*t); % 300 Hz sine wave example signal xh = filter(bb,1,x); % Hilbert transform of x
The analytic signal corresponding to x
is
the complex signal that has x
as its real part
and the Hilbert transform of x
as its imaginary
part. For this FIR method (an alternative to the hilbert
function),
you must delay x
by half the filter order to create
the analytic signal:
xd = [zeros(10,1); x(1:length(x)-10)]; % Delay 10 samples xa = xd + j*xh; % Analytic signal
This method does not work directly for filters of odd order,
which require a noninteger delay. In this case, the hilbert
function,
described in Hilbert Transform, estimates the analytic signal. Alternatively,
use the resample
function to delay the signal by
a noninteger number of samples.
Differentiators
Differentiation of a signal in the time domain is equivalent
to multiplication of the signal's Fourier transform by an imaginary
ramp function. That is, to differentiate a signal, pass it through
a filter that has a response H(ω) =
jω.
Approximate the ideal differentiator (with a delay) using firpm
or firls
with
a 'd'
or 'differentiator'
option:
b = firpm(21,[0 1],[0 pi],'d');
For a type III filter, the differentiation band should stop short of the Nyquist frequency, and the amplitude vector must reflect that change to ensure the correct slope:
bb = firpm(20,[0 0.9],[0 0.9*pi],'d');
In the 'd'
mode, firpm
weights
the error by 1/ω in nonzero amplitude bands to minimize the
maximum relative error. firls
weights
the error by (1/ω)^{2} in nonzero amplitude
bands in the 'd'
mode.
The plot shows the magnitude responses for the differentiators.
Constrained Least Squares FIR Filter Design
The Constrained Least Squares (CLS) FIR filter design functions implement a technique that enables you to design FIR filters without explicitly defining the transition bands for the magnitude response. The ability to omit the specification of transition bands is useful in several situations. For example, it may not be clear where a rigidly defined transition band should appear if noise and signal information appear together in the same frequency band. Similarly, it may make sense to omit the specification of transition bands if they appear only to control the results of Gibbs phenomena that appear in the filter's response. See Selesnick, Lang, and Burrus [2] for discussion of this method.
Instead of defining passbands, stopbands, and transition regions, the CLS method accepts a cutoff frequency (for the highpass, lowpass, bandpass, or bandstop cases), or passband and stopband edges (for multiband cases), for the response you specify. In this way, the CLS method defines transition regions implicitly, rather than explicitly.
The key feature of the CLS method is that it enables you to define upper and lower thresholds that contain the maximum allowable ripple in the magnitude response. Given this constraint, the technique applies the least square error minimization technique over the frequency range of the filter's response, instead of over specific bands. The error minimization includes any areas of discontinuity in the ideal, "brick wall" response. An additional benefit is that the technique enables you to specify arbitrarily small peaks resulting from the Gibbs phenomenon.
There are two toolbox functions that implement this design technique.
Description | Function |
---|---|
Constrained least square multiband FIR filter design | |
Constrained least square filter design for lowpass and highpass linear phase filters |
For details on the calling syntax for these functions, see their reference descriptions in the Function Reference.
Basic Lowpass and Highpass CLS Filter Design
The most basic of the CLS design functions, fircls1
,
uses this technique to design lowpass and highpass FIR filters. As
an example, consider designing a filter with order 61 impulse response
and cutoff frequency of 0.3 (normalized). Further, define the upper
and lower bounds that constrain the design process as:
Maximum passband deviation from 1 (passband ripple) of 0.02.
Maximum stopband deviation from 0 (stopband ripple) of 0.008.
To approach this design problem using fircls1
,
use the following commands:
n = 61; wo = 0.3; dp = 0.02; ds = 0.008; h = fircls1(n,wo,dp,ds);
Display the magnitude response. Note that the y-axis shown is in Magnitude Squared.
Multiband CLS Filter Design
fircls
uses the same technique to design FIR filters with a specified
piecewise constant magnitude response. In this case, you can specify a vector of
band edges and a corresponding vector of band amplitudes. In addition, you can
specify the maximum amount of ripple for each band.
For example, assume the specifications for a filter call for:
From 0 to 0.3 (normalized): amplitude 0, upper bound 0.005, lower bound –0.005
From 0.3 to 0.5: amplitude 0.5, upper bound 0.51, lower bound 0.49
From 0.5 to 0.7: amplitude 0, upper bound 0.03, lower bound –0.03
From 0.7 to 0.9: amplitude 1, upper bound 1.02, lower bound 0.98
From 0.9 to 1: amplitude 0, upper bound 0.05, lower bound –0.05
Design a CLS filter with impulse response order 129 that meets the specifications. Display the magnitude response.
n = 129; f = [0 0.3 0.5 0.7 0.9 1]; a = [0 0.5 0 1 0]; up = [0.005 0.51 0.03 1.02 0.05]; lo = [-0.005 0.49 -0.03 0.98 -0.05]; h = fircls(n,f,a,up,lo);
Note that the y-axis shown is in Magnitude Squared.
Weighted CLS Filter Design
Weighted CLS filter design lets you design lowpass or highpass
FIR filters with relative weighting of the error minimization in each
band. The fircls1
function enables you to specify
the passband and stopband edges for the least squares weighting function,
as well as a constant k
that specifies the ratio
of the stopband to passband weighting.
For example, consider specifications that call for an FIR filter with impulse response order of 55 and cutoff frequency of 0.3 (normalized). Also assume maximum allowable passband ripple of 0.02 and maximum allowable stopband ripple of 0.004. In addition, add weighting requirements:
Passband edge for the weight function of 0.28 (normalized)
Stopband edge for the weight function of 0.32
Weight error minimization 10 times as much in the stopband as in the passband
To approach this using fircls1
, type
n = 55; wo = 0.3; dp = 0.02; ds = 0.004; wp = 0.28; ws = 0.32; k = 10; h = fircls1(n,wo,dp,ds,wp,ws,k);
Note that the y-axis shown is in Magnitude Squared.
Arbitrary-Response Filter Design
The cfirpm
filter design
function provides a tool for designing FIR filters with arbitrary
complex responses. It differs from the other filter design functions
in how the frequency response of the filter is specified: it accepts
the name of a function which returns the filter response calculated
over a grid of frequencies. This capability makes cfirpm
a
highly versatile and powerful technique for filter design.
This design technique may be used to produce nonlinear-phase FIR filters, asymmetric frequency-response filters (with complex coefficients), or more symmetric filters with custom frequency responses.
The design algorithm optimizes the Chebyshev (or minimax) error using an extended Remez-exchange algorithm for an initial estimate. If this exchange method fails to obtain the optimal filter, the algorithm switches to an ascent-descent algorithm that takes over to finish the convergence to the optimal solution.
Multiband Filter Design
Consider a multiband filter with the following special frequency-domain characteristics.
Band | Amplitude | Optimization Weighting |
---|---|---|
[–1 –0.5] | [5 1] | 1 |
[–0.4 +0.3] | [2 2] | 10 |
[+0.4 +0.8] | [2 1] | 5 |
A linear-phase multiband filter may be designed using the predefined
frequency-response function multiband
, as follows:
b = cfirpm(38, [-1 -0.5 -0.4 0.3 0.4 0.8], ... {'multiband', [5 1 2 2 2 1]}, [1 10 5]);
For the specific case of a multiband filter, we can use a shorthand
filter design notation similar to the syntax for firpm
:
b = cfirpm(38,[-1 -0.5 -0.4 0.3 0.4 0.8], ...
[5 1 2 2 2 1], [1 10 5]);
As with firpm
, a vector of band edges is passed to
cfirpm
. This vector defines the frequency bands over
which optimization is performed; note that there are two transition bands, from
–0.5 to –0.4 and from 0.3 to 0.4.
In either case, obtain the frequency response and plot using linear scale. Note that the range
of data is (-pi,pi)
.
The filter response for this multiband filter is complex, which is expected because of the asymmetry in the frequency domain. The impulse response is shown below.
Filter Design with Reduced Delay
Consider the design of a 62-tap lowpass filter with a half-Nyquist
cutoff. If we specify a negative offset value to the lowpass
filter
design function, the group delay offset for the design is significantly
less than that obtained for a standard linear-phase design. This filter
design may be computed as follows:
b = cfirpm(61,[0 0.5 0.55 1],{'lowpass',-16});
The resulting magnitude response is
freqz(b,1)
The y-axis is in Magnitude Squared, which you can set by right-clicking on the axis label and selecting Magnitude Squared from the menu.
The group delay of the filter reveals that the offset has been reduced from
N/2
to N/2-16
(i.e., from
30.5
to 14.5
). Now, however, the group
delay is no longer flat in the passband region. To create this plot, click the
Group Delay Response button on the toolbar.
If we compare this nonlinear-phase filter to a linear-phase
filter that has exactly 14.5 samples of group delay, the resulting
filter is of order 2*14.5, or 29. Using b = cfirpm(29,[0 0.5 0.55 1],'lowpass')
, the passband
and stopband ripple is much greater for the order 29 filter. These
comparisons can assist you in deciding which filter is more appropriate
for a specific application.