Least-squares linear-phase FIR filter design
b = firls(n,f,a,w)
b = firls(n,f,a,
b = firls(n,f,a,w,
firls designs a linear-phase FIR filter
that minimizes the weighted, integrated squared error between an ideal
piecewise linear function and the magnitude response of the filter
over a set of desired frequency bands.
b containing the
of the order
n FIR filter whose frequency-amplitude
characteristics approximately match those given by vectors
The output filter coefficients, or "taps," in
the symmetry relation.
These are type I (
n odd) and type II (
linear-phase filters. Vectors
the frequency-amplitude characteristics of the filter:
f is a vector of pairs of frequency
points, specified in the range between 0 and 1, where
1 corresponds to the Nyquist frequency. The frequencies must be in
increasing order. Duplicate frequency points are allowed and, in fact,
can be used to design a filter exactly the same as those returned
fir2 functions with a rectangular (
a is a vector containing the desired
amplitude at the points specified in
The desired amplitude function at frequencies between pairs of points (f(k), f(k+1)) for k odd is the line segment connecting the points (f(k), a(k)) and (f(k+1), a(k+1)).
The desired amplitude function at frequencies between pairs of points (f(k), f(k+1)) for k even is unspecified. These are transition or "don't care" regions.
the same length. This length must be an even number.
firls always uses an even filter order for
configurations with a passband at the Nyquist frequency. This is because
for odd orders, the frequency response at the Nyquist frequency is
necessarily 0. If you specify an odd-valued
it by 1.
The figure below illustrates the relationship between the
in defining a desired amplitude response.
b = firls(n,f,a, and
b = firls(n,f,a,w, specify
a filter type, where
'hilbert' for linear-phase filters
with odd symmetry (type III and type IV). The output coefficients
b obey the relation
type III and type IV filters, using a special weighting technique. For
nonzero amplitude bands, the integrated squared error has a weight
of (1/f)2 so that the
error at low frequencies is much smaller than at high frequencies. For
FIR differentiators, which have an amplitude characteristic proportional
to frequency, the filters minimize the relative integrated squared
error (the integral of the square of the ratio of the error to the
Design an FIR lowpass filter of order 255 with transition band between
fvtool to display the magnitude and phase responses of the filter.
b = firls(255,[0 0.25 0.3 1],[1 1 0 0]); fvtool(b,1,'OverlayedAnalysis','phase')
An ideal differentiator has a frequency response given by . Design a differentiator of order 30 that attenuates frequencies above . Include a factor of in the amplitude because the frequencies are normalized by . Display the zero-phase response of the filter.
b = firls(30,[0 0.9],[0 0.9*pi],'differentiator'); fvtool(b,1,'MagnitudeDisplay','zero-phase')
Design a 24th-order antisymmetric filter with piecewise linear passbands.
F = [0 0.3 0.4 0.6 0.7 0.9]; A = [0 1.0 0.0 0.0 0.5 0.5]; b = firls(24,F,A,'hilbert');
Plot the desired and actual frequency responses.
[H,f] = freqz(b,1,512,2); plot(f,abs(H)) hold on for i = 1:2:6, plot([F(i) F(i+1)],[A(i) A(i+1)],'r--') end legend('firls design','Ideal') grid on xlabel('Normalized Frequency (\times\pi rad/sample)') ylabel('Magnitude')
Design an FIR lowpass filter. The passband ranges from DC to rad/sample. The stopband ranges from rad/sample to the Nyquist frequency. Produce three different designs, changing the weights of the bands in the least-squares fit.
In the first design, make the stopband weight higher than the passband weight by a factor of 100. Use this specification when it is critical that the magnitude response in the stopband is flat and close to 0. The passband ripple is about 100 times higher than the stopband ripple.
bhi = firls(18,[0 0.45 0.55 1],[1 1 0 0],[1 100]);
In the second design, reverse the weights so that the passband weight is 100 times the stopband weight. Use this specification when it is critical that the magnitude response in the passband is flat and close to 1. The stopband ripple is about 100 times higher than the passband ripple.
blo = firls(18,[0 0.45 0.55 1],[1 1 0 0],[100 1]);
In the third design, give the same weight to both bands. The result is a filter with similar ripple in the passband and the stopband.
b = firls(18,[0 0.45 0.55 1],[1 1 0 0],[1 1]);
Visualize the magnitude responses of the three filters.
hfvt = fvtool(bhi,1,blo,1,b,1,'MagnitudeDisplay','Zero-phase'); legend(hfvt,'bhi: w = [1 100]','blo: w = [100 1]','b: w = [1 1]')
One of the following diagnostic messages is displayed when an incorrect argument is used:
F must be even length. F and A must be equal lengths. Requires symmetry to be 'hilbert' or 'differentiator'. Requires one weight per band. Frequencies in F must be nondecreasing. Frequencies in F must be in range [0,1].
A more serious warning message is
Warning: Matrix is close to singular or badly scaled.
This tends to happen when the product of the filter length and transition width grows large. In this case, the filter coefficients b might not represent the desired filter. You can check the filter by looking at its frequency response.
Reference  describes the
theoretical approach behind
firls. The function
solves a system of linear equations involving an inner product matrix
of size roughly
n/2 using the MATLAB®
This function designs type I, II, III, and IV linear-phase filters. Type
I and II are the defaults for n even and odd respectively, while the
produce type III (n even) and IV (n odd) filters. The various filter
types have different symmetries and constraints on their frequency
responses (see  for details).
|Linear Phase Filter Type||Filter Order||Symmetry of Coefficients||Response H(f), f = 0||Response H(f), f = 1 (Nyquist)|
|H(1) = 0|
H(0) = 0
H(1) = 0
H(0) = 0
 Parks, Thomas W., and C. Sidney Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987, pp. 54–83.
 Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999.