fir1

Window-based FIR filter design

Syntax

b = fir1(n,Wn)

b = fir1(n,Wn,ftype)

b = fir1(___,window)

b = fir1(___,scaleopt)

Description

b = fir1(n,Wn) uses a Hamming window to design an nth-order lowpass, bandpass, or multiband FIR filter with linear phase. The filter type depends on the number of elements of Wn.

b = fir1(n,Wn,ftype) designs a lowpass, highpass, bandpass, bandstop, or multiband filter, depending on the value of ftype and the number of elements of Wn.

b = fir1(___,window) designs the filter using the vector specified in window and any of the arguments from previous syntaxes.

b = fir1(___,scaleopt) additionally specifies whether or not the magnitude response of the filter is normalized.

Note:   Use fir2 for windowed filters with arbitrary frequency response.

Examples

FIR Bandpass Filter

Design a 48th-order FIR bandpass filter with passband $$0.35\pi \le \omega \le 0.65\pi$$ rad/sample. Visualize its magnitude and phase responses.

b = fir1(48,[0.35 0.65]);
freqz(b,1,512)

FIR Highpass Filter

Load chirp.mat. The file contains a signal, y, that has most of its power above Fs/4, or half the Nyquist frequency. The sample rate is 8192 Hz.

Design a 34th-order FIR highpass filter to attenuate the components of the signal below Fs/4. Use a cutoff frequency of 0.48 and a Chebyshev window with 30 dB of ripple.

load chirp

t = (0:length(y)-1)/Fs;

bhi = fir1(34,0.48,"high",chebwin(35,30));
freqz(bhi,1)

Filter the signal. Display the original and highpass-filtered signals. Use the same y-axis scale for both plots.

outhi = filter(bhi,1,y);

tiledlayout("flow")
nexttile
plot(t,y)
title("Original Signal")
ys = ylim;

nexttile
plot(t,outhi)
title("Highpass Filtered Signal")
xlabel("Time (s)")
ylim(ys)

Design a lowpass filter with the same specifications. Filter the signal and compare the result to the original. Use the same y-axis scale for both plots.

blo = fir1(34,0.48,chebwin(35,30));

outlo = filter(blo,1,y);

tiledlayout("flow")
nexttile
plot(t,y)
title("Original Signal")
ys = ylim;

nexttile
plot(t,outlo)
title("Lowpass Filtered Signal")
xlabel("Time (s)")
ylim(ys)

Multiband FIR Filter

Design a 46th-order FIR filter that attenuates normalized frequencies below $$0.4\pi$$ rad/sample and between $$0.6\pi$$ and $$0.9\pi$$ rad/sample. Call it bM. Compute its frequency response.

ord = 46;

low = 0.4;
bnd = [0.6 0.9];

bM = fir1(ord,[low bnd]);
[hbM,f] = freqz(bM,1);

Redesign bM so that it passes the bands it was attenuating and stops the other frequencies. Call the new filter bW. Display the frequency responses of the filters.

bW = fir1(ord,[low bnd],"DC-1");

[hbW,~] = freqz(bW,1);
plot(f/pi,mag2db(abs(hbM)),f/pi,mag2db(abs(hbW)))
legend("bM","bW",Location="best")
ylim([-75 5])
grid

Redesign bM using a Hann window. (The "DC-0" is optional.) Compare the magnitude responses of the Hamming and Hann designs.

hM = fir1(ord,[low bnd],'DC-0',hann(ord+1));

hhM = freqz(hM,1);
plot(f/pi,mag2db(abs(hbM)),f/pi,mag2db(abs(hhM)))
legend("Hamming","Hann",Location="northwest")
ylim([-75 5])
grid

Redesign bW using a Tukey window. Compare the magnitude responses of the Hamming and Tukey designs.

tW = fir1(ord,[low bnd],'DC-1',tukeywin(ord+1));

htW = freqz(tW,1);
plot(f/pi,mag2db(abs(hbW)),f/pi,mag2db(abs(htW)))
legend("Hamming","Tukey",Location="best")
ylim([-75 5])
grid

Input Arguments

n — Filter order

integer scalar

Filter order, specified as an integer scalar.

For highpass and bandstop configurations, fir1 always uses an even filter order. The order must be even because odd-order symmetric FIR filters must have zero gain at the Nyquist frequency. If you specify an odd n for a highpass or bandstop filter, then fir1 increments n by 1.

Data Types: double

Wn — Frequency constraints

scalar | two-element vector | multi-element vector

Frequency constraints, specified as a scalar, a two-element vector, or a multi-element vector. All elements of Wn must be strictly greater than 0 and strictly smaller than 1, where 1 corresponds to the Nyquist frequency: 0 < Wn < 1. The Nyquist frequency is half the sample rate or π rad/sample.

• If Wn is a scalar, then fir1 designs a lowpass or highpass filter with cutoff frequency Wn. The cutoff frequency is the frequency at which the normalized gain of the filter is –6 dB.

• If Wn is the two-element vector [w1 w2], where w1 < w2, then fir1 designs a bandpass or bandstop filter with lower cutoff frequency w1 and higher cutoff frequency w2.

• If Wn is the multi-element vector [w1 w2 ... wn], where w1 < w2 < … < wn, then fir1 returns an nth-order multiband filter with bands 0 < ω < w1, w1 < ω < w2,  …, wn < ω < 1.

Data Types: double

ftype — Filter type

'low' | 'bandpass' | 'high' | 'stop' | 'DC-0' | 'DC-1'

Filter type, specified as one of the following:

• 'low' specifies a lowpass filter with cutoff frequency Wn. 'low' is the default for scalar Wn.

• 'high' specifies a highpass filter with cutoff frequency Wn.

• 'bandpass' specifies a bandpass filter if Wn is a two-element vector. 'bandpass' is the default when Wn has two elements.

• 'stop' specifies a bandstop filter if Wn is a two-element vector.

• 'DC-0' specifies that the first band of a multiband filter is a stopband. 'DC-0' is the default when Wn has more than two elements.

• 'DC-1' specifies that the first band of a multiband filter is a passband.

window — Window

vector

Window, specified as a vector. The window vector must have n + 1 elements. If you do not specify window, then fir1 uses a Hamming window. For a list of available windows, see Windows.

fir1 does not automatically increase the length of window if you attempt to design a highpass or bandstop filter of odd order.

Example: kaiser(n+1,0.5) specifies a Kaiser window with shape parameter 0.5 to use with a filter of order n.

Example: hamming(n+1) is equivalent to leaving the window unspecified.

Data Types: double

scaleopt — Normalization option

'scale' (default) | 'noscale'

Normalization option, specified as either 'scale' or 'noscale'.

• 'scale' normalizes the coefficients so that the magnitude response of the filter at the center of the passband is 1 (0 dB).

• 'noscale' does not normalize the coefficients.

Output Arguments

b — Filter coefficients

row vector

Filter coefficients, returned as a row vector of length n + 1. The coefficients are sorted in descending powers of the Z-transform variable z:

B(z) = b(1) + b(2)z^–1 + … + b(n+1)z^–n.

Algorithms

fir1 uses a least-squares approximation to compute the filter coefficients and then smooths the impulse response with window.

References

[1] Digital Signal Processing Committee of the IEEE Acoustics, Speech, and Signal Processing Society, eds. Programs for Digital Signal Processing. New York: IEEE Press, 1979, Algorithm 5.2.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The fir1 function supports GPU array input with these usage notes and limitations:

• Running this function on the GPU might be slower than on the CPU.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2025a: Use gpuArray objects

The fir1 function supports gpuArray objects. You must have Parallel Computing Toolbox™ to use this functionality.