A Matlab Function for FIR Half-Band Filter Design

• Matlab

 Half-band filters have -6 dB frequency of 1/4 the sample rate, and odd symmetry of the frequency response about 1/4 the sample rate.  Given the odd-symmetry of the response, the passband and stopband edge frequencies are symmetric with respect to fs/4.  This symmetry makes the halfband filter ideal for decimation by 2 or interpolation by 2.  And, remarkably, the odd-indexed coefficients of FIR half-band filters are zero, except for the main tap coefficient.

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FIR Half-band filters are not difficult to design.  In an earlier post [1], I showed how to design them using the window method.  Here, I provide a short Matlab function halfband_synth that uses the Parks-McClellan algorithm (Matlab function firpm [2]) to synthesize half-band filters.  Compared to the window method, this method uses fewer taps to achieve a given performance.  The function’s code is listed at the end of the article.  The function call is as follows:

    b= halfband_synth(fpass,fs,ntaps)

where

fpass = desired passband edge frequency, Hz.  fpass must be less than fs/4.

fs = sample rate, Hz.

ntaps = desired number of filter taps

ntaps + 1 must be an integer multiple of 4, so ntaps = 7, 11, 15, …

ntaps = 3 is not allowed.(For ntaps = 3, the coefficients are [1 2 1]/4).

b = vector of filter coefficients.  Given indexing [b₀ b₁ b₂ …], the odd-indexed coefficients are zero, except for the main tap.

Following are two of examples of half-band filter synthesis using the function.

Example 1.

Let fs = 1 Hz, fpass = 0.15*fs, and ntaps = 11.  The following Matlab code computes the half-band coefficients.

    fs= 1;
    fpass= 0.15*fs;
    ntaps = 11;
    b= halfband_synth(fpass,fs,ntaps);

The function prints fstop to the workspace: 

    fstop = 0.3500

and it computes the coefficients:

    b= [.0163 0 -.0683 0 .3038 .5 .3038 0 -.0683 0 .0163]

The filter has (ntaps + 3)/2 = 7 non-zero coefficients.  The coefficients are plotted in Figure 1 (top).  The frequency response can be computed as follows:

    [H,f]= freqz(b,1,256,fs);
    Hmag = abs(H);
    HdB= 20*log10(Hmag);

The magnitude of H is plotted in Figure 1 (bottom), showing the odd symmetry with respect to f_s/4.  Note |H| = 0.5 at f_s/4.  The dB-magnitude response is plotted in Figure 2, where we see that the response is equiripple in the stopband (top) and passband (bottom), as expected from the Parks-McClellan algorithm.

Figure 1.  Top: 11-tap half-band filter coefficients.   Bottom: Magnitude response.

Figure 2.  Top: 11-tap half-band filter dB-magnitude response.

Bottom: dB-magnitude response in the passband.

Example 2.

Let fs = 200 Hz, fpass = 40 Hz, and ntaps = 35.  The following Matlab code computes the half-band coefficients.

    fs = 200;
    fpass= 40;
    ntaps = 35;
    b= halfband_synth(fpass,fs,ntaps)

The filter has (ntaps + 3)/2 = 19 non-zero coefficients.  The coefficients and magnitude response are plotted in Figure 3.  The dB-magnitude response is plotted in Figure 4.

Figure 3.  Top: 35-tap half-band filter coefficients.   Bottom: Magnitude response.

Figure 4.  35-tap half-band filter dB-magnitude response.

How It Works

The simplest method for designing an equiripple half-band FIR filter would be to compute the stopband edge frequency that is symmetrical about fs/4 with the passband edge frequency, and set a goal function and frequencies as follows, where fpass is passband edge frequency and fs is sample frequency:

    fstop = fs/2 - fpass              % Hz stopband edge frequency
    a= [1 1 0 0];                     % amplitude goal function
    f= [0 fpass fstop fs/2]/(fs/2);   % normalized frequencies

The coefficients would be computed using firpm.  The odd-index coefficients from firpm would be close to zero, but not exactly zero.  These coefficients would then be set to exactly 0.

Here, we use a more accurate method described by Vaidyanathan and Nguyen [3].  To illustrate the method, we start by considering the half-band filter coefficients from Example 1:

    b= [.0163 0 -.0683 0 .3038 .5 .3038 0 -.0683 0 .0163]

Now define coefficients g:

    g = 2*b(1:2:end)            (1)
      = 2*[.0163 -.0683 .3038 .3038 -.0683 .0163]

The magnitude response |H(f)| of coefficients b is plotted in the top of Figure 5, and the magnitude response |G(f)| of coefficients g is plotted in the bottom.  The passband edge of |G| is exactly 2*fpass, where fpass is the passband edge of |H|, and |G| is zero at fs/2.  We call g a one-band filter.

Now let’s reverse this process by first synthesizing the one-band filter g and using it to compute the half-band filter coefficients b.  The passband edge is 2*fpass and the stopband is just the single frequency fs/2, so we can specify an ideal |G(f)| as follows:

    a = [1 1 0 0];                        % amplitude goal function
    f = [0 2*fpass fs/2 fs/2]/(fs/2);     % normalized frequencies

The desired filter b has ntaps = 11, and the length of g is:

    gtaps = (ntaps + 1)/2 = 6          (2)

Now g can be synthesized using firpm:

    g = firpm(gtaps-1,f,a);

The half-band filter b is realized by inserting zeros between the elements of g/2, which is the inverse of the process shown in Equation 1.  Finally, the main tap is set to 0.5.  The synthesis process requires only gtaps = (ntaps + 1)/2 coefficients, vs ntaps for the conventional approach. This results in quicker computation and more accurate coefficients.

Figure 5.  Top: Magnitude response |H| of coefficients b.

 Bottom:  Magnitude response |G| of coefficients g.

Matlab Function halfband_synth

The code for the function halfband_synth is listed below.  The number of taps is called ntaps.  ntaps + 1 must be an integer multiple of 4, so ntaps = 7, 11, 15, …  It is possible to design a half-band filter with an even number of taps, but this results in a filter with no zero-valued coefficients.  The cases ntaps = 9, 13, 17, …, are not used because the end coefficients are zero.  Finally, note that Matlab has a function to design half-band filters in the DSP System Toolbox [4].

Disclaimer:  I believe this code to be correct, but it may contain errors.  Be sure to verify the coefficients before using them in a design.

    % halfband_synth.m 7/2/25 Neil Robertson
    % Half-band filter synthesis function using firpm,
    % based on Vaidyanathan and Nguyen [3].
    %
    % b = halfband_synth(fpass,fs,ntaps)
    % fs = sample rate, Hz.
    % fpass = passband edge frequency, Hz.  fpass < fs/4.
    % ntaps = number of taps. ntaps = 7, 11, 15, ...
    % b = vector of filter coefficients.
    %
    function b = halfband_synth(fpass,fs,ntaps)
    if fpass >= fs/4
        error('fpass must be less than fs/4')
    end
    if mod((ntaps+1),4) ~= 0
        error('ntaps+1 must be an integer multiple of 4')
    end
    a = [1 1 0 0];                        % amplitude goal function
    f = [0 2*fpass fs/2 fs/2]/(fs/2);     % frequencies for optimization
    gtaps = (ntaps + 1)/2;                % one-band filter number of taps
    g = firpm(gtaps-1,f,a);               % one-band filter coefficients
    b = zeros(1,ntaps);
    b(1:2:end) = g/2;                     % half-band coefficients
    b(gtaps) = .5;                        % center tap coefficient
    fstop = fs/2 - fpass                  % Hz stopband edge frequency

References

1. Robertson, Neil, “Simplest Calculation of Half-band Filter Coefficients”, DSPRelated.com, Nov., 2017,

https://www.dsprelated.com/showarticle/1113.php

2.Matlab website, “firpm”, https://www.mathworks.com/help/signal/ref/firpm.html

3.Vaidyanathan, P. P., and Nguyen, Truong Q., “A ‘Trick’ for the Design of FIR Half-Band Filters”, IEEE Transactions on Circuits and Systems, VOL CAS-34, No. 3, March 1987. https://authors.library.caltech.edu/records/czbzh-df707

4.Matlab website, “FIR Halfband Filter Design”,https://www.mathworks.com/help/dsp/ug/fir-halfband-filter-design.html

Neil Robertson, July 2025

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