Signal Processing Magic (6) -- Half Band Filter

Half-band filter is a FIR filter often used in decimation by 2 filtering operation. Benefit of this filter is that half of the filter coefficients are 0's, which reduces the complexity of filtering operation. The name of "half-band" comes from that the filter's cut-off frequency is roughly fs/4, thus the filter's pass band is around half of the total.

To design a half-band filter, it starts from a sinc function as \[s[n]=\frac{sin(\pi*n/2)}{\pi*n/2},n=-N,-(N-1),...,N\]
We know that when N approaches infinite, freq response of this sinc function is rectangle ranging from -fs/4 to fs/4. But in real world, N is always a limited number, and when that occurs, the filter does not look great. Ripples occur in pass band and rejection in stop band is often not good enough either.

How to improve this? Windowing method comes for rescue. As introduced before, windowing method enhances the pass band (main lobe) and suppresses the stop band (side lobes). Assuming window is w[n], after adding window, the new filter becomes
\[h[n]=s[n]*w[n]\]
Since half of \(s[n]\)'s coefficients are 0's, half of \(h[n]\)'s coefficients are 0's too.

Matlab script to generate a 21-tap half-band filter with Hamming window is shown below:

N = 10;
x = [-10 : 10]/2;
s = sinc(x)
w = hamming(2*N+1);
h = s.*w.';

figure;
subplot(2,1,1);stem(s);title('Sinc');
subplot(2,1,2);stem(h);title('Half-band');

[H, F] = freqz(s, 1, 'whole', 2^15);
[Hf, Ff] = freqz(h, 1, 'whole', 2^15);

figure; hold on;
plot(F(1:end/2)/pi/2, 20*log10(abs(H(1:end/2))));
plot(Ff(1:end/2)/pi/2, 20*log10(abs(Hf(1:end/2))));
grid;
ylabel('Magnitude (dB)'); xlabel('Freq (fs)');
legend('Sinc','Half-band');

Filter coefficients for Sinc and half-band filter

Freq response for Sinc and half-band filter