[FPGA] FIR filter - half band filter

1. Definition of half band filter

     Half band filter is a special FIR filter , Its The order can only be even , Odd length . The filter coefficient except the intermediate value is 0.5 Outside , rest The coefficients of even numbers are 0（ Therefore, the multiplication and addition operations during filtering are greatly saved ）, Half band filter is a special low-pass FIR Digital filter . This filter is due to Passband and stopband are relative to one-half Nyquist Frequency symmetry , Therefore, nearly half of the filter coefficients are accurate to zero .
     Half band FIR General frequency response of filter ：

The amplitude frequency response is about π/ 2(fs/4) symmetry , The sampling rate is normalized to 2π radians/sec, The relationship between passband and stopband frequency is ：

The passband and stopband ripple are equal , These characteristics are reflected in the impulse response of the filter ; for example , For filters with an odd number of tap coefficients , About half of the coefficients are zero and interlaced between non-zero coefficients . The figure below 3-20, Yes. 11 A tapped half band filter ：

When realizing half band filter , The effective implementation structure obtained by using this feature is shown in the figure below ：

Particular attention ：
Digital filters are 2π The normalized ,2π Corresponding fs Sampling rate . This and FFT The result is the same . Sampling time discrete series , Spectrum to 2π For cycles ,（ Also with Fs For cycles ）

2 Usage example of half band filter

requirement ： The signal frequency is 25hz and 50hz, The sampling rate is 120hz, Design a half band filter to filter out 50hz The frequency component of
Filter design process ：

% The signal frequency is 25hz and 50hz, The sampling rate is 120hz, Design a half band filter to filter out 50hz The frequency component of 
clear all;
clc;
fs=120;
f0=25;
f1=50;
N=256;
n=0:(N-1);
% signal = sin(2*pi*f0/fs*n);
signal = sin(2*pi*f0/fs*n)+sin(2*pi*f1/fs*n);
coff = [0,0.0671297764639564,0,-0.0953987484367994,0,0.314556319097219,0.500000000000000,0.314556319097219,0,-0.0953987484367994,0,0.0671297764639564,0];
signal_f = filter(coff,1,signal);
fft_plot(signal,fs,' Signal before filtering ');
fft_plot(signal_f,fs,' Filtered signal ');


function fft_plot(y,fs,s_name)
    L_i = length(y)*100;
    s_i_fft = fft(y,L_i);
    s_i_fftshfit = fftshift(s_i_fft);
    P = abs(s_i_fftshfit)/length(y)*2;   % For non DC components or 0 It's divided by N multiply 2, For the DC component, it is directly divided by N 
%     fshift = (-L_i/2:L_i/2-1)*(fs/L_i);
    fshift = linspace(-fs/2,fs/2,L_i);
    figure;
    plot(fshift,P);
    title([s_name,' Bilateral spectrum of  ']);
    xlabel('f (Hz)');
    ylabel('|P(f)|');
end

The results are shown in the following figure ：

Reference material ：
【1】FPGA Signal processing series ——FIR Half band decimation filter
【2】MATLAB Generate FPGA COE Document XILINX FPGA Filter coefficients
【3】fdatool Toolbox design FIR Digital filter