FIR filter implementation

FIR (finite impulse response) filter is a digital filter without feedback. Absence of feedback makes impulse filter response to be finite. Digital filters can be with infinite impulse response (IIR filters).
The advantages of FIR filters compared to IIR filters:
1. FIR filter can be designed with linear phase.
2. FIR filter is simple to implement.

 3. FIR filter can be easily implemented on finite-precision arithmetic (a lot of microcontrollers can operate with 16-bit words, but for IIR filter correct working, in some cases, you need 32 bits to store "Y" coefficients. So there is much more problems with IIR filter implementation on 16-bit MCU, than with FIR filter implementation).

The disadvantages of FIR filters compared to IIR filters:
1. FIR sometimes requires more memory and calculations to achieve a desired response characteristic.
2. Some responses are not practical to implement.

The equation of the FIR filter is as follows:

Input x(n) and output y(n) signals are related via the convolution operation. h(k) - filter coefficients. N is a filter length.

Block diagram of the FIR filter:

Coefficients h(n):

Input data x(n):

Let N = 3 then:

Coefficients h(n):

 

Input data x(n): 

The equation in this case is as follows: 

A detailed description of the entry and modification of data in the filter is given below:

Consider the filter implementation in C language. The input signal is a unit impulse at zero time. Thus, finding the Fourier transform of the output signal we obtain the amplitude-frequency characteristic of the filter. The length of the input signal is 1000 points. After filtering the signal, the result is stored in the file Y.dat using WinAPI functions.

int main()
{

    float X[1000] = {0};
    X[0] = 1;

    float Y[1000] = {0};


    float yn = 0;
    const int N = 3;
    float h[N] = {0.32590173795979338000,
                  0.34819652408041324000,
                  0.32590173795979338000};
    float x[N] = {0,0,0};


    for(int i=0; i<1000; i++)
    {
                                                 //  Alternative implementation
        for(int k=0; k < N-1; k++)               //  for(int k=N-1; k>0; k--)
        {                                        //  {                      
          x[N-k-1] = x[N-k-2];//shift the data   //    x[k] = x[k-1];
        }                                        //  }              

              
        x[0] = X[i]; // move input sample to buffer
        yn = 0; // clear output sample

        for(int k=0; k < N; k++)
        {
            yn += h[k]*x[k]; // multiply data on coefficients with accumulation
        }

        Y[i] = yn;
    }

    HANDLE hFile = CreateFile("Y.dat", GENERIC_READ | GENERIC_WRITE, 0, NULL, 
                               CREATE_ALWAYS, 0, NULL);

    DWORD dwCount;
    WriteFile(hFile,Y,sizeof(Y),&dwCount,NULL);
    CloseHandle(hFile);

    return 0;
}

Filter response can be plotted in MATLAB. To do this, open the file Y.dat, read the response of the filter and apply Fourier transform.

clc;
close all;
clear all;

fid = fopen('Y.dat','r'); %open file
F = fread(fid,'float');   %read file
fclose(fid);              %close file 

N = length(F);            %get signal length (number of samples)
Fs = 500;                 %set sample frequency
dt = 1/Fs;                %get sample period in time domain
df = Fs/N;                %get sample period in frequency domain
t = (1:N)*dt;             %get time vector 
f = (1:N)*df;             %get frequency vector 

plot(t,F);                %plot signal in time 

figure;
S = abs(fft(F));          %get frequency response of filter 
plot(f(1:N/2),20*log10(S(1:N/2)/max(S(1:N/2))));  %plot frequency response
xlabel('Hz');
ylabel('S, dB');

Frequency response of the filter:

Sources:

1. Emmanuel C. Ifeachor and Barrie W. Jervis Digital Signal Processing: A Practical Approach, 2nd Edition. Hardcover. from English. - Moscow: Publishing House "Williams", 2008. - 992 pp.