In this lesson I will show how to code a finite impulse response (FIR) digital filter in the C programming language using floating point operations. Understanding the basic FIR filter is a stepping stone to understanding more complicated filters such as polyphase FIR filters and adaptive FIR filters.
A FIR filter is one in which the output depends only on the inputs (and not on previous outputs). The process is a discretetime convolution between the input signal and the impulse response of the filter to apply. I will call this impulse response “the filter coefficients.” The following equivalent equations describe the convolution process:
equation 1:
equation 2:
where h is the impulse response, x is the input, y[n] is the output at time n and N is the length of the impulse response (the number of filter coefficients). For each output calculated, there are N multiplyaccumulate operations (MACs). There is a choice of going through the filter coefficients forward and the input backwards (equation 1), or through the filter coefficients backwards and the input forwards (equation 2). In the example that follows, I will use the form in equation 1.
In my code example, samples will be stored in a working array, with lower indices being further back in time (at the “low” end). The input samples will be arranged in an array long enough to apply all of the filter coefficients. The length of the array required to process one input sample is N (the number of coefficients in the filter). The length required for M input samples is N – 1 + M.
A function will be defined that filters several input samples at a time. For example, a realtime system for processing telephony quality speech might process eighty samples at a time. This is done to improve the efficiency of the processing, since the overhead of calling the function is eighty times smaller than if it were called for every single sample. The drawback is that additional delay is introduced in the output.
The basic steps for applying a FIR filter are the following:

Arrange the new samples at the high end of the input sample buffer.

Loop through an outer loop that produces each output sample.

Loop through an inner loop that multiplies each filter coefficient by an input sample and adds to a running sum.

Shift the previous input samples back in time by the number of new samples that were just processed.
The code example defines an initialization function and a filtering function, and includes a test program. Download the PDF from the following link to see the code, or view it inline below:
#include <stdio.h> #include <stdint.h> ////////////////////////////////////////////////////////////// // Filter Code Definitions ////////////////////////////////////////////////////////////// // maximum number of inputs that can be handled // in one function call #define MAX_INPUT_LEN 80 // maximum length of filter than can be handled #define MAX_FLT_LEN 63 // buffer to hold all of the input samples #define BUFFER_LEN (MAX_FLT_LEN  1 + MAX_INPUT_LEN) // array to hold input samples double insamp[ BUFFER_LEN ]; // FIR init void firFloatInit( void ) { memset( insamp, 0, sizeof( insamp ) ); } // the FIR filter function void firFloat( double *coeffs, double *input, double *output, int length, int filterLength ) { double acc; // accumulator for MACs double *coeffp; // pointer to coefficients double *inputp; // pointer to input samples int n; int k; // put the new samples at the high end of the buffer memcpy( &insamp[filterLength  1], input, length * sizeof(double) ); // apply the filter to each input sample for ( n = 0; n < length; n++ ) { // calculate output n coeffp = coeffs; inputp = &insamp[filterLength  1 + n]; acc = 0; for ( k = 0; k < filterLength; k++ ) { acc += (*coeffp++) * (*inputp); } output[n] = acc; } // shift input samples back in time for next time memmove( &insamp[0], &insamp[length], (filterLength  1) * sizeof(double) ); } ////////////////////////////////////////////////////////////// // Test program ////////////////////////////////////////////////////////////// // bandpass filter centred around 1000 Hz // sampling rate = 8000 Hz #define FILTER_LEN 63 double coeffs[ FILTER_LEN ] = { 0.0448093, 0.0322875, 0.0181163, 0.0087615, 0.0056797, 0.0086685, 0.0148049, 0.0187190, 0.0151019, 0.0027594, 0.0132676, 0.0232561, 0.0187804, 0.0006382, 0.0250536, 0.0387214, 0.0299817, 0.0002609, 0.0345546, 0.0525282, 0.0395620, 0.0000246, 0.0440998, 0.0651867, 0.0479110, 0.0000135, 0.0508558, 0.0736313, 0.0529380, 0.0000709, 0.0540186, 0.0766746, 0.0540186, 0.0000709, 0.0529380, 0.0736313, 0.0508558, 0.0000135, 0.0479110, 0.0651867, 0.0440998, 0.0000246, 0.0395620, 0.0525282, 0.0345546, 0.0002609, 0.0299817, 0.0387214, 0.0250536, 0.0006382, 0.0187804, 0.0232561, 0.0132676, 0.0027594, 0.0151019, 0.0187190, 0.0148049, 0.0086685, 0.0056797, 0.0087615, 0.0181163, 0.0322875, 0.0448093 }; void intToFloat( int16_t *input, double *output, int length ) { int i; for ( i = 0; i < length; i++ ) { output[i] = (double)input[i]; } } void floatToInt( double *input, int16_t *output, int length ) { int i; for ( i = 0; i 32767.0 ) { input[i] = 32767.0; } else if ( input[i] < 32768.0 ) { input[i] = 32768.0; } // convert output[i] = (int16_t)input[i]; } } // number of samples to read per loop #define SAMPLES 80 int main( void ) { int size; int16_t input[SAMPLES]; int16_t output[SAMPLES]; double floatInput[SAMPLES]; double floatOutput[SAMPLES]; FILE *in_fid; FILE *out_fid; // open the input waveform file in_fid = fopen( "input.pcm", "rb" ); if ( in_fid == 0 ) { printf("couldn't open input.pcm"); return; } // open the output waveform file out_fid = fopen( "outputFloat.pcm", "wb" ); if ( out_fid == 0 ) { printf("couldn't open outputFloat.pcm"); return; } // initialize the filter firFloatInit(); // process all of the samples do { // read samples from file size = fread( input, sizeof(int16_t), SAMPLES, in_fid ); // convert to doubles intToFloat( input, floatInput, size ); // perform the filtering firFloat( coeffs, floatInput, floatOutput, size, FILTER_LEN ); // convert to ints floatToInt( floatOutput, output, size ); // write samples to file fwrite( output, sizeof(int16_t), size, out_fid ); } while ( size != 0 ); fclose( in_fid ); fclose( out_fid ); return 0; }
First is the definition of the working array to hold the input samples. I have defined the array length to handle up to 80 input samples per function call, and a filter length up to 63 coefficients. In other words, the filter function that uses this working array will support between 1 to 80 input samples, and a filter length between 1 and 63.
Next is the initialization function, which simply zeroes the entire input sample array. This insures that all of the inputs that come before the first actual input are set to zero.
Then comes the FIR filtering function itself. The function takes pointers to the filter coefficients, the new input samples, and the output sample array. The “length” argument specifies the number of input samples to process (must be between 0 and 80 inclusive). The “filterLength” argument specifies the number of coefficients in the filter (must be between 1 and 63 inclusive). Note that the same filter coefficients and filterLength should be passed in each time the firFloat function is called, until the entire input is processed.
The first step in the function is the storage of the new samples. Note closely where the samples are placed in the array.
Next comes the outer loop that processes each input sample to produce one output sample. The outer loop sets up pointers to the coefficients and the input samples, and zeroes an accumulator value for the multiplyaccumulate operation in the inner loop. It also stores each calculated output. The inner loop simply performs the multiplyaccumulate operation.
Finally comes the step to move the input samples back in time. This is probably the most difficult step in the FIR filter and the part that always takes me the longest to work out. Pay close attention to the position of the samples to move, and the number of samples moved. The amount of time shift is equal to the number of input samples just processed. The number of samples to move is one less than the number of coefficients in the filter. Also note the use of the memmove function rather than the memcpy function, since there is a potential for moving overlapping data (whether or not the data overlaps depends on the relative lengths of the input samples and number of coefficients).
That concludes the code example for the FIR filter. In Part 2, I will show how to do the same thing using fixed point operations.