EXAMPLE: A DIGITAL FIR FILTER
One of the most widely used digital signal-processing elements is the finite impulse response, or FIR, filter. Designers use filters to alter the magnitude or frequency content of a data signal, usually to isolate or accentuate a particular region of interest within the sample data spectrum. In this regard, you can think of filters as a method of preconditioning a signal. In a typical filter application, incoming data samples combine with filter coefficients through carefully synchronized mathematical operations, which are dependent on the filter type and implementation strategy, and then move on to the next processing stage. If the data source and destination are analog signals, then the samples must first pass through an A/D converter, and the results fed through a D/A converter.
The simplest form of a FIR filter is implemented through a series of delay elements, multipliers and an adder tree or chain.
Mathematically, this equation describes the single-channel FIR filter:
You can think of the terms in the equation as input samples, output samples and coefficients. If S is a continuous stream of input samples and Y is the resulting filtered stream of output samples, then n and k correspond to a particular instant in time. Thus, to compute the output sample Y(n) at time n, a group of samples at N different points in time, or s(n), s(n-1), s(n-2), … s(n-N+1), is required. The group of N input samples is multiplied by N coefficients and summed together to form the final result, Y.
Figure 2 – FIR filter of length 31
Figure 2 is a block diagram for a simple 31-tap FIR filter (length N = 31).
Various design tools are available to help select the ideal length of a filter and the coefficient values. The goal is to select the appropriate parameters to achieve the required filter performance. The most popular design tool for choosing these parameters is MATLAB. Once you have selected the filter parameters, the implementation follows the mathematical equation.
The basic steps for implementation of an FIR filter are:
1. Sample the incoming data stream.
2. Organize the input samples in a buffer so that each captured sample may be multiplied by each filter coefficient.
3. Multiply each data sample by each coefficient and accumulate the result.
4. Output filtered result.
A typical C program for implementing this FIR filter on a processor using a multiply–accumulate approach is shown in the code below.