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m01 - FIR Digital Filters Overview

Module by: C. Sidney Burrus

Summary: The finite duration impulse response (FIR) digital filter in one of two structures of linear, time invariant digital filters. It is straight forward to design, capable of linear phase, always stable, and can be implemented using the FFT.

FIR Digital Filters

There are two types of linear, time-invariant digital filters. We will investigate digital filters with a finite-duration impulse response (FIR) in this section and those with an infinite-duration impulse response (IIR) in another document. FIR filters have characteristics that make them useful in many applications [1,2].

  1. FIR filters can achieve an exactly linear phase frequency response
  2. FIR filters cannot be unstable.
  3. FIR filters are generally less sensitive to coefficient round-off and finite-precision arithmetic than IIR filters.
  4. FIR filters design methods are generally linear.
  5. FIR filters can be efficiently realized on general or special-purpose hardware.

However, frequency responses that have rapid transition between pass and stop bands are often better realized with IIR filters.

It is the purpose of this section to examine and evaluate these characteristics which are important in the design of the four basic types of linear-phase FIR filters.

Because of the usual methods of implementation, the Finite Impulse Response (FIR) filter is also called a nonrecursive filter or a convolution filter. From the time-domain view of this operation, the FIR filter is sometimes called a moving-average or running-average filter. All of these names represent useful interpretations that are discussed in this section; however, the name, FIR, is most commonly seen in filter-design literature and is used in these notes.

The duration or sequence length of the impulse response of these filters is by definition finite; therefore, the output can be written as a finite convolution sum by

y ( n ) = ∑ m = 0 N − 1 h ( m ) x ( n − m ) y ( n ) = ∑ m = 0 N − 1 h ( m ) x ( n − m ) (1)
where n n and m m are integers, perhaps representing samples in time, and where x ( n ) x ( n ) is the input sequence, y ( n ) y ( n ) the output sequence, and h ( n ) h ( n ) is the length-N impulse response of the filter. With a change of index variables, this can also be written as
y ( n ) = ∑ m = n n − N + 1 h ( n − m ) x ( m ) . y ( n ) = ∑ m = n n − N + 1 h ( n − m ) x ( m ) . (2)

If the FIR filter is interpreted as an extension of a moving sum or as a weighted moving average, some of its properties can easily be seen. If one has a sequence of numbers, e.g., prices from the daily stock market for a particular stock, and would like to remove the erratic variations in order to discover longer term trends, each number could be replaced by the average of itself and the preceding three numbers, i.e., the variations within a four-day period would be ``averaged out" while the longer-term variations would remain. To illustrate how this happens, consider an artificial signal x ( n ) x ( n ) containing a linear term, K 1 n K 1 n , and an undesired oscillating term added to it, such that

x ( n ) = K 1 n + K 2 cos ( π n ) x ( n ) = K 1 n + K 2 cos ( π n ) (3)
If a length-2 averaging filter is used with
h ( n ) = { 1 / 2  for  n = 0 , 1 0  otherwise h ( n ) = { 1 / 2  for  n = 0 , 1 0  otherwise (4)
it can be verified that, after two outputs, the output y ( n ) y ( n ) is exactly the linear term x ( n ) x ( n ) with a delay of one half sample interval and no oscillation.

This example illustrates the basic FIR filter-design problem: determine N, the number of terms for h ( n ) h ( n ) , and the values of h ( n ) h ( n ) for achieving a desired effect on the signal. The reader should examine simple examples to obtain an intuitive idea of the FIR filter as a moving average; however, this simple time-domain interpretation will not suffice for complicated problems where the concept of frequency becomes more valuable.

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