Page 380 - Analog and Digital Filter Design

16
CHAPTER

DIGITAL FIR FILTER DESIGN

Digital FIR filters were briefly introduced in Chapter 15. This chapter builds
on that introduction. In particular, windows to shape the digital filter’s fre-
quency response are described in more detail, with equations for all the popular
types.

You may recall from Chapter 15 that a digital filter works by processing a
signal in the time domain. The time domain representation of a “brick wall”
filter has a sinc(x) function. Multiplying a digitized signal by the sinc(x) func-
tion produces a filtering effect. Unfortunately, the sinc(xj function extends to
infinity, so it is truncated (cut short) in a practical filter. and this limits the
number of taps that are needed. To prevent a sudden change in the time-domain
response, a window is used to gradually reduce the amplitude of the sinc(x) func-
tion at its limits. The window changes the amplitude of some of the filter tap
coefficients and results in a nonperfect frequency domain response. but at least
it is practical.

An FIR filter comprises an array of delay elements connected in series. A tap
is taken after each element, and, at any sample instance, the value of the sample
is multiplied by a filter coefficient. Thus a multiplier is needed for each delay
element. Finally, the outputs of all the multipliers are added together to give the
output.

The number of taps is given by N, but there are N-1 delay elements; the
term N-1 is sometimes referred to as the filter order. It is common to use an
odd number of taps, which results in an even number of delay elements.
An example of a 7-tap FIR filter, which has an order of 6, is given in Figure
16.1.