Page 387 - Analog and Digital Filter Design

384 Analog and Digital Filter Design

The sinc function envelope to give the lowpass response, and the variants for
other responses, extends to plus and minus infinity-a little impractical! Trun-
cating the envelope, by limiting its extent to a certain time limit, causes ripple
in the frequency response passband and stopband, and limits the achievable
stopband attenuation. Truncation can be applied gradually using specially
designed window functions; these reduce the ripple effects and improve the stop-
band attenuation. Windows are applied by multiplying the window coefficients
by the sinc(x) coefficients.

Windows

There are two types of FIR iilter design methods. The first uses the relationship
between the time and frequency domains and is known as the Fourier Trans-
form method. The Fourier Transform of the “brick wall” frequency response
gives the sin(x)/x time domain response. The second FIR filter design method
uses a mathematical process known as the Remez exchange algorithm, which is
described later in this chapter. Basically, the filter coefficients are found by
varying them until the desired frequency response is produced.

Fourier Method of FIR Filter Design
Windows are designed to truncate the sinc(x) function to a certain number of
taps. The simplest is a rectangular window; each coefficient, Iz(n), has a value of
1 for N taps. The next simplest is the triangular window that has a maximum
value for the center value, but tapers down to zero at either side. Windows more
complicated than those just described use cosine functions to shape the final
frequency domain response. Cosine functions give either greater stopband
attenuation or a steeper skirt, as required.

Before describing the window functions, a brief note about the terminology is
required. The midway point in an odd number of N taps is referred to as the
zero time sample. Signals are then considered to exist in the filter between
-(N - 1)/2 and +(N - 1)/2 sample periods, relative to the zero time sample. The
reason for this is that the ideal frequency domain “brick wall” becomes a sinc(x)
function in the time domain, centered on zero and with a response from minus
infinity to plus infinity. The FIR filter approximates to a sinc(x) function but
with a truncated time domain, so the central sinc(x) coefficient in the filter is
still referred to as “zero time.” In practice, all signals are delayed by (N - 1)/2,
so that the “zero time” occurs at the (N- 1)/2 sample time; that is, with a sample
every 1 ms through a 21-tap filter, the “zero time” is at 10ms.

A delay has to be introduced in order to make the filter realizable, since negative
time is not allowed! Most windows are symmetrical, thus all that is necessary is