Page 88 - DSP Integrated Circuits

3.10 Frequency Response 73

It is customary to distinguish between LSI niters (systems) that have an
impulse response of finite or infinite duration. The first type is called FIR (finite-
length impulse response) filters since the impulse response becomes zero after a
finite number of samples. The latter type is called IIR (infinite-length impulse
response) filters since, even though the impulse response decays toward zero, it
theoretically never reaches zero.
If bk = 0 for all k, the system has a finite impulse response and is therefore an
FIR system. Normally, the system is an IIR system if b^ * 0 for at least one k, but
there are some exceptions. If any 6& * 0, the difference equation represents a recur-
sive algorithm in which some of the previously computed output values are used to
compute the next output value. It can be shown that IIR filters only can be realized
by using recursive algorithms while FIR filters can be realized by using recursive
or nonrecursive algorithms. However, it is not generally recommended that recur-
sive algorithms be used to realize FIR filters, because of stability problems.

3.10 FREQUENCY RESPONSE

A useful behavioral characterization of an LSI system is to describe the system
response for typical input signals. Naturally, of great interest for frequency selec-
tive systems is their response to periodic inputs.
The frequency response, H(eJ K>T }, is obtained with a complex sinusoidal input
signal, x(n) = eJ ncoT . From Equation (3.11) we get for an LSI system

The frequency response of an LSI system can also be determined from the cor-
responding difference equations by taking the Fourier transform of both sides of
Equation (3.11). We get

and