digital filter
non-recursive filters
Finite Impulse Response
FIR
Infinite Impulse Response
IIR
Chapter 2
Digital Filters
For the purpose of this book we call digital filter any linear, time-invariant
system operating on discrete-time signals. As we saw in chapter 1, such a system
is completely described by its impulse response or by its (rational) transfer
function. Even though the adjective digital refers to the fact that parameters
and signals are quantized, we will not be too concerned about the effects of
quantization, that have been briefly introduced in sec. 1.6. In this chapter, we
will face the problem of designing impulse responses or transfer functions that
satisfy some specifications in the time or frequency domain.
Traditionally, digital filters have been classified into two large families: those
whose transfer function doesn't have the denominator, and those whose transfer
function have the denominator. Since the filters of the first family admit a
realization where the output is a linear combination of a finite number of input
samples, they are sometimes called non-recursive filters
. For these systems, it is
more customary and correct to refer to the impulse response, which has a finite
number of non-null samples, thus calling them Finite Impulse Response (FIR)
filters. On the other hand, the filters of the second family admit only recursive
realizations, thus meaning that the output signal is always computed by using
previous samples of itself. The impulse response of these filters is infinitely long,
thus justifying their name as Infinite Impulse Response (IIR) filters.
2.1
FIR Filters
An FIR filter is nothing more than a linear combination of a finite number of
samples of the input signal. In our examples we will treat causal filters, therefore
we will not process input samples coming later than the time instant of the
output sample that we are producing.
The mathematical expression of an FIR filter is
y(n) =
N
m=0
h(m)x(n - m) .
(1)
In eq. 1 the reader can easily recognize the convolution (26), here specialized to
1
Strictly speaking, this definition is not correct because the same transfer functions can be
realized in recursive form
19