Definition. A real digital filter
is defined as any real-valued function
of a signal for each integer
.

Thus, a real digital filter maps every real, discrete-time signal to a
real, discrete-time signal. A complex filter, on the other
hand, may produce a complex output signal even when its input signal
is real.

We may express the input-output relation of a digital filter by the
notation

(5.1)

where denotes the entire input signal, and is the
output signal at time . (We will also refer to as
simply .) The general filter is denoted by
, which
stands for any transformation from a signal to a sample value at
time . The filter can also be called an operator on
the space of signals . The operator maps every signal
to some new signal
. (For simplicity, we take
to be the space of complex signals whenever is
complex.) If is linear, it can be called a linear
operator on . If, additionally, the signal space
consists only of finite-length signals, all samples long, i.e.,
or
, then every linear filter
may be called a linear transformation, which is
representable by constant matrix.

In this book, we are concerned primarily with single-input,
single-output (SISO) digital filters. For
this reason, the input and output signals of a digital filter are
defined as real or complex scalars for each time index (as opposed
to vectors). When both the input and output signals are
vector-valued, we have what is called a
multi-input, multi-out (MIMO) digital filter. We look at MIMO allpass filters in
§D.3 and MIMO state-space filter forms in Appendix E,
but we will not cover transfer-function analysis of MIMO filters using
matrix fraction descriptions [37].