5.1.1 Filter Characteristics
A digital filter is said to be linear if the output due to the application of input,
is equal to
are arbitrary constants, and y
n and y
n are the filter outputs due to
the application of the inputs x
n and x
n, respectively. The important property of
linearity is that in the computation of y(n) due to x(n), we may decompose x(n) into a
summation of simpler components x
n. We then compute the response y
n due to
n. The summation of y
n will be equal to the output y(n). This property is
also called the superposition.
A time-invariant system is a system that remains unchanged over time. A digital filter
is time-invariant if the output due to the application of delayed input xn À m is equal
to the delayed output yn À m, where m is a positive integer. It means that if the input
signal is the same, the output signal will always be the same no matter what instant the
input signal is applied. It also implies that the characteristics of a time-invariant filter
will not change over time.
A digital filter is causal if the output of the filter at time n
does not depend on the
input applied after n
. It depends only on the input applied at and before n
. On the
contrary, the output of a non-causal filter depends not only on the past input, but also
on the future input. This implies that a non-causal filter is able to predict the input that
will be applied in the future. This is impossible for any real physical filter.
Linear, time-invariant filters are characterized by magnitude response, phase
response, stability, rise time, settling time, and overshoot. Magnitude response specifies
the gains (amplify, pass, or attenuate) of the filter at certain frequencies, while phase
response indicates the amount of phase changed by the filter at different frequencies.
Magnitude and phase responses determine the steady-state response of the filter. For an
instantaneous change in input, the rise time specifies an output-changing rate. The
settling time describes an amount of time for the output to settle down to a stable
value, and the overshoot shows if the output exceeds the desired output value. The rise
time, the settling time, and the overshoot specify the transient response of the filter in
the time domain.
A digital filter is stable if, for every bounded input signal, the filter output is bounded.
A signal x(n) is bounded if its magnitude jxnj does not go to infinity. A digital filter
with the impulse response h(n) is stable if and only if
jhnj < I:
Since an FIR filter has only a finite number of non-zero h(n), the FIR filter is always
stable. Stability is critical in DSP implementations because it guarantees that the filter
DESIGN AND IMPLEMENTATION OF FIR FILTERS