The impulse-invariant method converts analog filter transfer
functions to digital filter transfer functions in such a way that the
impulse response is the same (invariant) at the sampling
instants [346], [365, pp.
216-219]. Thus, if
denotes the
impulse-response of an analog (continuous-time) filter, then the
digital (discrete-time) filter given by the impulse-invariant method
will have impulse response
, where
denotes the
sampling interval in seconds. Moreover, the order of the filter is
preserved, and IIR analog filters map to IIR digital filters.
However, the digital filter's frequency response is an aliased
version of the analog filter's frequency
response.9.3
To derive the impulse-invariant method, we begin with the analog transfer function
where
We now sample at intervals of
Taking the z transform gives the digital filter transfer function designed by the impulse-invariant method:
We see that the
Note that the series combination of two digital filters designed by
the impulse-invariant method is not impulse invariant. In other
terms, the convolution of two sampled signals is not the same as the
sampled convolution of those two (continuous-time) signals. This is
easy to see when aliasing is considered. For example, let the signal
sinc
be the
impulse-response of an ideal lowpass-filter having cut-off frequency
tuned below half the sampling rate
, i.e.,
. Then the sampled version of this signal
contains no
aliasing. Let a second signal be defined as
. Then
for all
, and we obtain the discrete-time
convolution
. On the other hand, the
continuous-time convolution
for all
, and the
sampling of that yields only zeros.