Example of an LTI system with complex impulse response
Complex band pass filters are used to get a band and its quadrature in one efficiently computed step. The most simple such design is a single-pole resonant bandpass with the discrete transfer function

$$H(z)=\frac{(1-r)\exp(i \phi)}{z-r \exp(i \phi)}$$

where $r&lt;1$ determines the bandwidth and $\phi \in [0,2\pi[$ the band center frequency. If bandwidth and center frequency are chosen to avoid significant negative frequency amplitudes, the resulting signal is practically analytic.

Strongly bandlimited and analytic signals have a complex magnitude envelope that is well behaved and can be interpreted as the instantaneous amplitude of the band. They also have a well defined phase derivative that can be interpreted as instantaneous frequency. This can be useful in a number of applications that require the extraction of such parameters.