In general, I know that the impulse response $h(t)$ of an LTI system can be complex. However, all of physically realizable, useful systems I've come across have purely real impulse responses. I did a web search in an attempt to find a useful, non-pathological system with a complex impulse response, but was not immediately successful.
Could anyone provide an example of a system with a complex impulse response, and state what applications it has? Links to such examples would also be appreciated.
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 $r<1$ 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.
Consider continuous-time LTI systems $\mathcal H_1$ and $\mathcal H_2$, whose transfer functions are
$$H_1 (s) = \frac{1}{s - i \omega_0} \qquad \qquad \qquad H_2 (s) = \frac{1}{s + i \omega_0}$$
and whose complex-valued impulse responses are
$$h_1 (t) = \exp(i \omega_0 t) \qquad \qquad \qquad h_2 (t) = \exp(-i \omega_0 t)$$
The parallel connection of $\mathcal H_1$ and $\mathcal H_2$ is the LTI system whose transfer function is
$$H (s) = H_1 (s) + H_2 (s) = \frac{1}{s - i \omega_0} + \frac{1}{s + i \omega_0} = \frac{2 s}{s^2 + \omega_0^2}$$
and whose impulse response is the sinusoid
$$h (t) = 2 \cos (\omega_0 t)$$
Note that the imaginary parts of $h_1$ and $h_2$ canceled each other.
Lastly, you may want to read about analytic signals and the Hilbert transform.
