# Example of an LTI system with complex impulse response

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.

• When we have a complex signal in general, there are two degrees of freedom (real and imaginary part). As an example, a multipath wireless channel is more conveniently represented by a complex baseband impulse response, where the complex coefficients represent both attenuation and phase shift.
– msm
Oct 21, 2016 at 7:09
• Physical systems can have far more than 1 or 2 measurable degrees of freedom. Each variable alone can be a measurable real quantity. But pairwise, some might respond to certain stimuli together and behave as if they were a single complex quantity rather than two independent real quantities. Pressure and local air velocity in some portions of a musical instrument. Oct 21, 2016 at 16:02
• Also take a look at this answer to a related question. Oct 21, 2016 at 16:03
• @robertbristow-johnson, and real quantities are just a bunch of bits that the programmer is applying rules of real arithmetic to ... Oct 21, 2016 at 22:48
• You're missing the point @robertbristow-johnson. All mathematical constructs, including integers, reals, complex numbers, vectors, matrices, or even the boolean algebra are equally imaginary. There is no foundation for calling real numbers more "real" than complex ones. Oct 22, 2016 at 8:22

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.

• That's a good example, but is it practical, or just mathematical? Oct 21, 2016 at 17:21
• @aconcernedcitizen In practical terms, a complex-valued signal takes two wires. Oct 21, 2016 at 17:26