# 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 '16 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. – hotpaw2 Oct 21 '16 at 16:02
• Also take a look at this answer to a related question. – Matt L. Oct 21 '16 at 16:03
• this gets to be a little bit philosophical or meta-physical. some of us believe that, when measuring physical quantity of real physical processes, only real-valued quantity is measured. a real physical system is not even LTI, but may behave close enough to LTI for small enough stimulus. the inputs and outputs of such a system are real, whether they be impulses or impulse responses or not.  but in the mind of a computer or DSP, you can certainly have complex quantities. but they are a pair of real quantities that the programmer is applying rules of complex arithmetic to. – robert bristow-johnson Oct 21 '16 at 16:22
• @robertbristow-johnson, and real quantities are just a bunch of bits that the programmer is applying rules of real arithmetic to ... – Jazzmaniac Oct 21 '16 at 22:48

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? – a concerned citizen Oct 21 '16 at 17:21
• @aconcernedcitizen In practical terms, a complex-valued signal takes two wires. – Rodrigo de Azevedo Oct 21 '16 at 17:26