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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.

J. Sanders
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    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 07:09
  • 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
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    @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
  • well, @Jazz , even more fundamentally, those real quantities that are just a bunch of bits are having boolean rules applied to them. but the CPU or DSP designers have implemented boolean rules that accomplish the task of quantitative arithmetic on the word format of the real quantities. – robert bristow-johnson Oct 21 '16 at 22:57
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    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. – Jazzmaniac Oct 22 '16 at 08:22
  • i was talking (or typing) primarily about physical quantity. physical quantity does not merely exist in the imagination of beings like us. and, even if real numbers are a construct beings like us cook up to describe physical quantity, i am saying that all physical quantity are so described. we do not measure an instantaneous voltage $v(t)$ as an imaginary or complex quantity. – robert bristow-johnson Oct 22 '16 at 19:06

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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.

Jazzmaniac
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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.

Rodrigo de Azevedo
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