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I have a difficulty to understand the convolution integral itself. Can anyone please explain what are the differences between the following 3 convolutions?

x(t)*h(t), x(-t)*h(t), x(-t)*h(-t).

Thanks.

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  • \$\begingroup\$ Have you written out the integral form of each of these? Perhaps that will clarify. You should also think about what each of these mean in a broader sense, i.e. what does reversing the impulse do to the result? \$\endgroup\$
    – uint128_t
    Dec 5, 2016 at 15:38

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Let h(t) be the impulse response of an LTI system.

1) x(t)*h(t) The convolution result gives the response of an LTI system with h(t) as impulse response to which the input is x(t)

2) x(-t)*h(t) The convolution result gives the response of an LTI system with h(t) as impulse response to which the input is x(-t). x(-t) is the reflection of x(t) by a mirror along jw axis

3) x(-t)*h(-t) The convolution result gives the response of an LTI system with h(-t) as impulse response to which the input is x(-t). The system with impulse response h(-t) has its poles and roots of h(t) exchanged between positive and negative half planes. So a stable system would become unstable and vise-verse (except for the poles and roots along jw axis).

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