why we use convolution?

Hello,

I am beginner in signal processing and I have a fundamental question about signals convolution.

suppose we have signal that interact with a impulse response s(n) of a surface or with impulse response h(n) of a filter.

why we write it as convolution? why we don't write it as, for example, correlation or simple multipication?

thanks you!

Answer 1

[Walter Roberson]

Using just multiplication could work if the filter is a simple gain that adjusts the amplitude of every sample by the same portion.

However, filters usually work on frequencies. For example a filter might be designed to remove (nearly) all frequencies above 8000 Hz. To do that work, it has to somehow figure out how quickly values are changing, on several different scales. However, multiplication by a constant or even by a vector of constants the same length as the signal, can never figure out how quickly values are changing and so cannot adapt the values being multiplied in order to deal with the frequencies.

Continuous convolution in theory gathers information across the entire signal, and applies the information learned at a particular time point. Discrete convolution does the same thing -- except that it happens to turn out that in some cases you can do a good-enough approximation by looking only at enough "nearby" samples instead of having to look at the entire signal.

[majid]

what about correlation?

why we cant use it instead of convolution?

[Walter Roberson]

How would that work?

Suppose you have two sounds, each 5 seconds. One of them has someone singing without additional music, and the other one has someone playing a drum without singing. Those would be quite different waveforms. What would you correlate against to detect and remove frequencies above 8000 Hz ?

If you were working with simple sine or cosine waves, then you could potentially correlate against a sine wave of fixed frequency, but how would you correlate against "all" frequencies above 8000 Hz ? And how long would you make the correlation window?

[Image Analyst]

I think @majid might mean that corr and conv are similar except that the kernel is flipped in conv. So he wants to know why the flipping of the kernel is important/required in some situations.

[majid]

hello again,

yes, exactly!

[Walter Roberson]

https://www.researchgate.net/post/Difference_between_convolution_and_correlation

[majid]

thank you very much @Walter Roberson.