Explain the formula of convolution and convolution theorem

Question

As stated in the title, I have two questions

Why convolution is defined as $(f*g)(x) = \int_{-\infty}^\infty f(t) g(x - t) dt$ instead of just $\int_{-\infty}^\infty f(t) g(x + t) dt$ ? Why we need to flip $g$ ?
Is there any intuitive explanation of convolution theorem (Fourier transform) ?

I had voted to close this question as being too broad, not as a duplicate of a previous question. In fact, I do not agree that "Filpping...." is at all an answer to this question which is asking why convolutions are defined the way that they are instead of the other way (without flipping) which is commonly called correlation instead of convolution. So, I am voting to re-open this question. — Dilip Sarwate, Commented Dec 24, 2018 at 18:28

score 2 · Accepted Answer · 2018-12-23 16:18:56Z

2

Because the front part of the input hits the filter first.

The graphical diagrams shown in most text books with an exponentially decaying signal multiplied by a step function show this directly.

While Causal response is not required for valid convolution, it is essential that the filter output responds and evolves from the head of the input.

Convolution also commutes, so the head of the impulse function also hits the signal first.

Correlation overlays the pair of functions which why it is typically used in measurements associated with similarly measures.

answered Dec 23, 2018 at 15:45

user28715

$\begingroup$ Can you explain you answer in more details ? $\endgroup$
– HOANG GIANG
Commented Dec 23, 2018 at 15:47
$\begingroup$ i just did, can you look at a book. pictures convey more information than words $\endgroup$
– user28715
Commented Dec 23, 2018 at 15:49

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