"Go" Why do I flip one of the functions when I define a convolution? (The questions and answers that you know)

why the function in the definition of convolution g (Tau) to first flip to g (-τ) then pan to g (x-τ) Instead of just remembering g (τ-x) what is the benefit of doing this?

I know that asking a definition of a concept is like asking " mom " Why do you call " mom " The same. But I have always found this definition somewhat awkward. Want to know the meaning behind this.

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Neutrinos

Links: http://www.zhihu.com/question/20500497/answer/45708002

Source: Know

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Instead of trying to think about the meaning of "flipping" directly from the formula, go back to the origin of the problem and you'll be enlightened.

For example, throw a stone into the calm water. We think of the reaction of the water as a shock response. water in the t=0 time when the stone thrown into the height of the H (0) Ripple, but the water will not immediately be calm, with the passage of time, the ripple amplitude will be smaller, in the t=1 moment, the amplitude attenuation of H (1), in the t=2 moment, the amplitude attenuation of H (2) ... Until a certain period of time, the surface of the water repeatedly attributed to calm.

From the time axis, we only lost a stone at the moment of t=0, and nothing else was done at the moment, but in t=1,2 ... the water is not calm at the moment, because the role of the past (t=0 time) has continued until now.

So, here's the question:

What if we throw a pebble at the t=1 moment? At this time the influence of the t=0 time has not disappeared (the water has not yet calmed down) new stones have been thrown in again, so how high is the wave now aroused? The answer is the superposition of the current stirred waves and the remnants of the t=0 moment of influence. So what is the impact of the t=0 moment on the t=1 moment?

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For illustrative purposes, let's make a two hypothesis:

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1. The response of the water surface to the "unit stone" is fixed.

2. The wave height of a stone that is twice as large as a "unit stone" is twice times that of a stone (i.e. the system satisfies the principle of linear superposition).

Now let's calculate how high the waves are at each moment:

• T=0 moment:

Y (0) =x (0) *h (0);

• T=1 moment:

Effects of current stones X (1) *h (0);

T=0 the residual effect caused by the stone x (0) at the moment x (0) *h (1);

Y (1) =x (1) *h (0) + x (0) *h (1);

• T=2 moment:

Effects of current Stones X (2) *h (0);

T=0 the residual effect caused by the stone x (0) at the moment x (0) *h (2);

T=1 the residual effect caused by the stone X (1) At the moment X (1) *h (1);

Y (2) =x (2) *h (0) + x (1) *h (1) +x (0) *h (2);

......

• T=n moment:

Effects of current Stones X (N) *h (0);

T=0 the residual effect caused by the stone x (0) at the moment x (0) *h (N);

T=1 Time Stone x (1) The residual effect caused by X (1) *h (N-1);

Y (n) =x (n) *h (0) + x (N-1) *h (1) +x (N-2) *h (2) +...+x (0) *h (n);

That's the formula for the discrete convolution.

Understanding the above question, let's take a look at how "flipping" is going:

Every time we throw a pebble, we stand at the current point in time, and the system responds to us in H (0), after the timeline (H (1), H (2) .... ) is an impact on the future. The overall response is to add to the residual impact of the past on the present.

Now let's observe the moment of t=4.

standing at the t=0 moment to see his influence on the next (t=4) moment (4 seconds from now) is visible x (0) *h (4)

stand at t=1 moment to see his influence on the future (t=4) Moment (3 seconds from now), Visible is X (1) *h (3)

stand at t=2 moment to see his influence on the future (t=4) Moment (2 seconds from now), Visible is X (2) *h (2)

stand at t=3 moment to see his influence on the future (t=4) Moment (1 seconds from now), Visible is X (3) *h (1)

So the so-called flip just because you are standing is now the future of the past, and because H (t) is always the same, so H (1) is actually the first second of H (1), and the first second of H (1) is now, so from the current x (4) angle to the left, you see the role of the past. H (t) is not flipped before, when you look from H (0) to the right, what you see is the effect on the future, and when you flip H (t), from H (0) to the left, you see the effects of the more distant past on the present, and this effect is relative to the effect from the x=4 to the left (both of which are increasingly far away), The response to action and action corresponds, and all of this is due to the change in the point and direction of your standing time.

"Go" Why do I flip one of the functions when I define a convolution? (The questions and answers that you know)