Convolution Proof of time scaling property

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Discussion Overview

The discussion revolves around the proof of the time scaling property in convolution, focusing on the integration process involved and the definitions of the functions used. Participants are exploring the mathematical aspects of the convolution integral and its implications.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the integration process and requests clarification on its workings.
  • Another participant notes that the question may belong to a different section, indicating a potential misplacement of the topic.
  • A participant points out the lack of definition for "c(at)" and questions the validity of the integral's equality without that definition.
  • One contributor highlights the significance of the change in variable during the integration process, suggesting that it introduces the scaling factor.
  • Another participant agrees with the previous point, asserting that the integral defines a function c(at) based on the convolution integral, but emphasizes the need for clarity on the function's definition.

Areas of Agreement / Disagreement

Participants express differing views on the clarity of the integration process and the definitions involved. There is no consensus on the correctness of the presented arguments, and the discussion remains unresolved.

Contextual Notes

Participants note the importance of defining functions clearly and the implications of variable changes in integration, but these aspects remain unresolved in the discussion.

woohs1216
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Time scaling proof.png


Hello

I don't quiet understand how the integration in the picture works...
I must have forgotten something...

Can anyone explain what is used?
 
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Hi
welcome to PF

looks like a maths problem , not EE have asked for it to be moved

take care to post questions in the correct section :)

Cheers
Dave
 
I see no place, in what you have posted, where "c(at)" is defined! In any case, the part you have enclosed in red, on the left, is identical to the calculation on the right. Do you not have a problem with that? In any case, it is impossible to say why that integral is equal to c(at) without knowing how the fuction, c, is defined.
 
The interesting part happens between the very first integral and the one you've outlined in red. That's when the change in variable happens and when the scaling factor comes into the mix.

I'm pretty sure c(t) has the form of the outlined integral, as that is the standard convolution integral, and as such is just a convenience.

Of course I might be wrong :-)
 
I agree with Lord Crc. They have not actually integrated anything. They have just defined a function c by the equation
c(at) = \int_{-\infty}^{\infty} x(m)g(at - m)\, dm
Since the variable m is integrated over, the value of the integral is indeed only a function of at. They probably chose the letter c to mean "convolution integral".
 

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