Convolution Proof of time scaling property

In summary, the conversation is discussing the integration seen in the picture and trying to understand how it works. They are debating whether the function c(t) is defined correctly and if the integration is necessary. They also mention the importance of posting questions in the correct section.
  • #1
woohs1216
2
0
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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  • #2
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
 
  • #3
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.
 
  • #4
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 :-)
 
  • #5
I agree with Lord Crc. They have not actually integrated anything. They have just defined a function c by the equation
[tex]c(at) = \int_{-\infty}^{\infty} x(m)g(at - m)\, dm[/tex]
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".
 

What is convolution?

Convolution is a mathematical operation that combines two functions to produce a third function. It is commonly used in signal processing to represent the output of a linear, time-invariant system in response to an input signal.

What is the time scaling property of convolution?

The time scaling property of convolution states that if a function is scaled by a factor of α, then the convolution of that function with another function will also be scaled by the same factor α.

How is the time scaling property of convolution proven?

The time scaling property of convolution can be proven using the definition of convolution and the properties of integrals. By substituting the scaled function into the convolution formula and using the change of variable method, it can be shown that the result is indeed scaled by the same factor α.

What is the significance of the time scaling property in signal processing?

The time scaling property is important in signal processing because it allows us to analyze the effects of changing the scale of a signal. This can be useful in applications such as filtering and noise reduction.

Are there any limitations to the time scaling property of convolution?

While the time scaling property holds true in most cases, there are some situations where it may not apply. For example, if the function being scaled is not continuous, the time scaling property may not hold. Additionally, if the function being scaled is not absolutely integrable, the convolution may not exist.

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