Associative property of convolution

In summary, The conversation discusses the proof of the associative property of convolution for finite intervals. The problem is explained in a pdf file and the speaker is seeking help in understanding the use of Fubini's theorem in this context. The other person explains that the mistake in the proof is due to the incorrect application of Fubini's theorem and suggests applying it after equation (9). The speaker also asks for a source to read about Fubini's theorem when the bounds are the integration variable.
  • #1
sainistar
4
0
Hi There

The associative property of convolution is proved in literature for infinite interval. I want to prove the associative property of convolution for finite interval. I have explained the problem in the attached pdf file.

Any help is appreciated.

Regards
Aman
 

Attachments

  • convproblem.pdf
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  • #2
Integral (10) is obviously wrong. Your two integrals have [itex]\theta[/itex] in their bounds. But [itex]\theta[/itex] is an integration variable! This can't be correct.

Why do you obtain something wrong here. Because after equation (3) they applied Fubini and switched the both integrals, and THEN they did the substitution.

You must do something similar. Apply Fubini after (9). But Fubini will in this case not be simply switching the integral signs...
 
  • #3
Thanks micromass

I was looking for the Fubini theorem when the bound are the integration variable. I did not find any. Can you please let me know any source i can read.
It will also be helpful if you can suggest how can i apply Fubini after (9).

Regards
 

1. What is the associative property of convolution?

The associative property of convolution is a mathematical property that states that the order in which multiple functions are convolved does not change the result. In other words, if we have three functions A, B, and C, the result of convolving A and B, and then convolving the result with C, will be the same as convolving A with the result of convolving B and C.

2. How is the associative property of convolution used in signal processing?

The associative property of convolution is used in signal processing to simplify the computations involved in convolving multiple functions. By rearranging the order of convolutions, we can reduce the number of calculations needed and make the process more efficient.

3. Can the associative property of convolution be applied to any type of function?

Yes, the associative property of convolution can be applied to any type of function, as long as the function is continuous and integrable. This includes both continuous and discrete signals, as well as functions in the time and frequency domains.

4. What is the difference between the associative property and the commutative property of convolution?

The associative property of convolution states that the order of convolutions does not matter, while the commutative property states that the order of the functions being convolved does not matter. In other words, the associative property deals with the order of operations, while the commutative property deals with the order of operands.

5. Can the associative property of convolution be proven mathematically?

Yes, the associative property of convolution can be proven mathematically by using the convolution integral. By rearranging the order of operations, we can show that the result remains the same, thus proving the associative property.

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