Associative Property of Convolution?

In summary, the conversation discusses the use of convolution in algebraic properties and whether the associative and communicative properties still apply with three arbitrary functions. It is proven that convolution is both associative and communicative, and this can be seen by looking at the relationship of transforms where convolution becomes multiplication. However, it is also mentioned that the notation used in the conversation can be confusing and may involve a convolution and a product, which may affect the validity of the properties.
  • #1
DWill
70
0
Hi,

I have a quick question about certain algebraic properties of convolution. If I have 3 functions x(f), y(f) and z(f), is the following true?

[x(f) . g(f)] * z(f) = [x(f) * z(f)].g(f)

I looked on Wikipedia but there's only a property like this if one of the terms is a scalar, so most likely I can't do relation described above?

Thanks!
 
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  • #2
Your notation is confusing. (What is *, what is .?) However, in general, convolution is associative and communicative. The easiest way to see it is by looking at the relationship of the transforms, where convolution becomes multiplication, which is both associative and communicative.
 
  • #3
Hey DWill.

On top of what mathman said, you can prove it has these properties by resorting to the definition of convolution.

Also if you aren't convinced, take a look at probability theory for finding the cumulative distribution for X,Y,Z where they are all independent (but not necessarily identically distributed) which is given by the convolution of all three pdf's.

Because X + Y + Z = (X + Y) + Z = X + (Y + Z) = Y + X + Z = Z + X + Y and so on, you intuitively get the idea once you accept the theorem in probability that convolution must be associative and commutative.
 
  • #4
Ok thanks mathman and chiro! That makes sense. I just wondered if the associative property still applied with three arbitrary functions, because in all the places I've looked so far there is only two functions and a scalar used for the associative property. I'll think about it a bit further but I think this clears it up.

And sorry if my notation was confusing, I was just going by the convention of the "." being multiplication and "*" being convolution.
 
  • #5
DWill said:
Ok thanks mathman and chiro! That makes sense. I just wondered if the associative property still applied with three arbitrary functions, because in all the places I've looked so far there is only two functions and a scalar used for the associative property. I'll think about it a bit further but I think this clears it up.

And sorry if my notation was confusing, I was just going by the convention of the "." being multiplication and "*" being convolution.
Now I am very confused. I assumed you were interested in a three function convolution. However your notation, as you just defined it, seems to involved a convolution and a product.
 
  • #6
That's correct, the operation I was asking involves a convolution and a product. I looked at it more myself and tried it out on a few functions, and I think this might not be possible?

To further clarify, I was wondering if the product of the convolution of x(f) and g(f) with z(f) is equal to the product of the convolution of x(f) and z(f) with g(f)?
 
  • #7
DWill said:
That's correct, the operation I was asking involves a convolution and a product. I looked at it more myself and tried it out on a few functions, and I think this might not be possible?

To further clarify, I was wondering if the product of the convolution of x(f) and g(f) with z(f) is equal to the product of the convolution of x(f) and z(f) with g(f)?

Ohh! That might not work. I was under the impression that both the . and the * were convolutions.
 
  • #8
mathman said:
Your notation is confusing. (What is *, what is .?) However, in general, convolution is associative and communicative.

Indeed! Sometimes it tells us lot!

:smile: Sorry, I couldn't resist.
 
  • #9
LCKurtz said:
Indeed! Sometimes it tells us lot!

:smile: Sorry, I couldn't resist.

Tooshay - I need to proofread better.
 

1. What is the associative property of convolution?

The associative property of convolution is a mathematical property that states that the order in which a series of convolutions is performed does not affect the final result. In other words, the grouping of operations does not matter and the final outcome will be the same regardless of the order of operations.

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

In science, the associative property of convolution is used in signal processing, specifically in the analysis of signals in the frequency domain. It allows for the simplification of complex calculations and the ability to break down a signal into smaller, more manageable components.

3. What is the difference between the associative and commutative properties of convolution?

The associative property of convolution states that the order of operations does not matter, while the commutative property states that the order of operands does not matter. In other words, the associative property refers to the grouping of operations, while the commutative property refers to the order of the numbers being operated on.

4. Can the associative property of convolution be used with any type of signal?

Yes, the associative property of convolution can be used with any type of signal, as long as the signal can be represented as a mathematical function. This includes continuous signals, discrete signals, and even complex signals like images.

5. Are there any limitations to the associative property of convolution?

While the associative property of convolution is a useful tool in signal processing, it is not always applicable. It assumes that the signals being convolved are both absolutely integrable, which means that they have a finite area under the curve. Additionally, the signals must also satisfy certain conditions in order for the property to hold true.

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