Convolution and Impulse Signals

In summary, convolution is a mathematical operation that involves multiplying and integrating two signals. However, the commutative property may not always apply when trying to change the time-shift between two general functions or equations. This is because convolution is only an integral between an impulse signal and a generic signal, and not necessarily between two generic signals. While there are examples of convolving two different signals, the commutative property may not always hold true in these cases.
  • #1
cshum00
215
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I am a little confused about convolutions.

I know that convolution is the multiplication and then integral of the two signals. The confusion starts at the commutative property. If i try to change the time-shift from signal to another for any 2 general functions or equations the commutative property doesn't work out.

for example:
let x(t) = sin(t)
and h(t) = t^2

If you try to convolve the signals above with the commutative property you get 2 different results.

However, the convolution's commutative property does work out if h(t) were to be a impulse function. So, does it mean that convolution is only an integral between an impulse signal and a generic signal and not two generic signals? (which is the part i am confused because i have seen examples of convolving 2 different signals)
 
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  • #2
I'm not sure what you are referring to. This article shows how convolutions are symmetric
http://en.wikipedia.org/wiki/Convolution"
Scroll down to "Definitions" for the commutative property.
 
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FAQ: Convolution and Impulse Signals

1. What is a convolution signal?

A convolution signal is a mathematical operation that combines two signals, usually referred to as the input signal and the impulse response, to produce a third signal. This operation is commonly used in signal processing to model the behavior of a linear system.

2. How is convolution different from regular multiplication?

Convolution is different from regular multiplication because it takes into account the entire history of the two signals being multiplied, while regular multiplication only considers the current values of the signals. This allows convolution to model the effects of a system's past behavior on the current output.

3. What is an impulse signal?

An impulse signal, also known as a Dirac delta function, is a mathematical function that is zero everywhere except at one point, where it has an infinitely large value. It is often used in convolution operations as the input signal, as it represents an instantaneous and infinitely short signal.

4. How is an impulse signal used in convolution?

An impulse signal is used in convolution as the input signal, representing an instantaneous and infinitely short input. This allows the convolution operation to model the behavior of a system when it is subjected to a sudden and brief input.

5. What are some applications of convolution and impulse signals?

Convolution and impulse signals have various applications in signal processing, such as image and audio processing, filter design, and system analysis. They are also used in fields such as physics, engineering, and mathematics to model the behavior of systems and solve differential equations.

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