How to correctly convolve two delta functions?

In summary, the correct treatment for computing the convolution of two delta functions is to integrate their product over the entire domain, which results in a single delta function at the point where both delta functions are non-zero. In this case, the result is simply ##\delta(\omega)##.
  • #1
badkitty
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How do I correctly compute the convolution of two delta functions? For example, if I want to compute ##\delta(\omega)\otimes\delta(\omega)##, I should integrate $$\int_{-\infty}^\infty \delta(\omega-\Omega)\delta(\Omega) d\Omega$$
This integrand "fires" at two places: ##\Omega = 0## and ##\Omega = \omega##, and evaluates to ##\delta(\omega)## in either case. That leads me to say that ##\delta(\omega)\otimes\delta(\omega) = 2\delta(\omega)##, which somehow feels wrong... I want it to be just ##\delta(\omega)##.

Anyone know the correct treatment here?
 
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  • #2
In your example, ##\int_{-\infty}^\infty \delta(\omega-\Omega) \delta(\Omega) \, d\Omega## is only non-zero if both ## \omega - \Omega = 0 ## and ## \omega = 0 ##.
So for this, you end up with ##\delta(\omega)##.
 

FAQ: How to correctly convolve two delta functions?

1. What is the purpose of convolving two delta functions?

The purpose of convolving two delta functions is to determine the output of a system when given an input signal and the system's impulse response. This is a common operation in signal processing and helps to understand how a system will respond to different signals.

2. How do you convolve two delta functions mathematically?

The mathematical process of convolving two delta functions involves taking the integral of the product of the two functions over all possible values. This can be represented as the convolution integral formula:
f(t)*g(t) = ∫f(τ)g(t-τ)dτ
where f(t) and g(t) are the two delta functions.

3. Can two delta functions with different amplitudes be convolved?

Yes, two delta functions with different amplitudes can be convolved. The resulting value will be the product of the two amplitudes. This is because delta functions are treated as impulses in the system and the amplitude of the impulse determines the output.

4. What is the difference between convolving two delta functions and multiplying them?

Convolving two delta functions is a mathematical operation that involves taking the integral of the product of the two functions, while multiplying them is simply multiplying their amplitudes. Convolution takes into account the entire signal, while multiplication only considers the amplitude at a specific point.

5. Can a delta function be convolved with itself?

Yes, a delta function can be convolved with itself. The resulting output will be a scaled and shifted version of the original delta function. This is because the convolution operation essentially sums the two functions together, and in this case, it results in a single impulse with a different amplitude and location.

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