How do I convolve delta functions in Fourier transform calculations?

In summary, the conversation discusses finding the Fourier transform of Cos(10t)sin(t) and the use of "half sum difference" identities to simplify the trig expression. The individual also realizes that convolving delta functions is simply adding frequency shifts.
  • #1
jackdaniel
2
0
Hi everyone, I need help finding the Fourier transform of Cos(10t)sin(t)

i know that i need to find the transform of cos and sin and then convolve them, but i m not sure how to convolve delta function. I would really appreciate any helps.
 
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  • #2
No it's better to do it by simplifying the trig expression first using the "half sum difference" identities.

That is, cos(10t)sin(t) = 1/2 ( sin(11t) - sin(9t) )
 
  • #3
thanks, the way you described made it a lot easier. By using your method I realized that convolving delta functions is simply adding the frequency shifts.

thanks
 

What is the convolution of a delta function?

The convolution of a delta function is a mathematical operation that combines two functions to create a third function. The resulting function represents the amount of overlap between the two original functions at each point.

What is the significance of a delta function in convolution?

A delta function is often used in convolution because it represents a point source or impulse. This allows for a simpler and more intuitive representation of the overlap between the two functions.

How is the convolution of a delta function calculated?

The convolution of a delta function is calculated by integrating the product of the two functions, with one of the functions being shifted by a variable amount. This integration is performed over the entire range of the shifted function.

What are some real-world applications of convolution with delta functions?

Convolution with delta functions is commonly used in signal processing, image processing, and filtering. It is also used in physics and engineering for modeling and analyzing systems and their responses.

Can the convolution of a delta function be expressed in a closed form?

In some cases, the convolution of a delta function can be expressed in a closed form using mathematical formulas. However, in most cases, it requires numerical integration or approximation methods to obtain an accurate value.

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