How do I Convolve Impulses?

In summary, the conversation is about the difficulty of understanding how to do convolution of impulses, specifically with two functions consisting of multiple impulses located at different frequencies and amplitudes. The concept of using delta functions to represent the impulses is mentioned, and the formula for convolution of delta functions is provided as a possible solution. The person is asking for clarification and help on how to perform this type of convolution.
  • #1
DWill
70
0
Hi guys,

I am just having a bit of difficulty figuring out how to do convolution of impulses. Suppose I have a function consisting of impulses located at -2ω0, 0, and +2ω0 (in frequency domain) with some arbitrary amplitude A. I want to convolve this function with another function consisting of two impulses located at -1ω0 and +1ω0 with some other arbitrary amplitude B.

I'm mainly confused because I'm not sure how the multiplication of two impulses would work.

Can anyone show me how this is done?

Thank you very much!
 
Mathematics news on Phys.org
  • #2
The convolution of periodic functions ##f,g## with period ##T## is defined as ##(f\ast g)(t)=\int_a^{a+T}f(\tau)g(t-\tau)\,d\tau ## so the only problem is to describe your impulses by e.g. a sine function.
 
  • #3
It ia common for engineers to use the word "impulses" for delta functions. If this is what the OP is referring to, then the convolution of impulses follows the rule
## \delta(\omega-\omega_0) \ast \delta(\omega-\omega_1) = \delta(\omega-\omega_0-\omega_1)
##
 

1. What is the concept of convolution of impulses?

The concept of convolution of impulses is a mathematical operation used to describe the output of a linear time-invariant (LTI) system when the input is a series of impulses. It is a fundamental operation in signal processing and is used to analyze the response of a system to any input signal.

2. How is the convolution of impulses calculated?

The convolution of impulses is calculated by taking the integral of the product of two functions, one representing the input signal and the other representing the impulse response of the system. This integral is taken over all time and represents the output of the system at a specific time.

3. What is the significance of convolution of impulses in real-world applications?

Convolution of impulses is a powerful tool used in many real-world applications, such as image and audio processing, communication systems, and control systems. It allows us to analyze the behavior of a system and predict its response to different input signals, making it an essential concept in engineering and science.

4. Can convolution of impulses be applied to non-linear systems?

No, convolution of impulses can only be applied to linear time-invariant systems. This is because the impulse response of a non-linear system is dependent on the input signal, making it impossible to separate the input and output signals as required for convolution.

5. How does the convolution of impulses relate to the Fourier transform?

The convolution of impulses and Fourier transform are closely related and are often used together in signal processing. The Fourier transform of the impulse response of a system is known as the system's frequency response, and the convolution of the input signal with the frequency response gives the output signal in the frequency domain. This relationship is known as the convolution theorem.

Similar threads

  • Engineering and Comp Sci Homework Help
Replies
4
Views
1K
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
14
Views
691
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Replies
4
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
4
Views
975
Replies
2
Views
3K
  • Calculus
Replies
1
Views
1K
  • Electrical Engineering
Replies
1
Views
873
Back
Top