The convolution of two functions with different parameters

In summary, the result of this (linear) convolution is that the Dirac delta function and the continuous signal s(t) are both shifted by tau_p. This is because anything convolved with the Dirac delta function remains the same, and in this case, the Dirac delta function is a function of tau shifted by tau_p while s(t) is a function of t. However, it is possible that using tau as a variable in this context could be a typo or a specific usage in a different context.
  • #1
EngWiPy
1,368
61
Hello all,

What is the result of this (linear) convolution:

[tex]s(t)\star\delta(\tau-\tau_p)[/tex]

where s(t) is a continuous signal, δ is the Dirac delta function, and \tau_p is a constant.

Thanks in advance
 
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  • #2
Anything convoluted with the Dirac is its self. So give that you've shifted the dirace by tau the result would be s shifted by tau as well
 
  • #3
cpscdave said:
Anything convoluted with the Dirac is its self. So give that you've shifted the dirace by tau the result would be s shifted by tau as well

You mean shifted by tau_p, right? But here the Dirac is a function of tau shifted by tau_p while s(t) is a function of t! Does this matter?
 
  • #4
Hmmmm I did't notice that the dirac was using tau... I would think that is a typo in the question as it doesn't make sense. However to be fair just cause I haven't seen soemthing doesn't mean it doesn't make sense in some context...

With convolution the only time I've seen tau used as a varible is with the integral definition of the convolution operation...

Sorry I guess I should've read the question more carefully :(
 
  • #5
for any insights!

The result of this convolution would be a shifted and scaled version of the signal s(t). The Dirac delta function acts as a weighting function, with its peak at τ_p, causing the signal to be shifted by τ_p. The parameter τ_p also acts as a scaling factor, which can stretch or compress the signal depending on its value. This convolution is commonly used in signal processing to model the effects of a system on a signal, where s(t) represents the input signal and δ(τ-τ_p) represents the impulse response of the system at time τ_p. Overall, the result of this convolution represents the output of the system at time τ_p.
 

Related to The convolution of two functions with different parameters

1. What is the definition of convolution in mathematics?

The convolution of two functions with different parameters is a mathematical operation that combines the two functions to create a third function. It is represented by the symbol * and is calculated by integrating the product of the two functions over all possible values of the parameters.

2. How is convolution used in signal processing?

In signal processing, convolution is used to combine two signals, such as audio or image signals, to create a new signal with specific characteristics. This is often used in filtering and smoothing applications.

3. What is the difference between convolution and correlation?

While convolution and correlation are similar operations, they differ in the order in which the functions are multiplied and integrated. Convolution involves flipping one of the functions before multiplying, while correlation does not.

4. Can convolution be applied to functions with infinite parameters?

Yes, convolution can be applied to functions with infinite parameters as long as the integral that defines the convolution converges.

5. How is convolution related to the Fourier transform?

The convolution theorem states that convolution in the time domain is equivalent to multiplication in the frequency domain. This means that the Fourier transform of the convolution of two functions is equal to the product of their individual Fourier transforms.

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