Convolution Properties and Fourier Transform

In summary, the task is to determine the truth of two assertions and explain why they are true or false. The first assertion states that if the convolution of two functions is equal to one of the functions, then the other function must be an impulse function. The second assertion states that if the convolution of two functions is identically zero, then one or both of the functions must also be identically zero. The Fourier Transform is used to support these assertions. The first assertion is found to be true, while the second assertion requires further consideration. Possible counterexamples are suggested for the second assertion.
  • #1
Pewgs
4
0

Homework Statement



Determine whether the assertions are true or false, explain.
(a) If (f * g)(t) = f(t), then g(t) must be an impulse, d(t).
(b) If the convolution of two functions f1(t) and f2(t) is identically zero,
(f1 * f2)(t) = 0
then either f1(t) or f2(t) is identically zero, or both are identically zero.

Homework Equations



Fourier Transform implies that Xf1(jw)*Xf2(jw)=transform on the convolution.

The Attempt at a Solution


a) If Xf1(jw)*Xf2(jw)=Xf1(jw) then Xf2(jw) must be equal to 1, which is the same as d(t).

Not sure about b...
 
Physics news on Phys.org
  • #2
Welcome to PF, Pewgs! :smile:

Your (a) looks good!

For (b) you should perhaps consider 2 transforms that multiply to zero, but are not zero themselves.
For instance a Heaviside function and its mirror.
 

1. What is convolution and how is it used in signal processing?

Convolution is a mathematical operation that combines two signals to produce a third signal. It is typically used in signal processing to analyze signals and extract useful information from them. It can also be used to filter signals, remove noise, and perform other types of signal manipulations.

2. What are the properties of convolution?

The properties of convolution include commutativity, associativity, and distributivity. Commutativity means that the order of the signals being convolved does not affect the result. Associativity means that the order in which convolutions are performed does not affect the final result. Distributivity means that convolution can be distributed over addition and subtraction.

3. How does convolution relate to the Fourier transform?

The Fourier transform is a mathematical operation that decomposes a signal into its constituent frequencies. It is closely related to convolution because convolution in the time domain is equivalent to multiplication in the frequency domain. This means that performing convolution on signals in the time domain is equivalent to performing multiplication on their Fourier transforms in the frequency domain.

4. What are the benefits of using the Fourier transform in signal processing?

The Fourier transform allows us to analyze signals in the frequency domain, which can provide valuable insights into their characteristics. It also allows us to filter signals, remove noise, and perform other manipulations that would be difficult to do in the time domain. Additionally, the Fourier transform has many efficient algorithms that make it a powerful tool for signal processing.

5. Are there any limitations or drawbacks to using convolution and Fourier transform in signal processing?

While convolution and Fourier transform have many benefits, there are also some limitations and drawbacks. One limitation is that they are best suited for linear systems, meaning that they may not accurately model or handle nonlinear signals. Additionally, they can be computationally intensive, especially for large or complex signals. It is also important to choose appropriate parameters and techniques when using convolution and Fourier transform to avoid artifacts or errors in the analysis.

Similar threads

Fourier transform: duality property and convolution
    • Calculus and Beyond Homework Help
Replies
1
Views
776
Fourier transform of wave packet
    • Calculus and Beyond Homework Help
Replies
3
Views
747
Help finding ths Fourier transform
    • Calculus and Beyond Homework Help
Replies
1
Views
926
Show uniqueness in Dirichlet problem in unit disk
    • Calculus and Beyond Homework Help
Replies
6
Views
367
Fourier transform: signal with filter
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
What is the inverse Fourier transform of e^3iωt for solving ut+3ux=0?
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
Laplace transform of derivative of convolution
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Applying Convolution to a PDE with a Fourier Transform
    • Calculus and Beyond Homework Help
Replies
12
Views
1K
Fourier series and the shifting property of Fourier transform
    • Calculus and Beyond Homework Help
Replies
6
Views
906
What is the Inverse Fourier Transform of (3jw+9)/((jw)^2+6jw+8)?
    • Calculus and Beyond Homework Help
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top