Fourier transform of rect(x/2)*comb(x) + sketch

In summary, the problem is asking for the Fourier Transform of a convolution between the rectangle function and the Dirac comb. The solution involves taking the FT of each individual function and multiplying them together. The resulting graph is a sampling of points at the period of the Dirac comb, with a point of 2 at u=0 and all other points being 0. This may seem like a simple answer, but it is correct.
  • #1
scholzie
75
0

Homework Statement



Take the Fourier Transform of f(x)=rect(x/2)*comb(x) where rect is the rectangle function and comb is the Dirac comb. Sketch the results.

Homework Equations


The FT of a convolution is the product of the individual FTs.


The Attempt at a Solution


Taking the FT is pretty simple, but it's the graph that I'm a little confused about. If we take the FT of f(x)<->F(u)

F(u)=2sinc(2u)comb(u), if we define sinc(u)=sin(pi u)/(pi u), rather than the other way which leaves out pi.

The graph of such a function is simply the sampling of points along 2sinc(2u) at the period of comb(u), which should just be 1. Therefore at u=0, you'd have a point of 2 (although 2sinc(2u) isn't defined at u=0, though I believe it's commonly considered to be defined as its limit, which in this case is 2). All of the other integer values of 2sinc(2u) are simply 0.

Am I correct in assuming that the graph of 2sinc(2u)comb(u) is just {2,0,0,...,0} for all n>=0?

This seems like a silly answer for a problem, but maybe that's the point. I just have a tendency to second guess myself when given a stupid problem with a simple answer.
 
Physics news on Phys.org
  • #2
Sorry, this thread should be in Calculus & Beyond. I thought I was in that forum before I posted. If a mod could move it, I'd appreciate not having to re-post it.
 

1. What is the Fourier transform of rect(x/2)*comb(x)?

The Fourier transform of rect(x/2)*comb(x) is a combination of the Fourier transforms of the rectangular function (rect) and the comb function. It results in a series of evenly spaced spikes in the frequency domain, with the height of each spike determined by the amplitude of the original functions.

2. What is the significance of the rect(x/2)*comb(x) function in signal processing?

The rect(x/2)*comb(x) function is often used in signal processing as a windowing function. It helps to isolate specific frequencies in a signal and reduce the effect of noise or other unwanted frequencies. It is commonly used in spectrum analysis and filtering applications.

3. Can you explain the term "rect" in this context?

The term "rect" refers to the rectangular function, which is defined as 1 for values within a certain range and 0 for values outside that range. In this case, the range is x/2 to x/2, resulting in a rectangular pulse centered at x=0.

4. What does the "comb" function represent in this Fourier transform?

The "comb" function, also known as the Dirac comb, is a series of delta functions spaced at regular intervals. In this Fourier transform, it represents the periodic nature of the rectangular pulse, with the spacing between the delta functions determined by the period of the rectangular pulse (in this case, x/2).

5. How would you sketch the Fourier transform of rect(x/2)*comb(x)?

To sketch the Fourier transform of rect(x/2)*comb(x), you would plot a series of evenly spaced spikes in the frequency domain, with the height of each spike determined by the amplitude of the original functions. The spacing between the spikes would be determined by the period of the rectangular pulse (x/2). The sketch would also show that the transform is symmetric about the origin due to the periodic nature of the comb function.

Similar threads

  • Advanced Physics Homework Help
Replies
11
Views
989
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
6
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
676
  • Advanced Physics Homework Help
Replies
2
Views
4K
  • Advanced Physics Homework Help
Replies
2
Views
747
  • Calculus and Beyond Homework Help
Replies
5
Views
222
  • Advanced Physics Homework Help
Replies
17
Views
2K
  • Advanced Physics Homework Help
Replies
6
Views
1K
Back
Top