Finding the fourier spectrum of a function

  • #1
323
15

Homework Statement


Find the Fourier spectrum ##C_k## of the following function and draw it's graph:
Capture.JPG


Homework Equations


3. The Attempt at a Solution [/B]
I know that the complex Fourier coefficient of a rectangular impulse ##U## on an interval ##[-\frac{\tau}{2}, \frac{\tau}{2}]## is ##C_k = \frac{U\tau}{T}\frac{\sin {kw\frac{\tau}{2}}}{kw\frac{\tau}{2}}## and since ##f(t)=U\cos {w_ot}## i can say that ##f(t)=\frac{U}{2}(e^{jw_ot}-e^{-jw_ot})## which if i use the property of the fourier series get:
##C_k = \frac{U\tau}{T}\frac{\sin {k(w+w_o)\frac{\tau}{2}}}{k(w+w_o)\frac{\tau}{2}} - \frac{U\tau}{T}\frac{\sin {k(w-w_o)\frac{\tau}{2}}}{k(w-w_o)\frac{\tau}{2}}##. Is this correct. How would i draw a graph of this?
 

Attachments

  • Capture.JPG
    Capture.JPG
    7.1 KB · Views: 514

Answers and Replies

  • #2
456
97
I think you have the solution. You can also think of it as follows: Your function is the product of a rectangle and a cosine. The FT of the rectangle is the sinc-function that you have in your solution. The FT of the cosine consists of two delta-functions (at plus and minus the frequency of the cosine). The FT of the product is the convolution of the two seperate FTs and that's what you write. The graph should show two sinc-functions centered around the positions of the deltas.
 
  • Like
Likes Merlin3189 and diredragon

Related Threads on Finding the fourier spectrum of a function

Replies
5
Views
2K
Replies
4
Views
1K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
11
Views
943
  • Last Post
Replies
0
Views
412
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
3K
Replies
1
Views
799
  • Last Post
Replies
14
Views
2K
Replies
4
Views
2K
Top