Fourier transform for cosine function

Click For Summary
SUMMARY

The discussion focuses on the Fourier Transform of the cosine function defined as f(t)=cos(at) for |t|<1 and f(t)=0 for |t|>1. The solution presented is F(w)=[sin(w-a)/(w-a)]+[sin(w+a)/(w+a)], derived using the identities for cosine and sine in terms of exponential functions. The participants emphasize the need to post such questions in the homework forum with the appropriate template filled out for clarity and organization.

PREREQUISITES
  • Understanding of Fourier Transform concepts
  • Familiarity with trigonometric identities
  • Knowledge of complex exponential functions
  • Basic calculus skills for evaluating integrals
NEXT STEPS
  • Study the properties of Fourier Transforms
  • Learn about the application of Fourier Transform in signal processing
  • Explore the derivation of Fourier series for periodic functions
  • Investigate the use of Fourier Transform in solving differential equations
USEFUL FOR

Students studying signal processing, mathematicians interested in Fourier analysis, and educators looking for examples of Fourier Transform applications.

Soumitra
Messages
5
Reaction score
0
Fourier Transform problem with f(t)=cos(at) for |t|<1 and same f(t)=0 for |t|>1. I have an answer with me as F(w)=[sin(w-a)/(w-a)]+[sin(w+a)/(w+a)]. But I can't show it.
 
Physics news on Phys.org
Try the following identities
<br /> \cos x \equiv \frac{e^{ix} + e^{-ix}}{2} \\<br /> \sin x \equiv \frac{e^{ix} - e^{-ix}}{2i}
 
This question needs to be posted in the homework forum, with the homework template filled.

Thread closed.
 

Similar threads

Graduate Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w))
  • · Replies 3 ·
Replies
3
Views
12K
Graduate How do I know "what Fourier transform" to use?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Peak of Analytical Fourier Transform
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Proof of Parseval's Identity for a Fourier Sine/Cosine transform
  • · Replies 12 ·
Replies
12
Views
12K
Undergrad The Relationship Between Angular and Cyclic Frequency in Fourier Transform
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines
  • · Replies 139 ·
5
Replies
139
Views
11K
MHB Fourier and inverse fourier transform
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad The domain of the Fourier transform
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Fourier transform, same frequencies, different amplitudes
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Compact summary of Fourier series equations
  • · Replies 11 ·
Replies
11
Views
2K