How Can I Prove This Fourier Transform Pair for a Rectangular Function?

In summary, The conversation is about proving a Fourier transform pair and checking its correctness. The given function is s(t) = A Sin[w0 t] * rect[t/T - T/2] and its Fourier transform is S(f) = exp(-j w T)*T/2*A* {Sinc[(w+w0)T/2/Pi] + Sinc[(w-w0)T/2/Pi]}, where rect is a rectangular function. The Homework Equations mention the use of convolution theorem and the attempt at a solution involves using integration to simplify the problem. The range of t is determined to be between (T+T^2)/2 and (T^2-T)/2.
  • #1
thedean515
11
0

Homework Statement



I'd like to prove a F/T pair and to confim if they are correct.

s(t) = A Sin[w0 t] * rect[t/T - T/2] ... (1)

it's Fourier transform is

S(f) = exp(-j w T)*T/2*A* {Sinc[(w+w0)T/2/Pi] + Sinc[(w-w0)T/2/Pi]} ...(2)

where rect is rectangular function

Homework Equations



I can prove rect[t/T] -> T Sinc[Pi f T]

The Attempt at a Solution



I tried to use mathematica, but it didn't give me as good results as (2)

Somebody can prove it?
 
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  • #2
You know what the FT of rect and sin(wt) is. Use convolution theorem to get the FT of rect*sin.
 
  • #3
Hi, thanks chistianjb. I was going to using convolution, but seems too much maths involved. Because rectangular has only value within a range, this will simplfy the integration lots.

I worked out the range of t is between (T+T^2)/2 and (T^2-T)/2, am I right?

sb can try to integrate[Sin[w0 t], {t, (T+T^2)/2, (T^2-T)/2}]?
 

1. What is a Fourier transform pair?

A Fourier transform pair is a mathematical concept that describes the relationship between a function and its representation in the frequency domain. It consists of a Fourier transform, which is a method for representing a function as a combination of sinusoidal waves, and its inverse transform, which converts the function back to its original form.

2. What is the purpose of using a Fourier transform pair?

The purpose of using a Fourier transform pair is to analyze and manipulate signals or functions in the frequency domain. This allows for easier identification of specific frequencies and the removal of unwanted noise or components from a signal.

3. How is a Fourier transform pair calculated?

A Fourier transform pair is calculated using integral calculus. The Fourier transform of a function is calculated by taking the integral of the function multiplied by a complex exponential function. The inverse Fourier transform is calculated by taking the integral of the Fourier transform multiplied by a different complex exponential function.

4. What is the difference between a Fourier transform pair and a Laplace transform pair?

While both Fourier and Laplace transform pairs are used to analyze signals or functions, they differ in their respective domains. A Fourier transform pair operates in the frequency domain, while a Laplace transform pair operates in the complex frequency domain. Additionally, the Laplace transform can handle a wider range of functions and is often used for more complex systems.

5. What are some real-world applications of a Fourier transform pair?

Fourier transform pairs are commonly used in various fields such as signal processing, image processing, and data analysis. They are also used in engineering, physics, and other scientific disciplines to analyze and manipulate signals and functions in the frequency domain. Some specific applications include image and sound compression, filtering and noise removal, and solving differential equations.

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