Can the Dirac Delta Function be Used to Find the Fourier Transform of sin(at)?

In summary, the conversation discusses finding the Fourier transform of sin(at). The individual initially struggles with finding the solution, but eventually realizes that they can use the definition of the Dirac delta function to solve it. The final result is i π δ(ω+α) - i π δ(ω-α).
  • #1
Jncik
103
0

Homework Statement


find the Fourier transform of sin(at)


Homework Equations





The Attempt at a Solution



I'm not sure about the solution but

it is known that

\frac{i%28e^{-iat}%20-%20e^{iat}%29}{2}%20=%20\frac{i}{2}%20e^{-iat}%20-%20\frac{i}{2}%20e^{iat}.gif


now I tried using the formula of Fourier transform but I couldn't find anything

my question is this:

can I use the definition of the dirac delta function in order to find it?

If I remember correctly we have

[URL]http://latex.codecogs.com/gif.latex?e^{i\omega_{0}%20t}%20%3C-%3E%202\pi%20\delta%20%28\omega%20-%20\omega_{0}%29[/URL]

hence the result would be

i π δ(ω+α) - i π δ(ω-α)

is this correct?
 
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  • #3
looks right to me. sin(at) is one of the functions that is listed in tables in any textbook with Fourier transforms so it would be easy to check your answer.
 
  • #4
thanks for your help :)
 

1. What is the Fourier transform of sin(at)?

The Fourier transform of sin(at) is a mathematical operation that decomposes the function into its constituent frequencies. It represents the function as a sum of sine and cosine waves of different frequencies, amplitudes, and phases.

2. How is the Fourier transform of sin(at) calculated?

The Fourier transform of sin(at) can be calculated using the Fourier transform formula, which involves integrating the function over all time and multiplying it by a complex exponential function.

3. What is the physical significance of the Fourier transform of sin(at)?

The Fourier transform of sin(at) has applications in many fields, including signal processing, communication systems, and image processing. It allows us to analyze the frequency components of a signal and can also be used to filter out unwanted frequencies.

4. What are the properties of the Fourier transform of sin(at)?

The Fourier transform of sin(at) has several properties, including linearity, time-shift, time-scaling, and frequency-shift properties. These properties allow us to manipulate the function and analyze its frequency components easily.

5. Can the Fourier transform of sin(at) be inverted?

Yes, the Fourier transform of sin(at) can be inverted using the inverse Fourier transform formula. This allows us to reconstruct the original function from its frequency components.

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