Fourier Transform and Dirac Delta Function

In summary, the conversation discusses the use of FT and dirac delta function to analyze a given signal and extract its frequency components. The dirac delta function is used to represent spikes or impulses in the signal, and by multiplying the FT with a dirac delta function at a specific frequency, the amplitude of that frequency component can be obtained.
  • #1
lkh1986
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Homework Statement



I am new to FT and dirac delta function. Given the following signal:

[tex]x\left(t\right)=cos\left(2\pi5t\right)+cos\left(2\pi10t\right)+cos\left(2\pi20t\right)+cos\left(2\pi50t\right)[/tex]

I use the online calculator to find me the FT of the signal, which is:

[tex]F\left(s\right)=\pi\left(dirac\left(s-10\pi\right)+dirac\left(s+10\pi\right)\right)
+\pi\left(dirac\left(s-20\pi\right)+dirac\left(s+20\pi\right)\right
+\pi\left(dirac\left(s-40\pi\right)+dirac\left(s+40\pi\right)\right
+\pi\left(dirac\left(s-100\pi\right)+dirac\left(s+100\pi\right)\right[/tex]

I then compare the FT with the graph, which shows 4 spikes at the 4 different freuency component, i.e. 5, 10, 20 and 50 Hz.

My question is, how do we get the 5, 10, 20, and 50 from the FT [tex]F\left(s\right)[/tex]?

Homework Equations


The Attempt at a Solution



I use the linearity property of FT and try to do simplify the question. I mean, if I can get 5 from the first part, I can apply the similar concept and extend it to obtain the remaining 10, 20 and 50 Hz.

[tex]x\left(t\right)=cos\left(2\pi5t\right)[/tex]
will give me the FT in the form of
[tex]F\left(s\right)=\pi\left(dirac\left(s-10\pi\right)+dirac\left(s+10\pi\right)\right)[/tex]

Now, I know I need to "extract" number 5 from the FT. How do I accomplish that?

Is my interpretation for this one correct?
[tex]dirac\left(s-10\pi\right)[/tex] means that the function is 0 for all values but it's non-zero when the value is [tex]s-10\pi[/tex]?

Thanks.
 
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  • #2


Hello,

Your interpretation of the FT and dirac delta function is correct. The dirac delta function is often used to represent impulses or spikes in a signal, which is why you see spikes at the frequencies of 5, 10, 20, and 50 Hz in the FT.

To extract the frequencies from the FT, you can use the property of the dirac delta function that states:

∫f(t)δ(t-a)dt = f(a)

This means that when you multiply the FT with a dirac delta function at a specific frequency, the resulting value will be the amplitude of that frequency component in the signal.

For example, to extract the frequency of 5 Hz, you can multiply the FT with dirac(s-10π):

F(s) * dirac(s-10π) = π(dirac(s-10π) + dirac(s+10π)) * dirac(s-10π) = π(dirac(0) + dirac(20π)) = π

This means that the amplitude of the 5 Hz frequency component is π. You can apply the same concept to extract the other frequencies from the FT.

I hope this helps. Let me know if you have any further questions.
 

What is a Fourier Transform?

A Fourier Transform is a mathematical operation that decomposes a function into its frequency components. It allows us to analyze the frequency content of a signal or function.

What is the relationship between Fourier Transform and Dirac Delta Function?

The Dirac Delta Function is a mathematical construct that is used in the representation of Fourier Transforms. It represents an infinitely narrow and tall pulse, which is useful in representing signals with infinitely small duration.

How is the Fourier Transform used in signal processing?

The Fourier Transform is an essential tool in signal processing as it allows us to analyze the frequency components of a signal. It is used in a wide range of applications, including noise reduction, filtering, and compression.

What is the inverse Fourier Transform?

The inverse Fourier Transform is the opposite operation of the Fourier Transform. It takes the frequency components of a signal or function and reconstructs the original signal or function.

What are some real-world applications of Fourier Transform and Dirac Delta Function?

Fourier Transform and Dirac Delta Function have numerous applications in various fields, including engineering, physics, and mathematics. Some examples include image and sound processing, data compression, and solving differential equations.

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