- #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.