DTFT derivation from first principles

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jonnybgood

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Hi,
I am trying to derive a signal in time domain to DTFT.

Can you check for any mistake in my derivation (attached) because I cannot agree with the answer.

thanks
 

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  • DTFT derivation.pdf
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I dont know the solution yet. But here are some hints. I think that you need to split e^-jwm in e^n * e^-jw. And determine what is the sumation of e^n.
Also you can go backwards from the solution and see from what summation does it come.
Other thing I would check is to apply the properties of the DTFT for time reversal and time shift.
By the way n is missing in the 2nd step.
If you still need that I may take a another look at it.
 

I tried to follow your steps and this is the best I could do. I appreciate if you can point out other errors.

thanks
 

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  • DTFT derivation.pdf
    1.2 MB · Views: 130

The solution given to you is exactly the same as for DTFT{a^n*u[n]} for mag(a) < 1. ( you can see it at wikipedia).
whose derivation is (latex commands)
\sum\limits_{n=0}^\infty \alpha^n $e^{-J\omega n}
\sum\limits_{n=0}^\infty $(\alpha $e^{-J\omega })^n
which is a geom series and its sum is your solution.

So I think that your problem is to transform your time delayed and reversed signal into u[n]
I guess that the error is when you change the limits of the sum and you subtract the n=0 term.
But I am not a mathematician.
I am still trying.
 

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