Revision as of 02:47, 19 October 2010

Practice Question 1, ECE438 Fall 2010, Prof. Boutin

On Computing the DFT of a discrete-time periodic signal

Compute the discrete Fourier transform of the discrete-time signal

$ x[n]= e^{-j \frac{2}{3} \pi n} $.

How does your answer related to the Fourier series coefficients of x[n]?

Post Your answer/questions below.

$ X [k] = \sum_{k=0}^{N-1} x[n].e^{-j.2\pi k n/N} $

$ N=3 $ That's correct! -pm

$ x[n]= e^{-j \frac{2}{3} \pi n} $

$ X [k] = \sum_{k=0}^{2}e^{-j(n)(\frac{2}{3}\pi)(1+k)} $ You are using the long route, instead of the short route. -pm

$ X [k] = 1+ e^{-j(1)(\frac{2}{3}\pi)(1+k)} +e^{-j\frac{4}{3}\pi(1+k)} $ This gives you a very complicated answer. -pm

- AJFunche Nice effort! -pm----

$ x[n] = \frac{1}{N}\sum_{k=0}^{N-1} X[k]e^{j2\pi k\frac{n}{N}} $

$ x[n] = \frac{1}{3}\sum_{k=0}^{2} X[k]e^{j\frac{2\pi}{3}kn} $

$ x[n] = \frac{1}{3} \cdot (X[0] + X[1]e^{j(\frac{2\pi}{3}n + 2\pi)} + X[2]e^{j(\frac{2\pi}{3}(2)n + 2\pi)}) $

Notice that all the powers of e in this expression are positive, but the signal x[n] is expressed as a negative power of e, so you cannot compare just yet. -pm

whoops, I was doing the homework. is that correct? - ksoong

Tecnically yes, but not realy useful for computing the DFT. Instead, use the fact that $ e^{ 2 \pi j}=1 $ to rewrite x[n] as a positive power of e. (Just add $ 2 \pi j $ to the exponent of e). -pm

$ x[n] = \frac{1}{3} \cdot (X[0] + X[1]e^{-j(\frac{2\pi}{3}n)} + X[2]e^{-j(\frac{2\pi}{3}(2)n)}) $

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