# Need help with Discrete Fourier Transform

Yes, this is a homework.

I've been told to perform Fourier transform on the following sequence of values:

a=[0 2 -1 3]


I think I'm supposed to use Discrete Fourier Transform and individually perform transform on each of the values. I've got this formula:

$$F(u) = \sum\limits_{x=0}^{M-1} f(x) e^{-j2{\pi}ux/M}, u = 0,1,..,M-1$$ $$f(x) = \frac 1 M\sum\limits_{u=0}^{M-1} F(u) e^{j2{\pi}ux/M}, x = 0,1,..,M-1$$

Okay, so what are all these variables? I guess that M would be 4 in my case? But what are j and u?

How am I supposed to use these formulas, as both formulas include the other, thus creating an infinite loop?

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## 2 Answers

Put $M=4$ (since there are 4 components in $a$) and let the components of $a$ be the values of $f$: $f(0)=0$, $f(1)=2$, $f(2)=-1$, $f(3)=3$. Compute $F$, then use the inverse formula. You should recover the $f$ you started with.

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So $F(1)$ would be: $$F(1) = \sum\limits_{x=0}^{M-1} f(x) e^{-j2{\pi}x/4}$$ Which is: $$f(0)e^{0} + f(1)e^{-j1/2{\pi}} + f(2)e^{-j{\pi}} + f(3)e^{-j3/2{\pi}}$$ Which is: $$2e^{-j1/2{\pi}} -e^{-j{\pi}} + 3e^{-j3/2{\pi}}$$ Which is... what? – Habba Apr 17 '12 at 20:47
@Habba: which is...that! That's a complex number (sum of complex numbers in polar form), you're supposed to konw about it before doing Fourier transforms. en.wikipedia.org/wiki/Euler's_formula – leonbloy Apr 17 '12 at 21:25
Well, I guess it answers the question asked. Doesn't mean I really get it, but it will do. I'll need to do some more reading before I come back with more questions. – Habba Apr 17 '12 at 21:32

I guess that M would be 4 in my case?

Correct

But what are j and u?

$j$ is the imaginary unit (more commonly denoted as $i$)

$u$ is the discrete variable that takes value on $0..M-1$ and corresponds to the indexes of the transformed signal. In the same way, $x$ correspond to the original signal (which you called $a$, and you should regards as $f(x)$. You need to compute $F(u)$ , which must be evaluated in four values of $u$.

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