Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm finding it hard to find a derivation of the DFT from books, was wondering if someone can point me to some resources or attempt to explain it here.

share|improve this question
1  
Derivation... from what? For instance, there is a way to see DFT as an approximation to Fourier series. There's also a way to see DFTs as a special kind of change-of-basis of a finite-dimensional vector space. –  Zhen Lin Mar 14 '11 at 19:11
    
I would like the derivation of the DFT as an approximation to the Fourier Series. Thanks –  chutsu Mar 18 '11 at 15:06
add comment

1 Answer

Sorry. I really can't answer the question on dummies' level. But here is a possible explanation.

A periodic function $f:[0,2\pi]\to \mathbb{R}$ is sampled at $N+1$ equidistant nodes $x_j=j{2\pi\over N},j=0,1,\ldots ,N$: $v_j=f(x_j)$. We try to write $f(x)$ as Fourier series: $f(x)\approx {1\over 2\pi}\sum_{k=-N/2}^{N/2} \hat{v}_k e^{\imath kx}$.

Now I want to show you $\hat{v}_k={2\pi\over N}\sum_{j=1}^N v_j e^{-\imath kx_j}$, which is exactly the discrete Fourier transform.

Take $x=x_j$ and summation of $j$ from $1$ to $N$, then use the discrete orthogonality from here, you will get it.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.