So, let us compute the discrete Fourier transform of a zero-padded signal.

Let's call it XM[h], where h is a frequency.

It is a sum from 0 to M minus 1 of a signal x-prime,

which is the extension of the initial signal x, and the usual expression.

So, this is a sum from 0 to N-1 of xn, e to the -j 2pi over M,

that's an important point, n times h.

We do the usual trick, by now you should be familiar with this one.

So, we have the sum over n, and then we replace x n by the inverse DFT of XN.

It's the usual expression.

And then we reorder the summations.

So we take out the summation over k in front, with the x sub n k.

And between parentheses we have this expression, which now looks familiar.

It's a sum from 0 to capital M minus 1.

And it has exactly the same expression as the DTFT of a finite-length signal.

But, instead of having omega, we have two pi over capital M times h.

And so, this is simply the expression of X over bar e to the j omega,

evaluated at the location omega is equal to two pi over capital M times h.

The exercise we had done before.