The discrete Fourier transform is one of the most important computational tools in signal processing. This lesson briefly introduces you to some of the applications for the discrete Fourier transform, its definition, and develops the relationship between the discrete Fourier transform and the discrete-time Fourier transform. An understanding of this relationship is essential to proper use of the discrete Fourier transform in signal processing.

## Prerequisites - Discrete Fourier Transform

## Video

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Chandan Jaiswal says

Can you please explain how we are getting an impulse train SUMMATION -Infinity to +infinity of detlta[n-kN] by inverse DTFT of S(exp(jw)) ?

Barry Van Veen says

Good question – I didn't have time dive in to this on the video.

First, I think it is simpler to assume the inverse DTFT is computed on 0 to here than the more usual to . First suppose that as the graph seems to suggest. Now we know that the inverse DTFT of is . So . Note that is exactly 1 whenever for any integer () and all . In this case we are summing 1 a total of times and the sum adds up to . It turns out the sum is exactly zero whenever , something you can show by using the formula for a finite geometric series () with . (In this case for all , hence the sum is zero.) In the video I ignored the scale factor because I was more interested in conveying the concept than diving into the low level details. I should have stated that each of the impulses in has strength , then that factor would cancel out in the math and would end up exactly as I stated. But I didn't specify other than in the graph, leaving room for this ambiguity. Note that this scale factor parallels that of sampling in time, where the replicates in frequency are scaled by .

mohammad hussien says

is there a relation between Fourier series coefficients and discrete Fourier transform ?

Barry Van Veen says

This is a great question. Let's first write out the expression for the FS and DFT coefficients. For the FS coefficients,

while for the DFT coefficients

where the period is for the FS and for the DFT. Both expressions integrate(sum) the signal times a complex sinusoid over one period.

Now consider sampling at intervals of . We assume that and that there are exactly samples in one period, i.e., . If we approximate the integral in the expression for the Fourier series using a Riemann sum, then

Substitute and (into the exponent) to obtain

or, . The approximation improves as decreases as then the Riemann sum is a better approximation to the integral.

Thus the DFT coefficients can be viewed as FS coefficients computed using a Riemann sum approximation to the FS integral.