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The Discrete Fourier Transform

Examine some transform pairs, drawn in a circular format, since both time and frequency domains are implicitly periodic.

Brief Explanation

The plain old Fourier transform expresses a function of time as an infinite sum (an integral) of sine waves. In the experimental realm we have only a finite set of samples, and we use the discrete Fourier transform (DFT) instead. These pages will help you develop some intuition about how the DFT works, and what its limitations are in representing the “real” world.

If you are thinking that you need to learn about the FFT, fear not: it is numerically identical to the DFT; it is just a (very ingenious) more efficient algorithm. We will discuss it later.

Below are two sets of functions: on the left, the real (top) and imaginary (bottom) parts of a 64-point function of time. Since time is a discrete variable, we index it with the Greek letter 𝜏 (tau) instead of t, following Bracewell’s convention. On the right is the spectrum of the time function, that is, its forward transform. The discrete frequency variable is 𝜈 (nu).

By clicking the buttons, you can graph some familiar functions in time and frequency. In later pages, you will be able to drag a slider to watch changes happen in real time.

Try It Yourself

Set the time domain function by clicking a button. Hold down the SHIFT key while clicking to assign the selection to the frequency domain function. In each case, the other graph will be updated so that the graphs form a transform pair.



The graphs are drawn in a circle because with the DFT, both domains are implicitly periodic. That fact is part of a larger pattern:

The τ=0 and 𝜈=0 data points and graph axes are in front, and the ±32 data points are in the rear, where the radial line is drawn. A physical measurement should not have any data in the “negative time” region, because that is non-causal. This may seem like a limitation, but it is analogous to the Nyquist criterion in the frequency domain: no components above half the sampling frequency. In the frequency domain graphs, the data point at 𝜈 = ±32 represents half the sampling frequency.

This page could generalize or specialize in many directions! Some of the patterns you can see are:

The next page will let you interactively select a component in the frequency domain, and watch how the function of time changes.

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