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 impulse δ[τ].
  • Shifts the impulse.
• A constant value (DC component).
• The analytic impulse Δ[τ].
• Some favorite waves.
• A (non-causal) rectangular window.
  • Wider windows.
• A (non-causal) triangular window. ←ℱ→ Hann
  • A causal tent window.
  • See the pattern?
  • The causal function or window is zero for negative times or frequencies.
• Hann (raised sinusoid, sinusoid squared) ←ℱ→ Tent
  • Treating time as running 0 to N
  • Treating time as running -N/2 to N/2

Remarks

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 Fourier transform applies to continuous functions defined along the whole real axis, from -∞ to +∞, and yields spectra with the same domain.
• The Z-transform applies to sampled time functions (t → τ): the time domain is still infinite but now is countable. The spectrum is periodic (it lies on the unit circle of the complex plane) but is still continuous.
• With the DFT, both domains are the same again: discrete and periodic.

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:

• If a time function is symmetric (aka “even”, f[τ] = f[-τ]) its spectrum will be purely real.
• If a time function is causal, its spectrum will be complex, and Hermitian. (The real part of the spectrum will be even; the imaginary part will be odd.)
• The more narrowly constrained a function is in one domain, the more widely distributed it is in the other domain. This is Heisenberg’s Uncertainty Principle.
• If a time function is real, its spectrum is complex, and vice-versa.
• There is a vital connection between the causal window and the analytic impulse.

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