In the last two posts in my Fourier transform series I discussed the continuous-time Fourier transform. Today I want to start getting "discrete" by introducing the discrete-time Fourier transform (DTFT).

The DTFT is defined by this pair of transform equations:

Here x[n] is a discrete sequence defined for all n:

I am following the notational convention (see Oppenheim and Schafer, Discrete-Time Signal Processing) of using brackets to distinguish between a discrete sequence and a continuous-time function. n is unitless. The frequency-domain variable, , is continuous with units of radians.

Note that is periodic with period .

Here are a few common transform pairs:

Unit Impulse

DTFT of Unit Impulse

Rectangular Pulse

DTFT of Rectangular Pulse

Note that the DTFT of a rectangular pulse is similar to but not exactly a sinc function. It resembles the sinc function between and , but recall that is periodic, unlike the sinc function.

Cosine

DTFT of Cosine

The DTFT of a discrete cosine function is a periodic train of impulses:

I updated the above plot on 6-Jan-2010 to show the location of the impulses. -SE

Because of the periodicity of it is very common when plotting the DTFT to plot it over the range of just one period: . For example, the DTFT of the rectangular pulse will most often be shown like this:

Next time I'll discuss the relationship between the continuous-time and the discrete-time Fourier transforms. Until then, Happy New Year everyone!

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