Previously in my Fourier transforms series I've talked about the continuous-time Fourier transform and the discrete-time Fourier transform. Today it's time to start talking about the relationship between these two.

Let's start with the idea of sampling a continuous-time signal, as shown in this graph:

Mathematically, the relationship between the discrete-time signal and the continuous-time signal is given by:

(When I write equations involving both continuous-time and discrete-time quantities, I will sometimes use a subscript "c" to distinguish them.)

The sampling frequency is (in Hz) or (in radians per second).

The discrete-time Fourier transform of is related to the continuous-time Fourier transform of as follows:

But what does that mean? There are two key pieces to this equation. The first is a scaling relationship between and : . This means that the sampling frequency in the continuous-time Fourier transform, , becomes the frequency in the discrete-time Fourier transform. The discrete-time frequency corresponds to half the sampling frequency, or .

The second key piece of the equation is that there are an infinite number of copies of spaced by .

Let's look at a graphical example. Suppose looks like this:

Note that equals zero for all frequencies . This is what we mean when we say a continuous-time signal is band-limited. The frequency is called the bandwidth of the signal.

The discrete-time Fourier transform of looks like this:

where . As I mentioned before, normally only one period of is shown:

For this example, then, between and looks just like a scaled version of .

Next time we'll consider what happens when doesn't look like . In other words, we're about to tackle aliasing.