Relationship between continuous-time and discrete-time Fourier transforms
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.
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