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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Steve, I don’t think the terms for Xc have Omega scaled by T otherwise the spectral images would be wider (or narrower) than the base-band spectrum.
Mark—The frequency scaling term is correct. That’s what makes the sampling frequency in the Omega domain map to 2*pi in the omega domain.
But doesn’t Omega (big-omega) have units of s^-1 so that Omega/T now has units of s^-2? Or should that be omega (small omega) on the right hand side of the equality?
Whoops, there is something wrong with that equation … omega on the left of the equals and Omega on the right, and the periodicity is off. I’ll fix it later on tonight.
Mark—Sorry about my confusion. I corrected the equation and the following two paragraphs.
Steve, Shouldn’t you categorize this post under “Fourier transforms”? Thanks anyway!
Sujith—Thanks for pointing that out.