Steve on Image Processing

Relationship between continuous-time and discrete-time Fourier transforms 7

Posted by Steve Eddins,

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.

Get the MATLAB code

Published with MATLAB® 7.9

7 CommentsOldest to Newest

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.

Steve, Shouldn’t you categorize this post under “Fourier transforms”? Thanks anyway!

These postings are the author's and don't necessarily represent the opinions of MathWorks.