Steve on Image Processing

Continuous-time Fourier transform basics 12

Posted by Steve Eddins,

In my planned (well, partially planned) discussion on Fourier transforms, I'll focus on three of the four types I listed in my November 23 post:

  • Continuous-time Fourier transform
  • Discrete-time Fourier transform
  • Discrete Fourier transform

The existence of multiple transform flavors, as well as the details of their relationships, is at the heart of much of the confusion on this topic.

Let's start with the continuous-time Fourier transform. (When the context makes it clear whether I'm talking about the continuous-time or the discrete-time flavor, I'll often just use the term Fourier transform.)

The continuous-time Fourier transform is defined by this pair of equations:

There are various issues of convention and notation in these equations:

  • You may see a different letter used for the frequency domain ( or f, for example). I am in the habit of using for the continuous-time Fourier transform and for the discrete-time Fourier transform.
  • You may see i instead of j used to represent . I tend to follow the electrical engineering tradition of using j.
  • You may see terms appearing in the exponent of e and not in front of the inverse transform integral.
  • You may see the signs of the exponent terms switched in the transform equations (that is, the minus sign in the exponent appears in the inverse transform instead of the forward transform).

All these variations exist because we don't already have enough to worry about in the rest of our lives.

With the equations I use, the frequency domain unit is angular frequency (radians/second).

There are so many useful Fourier transform properties and transform pairs that it's hard for to pick the bare minimum necessary for the ideas I want to convey. Today I'll just show you two of the most essential Fourier transform pairs in signal processing applications.

Here's a cosine signal:

Plots corrected December 14 thanks to help from Mark Andrews.

And here is its Fourier transform:

This is what most people who have some knowledge of the Fourier transform expect to see. A signal containing a single frequency (here the frequency is 1 rad/s) has all its frequency domain energy concentrated at that single frequency.

The second pair is a rectangular pulse in the time domain and a sinc function in the frequency domain.

I'd like to call your attention especially to the dots ("...") at the left and right on the cosine and sinc function plots. The dots are there to remind you that these functions have infinite extent. That is, they are nonzero over the entire domain.

That's an important thing to keep in mind. I'll come back to that point next time.

One final note: I've started using the category "Fourier transforms" for posts on this topic. You can see all the related posts by clicking on the category link on the right side of the page.


Get the MATLAB code

Published with MATLAB® 7.9

12 CommentsOldest to Newest

Steve:
Nice preliminaries, however I would like to know the difference between continous time omega and discrete-time OMEGA.

The variations in the forms of the equations happened because different disciplines used different forms for the integral. Optics, Physics, Electrical Engineering, Mathematics all had their favorite forms.

The different forms of the Fourier forms will persist for years because it is so entrenched in our culture and traditions. However, in other areas mathematical notation is converging on a more standard notation, thanks to the advent of the internet, and access to journal articles from other disciplines.

Steve, some minor points:

1. The delta functions should each have an area of pi, not 0.5.

2. The spectrum of the square pulse has its first zero crossings at +/- 2pi, not +/- pi.

Did you do all the plots in MATLAB? When teaching signal processing I find the biggest chore is preparing (electronically) diagrams that contain piecewise signals and/or delta functions. Do you have any tips?

Mark—I did the plots using a combination of MATLAB and a drawing program. (I used Visio because that’s what I have on my work computer. At home I have Illustrator.) I start in MATLAB when there’s some sort of curve that I want to represent precisely in the diagram. I plot the curve and then superimpose lines for the x-axis, y-axis, and any desired ticks. I turn off the MATLAB axes. Then I select Edit/Copy Figure and then paste into the drawing program. I ungroup the pasted graphic in the drawing program, at which point I’m ready to finish up the diagram by adding arrowheads, text labels, etc. Here’s the MATLAB script I started with to generate the sinc plot in this post:

omega = linspace(-10.5*pi, 10.5*pi, 100);
X = sin(omega/2) ./ (omega/2);
plot(omega,X,'k')
hold on
plot([-11*pi 11*pi], [0 0], 'k')
plot([0 0], [-.25 1.1], 'k')
for k = -10:2:10
    plot([k*pi k*pi], [-0.025 0.025], 'k')
end
hold off
axis off

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