8/18/2019 Continuous-time Fourier Transform Basics » Steve on Image Processing

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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

e 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.

t's start with the continuous-time Fourier transform. (When the context makes it clear whether I'm talking about the continuous-time or the d

e flavor, I'll often just use the term Fourier transform.)

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

ere 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 Fo

transform and for the discrete-time Fourier transform.

You may see i instead of j used to represent . I tend to follow the electrical engineerin g 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

transform instead of the forward transform).

these varia tions exist becau se we don't alre ady have enou gh to w orry abo ut i n the rest o f ou r l ives.

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

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

re's a cosine signal:

http://blogs.mathworks.com/steve/2009/11/23/fourier-transforms/

8/18/2019 Continuous-time Fourier Transform Basics » Steve on Image Processing

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ots corrected December 14 thanks to help from Mark Andrews.

d here is its Fo urier transform:

is is what most people who have some knowledge of the Fourier transform expect to see. A signal containing a single frequency (here the

quency is 1 rad/s) has all its frequency domain energy concentrated at that single frequency.

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

8/18/2019 Continuous-time Fourier Transform Basics » Steve on Image Processing

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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

ese functions have infinite extent. That is, they are nonzero over the entire domain.

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

e 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

k on the right side of the page.

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