One and Two Dimensional Fourier Analysis

One and Two Dimensional Fourier Analysis

Tolga Tasdizen ECE

University of Utah

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Fourier Series

? J. B. Joseph Fourier, 1807

? Any periodic function can be expressed as a weighted sum of sines and/or cosines of different frequencies.

? 1992?2008 R. C. Gonzalez & R. E. Woods

What is the period of this function?

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Fourier Series

? f(t) periodic signal with period T

? Frequency of sines and cosines

The complex exponentials form an orthogonal basis for the range [-T/2,T/2] or any other interval with length T such as [0,T]

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Types of functions

Continuous f(t)

Discrete f(n)

Periodic

Fourier series Discrete Fourier series

Non-periodic Fourier transform

Discrete Fourier transform

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Fourier Transform Pair

? The domain of the Fourier transform is the frequency domain.

? If t is in seconds, mu is in Hertz (1/seconds)

? The function f(t) can be recovered from its Fourier transform.

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