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Fourier Transforms
book

Fourier Transforms

by Eric W. Hansen
September 2014
Intermediate to advanced content levelIntermediate to advanced
776 pages
24h 3m
English
Wiley
Content preview from Fourier Transforms

CHAPTER 5 THE FOURIER TRANSFORM

So far, we have studied Fourier representations for finite sequences (the discrete Fourier transform), infinite sequences (the discrete-time Fourier transform), and functions on a finite real interval (the Fourier series).

Transform Time domain Frequency domain
Discrete Fourier transform Discrete, bounded Discrete, bounded
n ∈ {0, 1, …, N − 1} m ∈ {0, 1, …, N − 1}
Fourier series Continuous, bounded Discrete, unbounded
x ∈ [0, L]
Discrete-time Fourier transform Discrete, unbounded Continuous, bounded
θ ∈ [ − π, π)
Fourier transform Continuous, unbounded Continuous, unbounded

Matching these up, there is a pattern: if one domain is bounded, the other domain is discrete, and if one domain is unbounded, the other domain is continuous. The one transform we have yet to explore should map a continuous, unbounded time domain to a continuous, unbounded frequency domain. That fourth member of the Fourier family is the (continuous-time) Fourier transform:

This chapter begins with a heuristic development of the Fourier transform ...

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ISBN: 9781118479148Purchase book