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# The Continuous Fourier Transform

DOI link for The Continuous Fourier Transform

The Continuous Fourier Transform book

# The Continuous Fourier Transform

DOI link for The Continuous Fourier Transform

The Continuous Fourier Transform book

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## ABSTRACT

A continuous, aperiodic, or transient waveform can be represented in the frequency domain by the Fourier integral, giving the CFT of the waveform. e CFT can be considered as the limiting case of the Fourier series for a periodic waveform, each cycle of which is the transient f(t) of concern. When a waveform is periodic, we can nd the complex Fourier coecients, cn, using Equation (5.1). In the limit as T → ∞, we can write:

lim lim ( ) exp( ) ,

T n T f t jn t t

= − ≡

2c r s/ −∞

∞∫ (5.1) In the limit, nωo approaches the continuous radian frequency variable, ω.us, we can nally write

lim ( ) ( ) T n

→ ≡ ∫c F ω ωe d (5.2) Equation (5.2) denes the CFT of a transient (single) time function [note that the CFT can also be taken on a one-dimensional (1-D) spatial function, f(x)]. F(ω) is in general a complex quantity (a 2-D vector with real and imaginary parts), and thus can be written in exponential or polar form

F(ω) = ∣F(ω)∣ejφ(ω) = ∣F(ω)∣ < φ(ω) (5.3)

e Fourier inversion integral allows us to recover f(t), given its F(ω). It can be derived from the complex form of the Fourier series given T → ∞

lim ( ) exp( ) ( ) T n

jf t T

T jn t e →∞

= ≡∑1 2 12pi ω ω pi ω ωc Fo o t dω −∞

∞∫ (5.4) e continuous Fourier inversion integral, Equation (5.4), is sometimes written, noting that ω = 2πf r/s, as

f t f j ft f f( ) ( )exp( ) , /≡ =

∞∫ F 2 2 2pi pi ω pid (5.5)

e CFT has many interesting properties, the more useful of which are given in Table 5.1. We consider the CFT of a real f(t), where t can be negative.