Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 786

¼ ¼ 1:061 and

10

jp 1

1:061
j

jc 3 j ¼

3

10 ¼ 1:061

j

¼ 1:061

jp 1

jc 3 j ¼

3

B.1.5 FOURIER TRANSFORM
The Fourier transform is a mathematical function that provides frequency spectral analysis for
a nonperiodic signal. The Fourier transform pair is defined as
Fourier transform:

Z N

x t e jut dt

XðuÞ¼ (B.20)

N

Inverse Fourier transform:

Z N

1
2 p

XðuÞe jut du xðtÞ¼ (B.21)

N

where xðtÞ is a nonperiodic signal and XðuÞ is a two-sided continuous spectrum versus the continuous
frequency variable u , where N < u < N . Again, the spectrum is a complex function that can be
further written as

XðuÞ¼jXðuÞj : fðuÞ (B.22)

where jXðuÞj is the continuous amplitude spectrum, while : fðuÞ designates the continuous phase

spectrum.

EXAMPLE B.5
Let xðtÞ be a single rectangular pulse, shown in Figure B.8 , where the pulse width is s ¼ 0:5 second. Find its
Fourier transform and sketch the amplitude spectrum.

FIGURE B.8
Rectangular pulse in Example B.5.