This document contains a summary of key concepts from a chapter on Fourier transforms and their properties. It begins with an overview of the motivation for Fourier transforms as an extension of Fourier series to allow representation of aperiodic signals. It then provides examples of Fourier transforms for common functions like a rectangular pulse and exponential. The remainder summarizes important properties of Fourier transforms including: time-frequency duality, symmetry of direct and inverse transforms, scaling which relates time/bandwidth compression, time-shifting which causes phase change, and frequency-shifting which translates the spectrum.

1. Communication System
Ass. Prof. Ibrar Ullah
BSc (Electrical Engineering)
UET Peshawar
MSc (Communication & Electronics Engineering)
UET Peshawar
PhD (In Progress) Electronics Engineering
(Specialization in Wireless Communication)
MAJU Islamabad
E-Mail: ibrar@cecos.edu.pk
Ph: 03339051548 (0830 to 1300 hrs)
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2. Chapter-3
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Aperiodic signal representation by Fourier integral
(Fourier Transform)
Transforms of some useful functions
Some properties of the Fourier transform
Signal transmission through a linear system
Ideal and practical filters
Signal; distortion over a communication channel
Signal energy and energy spectral density
Signal power and power spectral density
Numerical computation of Fourier transform
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3. Fourier Transform
Motivation
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The motivation for the Fourier transform comes from the study of
Fourier series.
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In Fourier series complicated periodic functions are written as
the sum of simple waves mathematically represented by sines
and cosines.
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Due to the properties of sine and cosine it is possible to recover
the amount of each wave in the sum by an integral
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In many cases it is desirable to use Euler's formula, which states
that e2πiθ = cos 2πθ + i sin 2πθ, to write Fourier series in terms of
the basic waves e2πiθ.
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From sines and cosines to complex exponentials makes it
necessary for the Fourier coefficients to be complex valued.
complex number gives both the amplitude (or size) of the wave
present in the function and the phase (or the initial angle) of the
wave.
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4. Fourier Transform
• The Fourier series can only be used for periodic
signals.
• We may use Fourier series to motivate the Fourier
transform.
• How can the results be extended for Aperiodic
signals such as g(t) of limited length T ?
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5. Fourier Transform
Toois made long enough
T is made long enough
to avoid overlapping
to avoid overlapping
between the repeating
between the repeating
pulses
pulses
The pulses in the periodic signal
repeat after an infinite interval
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6. Fourier Transform
Observe the nature of the spectrum changes as To increases. Let
define G(w) a continuous function of w
Fourier coefficients Dnnare
Fourier coefficients D are
1/Tootimes the samples of
1/T times the samples of
G(w) uniformly spaced at
G(w) uniformly spaced at
woorad/sec
w rad/sec
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7. Fourier Transform
is the envelope for the coefficients Dn
Let To → ∞ by doubling To repeatedly
Doubling Toohalves the
Doubling T halves the
fundamental frequency
fundamental frequency
wooand twice samples in
w and twice samples in
the spectrum
the spectrum
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8. Fourier Transform
If we continue doubling To repeatedly, the spectrum becomes
denser while its magnitude becomes smaller, but the relative
shape of the envelope will remain the same.
To → ∞
wo → 0
Dn → 0
Spectral components are spaced at
Spectral components are spaced at
zero (infinitesimal) interval
zero (infinitesimal) interval
Then Fourier series can be expressed as:
⇒
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9. Fourier Transform
As
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10. Fourier Transform
⇒
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11. Fourier Transform
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12. Fourier Transform
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13. Fourier Transform
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14. Example 3.1
Solution:
G ( w) =
∞
g ( t ) e − jwt dt
∫
−∞
⇒
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15. Example 3.1
Fourier spectrum
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16. Compact Notation for some useful
Functions
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17. Compact Notation for some useful
Functions
2) Unit triangle function:
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18. Compact Notation for some useful
Functions
3) Interpolation function sinc(x):
The function sin x “sine over argument” is called sinc
x
function given by
sinc function plays an
sinc function plays an
important role in signal
important role in signal
processing
processing
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19. Fourier
Fourier series:
and
Fourier transform:
and
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20. Some useful Functions
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21. Some useful Functions
2) Unit triangle function:
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22. Some useful Functions
3) Interpolation function sinc(x):
The function sin x “sine over argument” is called sinc
x
function given by
sinc function plays an
sinc function plays an
important role in signal
important role in signal
processing
processing
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23. Example 3.2
Consider
⇒
Fourier transform
Fourier transform
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24. Example 3.2
Therefore
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25. Example 3.2
Spectrum:
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26. Example 3.3
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27. Example 3.4
Spectrum of aaconstant signal g(t) =1 is an
Spectrum of constant signal g(t) =1 is an
impulse
impulse
2πδ ( w )
Fourier transform of g(t) is spectral representation of everlasting exponentials
Fourier transform of g(t) is spectral representation of everlasting exponentials
components of of the form e jwt . .Here we need single exponential e jwt
components of of the form
Here we need single exponential
component with w = 0, results in a single spectrum at a single frequency
component with w = 0, results in a single spectrum at a single frequency
w=0
w=0
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28. Example 3.5
Spectrum of the everlasting exponential
Spectrum of the everlasting exponential
e jw o t
is aasingle impulse at w = w o
is single impulse at
Similarly we can represent:
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29. Example 3.6
According to Euler formula:
As
and
and
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30. Example 3.6
Spectrum:
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31. Some properties of Fourier transform
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32. Some properties of Fourier transform
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33. Some properties of Fourier transform
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34. Example 3.5
Spectrum of the everlasting exponential
Spectrum of the everlasting exponential
e jw o t
is aasingle impulse at w = w o
is single impulse at
Similarly we can represent:
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35. Example 3.6
According to Euler formula:
As
and
and
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36. Example 3.6
Spectrum:
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37. Some properties of Fourier transform
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38. Some properties of Fourier transform
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39. Some properties of Fourier transform
Symmetry of Direct and Inverse Transform Operations—
1- Time frequency duality:
•g(t) and G(w) are remarkable similar.
•Two minor changes, 2π and opposite
signs in the exponentials
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40. Some properties of Fourier transform
2- Symmetry property
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41. Some properties of Fourier transform
Symmetry property on pair of signals:
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42. Some properties of Fourier
transform
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43. Some properties of Fourier transform
3- Scaling property:
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44. Some properties of Fourier
transform
The function g(at) represents the function g(t) compressed in
time by a factor a
The scaling property states that:
Time compression
→ spectral expansion
Time expansion
→
spectral compression
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45. Some properties of Fourier transform
Reciprocity of the Signal Duration and its Bandwidth
As g(t) is wider, its spectrum is narrower and vice versa.
Doubling the signal duration halves its bandwidth.
Bandwidth of a signal is inversely proportional to the signal duration
or width.
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46. Some properties of Fourier transform
4- Time-Shifting
Property
Delaying aasignal by
Delaying signal by
its spectrum.
its spectrum.
to does not change
does not change
Phase spectrum is changed by
Phase spectrum is changed by
− wt o
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47. Some properties of Fourier transform
Physical explanation of time shifting property:
Time delay in a signal causes linear phase shift in its spectrum
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48. Some properties of Fourier transform
5- Frequency-Shifting
Property:
Multiplication of aasignal by aafactor of
Multiplication of signal by factor of
e jwo t
shifts its spectrum by
shifts its spectrum by
w = wo
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49. Some properties of Fourier transform
e jwot is not a real function that can be generated
In practice frequency shift is achieved by multiplying g(t) by a
sinusoid as:
Multiplying g(t) by aasinusoid of frequency
Multiplying g(t) by sinusoid of frequency
shift the spectrum G(w) by ± wo
shift the spectrum G(w) by
Multiplication of sinusoid by g(t) amounts to modulating the sinusoid
amplitude. This type of modulation is called amplitude modulation.
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wo