Chapter 3

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Communication Systems
Instructor: Engr. Dr. Sarmad Ullah Khan
Assistant Professor
Electrical Engineering Department
CECOS University of IT and Emerging Sciences
Sarmad@cecos.edu.pk
Chapter 3
Analysis and Transmission of
Dr. Sarmad Ullah Khan
y
Signals
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Outlines
• Aperiodic signal representation by Fourier
integral (Fourier Transform)
• Transforms of some useful functions
• Some properties of the Fourier transform
Si l i i h h li
• 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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Outlines
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Fourier Transform
• The motivation for the Fourier transform comes from
the study of Fourier series.
• In Fourier series complicated periodic functions are
written as the sum of simple waves mathematically
represented by sines and cosines.
p y
• Due to the properties of sine and cosine it is possible to
recover the amount of each wave in the sum by an
integral
• 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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Fourier Transform
• 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
function and the phase (or the initial angle) of the
wave.
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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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Fourier Transform
• How can the results be extended for Aperiodic signals
such as g(t) of limited length T ?
• Idea
(1) Transformation to periodic signal
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Fourier Transform
• Idea
(1) Transformation to periodic signal
(2) The period T0 is made long enough to avoid overlap
(3) If , periodic signal can be represented by an
exponential Fourier series where each pulse repeat
after infinite interval
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Fourier Transform
• Therefore
Fourier series representing will also represent g(t)
• Where
• And
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Fourier Transform
• Integrating over -T0/2 to T0/2 is same as
integrating g(t) from - to +
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Fourier Transform
• To see changes in nature of spectrum, consider G(f) a
continuous function of ω.
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Fourier Transform
is the envelope for the coefficients Dn
13
Let To by doubling To repeatedly
Doubling To halves the
fundamental frequency ω0
and twice samples in the
spectrum

Fourier Transform
• If we continue doubling T0 repeatedly, the spectrum
becomes denser while its magnitude becomes
smaller, but the relative shape of the envelope will
remain the same
• Then Fourier series can be expressed as:
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oT 0ow0nD
Fourier Transform
• Then Fourier series can be expressed as:
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Fourier Transform
• As
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Fourier Transform
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
Fourier Transform
• G(f) is called direct Fourier Transform of g(t)
• g(t) is called inverse Fourier Transform of G(f)
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Fourier Transform
• G(f) is complex then it will have magnitude and angle
spectra
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Fourier Transform
• G(ω) is called the Fourier spectrum of g(t)
• |G(ω)|2 is called the power spectrum of g(t)
• The spectrum of an aperiodic function g(t) contains
p p g( )
infinite number of sinusoids starting from –infinity to
+ infinity and continue forever
• The amplitude and phases are such that they add up
exactly to g(t) over a finite interval and add up to zero
outside this interval
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Fourier Transform
• If g(t) is a real function then the amplitude spectrum is an
even function and angle spectrum is an odd function
• This property is called Conjugate Symmetry Property
|G( f)| |G(f)| d ϴ ( f) ϴ (f)
|G(-f)| = |G(f)| and ϴg (-f) = -ϴg (f)
G(-f) = G*(f)
• The transform G(f) is the frequency domain specification
of g(t)
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Fourier Transform
• Fourier transform is linear if
g1(t) G1(f) and g2(t) G2(f)
• Then for all constants a1 and a2, we have
a1g1(t) + a2g2(t) a1G1(f) + a2G2(f)
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Example 3.1
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Solution:
dtetgwG jwt




 )()(
Example 3.1
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Fourier spectrum

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Outlines
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Transforms of Useful Functions
• Some useful functions definitions
Gate function
It is a function of unit length with unit height
It is also called rectangular function, pi function, unit
pulse or normalized boxcar function
It is defined as
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Unit Triangular function
It is a function of unit length with unit height having
centre at origin
It is also called triangle function, hat function, tent
function
It is defined as
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Sinc function
It is a function of Sin(x)/x, “Sine over argument”
It is also called cardinal sine function
It is denoted by sinc(x) and defined as
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sinc function plays an
important role in signal
processing
Sinc function
It is an even function
sinc(x)=0 when sin(x)=0 except at x=0
Using L’Hopital rule, sinc(0) = 1
Period of 2Π
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Example 3.2
Consider

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Fourier transform

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Example 3.2
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Therefore
Example 3.2
Spectrum:
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Example 3.3
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Spectrum of a constant signal g(t) =1 is an impulse Spectrum of a constant signal g(t) =1 is an impulse
)(2 w
Example 3.4
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Fourier transform of g(t) is spectral representation of everlasting exponentials
components of of the form          . Here we need single exponential component with
w = 0, results in a single spectrum at a single frequency
w = 0
Fourier transform of g(t) is spectral representation of everlasting exponentials
components of of the form          . Here we need single exponential component with
w = 0, results in a single spectrum at a single frequency
w = 0
jwt
e jwt
e
Example 3.5
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Spectrum of  the everlasting exponential               is a single impulse atSpectrum of  the everlasting exponential               is a single impulse attjwo
e oww 
Similarly we can represent:
Example 3.6
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According to Euler formula:
and
As

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Spectrum:
Example 3.6
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Outlines
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Properties of Fourier transform
Fourier series:
and
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Fourier transform:
and
• Some useful Properties of Fourier Transform are
Time-Frequency Duality
Duality Property
Time Scaling Property
Time Shifty Property
Frequency Shifting Property
Convolution Theorem
Time Differentiation and Time Integration
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Photograph and Negative
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Duality Property
If Fourier transform of g(t) is G(f) then the Fourier transform of
G(t), with ‘f’ replaced by ‘t’, is g(-f) which is original time domain
signal with t replace by –f.
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For positive real constant a,
If a<1, then
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Reciprocity of the Signal Duration and its Bandwidth
As g(t) is wider, its spectrum is narrower and vice
s g(t) s w de , ts spect u s a owe a d v ce
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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Time Shifting Property
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Delaying a signal by does not change its
spectrum.
Phase spectrum is changed by
Delaying a signal by does not change its
spectrum.
Phase spectrum is changed by
ot
owt
Time Shifting Property
Time delay in a signal causes linear phase shift in its
spectrum
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Multiplication of a signal by a factor of shifts its spectrum byte ojw
oww 
ejωt is not a real function that can be generated
In practice frequency shift is achieved by multiplying
g(t) by a sinusoid as:
g(t) by a sinusoid as:
Multiplication of sinusoid by g(t) amounts to
modulating the sinusoid amplitude. This type of
modulation is called amplitude modulation
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Convolution Property
The convolution of two signals g(t) and w(t) is represented by
g(t)*w(t) and defined by
The time convolution property and its dual, the frequency
convolution property state that if
Then
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Time Differentiation and Time Integration
Differentiation property
Integration property
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Chapter 3

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