Let%20g(t)%20be%20periodic;%20period%20=%20To%20.%20Fundamental%20frequency%20=%20fo%20=%201/%20To%20Hz%20or%20?o%20=%202?/%20To%20rad/sec.

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Communication Systems Prof. Ravi Warrier
FOURIER SERIES
Let g(t) be periodic period To . Fundamental
frequency fo 1/ To Hz or ?o 2?/ To
rad/sec. Harmonics n fo , n 2,3 4, . . .
Trigonometric forms
EXAMPLE
g(t)
2
0.2
-0.2
0.6
1
-0.6
-1
0
t
a) For R 1 M? and C1 µF , what is go(t) ? b)
For R 1 M? and C0.1 µF , what is go(t) ?
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Communication Systems Prof. Ravi Warrier
EXERCISE
g(t)
2
3
5
-5
0
-1
1
-3
t (sec)
-2
Filter H(?)
g(t)
go(t)
What is go(t) if the frequency response of the
filter is as shown ?
EXERCISE a)Find the Fourier series in
trigonometric compact form.
g(t)
2
2
4
-4
0
-1
1
-2
t (sec)
-2
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Communication Systems Prof. Ravi Warrier
EXAMPLE
g(t)
2
0
t
Sketch the Fourier spectra.
Suppose that g(t) is passed through a filter of
frequency response as shown. What is the output
signal ? (Both positive and negative frequencies
are shown here)
H(j?)
1
H(j?)
g(t)
go(t)
0
3.2
-3.2
?
H(j?)
?
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Communication Systems Prof. Ravi Warrier
ENERGY AND POWER Energy of g(t)
g(t) is an energy
signal if Eg lt ?. Power of g(t)
g(t) is a power signal if 0lt P lt ?. EXAMPLES
Let g(t) be as shown . The energy of g(t) is
Eg54 J. Let g(t) be a unit step function g(t)
u(t). Is this a power or an energy signal ?
g(t)
3
?
-1
4
5
t
-3
g(t)
1
Not an energy signal.
?
t
A power signal.
Average Power of sinewaves
POWER OF ANY PERIODIC FUNCTION IN TERMS OF
FOURIER COEFFICIENTS
g(t)
2
EXAMPLE
0.2
-0.2
0
?
-0.6
-1
-0.6
0.6
t
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Communication Systems Prof. Ravi Warrier
Signal Comparison CORRELATION Let g1(t) and
g2(t) be two signals. Their correlation is
defined as
If g1(t) g2(t) g(t), this becomes
autocorrelation function, given by
We see that ?g(0)Eg we ge the signal energy.
That is, the signal energy autocorrelation at ?
0.
FOURIER TRANSFORMS
Definition Fg(t) G(?) F-1G(?)
g(t)
TRANSFORM EXAMPLES
g(t)
1
0
0
g(t)
A
0
0
g(t)
A?
A
0
g(t)
A
0
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Communication Systems Prof. Ravi Warrier
g(t)
0
g(t)
0
0
0
1
0
-1
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Communication Systems Prof. Ravi Warrier
PROPERTIES Fg(t) G(?) 1. Symmetry 2.
Scaling 3. Time-shifting 4. Frequency
shifting
FOR PROOF READ TEXT.
(The term represents a linear phase
in time domain it is delay).
What is
Here g(t) is modulating the sinusoid amplitude -
AMPLITUDE MODULATION. g(t) is the modulating
signal, cos(?ot) is called the carrier. EXAMPLE

We will find the Fourier transform of
g(t)cos(10t).
cos(10t)
g(t)cos(10t)
1
TIME domain
?

0
0
FREQUENCY domain
1
Note Multiplication in time-domain doesnt
transform to multiplication in frequency domain.
0
10
-10
Math
EXERCISE 1. What is 2.
Find the spectrum of a)
b)
Sketch the time functions and the spectra.
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Communication Systems Prof. Ravi Warrier
5. Differentiation
EXAMPLE We find the Fourier transform of the
triangular function shown using this property.
g(t)
A
0
0
0
6. Integration
7. Convolution
CONVOLUTION IN TIME DOMAIN (? )
MULTIPLICATION IN FREQUENCY DOMAIN
CONVOLUTION IN FREQUENCY DOMAIN
MULTIPLICATION IN TIME DOMAIN
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Communication Systems Prof. Ravi Warrier
EXAMPLE
Note g(t) has a pulse width of ?/2 sec
but has pulse width of ? sec. Convolution
increases the width of the function.
g(t)
1
0
0
EXERCISE Let g(t) sinc(50t). What is the
spectral width (bandwidth) of g(t) ? What is the
bandwidth of g2(t) ?
EXAMPLE What is the Fourier transform of a
periodic function ? Periodic functions can be
expressed in time domain as sum of a dc term and
sinusoids of fundamental frequency and harmonics.
Fourier transform of a sinusoid is a pair of
impulse functions. Therefore, the Fourier
transform of a periodic function is a sum of
impulse functions centered at zero frequency,
fundamental frequency and harmonics.
EXAMPLE We find the Fourier transform of the
periodic function shown.
The Fourier series of g(t) is given (in page 51,
text) by
g(t)
1
0
We have
The Fourier transform of g(t) is
G(?)
-5 -4 -3 -2 -1 0 1
2 3 4 5
?G(?)
-5 -4 -3 -2 -1 0 1
2 3 4 5
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Communication Systems Prof. Ravi Warrier
EXAMPLE Consider the periodic function g(t)
consisting of impulse functions at equal spaces
of To sec. We find the Fourier transform of
g(t). We can express g(t) as
g(t)
-4To -3To -2To -To 0 To 2To
3To 4To
G(?)
?o
-4?o -3?o -2?o - ?o 0 ?o 2?o
3?o 4?o
SIGNAL ENERGY AND ENEGY SPECTRAL DENSITY
We define signal energy as . If g(t) is
complex we can express energy as
Parsevals theorem Signal energy is
Energy Spectral Density (ESD) is called the
energy spectral density of g(t). The signal
energy is the integral of the energy spectral
density ( multiplied by 2?).
ESD provides a way of computing energy from the
Fourier transform of g(t).
EXAMPLE Consider g(t)e-0.5t u(t). Find the
energy and the ESD of g(t).
0
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Communication Systems Prof. Ravi Warrier
ENERGY OF MODULATED SIGNAL
Let g(t) be a baseband energy signal
bandlimited to B Hz. Let

Suppose that G(?) is as shown. Then
G(?)
0
?(?)
The signal energy is reduced by 1/2 when it is
multiplied by a sinewave of unit amplitude.
ESD OF A SYSTEM INPUT AND OUTPUT
H(?)
G(?)
Y(?)
EXAMPLE Find the input and output energies.
R200? and C0.01 F. g(t)sinc(t).
EXERCISE Redo the Example problem for R200?
and C0.001 F.
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Communication Systems Prof. Ravi Warrier
Autocorrelation Function and ESD For g(t) a
real function
Energy Spectral Density is the Fourier Transform
of the autocorrelation function.
SIGNAL POWER AND POWER SPECTRAL DENSITY(PSD)
Energy and energy spectral density are useful for
energy signals. For power signals we define power
and power spectral density as follows
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Communication Systems Prof. Ravi Warrier
EXAMPLE Let g(t)A cos(?ot) , a power signal.
0
Input signal power , output signal power Let
g(t) be a power signal applied to a system.
H(?)
Y(?)
G(?)
EXAMPLE Consider a noise signal n(t) with PSD
is
the input to a differentiator. What is the
output noise power ?
K
H(?)j ?
y(t)
n(t)
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Communication Systems Prof. Ravi Warrier
EXERCISE 1 Consider a noise signal n(t) with
PSD applied to a RC filter with RC1 sec.
Determine the input noise power and output noise
power. (Input power 2W, Output power0.25 W)
2. Suppose that the input to RC filter is
g(t)2cos(0.5t), what is the input and out put
signal powers ? (Ans power in2W, power
out1.79W)
3. Next consider the input filter to be
g(t)n(t). Find the (signal power)/(noise power)
at the input and output. This is called the
signal-to-noise ratio.
Distortionless Transmission The ideal goal of a
communication system is to make sure the received
and transmitted signals are the same. That is,
the received signal is not distorted. This means
the communication system transfer function should
have a constant magnitude and linear phase
characteristics, in the frequency region of
interest.
H(?)
Y(?)
G(?)
K
0
0
REVIEW 1) Fourier transform inverse Fourier
transform definitions 2) Properties Important
ones - Symmetry, time-delay /phase shift,
Modulation 3) Results Fourier transform of
periodic functions, Energy, ESD,
autocorrelation, power, PSD,
autocorrelation, input energy-output energy,
input power-output power. What is the
autocorrelation function of a sinewave ? What is
the PSD of a sinewave ? What is the average
power of a sinewave ? Does phase shift affect the
power and autocorrelation ? What is the
autocorrelation function of g(t) cos(20t) ? What
is the autocorrelation function of g(t)
sin(20t) ? READ TEXT BOOK A LOT.
EXERCISE Find the Fourier transform and sketch
the spectra of i)g(t) sinc(20t)cos(100t) ii)
g(t) sinc(20t)cos2(100t).
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Communication Systems Prof. Ravi Warrier
g(t)1 for binary 1, g(t) -1 for binary 0
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Communication Systems Prof. Ravi Warrier