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ECE-4433: DIGITAL

COMMUNICATION SYSTEMS-

SIGNAL REPRESENTATION

Dr. Raveendra K. Rao

ECE-4433

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SIGNAL BANDWIDTH

Bandwidthis a measure of significant spectral

content of the signal for positive frequencies

The term significant is mathematically imprecise.

Consequently, there is no universally accepted

definition of bandwidth

Low-pass signal: A signal is low-pass if its

spectral content is centered around zero.

Band-pass signal: A signal is bandpass if its

spectral content is centered around , a non-

zero carrier frequencycf

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SIGNAL BANDWIDTH

I: Low-pass Bandwidth: When the spectrum of

a signal is symmetric with MAIN LOBE bounded

by well-defined nulls (frequencies at which

spectrum is zero):

BW= Total width of Main Lobe

Example:

BW=

2

1

( ) )(sin)( fGfTcATTtArecttg ==

=

Hz

TT

12

2

1=

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SIGNAL BANDWIDTH

II: Null-to-Null (Bandpass)Bandwidth: When

the signalg(t)is multiplied by a carrier

frequency, the bandwidth is width of the main

lobe (null-to-null).

Example:

When

Bandwidth=

( ) [ ])()(2

12cos)( ccc ffGffGtftg ++

( )fTcATT

t

Arecttg sin)(

=

HzT

2

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SIGNAL BANDWIDTH

III: 3-dB Bandwidth (low-pass): The

separation between zero-frequency, where the

amplitude of the spectrum attains its peak value,

and the positive frequency at which amplitude

spectrum falls to of its peak value

IV: 3-dB Bandwidth (band-pass): The

separation between the two frequencies at which

the amplitude of the signal drops to of thepeak value at

2/1

2/1cf

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SIGNAL BANDWIDTH

V: 99% Bandwidth: The energy of the signal is given

by:

99% Bandwidth is determined by using:

99% Bandwidth (low-pass)=

99% Bandwidth (band-pass) =

+

= dffGE 2|)(|

+

=9 9

9 9

2|)(|99.0

f

f

dffGE

99f

992 f

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RANDOM SIGNALS

Consider an experiment such as rolling of a die.The sample spaceS(collection of all possible

outcomes of an experiment) is given by:

Sis also called as the probability space. Trial isa single performance of an experiment. Events

are subsets ofS.For example,

are events. , null set is also a subset ofS.Sis called the certain event and is called the

impossible event

},,,,,{ 654321 ffffffS=

},,{},,,{},,{ 32163132 fffCfffBffA ===}{

}{

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RANDOM SIGNALS

Axioms of Probability

If , then

For every eventA,

Proof:

1)(

0)(

=

SP

AP

}{=AB )()()( BPAPBAP +=+

1)(1)( = APAP

1)(1)(1)()(

1)()(},{,

==+

==+==+

APAPAPAP

SPAAPAASAA

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RANDOM SIGNALS

Prove:

for arbitrary eventsAandB,using axioms ofprobability.

InProbability theoryall set operations hold good.

For example, consider tossing of a die:

)()()()( ABPBPAPBAP +=+

},,,,,{ 654321 ffffffS=

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RANDOM SIGNALS

Let

Then,

},,{},,,{},,{ 32163142 fffCfffBffA ===

},{

},,,{

},,,{

31

6321

6531

ffCB

ffffCB

ffffA

==

=

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RANDOM SIGNALS

Joint Events & Joint Probabilities

Experiment #1: Outcomes are

Experiment #2: Outcomes are

Then,

is the joint probability of

occurrence of events

niAi ,...,2,1==mjBj ,...,2,1, =

},...,2,1;,...,2,1),,{( mjniBAS ji ===

1),(0 ji BAP

ji BA ,

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RANDOM SIGNALS

If s are mutually exclusive, then

If s are mutually exclusive, then

If all outcomes are mutually exclusive

iA

=

=n

i

jij BAPBP1

),()(

jB

=

=m

j

jii BAPAP1

),()(

= =

=n

i

m

j

ji BAP1 1

1),(

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RANDOM SIGNALS

Conditional Probability

Total Probability Theorem:If

is a partition ofSandBis an arbitrary event,

then

( ) 0)(,)(

)(| = MP

MP

AMPMAP

},...,,{ 21 nAAAM=

)()|(...)()|()()|()( 2211 nn APABPAPABPAPABPBP +++=

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RANDOM SIGNALS

Independent Events

Two eventsAandBare independent, if

Three events,A, B,andCare independent if

)()()( BPAPABP =

)()()(

)()()()()()(

)()()()(

APCPCAP

CPBPBCPBPAPABP

CPBPAPABCP

=

==

=

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RANDOM SIGNALS

Random Variable (RV)

Consider the Sample SpaceS= ,

where

s are outcomes of the experiment.An RV is a process of assigning a number

to every outcome of the experiment .The

resulting function must satisfy the following two

conditions:

The set is an event for every

},...,,{ 21 nfff

if

f

}{ xX x

)(fXX

0}{}{ ===+= XPXP

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RANDOM SIGNALS

Distribution Function

The distribution function of the RV is the

function:

defined for every

Density Function

The derivative is called the density

function of .

}{)( xXPxFX =

X

x

dx

xdFxf XX

)()( =

X

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RANDOM SIGNALS

Properties of Distribution Function

)()(}{

)()(}{

1)(;0)(

),()(

1)(0

2121

==

=