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TE-7001

Digital Communication Systems

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Random SignalsRandom Variables

All useful message signals appear random; that is, the receiverdoes not know, a priori, which of the possible waveform havebeen sent.

Let a random variable X(A ) represent the functional relationship

between a random event A and a real number.

Notation - Capital letters, usually X or Y, are used to denoterandom variables. Corresponding lower case letters, x or y, areused to denote particular values of the random variables X or Y.

Example:

means the probability that the random variable X will take avalue less than or equal to 3.

( 3)P X

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Probability Mass Function for (discrete RVs)

Probability Mass Function (p(x)) specifies the probability ofeach outcome (x) and has the properties:

( ) 0

( ) 1

( ) ( )

x

p x

p x

P X x p x

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Cumulative Distribution Function

Cumulative Distribution Functionspecifies the probabilitythat the random variable will assume a value less than or

equal to a certain variable (x).

The cumulative distribution, F(x), of a discrete randomvariable X with probability mass distribution, p(x), is given

by:

( ) ( ) ( )x t

F x P X x p t

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Mean & Standard Deviation of a Discrete Random Variable X

Mean or Expected Value

Standard Deviation

xall

)x(xp

2

1

xall

2)x(p)x(

21

xall

22 )x(px

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Example

The probability mass function of X is

X p(x)

0 0.001

1 0.027

2 0.243

3 0.729

The cumulative distribution function of X is

X F(x)

0 0.0011 0.028

2 0.271

3 1.000

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p(x)

x

F(x)

x

1

0.5

0

Probability

MassFunction

CumulativeDistributionFunction

0 1 2 3 4

0 1 2 3 4

Example Contd.

1

0.5

0

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Example Contd.

)(XE

7.2

)729.0)(3(243.0)2(27.0)1(0

)(3

0x

xpx

)(2 XVar

,27.0

06561.011907.007803.000729.0

)()(3

0x

2

xpx

and

5196.0

0.27

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Probability Density Function

The function f(x) is a probability density functionfor the continuous random variableX, defined overthe set of real numbers R, if

1. f(x) 0 for all x R.

2.

3. P(a < X < b) =

.1dx)x(f

b

a

dxxf .)(

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Note: For any specific value of x, say x = c,

f(x) can be thought of as a smoothed relativefrequency histogram

0cXP

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Cumulative Distribution Function

x

.dt)t(f)xX(P)x(F

)()( xF

dx

dxf

The cumulative distribution function F(x) of acontinuous random variable X with densityfunction f(x) is given by:

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Probability Density Distribution:f(t)

t

Area = P(t1< T

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Special Continuous Probability Distributions

Uniform Distribution

Normal Distribution

Lognormal Distribution

Exponential Distribution

Weibull Distribution

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bxaforab

1)x(f

a b0

f(x)

x

1/(b-a)

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Probability Distribution Function

bxafor

ab

ax)()(

xXPxF

a b

0

F(x)

x

1

axfor0

bfor x1

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Uniform Distribution:

Mean & Standard Deviation

Mean

Standard Deviation

2

ba

12

ab

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Ensemble Averages

{ } ( )X X

m E X xp x dx

The first moment of a probabilitydistribution of a random variable

X is called mean value mX, orexpected value of a randomvariableX

The second moment of a

probability distribution is themean-square value of X

Central momentsare themoments of the differencebetweenX and mXand the

second central moment is thevariance ofX

Variance is equal to thedifference between the mean-square value and the square ofthe mean

2 2{ } ( )XE X x p x dx

2 2

2

var( ) {( ) }

( ) ( )

X

X X

X E X m

x m p x dx

2 2var( ) { } { }X E X E X

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1.3 Spectral Density

The spectral density of a signal characterizes the distribution of

the signals energy or power in the frequency domain.

This concept is particularly important when considering filtering in

communication systems while evaluating the signal and noise atthe filter output.

The energy spectral density (ESD) or the power spectral density

(PSD) is used in the evaluation.

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1. Energy Spectral Density (ESD)

Energy spectral density describes the signal energy per unit

bandwidth measured in joules/hertz.

Represented as x(f), the squared magnitude spectrum

(1.14)

According to Parsevalstheorem, the energy of x(t):

(1.13)

Therefore:

(1.15)

The Energy spectral density is symmetrical in frequency aboutorigin and total energy of the signal x(t) can be expressed as:

(1.16)

2( ) ( )x f X f

2 2

x

- -

E = x (t) dt = |X(f)| df

x

-

E = (f) dfx

x

0

E = 2 (f) df x

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2. Power Spectral Density (PSD)

Thepower spectral density (PSD) function Gx(f ) of the periodic

signalx(t) is a real, even, and nonnegative function of frequencythat gives the distribution of the power ofx(t) in the frequency

domain.

PSD is represented as:

(1.18)

Whereas the average power of a periodic signal x(t) is

represented as:

(1.17)

Using PSD, the average normalized power of a real-valuedsignal is represented as:

(1.19)

2

x n 0

n=-

G (f ) = |C | ( )f nf

0

0

/2

2 2

x n

n=-0 / 2

1P x (t) dt |C |

T

TT

x x x

0

P G (f)df 2 G (f)df

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Autocorrelation

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1.4 Autocorrelation1. Autocorrelation of an Energy Signal

Correlation is a matching process; autocorrelation refers to the

matching of a signal with a delayed version of itself.

Autocorrelation function of a real-valued energy signalx(t) is

defined as:

(1.21)

The autocorrelation function Rx() provides a measure of how

closely the signal matches a copy of itself as the copy is shifted

units in time.

Rx() is not a function of time; it is only a function of the time

difference between the waveform and its shifted copy.

xR ( ) = x(t) x (t + ) dt for - <

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1. Autocorrelation of an Energy Signal

The autocorrelation function of a real-valued energy signal has

the following properties:

symmetrical in about zerox xR ( ) =R (- )

The auto-correlation function is an even function

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maximum value occurs at the originx xR ( ) R (0) for all

Autocorrelation magnitude can never be larger than it is at zero shift

If a signal is time shifted its autocorrelation does not change.

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Autocorrelation and Energy

Spectral Density (ESD) form a

Fourier transform pair, as designated

by the double-headed arrows.

value at the origin is equal to the

energy of the signal

x xR ( ) (f)

2

xR (0) x (t)dt

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2. Autocorrelation of a Power Signal

Autocorrelation function of a real-valued power signalx(t) is

defined as:

(1.22)

When the power signalx(t) is periodic with period T0, the

autocorrelation function can be expressed as

(1.23)

/ 2

xT / 2

1R ( ) x(t) x (t + ) dt for - <

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2. Autocorrelation of a Power Signal

The autocorrelation function of a real-valuedperiodic signal hasthe following properties similar to those of an energy signal:

symmetrical in about zero

maximum value occurs at the origin

autocorrelation and PSD form a

Fourier transform pair

value at the origin is equal to theaverage power of the signal

x xR ( ) =R (- )

x xR ( ) R (0) for all

x xR ( ) (f)G

0

0

T / 2

2

x

0 T / 2

1R (0) x (t)dtT

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Random Processes A random processX(A, t) can be viewed as a function of two

variables: an event A and time.

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Random process

A random process is a collection of time functions, or signals,corresponding to various outcomes of a random experiment. For

each outcome, there exists a deterministic function, which is called

a sample function or a realization.

Sample functionsor realizations

(deterministic

function)

Random

variables

time (t)

Rea

lnumber

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Statistical Averages of a Random Process

A random process whose distribution functions are

continuous can be described statistically with a probabilitydensity function (pdf).

A partial description consisting of the mean and

autocorrelation function are often adequate for the needs of

communication systems.

Mean of the random process X(t) :

(1.30)

Autocorrelation function of the random process X(t)

(1.31)

{ ( )} ( ) ( )kk X X k

E X t xp x dx m t

1 2 1 2( , ) { ( ) ( )}XR t t E X t X t

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Specifying a Random Process

A random process is defined by all its joint CDFs

for all possible sets of sample times

t0 t1t2

tn

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Stationarity

A random process X(t) is said to be stat ionary in the str ict

senseif noneof its statistics are affected by a shift in thetime origin.

A random process is said to be wide-sense stat ionary (WSS)

if two of its statistics, its mean and autocorrelation function,

do not vary with a shift in the time origin.

(1.32)

(1.33)

{ ( )} constantXE X t m a

1 2 1 2( , ) ( )X XR t t R t t

Stationarity

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Stationarity

If time-shifts (any value T) do not affect its joint CDF

t0t1

t2tn

t0 + Tt1+T

t2+T

tn+T

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Weak/Wide Sense Stationarity (wss)

Keep only above two properties (2ndorder stationarity)

Dont insist that higher-order moments or higher order joint CDFs

be unaffected by lag T

With LTI systems, we will see that WSS inputs lead to WSS outputs,

In particular, if a WSS process with PSD SX(f) is passed through a linear

time-invariant filter with frequency response H(f), then the filter output is

also a WSS process with power spectral density |H(f)|2SX(f).

Gaussian w.s.s.= Gaussian stationary process (since it only has 2nd

order moments)

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Stationarity: Summary

Strictly stationary:If none of the statistics of the random process are affectedby a shift in the time origin.

Wide sense stationary (WSS):If the mean and autocorrelation function donot change with a shift in the origin time.

Cyclostationary:If the mean and autocorrelation function are periodic in time.

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Autocorrelation of a Wide-Sense Stationary

Random Process

For a wide-sense stationary process, the autocorrelationfunction is only a function of the time difference = t1t2;

(1.34)

Properties of the autocorrelation function of a real-valued wide-

sense stationary process are

( ) { ( ) ( )}XR E X t X t for

2

1. ( ) ( )

2. ( ) (0)

3. ( ) ( )

4. (0) { ( )}

X X

X X

X X

X

R R

R R for all

R G f

R E X t

Symmetrical in about zero

Maximum value occurs at the origin

Autocorrelation and power spectral

density form a Fourier transform pair

Value at the origin is equal to the

average power of the signal

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Ergodicity

Time averages = Ensemble averages

[i.e. ensemble averages like mean/autocorrelation can be computedas t ime-averages over a single realization of the random process]

A random process: ergodic in meanand autocorrelation(like w.s.s.)if

and

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Time Averaging and Ergodicity

When a random process belongs to a special class, known as anergodic process, its time averages equal its ensemble averages.

The statistical properties of such processes can be determinedby time averaging over a single sample function of the process.

A random process is ergodic in the mean if

(1.35)

It is ergodic in the autocorrelation function if

(1.36)

/ 2

/2

1lim ( )

T

Xx

T

m X t dt T

/ 2

/ 2

1( ) lim ( ) ( )

T

Xx

T

R X t X t dtT

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Power Spectral Density and Autocorrelation

A random processX(t) can generally be classified as a power

signal having a power spectral density (PSD) GX(f )

Principal features of PSD functions

3. ( ) ( )X XG f R

1. ( ) 0XG f 2. ( ) ( )X XG f G f

4. (0) ( )X XP G f df

and is always real valued

for X(t) real-valued

PSD and autocorrelation form a

Fourier transform pair

relationship between average

normalized power and PSD

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Summary of Relationships of Ergodicity of Random

Process to interchange the Time and EnsembleAverages

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Grey area contributesto negative values

Let X(t) is ergodic in theautocorrelation function

Autocorrelation function allows

to express signals PSD

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Noise in Communication Systems

The term noise refers to unwanted electrical signals that arealways present in electrical systems; e.g spark-plug ignitionnoise, switching transients, and other radiating electromagneticsignals.

The presence of noise superimposed on a signal tends toobscure or mask the signal; it limits the rate of information

transmission. Can describe thermal noise as a zero-mean Gaussian random

process.

A Gaussian process n(t) is a random function whose amplitudeatany arbitrary time t is statistically characterized by the Gaussian

probability density function

(1.40)21 1( ) exp

22

np n

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The normalized or standardized Gaussian density function of a

zero-mean process is obtained by assuming unit variance.

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A random signal is often represented as the sum of a Gaussian

noise random variable and a dc signal. That is,

z = a +n

where z is the random signal, a is the dc component, and n is the

Gaussian noise random variable

The pdf (z) is then expressed as

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White Noise

The primary spectral characteristic of thermal noise is that itspower spectral density is the same for all frequencies of interestin most communication systems

In other words, a thermal noise source emanates an equalamount of noise power per unit bandwidth at all frequencies ---from dc to about 1012Hz.

Therefore, a simple model for thermal noise assumes that itspower spectral density Gn(f) is flat for all frequencies, as shownon next slide and is denoted as

0

( ) /2n

N

G f watts hertz Where the factor of 2 is included to indicate that Gn(f) is a two-

sided power spectral density.

When the noise power has such a uniform spectral density werefer to it as white noise

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Uniform spectral

density

White Noise

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White Noise

Power spectral density Gn(f )

(1.42)

Autocorrelation function of white noise is given by the inverseFourier transform of the noise power spectral density

(1.43)

The average power Pnof white noise is infinite because itsbandwidth is infinite .

This can be seen by combining the following two equations

and

to get(1.44)

0

( ) /2n

N

G f watts hertz

1 0

( ) { ( )} ( )2n n

N

R G f

0( )

2

Np n df

Additive White Gaussian Noise

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Additive White Gaussian Noise

Si l T i i th h Li

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Signal Transmission through Linear

Systems

A system can be characterized equally well in the time domain or

the frequency domain, techniques will be developed in both

domains

The system is assumed to be linear and time invariant.as shown:

The signal applied to the input of the system can be described

either as a time-domain signal,x(t), or by its Fourier transformX(f).

The use of time-domain analysis yields the time-domain output

y(t), in response, h(t), the characteristic or impulse response of

the network will be defined.

Si l T i i th h Li

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Signal Transmission through Linear

Systems

When the input is considered in frequency domain, we shall

define a f requency transfer fun ct ion H(f) for the system, which

will determine the frequency-domain output Y(f).

It is also assumed that there is no stored energy in the system at

the time the input is applied

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Impulse Response

The impulse response h(t) of a LTIsystem at its input is unitimpulse which is nonrealizable signal, having infinite amplitude,

zero width, and unit area. The linear time invariant system or network is characterized in the

time domain by an impulse response h (t ), to an input unit impulse(t)

(1.45)

The response of the network to an arbitrary input signalx (t ) isfound by the convolution ofx(t ) with h (t )

(1.46)

The system is assumed to be causal, which means that there canbe no output prior to the time, t =0,when the input is applied.

Theconvolution integral can be expressed as:

(1.47a)

( ) ( ) ( ) ( )y t h t when x t t

( ) ( ) ( ) ( ) ( )y t x t h t x h t d

0

( ) ( ) ( )y t x h t d

I l R

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Impulse Response

F T f F i

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Frequency Transfer Function

The frequency-domain output signal Y(f ) is obtained by takingthe Fourier transform

(1.48)

Frequency transfer function or the frequency response is definedas:

(1.49)

(1.50)

The phase response is defined as:

(1.51)

( ) ( ) ( )Y f X f H f

( )

( )( )

( )

( ) ( ) j f

Y fH f

X f

H f H f e

1 Im{ ( )}( ) tanRe{ ( )}

H ff

H f

H(f) is complexand can bewritten as

R d P d Li S

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Random Processes and Linear Systems

If a random process forms the input to a time-invariant linear

system,the output will also be a random process.

The input power spectral density GX(f )and the output power

spectral density GY(f )are related as:

(1.53)2

( ) ( ) ( )Y XG f G f H f

Di i l T i i

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Distortionless TransmissionWhat is the required behavior of an ideal transmission line?

The output signal from an ideal transmission line may have sometime delay and different amplitude than the input

It must have no distortionit must have the same shape as theinput.

For ideal distortionless transmission:

(1.54)

(1.55)

(1.56)

Output signal in time domain

Output signal in frequency domain

System Transfer Function

0( ) ( )y t Kx t t

02

( ) ( ) j ft

Y f KX f e

02( ) j ft

H f Ke

What is the required behavior of an ideal transmission line?

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What is the required behavior of an ideal transmission line?

To achieve ideal distortionless transmission, the overall systemresponse must have a constant magnitude response

The phase shift must be linear with frequency

All of the signals frequency components must also arrive withidentical time delay in order to add up correctly

Time delay t0is related to the phase shift and the radianfrequency = 2f by:

t0(seconds) = (radians) / 2f (radians/seconds ) (1.57a)

Another characteristic often used to measure delay distortion of a

signal is called envelope delay or group delay:(1.57b)1 ( )

( )2

d ff

df

Id l Fil

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Ideal Filters

For the ideal low-pass filtertransfer function with bandwidth

Wf= fuhertz can be written as:

Figure1.11 (b) Ideal low-pass filter

(1.58)

Where

(1.59)

(1.60)

( )( ) ( ) j fH f H f e

1 | |( )

0 | |

u

u

for f fH f

for f f

02( ) j ftj fe e

Id l Filt

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Ideal Filters

The impulse response of the ideal low-pass filter:

0

0

1

2

2 2

2 ( )

0

0

0

( ) { ( )}

( )

sin 2 ( )2

2 ( )

2 sin 2 ( )

u

u

u

u

j ft

f

j ft j ft

f

f

j f t t

f

uu

u

u u

h t H f

H f e df

e e df

e df

f t tf

f t t

f nc f t t

Id l Filt

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Ideal Filters

For the ideal band-pass filtertransfer function

For the ideal high-pass filtertransfer function

Figure1.11 (a) Ideal band-pass filter Figure1.11 (c) Ideal high-pass filter

R liz bl Filt

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Realizable Filters

The simplest example of a realizable low-pass filter; an RC filter

1.63)

Figure 1.13

( )

21 1( )

1 2 1 (2 )

j fH f ej f f

R liz bl Filt r

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Realizable Filters

Phase characteristic of RC filter

Figure 1.13

Realizable Filters

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Realizable Filters

There are several useful approximations to the ideal low-passfilter characteristic and one of these is the Butterworth filter

(1.65)

Butterworth filters arepopular because they

are the bestapproximation to the

ideal, in the sense ofmaximal flatness in the

filter passband.

2

1( ) 1

1 ( / )n

n

u

H f nf f

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Example 1.2

Bandwidth Of Digital Data

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An easy way to translate the

spectrum of a low-pass or basebandsignal x(t) to a higher frequency is to

multiply or heterodyne the baseband

signal with a carrier wave cos 2fct

xc(t) is called a double-sideband

(DSB) modulated signalxc(t) = x(t) cos 2fct (1.70)

From the frequency shifting theorem

Xc(f) = 1/2 [X(f-fc) + X(f+fc) ] (1.71)

Generally the carrier wave

frequency is much higher than thebandwidth of the baseband signal

fc>> fm and therefore WDSB= 2fm

gBaseband versus Bandpass

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Bandwidth Dilemma

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Theorems of

communication and

information theory are

based on the

assumption of strictly

bandlimited channels

The mathematical

description of a real

signal does not permit

the signal to be strictlyduration limited and

strictly bandlimited.

Bandwidth Dilemma

Bandwidth Dilemma

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Bandwidth Dilemma

All bandwidth criteria have in common the attempt to specify a

measure of the width, W, of a nonnegative real-valued spectraldensity defined for all frequencies f <

The single-sided power spectral density for a single heterodyned

pulsexc(t) takes the analytical form:

(1.73)

2

sin ( )( )

( )

cx

c

f f TG f T

f f T

Different Bandwidth Criteria

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The power spectral density for a single heterodyned pulsexc(t) is

The plot of this PSD determines the bandwidth criteria asdepicted in below figure :

(a) Half-power bandwidth.This is the interval between frequencies at which Gx(f) hasdropped to half-power, or 3dB below the peak value.

(b) Equivalent rectangular or noise equivalent bandwidth.

This was originally conceived to permit rapid computation ofoutput noise power from an amplifier with a wideband noise inputi.e., the signal bandwidth.

(c) Null-to-null bandwidth.

The most popular measure of bandwidth for digitalcommunications is the width of the main spectral lobe, wheremost of the signal power is contained.


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(d) Fractional power containment bandwidth.

This bandwidth criterion has been adopted by FCC, which

states that the occupied bandwidth is the band that leavesexactly 0.5% of the signal power above the upper band limit andexactly 0.5% of the signal power below lower band limit. Thus99% of the signal power is inside the occupied band.

(e) Bounded power spectral density.

A popular method of specifying bandwidth is to. state thateverywhere outside the specified band, Gx(f) must have fallen atleast to a certain stated level below that found at the band center.Typical attenuation levels might be 35 or 50 dB

(f) Absolute bandwidth.

This is the interval between frequencies , outside of which thespectrum is supposed as zero.


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(a) Half-power bandwidth.

(b) Equivalent rectangular

or noise equivalent

bandwidth.

(c) Null-to-null bandwidth.

(d) Fractional power

containment

bandwidth.

(e) Bounded power

spectral density.

(f)Absolute bandwidth.

Example 1.4

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