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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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Probability Density Function
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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Uniform Distribution
Probability Density Function
bxaforab
1)x(f
a b0
f(x)
x
1/(b-a)
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Uniform Distribution
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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1. Autocorrelation of an Energy Signal
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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1. Autocorrelation of an Energy Signal
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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Noise in Communication Systems
The normalized or standardized Gaussian density function of a
zero-mean process is obtained by assuming unit variance.
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Noise in Communication Systems
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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Different Bandwidth Criteria
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.
Different Bandwidth Criteria
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Different Bandwidth Criteria
(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.
Different Bandwidth Criteria
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Different Bandwidth Criteria
(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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p