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Lecture #3
Review on Lecture 2
Random Signals Signal Transmission Through Linear Systems
Bandwidth of Digital Data
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Averages
time average:Averaged quantity of a single
system over a time interval.(Directly related
to the real experiment)
ensemble average:Averaged quantity ofmany identical systems at a certain time
(theoretical concept)
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1.1 Ensemble Averages
The first moment of a probability
distribution of a random variableX is called mean value mX, orexpected value of a randomvariableX
The second moment of aprobability distribution is the
mean-square value of X
Central momentsare themoments of the differencebetweenX and mXand thesecond central moment is thevariance ofX
Variance is equal to thedifference between the mean-square value and the square ofthe mean
mX= E{X} = xpx(x) dx-
E{X2} = x2px(x) dx-
Var(X)= E{(XmX)2}
= (xmX)2px(x) dx
-
Var(X)=E{X2}E{X}2
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2. 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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2.1 Statistical Averages of a Random
Process A random process whose distribution functions are continuous can be
described statistically with a probability density 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 processX(t) :
E{X(tk)} = x Px(x) dx= mx(tk) (1.30)
Autocorrelation function of the random processX(t)Rx(t1,t2) = E{X(t1) X(t2)} = xt1xt2Px(xt1,xt2) dxt1dxt2 (1.31)
-
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2.2 Stationarity
A random processX(t) is said to be stationary in the strict sense
if noneof its statistics are affected by a shift in the time origin.
A random process is said to be wide-sense stationary (WSS) if
two of its statistics, its mean and autocorrelation function, do notvary with a shift in the time origin.
E{X(t)} = mx= a constant (1.32)
Rx(t1,t2) = Rx(t1t2) (1.33)
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2.3 Autocorrelation of a Wide-Sense
Stationary Random Process
For a wide-sense stationary process, the autocorrelation
function is only a function of the time difference = t1t2;
Rx() = E{X(t) X(t + )} for - < < (1.34)
Properties of the autocorrelation function of a real-valued wide-
sense stationary process are
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3. Time Averaging and Ergodicity
When a random process belongs to a special class, known as an
ergodic 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
mx= lim 1/T x(t) dt (1.35-a)
It is ergodic in the autocorrelation function if
Rx() = lim 1/T x(t)x(t + ) dt (1.35-b)
-T/2
T/2
T -T/2
T/2
T
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5. Noise in Communication Systems
The term noise refers to unwanted electrical signals that are
always present in electrical systems; e.g spark-plug ignitionnoise, switching transients, and other radiating electromagnetic
signals or atmosphere, the sun, and other galactic sources.
Can describe thermal noise as a zero-mean Gaussian random
process.
A Gaussian process n(t) is a random function whose amplitudeat
any arbitrary time t is statistically characterized by the Gaussian
probability density function
(1.40)
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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.
Central limit theorem
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5.1 White Noise
The primary spectral characteristic of thermal noise is that its
power spectral density is the same for all frequencies of interestin most communication systems
Power spectral density Gn(f )
(1.42)
Autocorrelation function of white noise is (meaning) uncorrelated
(1.43)
The average power Pnof white noise is infinite
(1.44)
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The effect on the detection process of a channel with additivewhite Gaussian noise (AWGN) is that the noise affects each
transmitted symbol independently.
Such a channel is called a memoryless channel.
The term additive means that the noise is simply superimposed
or added to the signal
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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.
It is also assumed that there is no stored energy in the system
at the time the input is applied
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1. Impulse Response A.5
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)
h(t) = y(t) when x(t) = (t) (1.45)
The response of the network to an arbitrary input signalx (t )isfound by the convolution ofx (t )with h (t )
y(t) = x(t)*h(t) = x()h(t- )d (1.46)
The system is assumed to be causal,which means that there can
be no output prior to the time, t =0,when the input is applied.
convolution integral can be expressed as:
y(t) = x() h(t- )d (1.47a)
-
-
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2. Frequency Transfer Function
The frequency-domain output signal Y (f )is obtained by taking
the Fourier transform
Y(f) = H(f)X(f) (1.48)
Frequency transfer function or the frequency response is defined
as:H(f) = Y(f) / X(f) (1.49)
H(f) = |H(f)| ej (f) (1.50)
The phase response is defined as:(f) = tan-1 Im{H(f)} / Re{H(f)} (1.51)
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2.1. 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:
Gy(f)= Gx(f) |H(f)|2 (1.53)
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3. 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:
y(t) = K x( t - t0 ) (1.54)
Y(f) = K X(f) e-j2f t0 (1.55)
H(f) = K e-j2f t0 (1.56)
Output signal in time domain
Output signal in frequency domain
System Transfer Function
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What is the required behavior of an ideal transmission line?
The overall system response must have a constant magnituderesponse
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 asignal is called envelope delay or group delay:
(f) = -1/2 (d(f) / df) (1.57b)
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3.1. Ideal Filters
For the ideal low-pass filtertransfer function with bandwidth Wf=
fuhertz can be written as:
Figure1.11 (b) Ideal low-pass filter
H(f) = | H(f) | e-j (f) (1.58)
Where
| H(f) | = { 1 for |f| < fu
0 for |f| fu }
(1.59)
e-j (f) = e-j2f t0 (1.60)
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3.1. Ideal Filters
The impulse response of the ideal low-pass filter:
h(t) = F-1 {H(f)}
= H(f) e-j2f tdf (1.61)
= e-j2f t0e-j2f tdf
= e-j2f (t - t0)df
= 2fu* sin 2fu(tt0)/ 2fu(tt0)
= 2fu* sinc 2fu(tt0)
(1.62)
-
- fu
fu
- fu
fu
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3.1. Ideal Filters
For the ideal band-pass filter
transfer function
For the ideal high-pass filter
transfer function
Figure1.11 (a) Ideal band-pass filter Figure1.11 (c) Ideal high-pass filter
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3.2. Realizable Filters
The simplest example of a realizable low-pass filter; an RC filter
H(f) = 1/ 1+j2f RC = e-j (f)/ 1+ (2f RC )2 (1.63)
Figure 1.13
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3.2. Realizable Filters
Phase characteristic of RC filter
Figure 1.13
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3.2. RealizableFilters
There are several useful approximations to the ideal low-pass filter
characteristic and one of these is the Butterworth filter
| Hn(f) | = 1/(1+ (f/fu)2n)0.5
n 1 (1.65)
Butterworth filters arepopular because they
are the best
approximation to theideal, in the sense of
maximal flatness in thefilter passband.
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4. Bandwidth Of Digital Data4.1 Baseband versus Bandpass
An easy way to translate thespectrum of a low-pass or baseband
signal 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 signal
xc(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 the bandwidth ofthe baseband signal
fc>> fm and therefore WDSB= 2fm
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4.2 Bandwidth
Theorems ofcommunication 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.
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4.2 Bandwidth
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 pulsexc(t) takes the
analytical form:
(1.73)
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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.
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Digital Communications: A Discrete-Time Approach:
International Edition
View Larger
ImageMichael Rice
Publisher: Pearson HigherEducation
Copyright: 2009
Format: Paper; 800 pp
ISBN-10: 0138138222
ISBN-13: 9780138138226