DC Lecture3

8/12/2019 DC Lecture3

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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

Date post:	03-Jun-2018
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