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Analog Digital Communication :

Asad Abbas

Assistant Professor Telecom Department

Air University, E-9, Islamabad

Lec2_chpater01

10/29/2009 Lecture 1 2

Scope of the course ...

General structure of a communication systems

FormatterSource

encoder

Channel

encoderModulator

FormatterSource

decoder

Channel

decoderDemodulator

Transmitter

Receiver

SOURCE

Info.Transmitter

Transmitted

signal

Received

signalReceiver

Received

info.

Noise

ChannelSource User

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10/29/2009 Lecture 1 3

Random Variable

The random variable, X(A) represents a

functional relationship between random eventand a real number.

It is designated by X and functional

dependence upon A is considered as implicit.

Discrete Random Variable

If in any finite interval X assumes only finite number of

distinct values, it is discrete random variable, for example

tossing of dice, tossing of a coin

Continuous Random Variable

If X assumes any value within interval it is continuous, for

example noise, temperature etc

10/29/2009 Lecture 1 4

Distribution

Cumulative Distribution (CD):

Fx (x) = P ( X

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10/29/2009 Lecture 1 5

Distribution (contd..)

Probability Density Distribution (PDF)

px(x) = d/dx (Fx (x))It is rate of change of CD with respect to the random

variable

Properties

( ) ( )x

X XF x p x dx=

10/29/2009 Lecture 1 6

Ensemble Averages of Random Variable

Mean =

Nth moment of PD of a random variable=

The second central moment is given by

Second moment of PD of a random variable=

Mean of a Discreet RV = E{ X } =1

( )n

i

i

X xPx x

=

=

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10/29/2009 Lecture 1 9

Random Process..contd

The random processes at each a point in time

is random variable. X(tk) is random variable by observing random

process at time at t=k.

The values that X(tk) can take are

X1(tK)XN(tK)

The random processes have all the properties

of random variables, such as mean,

correlation, variances, etc

10/29/2009 Lecture 1 10

Statistical Properties of Random process

Mean

Autocorrelation of Random Process X(t)

It measures of the degree to which two time samples

of the same random process are related

It is function of two variables t1= t and t2= t+

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10/29/2009 Lecture 1 11

Random process Types

Stationary Process or Strictly Stationery Process:

If all of the statistical properties random process donot change with time it is called strictly stationery,that is:

Random process do not depends on time i.e

X(A,t) = X(A)

Its Probability Density Function do not change withtime, i.e

_

Mean= E{X(t)} = mx(t)= constant

Autocorrelation= Rx (t1, t2)= constant

Nth moment = E{X(t)n} = constant.

1 2 1 2, ,...... , ,......

t t t t tk X X tk X X Xp p p p p p

+ + +=

10/29/2009 Lecture 1 12

Random Process Types

Wide ( or Weak) sense stationary (WSS):

In WSS random process only two statistics (

mean and autocorrelation) do not change

with time

Mean of X(t)= E{X(t)} = mx(t) = Constant

Rx (t1, t2) = Rx (t1+, t2+) = Rx (t2 t1,0)= Rx ()

It means Autocorrelation only depends on difference

between t1 and t2. Thus all pairs of X(t) at times

separated by t2-t1 have same correlation value

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10/29/2009 Lecture 1 13

Random process Types (Contd..)

Ergodic process:A random process is ergodic if its

ensemble ( statistical) and time averages are same,that is:

10/29/2009 Lecture 1 14

Random Process

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10/29/2009 Lecture 1 15

Spectral density

Energy signals:

Energy spectral density (ESD):

Power signals:

Power spectral density (PSD):

Random process: Power spectral density (PSD):

10/29/2009 Lecture 1 16

Autocorrelation

Autocorrelation of an energy signal

Autocorrelation of a power signal

For a periodic signal:

Autocorrelation of a random signal

For a WSS process:

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10/29/2009 Lecture 1 17

Properties of an autocorrelation function

For real-valued (and WSS in case of random

signals):1. Autocorrelation and spectral density form a Fourier

transform pair.

2. Autocorrelation is symmetric around zero.

3. Its maximum value occurs at the origin.

4. Its value at the origin is equal to the average power or

energy.

10/29/2009 Lecture 1 18

Power SpectralDensity and

Autocorrelation of a

Low Rate Signal

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10/29/2009 Lecture 1 19

Power Spectral

Density and

Autocorrelation of a

High rate Signal

10/29/2009 Lecture 1 20

Noise

It is undesired signal interfering with the

desired signal.

External Sources

Atmospheric Noise ( Max Freq range: 30 Mhz)

lightening

Solar Noise

Cosmic Noise

( Source is sun and distant stars. Frequency range is 8Mhz-

1.43 GHz)

Industrial Noise (Freq Range = 1-600 MHz)

Ignition, motors, leakage from high voltage line, Fluorescent

tube

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10/29/2009 Lecture 1 23

White Noise

The Power Spectral Density ( Gn(f) ) of thermal

noise is same from DC to about 1012 Hz. ThusGn(f) is flat for all frequencies of interest

[w/Hz]

Power spectral

density

Autocorrelation

function

The autocorrelation of White noise is given by inverse Fourier Transform

of Gn(f)

10/29/2009 Lecture 1 24

Signal transmission through linearsystems

Frequency Transfer Function

Deterministic signals:

Random signals:

Input Output

Linear system

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10/29/2009 Lecture 1 25

Signal transmission - contd

Ideal distortion less transmission:

All the frequency components of a signal at input of a linearsystem are amplified (or attenuated ) and delayed by the

system equally.

= Group Delay

10/29/2009 Lecture 1 26

Frequency and Impulse Responses ofIdeal Filter

Ideal Lowpass filters: BW = fu - 0 = fu

Low-pass

Non-causal!

fu-fu 0

1

2 [2 ( )]u u of sinc f t t=

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10/29/2009 Lecture 1 27

Frequency and Impulse responses of

Ideal Filter

Band-pass

fufl-fl-fu

BW = fu-fl

High-pass

Ideal Bandpass filters:

Filter Bands Pass band

Transition Band

Stop Band

10/29/2009 Lecture 1 28

Frequency response of a Realizable Filter

Realizable filters: RC filters

3 0

2

1

2

1( )

1 ( )u

cut off dB u

f

f

f f f fRC

H f

= = = =

=+

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10/29/2009 Lecture 1 29

Frequency Response of Realizable Filter

R1 = R2 =1 Ohm

At cut- off frequency = ( ) 3

( ) .707

dBH f dB

H f

=

=

10/29/2009 Lecture 1 30

Bandwidth of a Passband signal

Different definition of bandwidth:

a) Half-power bandwidth

b) Noise equivalent bandwidth

c) Null-to-null bandwidth

d) Fractional power containment bandwidth

e) Bounded power spectral density

f) Absolute bandwidth

(a)

(b)

(c)

(d)

(e)50dB

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10/29/2009 Lecture 1 31

Bandwidth of signal(contd)

10/29/2009 Lecture 1 32

END

Thank You