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7/27/2019 Lec02 Final
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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