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Digital CommunicationsLec2

Review of Signals and Random

Variables

Dr. Asad Mahmood,

Grad Course Fall 2009,Centre for Advanced Studies in Engineering ,

Islamabad, Pakistan.

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Outline for Todays Lecture

Introduction to the Course Grading Policy

Modules / Learning Outcomes of this course

Introduction to Digital Communication Systems

History Analog / Digital Communication Systems

Advantages / Disadvanatges

Physical ( PHY ) Layer

Modern Digital Communication Systems

Review of Signals and their Representation

Characteristics / Representations

Role of Random Signals / Processes in Digital Communications

Random / Stochastic Variables and Processes

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Physical Layer

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Physical (PHY) Layer

Signal Processing in PHY Layer TX

Encryption Source Coding Channel Coding

Reduces the Error Probabilityat a given SNRat the expenseof Bandwidth / Throughput

Multiplexing System viewed as a Network Single-User Vs.Multiple-User

Modulation RX

Filtering Equalization Synchronization Demodulation De-multiplexing Channel Decoding Source Decoding Decryption

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Classification of signals

Deterministic and random signals

Deterministic signal:

No uncertainty with respect to the signal value at any time.

Modelled by explicit mathematical Equations e.g. x(t) =5cos(10t)

Random signal: Some degree of uncertainty in signal values before it

actually occurs.

Over a Long-time it may exhibit certain regularities/characteristics

Expressed in the form of probabilities/ statistical propertiesetc.

Thermal noise in electronic circuits

Reflection of radio waves

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Classification of signals

Periodic and non-periodic signals

Analog and discrete signals

A discrete signal

Analog signals

A non-periodic signalA periodic signal

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Classification of signals ..

Signal Energy and Power are important parameters for a

communication System The performance of a comm. Sys. depends on received signal energy.

Power= Rate at which energy is transmitted determines the voltagerequirements for a transmitter (TX). For modelling convenience

Energy and power signals

A signal is an energy signal if, and only if, it has nonzero but finite energy for alltime:

A signal is a power signal if, and only if, it has finite but nonzero power for alltime:

Periodic and random signals are generally classified as power signals.

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Spectral Density

Energy Spectral Density Parsevals Theorem

Distribution of signal energy infrequency domain

ESD for real valued signals

Power Spectral Density

Parseval theorem for a real-valuedperiodic signal

Distribution of power of x(t) in

the frequency domain PSD of periodic signal is a

discrete function of freq.

Average normalized power for asinusoid

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Autocorrelation Matching (Correlation) of a signal with a delayed version

of itself

Autocorrelation of an energy signal Properties

Symmetrical in about zero Maximum occurs at origin

Autocorrelation and ESD form a Fourier Transform pair

Value at origin is equal to energy of the signal

Autocorrelation of a power signal

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Random Signals

Huge importance in context of communications Statistics of the transmitted message

Statistics of the noise / interference

Random Variable (R.V) A random variable X(A) represent the functional relationship

between a random event A and a real number

Distribution Function

Probability Density Function

Expected Value

VarianceMeasure of randomness of a R.V

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Random process

Natural extension of RV when dealing with signals

Time-varying random signals in communication systems

Thermal noise

Wave propagation characteristics

Information source no need to transmit if already known

Modelling of signals as RV rather than deterministic functions

A Random processor Random Signalcan be viewed as a set ofpossible realizations of signal waveforms

Hence we have signals/functions instead of numbers in Randomprocesses as compared to a Random variable

Exp :Variable Freq. generator

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Random processes

A random process (RP) or stochastic process is a function that

maps all elements of a sample space into a collection or ensembleof time functions called sample functions.

The value a random process at any given time cannot be predictedin advance depends on the value of the initial outcome/sample

RP is a function of two variables event (A) and time (t), either orboth of them can be fixed

X(t,s) = X(t) is a RP

X(tj,s) = X(s) is a RV X(t,sk) = x(t) is a deterministic function of time or sample function

X(tj, sk) = x is a real number

Discrete RP, Continuous RP, Discrete-time RP = Random Vector

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Random processes

A random process (RP) orstochastic process is a functionthat maps all elements of asample space into a collection or

ensemble of time functions calledsample functions.

The value a random process atany given time cannot be

predicted in advance dependson the value of the initialoutcome/sample

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Random processes A random process whose distribution functions are continuous is

described statistically by a Probability density function

In general, the form of the pdf of a random process will bedifferent for different times

In most situation empirically determining the distribution functions isnot possible, however partial description consisting of the mean andautocorrelation function are often adequate for the needs of thecommunication system

Statistical Mean of the Random Process

Autocorrelation function of the Random Process