lec2 signals review
TRANSCRIPT
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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