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Communication Theory II
Lecture 8: Stochastic Processes
Ahmed Elnakib, PhD
Assistant Professor, Mansoura University, Egypt
1March 5th, 2015
Lecture Outlines
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o Stochastic processes What is a stochastic process?
Types:
o Stationary vs. nonstationary processes
o Strictly stationary processes vs. weakly stationary processes (e.g., Ergodic processes)
Parameters:
o Mean, correlation, Covariance
o Power spectral density
Transmission of a weakly stationary process in LTI system
Poisson and Gaussian Processes
Introduction to Stochastic Processes
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o Received signal at the wireless channel output varies randomly with time Processes of this kind are said to be random or stochastic
o It is not possible to predict the exact value of a signal drawn from astochastic process However, it is possible to characterize the process in terms of statistical
parameters such as average power, correlation, and power spectra
EncoderSource Channel Decoder
Signal with random changes
Stochastic Processes
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o A stochastic process may be presented as thesample space or ensemble composed offunctions of time
o Each sample point of the sample spacepertaining to a stochastic process is a functionof time
o The totality of sample points corresponding tothe aggregate of all possible realizations of thestochastic process is called the sample space
o A single realization of a process is a randomwaveform that evolves across time Each realization of a process is associated
with a sample point
Set (ensemble) of sample functions
o Consider a stochastic process specified by: outcomes s observed from some sample space S events defined on the sample space S probabilities of these events
o Suppose, we assign each sample point s a function in time in accordance with the rule: X(t,sj), -T≤ t ≤T
2T: total observation intervalo For a fixed sample point sj the graph of the function
X(t,sj) is called a realization or sample function of
the stochastic process
To simplify we denote this sample function
Xj(t)= X(t,sj) , -T≤ t ≤T
Stochastic Processes (cont’d)
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Set (ensemble) of sample functions
o A realization or sample function of the stochastic
process: Xj(t)= X(t,sj) , -T≤ t ≤T
o At particular instant of time, we deal with a R.Vsampled (observed) at that instant of time
o A random variable is constituted as the set ofnumbers observed at a fixed time tk inside theobservation interval
Stochastic Processes and R.Vs
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Set (ensemble) of sample functions
A stochastic process X(t,s) or X(t) is represented by the time indexed ensemble (family) of random variables {X(tk,s)} or {X(tk)}
Stochastic Processes and R.Vs (cont’d)
o Stochastic process the outcomes ofstochastic experiment is mapped into awaveform (function of time)
o R.V. the outcomes of a stochasticprocess is mapped to a number
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Set (ensemble) of sample functions
A stochastic process X(t) is an ensemble of timefunctions, which, together with a probabilityrule, assigns a probability to any meaningfulevent associated with an observation of one ofthe sample functions of the stochastic process
Important Types of Stochastic Processes
o Stationary and nonstationary
o Strictly stationary and weakly stationary (wide-sense stationary)
Ergodic processes (subsets of weakly stationary processes)
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Stationary Processes
o In dealing with stochastic processes encountered in the real world:
We often find that the statistical characterization of a process is independent ofthe time at which observation of the process is initiate
That is, if such a process is divided into a number of time intervals, the varioussections of the process exhibit essentially the same statistical properties
This process is said to be stationary. Otherwise, it is said to be nonstationary
o A stationary process arises from a stable phenomenon that has evolved into asteady-state mode of behavior, whereas a nonstationary process arises from anunstable phenomenon
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Strictly stationary
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Strictly stationary (cont’d)
oA stochastic process X(t), initiated at time t = -∞, is strictly stationaryif the joint distribution of any set of random variables obtained byobserving the process X(t) is invariant with respect to the location ofthe origin t = 0
oHow such a process is random?
The finite-dimensional distributions
depend on the relative time separation between random variables,
not on their absolute time
However, the stochastic process has the same probabilistic behavior throughout the global time t 11
Jointly Strictly Stationary Processes
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2.
Properties of Strictly Stationary Processes
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1.
Example (Multiple Spatial Windows)
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Weakly (Wide-sense) stationary Processes
o A stochastic process X(t) is said to be weakly stationary if its second-order moments satisfy the following two conditions:
1. The mean of the process X(t) is constant for all time t
2. The autocorrelation function of the process X(t) depends solely (alone)on the difference between any two times at which the process issampled
“auto” in autocorrelation refers to the correlation of the process withitself
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Statistical Parameters of Stochastic Processes
o Mean
o Correlation
o Cross-correlation
o Covariance
o Power Spectral Density
o Cross Spectral Density
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oThe mean of a real-valued stochastic process X(t) is the expectation of the randomvariable obtained by sampling the process at some time t
Where is the first-order probability density function of the process X(t),observed at time t
Note also that the use of single X as subscript in is intended to emphasize thefact that is a first-order moment
Mean of Stochastic Processes
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o The mean of a process is given by:
o For a process X(t), to satisfy the first condition of weak stationary,
The mean is a constant for all time (independent of time t)
Mean of Weakly stationary Processes
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Autocorrelation of A Stochastic Process
oAutocorrelation function of the stochastic process X(t) is the expectation of the product of two random variables, X(t1) and X(t2), obtained by sampling the process X(t) at times t1and t2:
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Where is the second-order probability density function
of the process X(t), observed at times t1and t2
Autocorrelation of A Weakly Stationary Process
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Equivalently, if :
The autocorrelation function of the process X(t) depends solely (alone) onthe difference between any two times at which the process is sampled
𝑡1 = 𝑡 and 𝑡2 = 𝑡1+ 𝜏
Autocovariance of A Weakly Stationary Process
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Mean and Autocorrelation of Weakly stationary Processes
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for the mean and the autocorrelation
Properties of the Autocorrelation Function
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o Mean Square value
o Symmetry
Thus the auto correlation may be defined also as:
o Maximum magnitude at zero shift: or
o Normalized autocorrelation function: 0≤ρXX≤1
o The autocorrelation function 𝑅𝑋𝑋 𝜏 provides a means of describing the
interdependence of two random variables obtained by sampling the stochastic process
X(t) at times 𝜏 seconds apart
o The more rapidly the stochastic process X(t) changes with time, the more rapidly will
the autocorrelation function 𝑅𝑋𝑋 𝜏 decrease from its maximum 𝑅𝑋𝑋 0 as increases
o At a decorrelation time 𝜏dec , such that, for 𝜏 > 𝜏dec: The magnitude of the autocorrelation function
𝑅𝑋𝑋 𝜏 remains below some prescribed value
Autocorrelation Function and Decorrelation Time
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The one percent decorrelation time 𝜏dec of a weakly stationary process of X(t) zero mean is
the time taken for the magnitude of the autocorrelation function 𝑅𝑋𝑋 𝜏 decrease, forexample, to 1% of its maximum value 𝑅𝑋𝑋 0
o Consider a sinusoidal signal with random phase, defined by
Example 1: Sinusoidal Wave with Random Phase
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o The random variable is equally likely to have any value in the interval [– ]
o Each value of corresponds to a point in the sample space S of the stochastic process X(t)
o The process X(t) represents a locally generated carrier in the receiver of a communication
system, which is used in the demodulation of a received signal
o Find and plot the autocorrelation function
o Write a Matlab© program showing that by increasing the number of samples from a
uniformly distributed R.V., the normalized histogram of these samples is closer to a
uniformly distributed mass function
normalized histogram at (X=xi)=frequency of X=xi
summation of all frequencies=
frequency of X=xi
Number of samples
o Group is allowed (3-5) using the oral grouping
o Post your solution by Email to [email protected]
o You can be asked individually about your solution
Homework
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Solution
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Questions
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