Esla Mod3@Azdocuments - in
Esla Mod3@Azdocuments - in
Esla Mod3@Azdocuments - in
Algebra
Module 3
Random Processes
Faculties Handling: -
1. Mrs. Anjani , 2. Mrs. Laxmi G
Asst Professor, Senior Asst Professor,
Dept. of ECE, Dept. of ECE,
MITE, Moodabidri. MITE, Moodabidri
Course Outcomes(CO)
After studying this course, students will be able to:
Identify and associate random variables and distributive functions associated to
CO1 Communication events.
Dept.of ECE,MITE,Moodabidri 4
LESSON PLAN
Date Date
Sl no TOPIC Remarks
planned engaged
5
LESSON PLAN
Date Date
Sl no TOPIC Remarks
planned engaged
10
Random Process
• A random process is a time-varying function that
assigns the outcome of a random experiment to each
time instant: X(t).
• For a fixed (sample path): a random process is a time
varying function, e.g., a signal.
• –For fixed t: a random process is a random variable.
• If one scans all possible outcomes of the underlying
• random experiment, we shall get an ensemble of
signals.
• Random Process can be continuous or discrete
Probability Density function
• Random variables x 1 , x 2 , . . . , x n represent
amplitudes of sample functions at t 5 t 1 , t 2 ,
...,tn.
• – A random process can, therefore, be viewed
as a collection of an infinite number of
random variables:
• Deterministic and random processes :
• both continuous functions of time (usually), mathematical concepts
• deterministic processes :
physical process is represented by explicit mathematical relation
• Example :
response of a single mass-spring-damper in free vibration in
laboratory
• Random processes :
result of a large number of separate causes. Described in probabilistic terms
and by properties which are averages
• random processes :
fX(x)
x(t)
time, t
• Ensemble averaging :
properties of the process are obtained by averaging over a collection
or ‘ensemble’ of sample records using values at corresponding times
• Time averaging :
properties are obtained by averaging over a single record in time
• Stationary random process :
• Ensemble averages do not vary with time
• Ergodic process :
stationary process in which averages from a single record are the same
as those obtained from averaging over the ensemble
x(t)
x
time, t T
1 T
x Lim x(t)dt
T T 0
• The mean value,x , is the height of the rectangular area having the
same area as that under the function x(t)
• Can also be defined as the first moment of the p.d.f. (ref. Lecture 3)
• Mean square value, variance, standard deviation :
x
x(t)
x
time, t T
1 T 2
mean square value, x Lim x (t)dt
2
T T 0
variance, 2
x
σ x(t) x Lim x(t) - x dt
2 1 T
T T 0
2
(average of the square of the deviation of x(t) from the mean value,x)
time, t T
• The autocorrelation, or autocovariance, describes the general
dependency of x(t) with its value at a short time later, x(t+)
1 T
x ( ) Lim x(t) - x . x(t τ) - x dt
T T 0
Rx(Ʈ) = E *X(t)X(t+Ʈ)]
Rxy(Ʈ) = E *X(t)Y(t+Ʈ)]
Addition of Random Process
W(t)= X(t)+ Y(t)
Then,
Rw(Ʈ)=E*{X(t)+Y(t)}{X(t+Ʈ) Y(t+Ʈ)}]
Rw(Ʈ) =Rx(Ʈ)Ry(Ʈ)
• Covariance :
• The covariance is the cross correlation function with the time delay, ,
set to zero
1 T
c xy (0) x(t).y(t) Lim x(t) - x . y(t) - y dt
T T 0
Note that here x'(t) and y'(t) are used to denote the
fluctuating parts of x(t) and y(t) (mean parts subtracted)
x' (t).y'(t)
ρ
σ x .σ y
z2
z1
The cross spectral density is twice the Fourier Transform of the cross-
correlation function for the processes x(t) and y(t)
If x(t) and y(t) are local fluctuating forces acting at different parts
of the structure, xy(n1) describes how well the forces are
correlated (‘synchronized’) at the structural natural frequency, n1
• Input - output relationships :
Input x(t) Output y(t)
Linear system
There are many cases in which it is of interest to know how an input random
process x(t) is modified by a system to give a random output process y(t)
Application : The input is wind force - the output is structural
response (e.g. displacement acceleration, stress). The ‘system’ is
the dynamic characteristics of the structure.
frequency, n