Example: Consider an RC circuit as shown. Find h(t)
for this circuit.
R
+
+
x(t)
i(t)
C

y(t)

Write KVL,
1
C
R i(t) +
t
i(t) dt = x (t)
Take FT of both the sides,
R I ( ) +
1
I ( ) = X ( )
j C
Dep
Chapter 7: Spectral Density
Although Fourier transform does not exist for r.p.s (infinite
energy), the autocorrelation and crosscorrelation functions
are nonperiodic and energy signals. Thus for th
Brief Review on Digital Filters:
Why Digital vs. Analog?
Advantages:
Less (no) sensitivity to temperature variations / aging.
Better reliability.
Can be made adaptive.
More compact and light weight.
L
Time Averages and Ergodicity:
In practice, we would like to deal with only a single sample
function rather than the ensemble of functions. For example,
we may wish to infer the probability law or cert
Measures on two (or more) r.p.s:
Let X(t) and Y(t) be r.p.s with ACFs RXX(t 1,t2 ) and RYY(t1,t 2),
respectively. The crosscorrelation between X(t) and Y(t),
which gives a measure of dependence betwe
Chapter 5: Random Processes
In engineering and science, we often encounter problems, which
involve random signals, e.g., bit stream in a binary communication
system (random message), noise or other in
Example 12: Given joint PDF
f X ,Y ( x, y ) = 0.4 ( x + ) ( y 2) + 0.3 ( x ) ( y 2) +
0.1 ( x ) ( y ) + 0.2 ( x 1) ( y 1)
Find , which minimizes the correlation rX,Y.
Find minimum r X,Y.
Are X, Y orth
Note that,
20 = E[( X X ) 2 ] = 2
X
2
02 = E[(Y Y ) 2 ] = Y
The other 2nd order joint central moment, which is very
important is 11. This is called as the Covariance of X and Y.
c XY = 11 = E[( X X )
Operations on Multiple r.v.s:
Joint Moments, Correlation and Covariance:
These describe the relationships between r.v.s.
(1) Moments about the origin:
The expected value of product XiYj is the joint m
Example 5: Given
1
f X ,Y ( x , y ) = r 2
0
x 2 + y2 r2
elsewhere
Find f X(xy).
Example (5): Use marginal density,
fY ( y ) =
r 2 y2
dx
2
r 2 y 2 r
f X ,Y ( x, y ) dx =
2 r 2 y 2
=
r2
0
y < r
y
Definition2 (Joint Probability Density Function):
For two r.v.s X and Y, the joint PDF is
f X ,Y ( x , y ) =
2 FX ,Y ( x, y )
x y
If X and Y are discrete r.v.s, we get
NM
f X ,Y ( x , y ) = P( xm ,
Chapter 3
Multiple Random Variables:
Many experiments involve dealing with multiple r.v.s and one
may be interested in their interactions or joint behavior, e.g.,
suppose in every high school, the SAT
(4) Poisson Distribution:
If a r.v. X is discrete, taking values at points k = 0, 1, with
P( X = k ) = e
a
ak
,
k!
a>0
Then X has a poisson density and distribution as,
f X ( x ) = P( X = k ) ( x k )
Important PDFs
(1) Normal (Gaussian) Distribution:
A r.v. X is called Gaussian if its PDF is of a form
1
f X (x ) =
2
2 X
e ( x x )
X: Mean
X: Standard Deviation
about the mean
2 2
X
2
FX (x)
fX (x)
1
Statistical Averages and Expectation
It is often desirable to characterize a r.v. by a few numbers
(deterministic). These are called moments of r.v. and are
obtained by certain averaging or expectatio
Transformation of a Random Variable
Let X be a r.v. with known PDF, fX (x), or CDF, FX (x). This
r.v., is transformed using transformation function T(.) to
r.v. Y = T(X)
X
Y = T (X)
fX (x)
T (.)
fY (y
CDF of Discrete r.v. :
A discrete r.v., X taking on one of the countable set of possible
values x1, x2 with probability P [X = xk], k [1, N]
forming a stairstep CDF with amplitude of each step being
Chapter 2
Random Variables:
Many random phenomena have outcomes that are real numbers,
e.g., the voltage, v(t) at time, t, across a noisy resistor, number
of people on a New York to Chicago train, etc
Independent Events:
Events A and B are said to be statistically independent if
P( A  B) = P( A)
i.e., the prob. of event A is the same whether event B has
occurred or not. Thus the equivalent conditi
Remark:
The total probability is still valid even if the union of Bn does
not equal S, provided that
N
A U Bn
n=1
Bayes Theorem:
Recall that conditional probability applies to any two events,
P( A  B
Remark:
The total probability is still valid even if the union of Bn does
not equal S, provided that
N
A U Bn
n=1
Bayes Theorem:
Recall that conditional probability applies to any two events,
P( A  B
(3) If A C = , then
Pr( A C  B ) = Pr( A  B ) + Pr( C  B)
Proof: Use the equation for Conditional Probability,
Pr( A C  B ) =
Pr( A C B)
Pr( B)
Now using distributive law,
( A C ) B = ( A B) ( C B
Probability Definition and Axioms
Probability is a nonnegative number assigned to an event, A,
and is denoted by Pr(A).
Axioms of Probability (Kolmogrov):
(1) Pr (A) 0
(2) Pr (S) = 1
, i.e., Certain.
Set Definitions
Set: Collection of objects, which are called set elements.
aA
a is element of set A
aA
a is not an element of set A
In study of probability, elements are outcomes of experiments.
= cf
Introduction to Probability
Deterministic vs. Random Phenomena:
Deterministic: Outcome is certain
Random: Outcome is uncertain
e.g.
Background hiss noise in radio broadcast, Motion of
electrons, etc.