Transcript No Slide Title - unix.eng.ua.edu

Stochastic (or Probabilistic)
Mathematics and its Relevance
to Thermal Noise
Stochastic (or Probabilistic)
Mathematics
• A new approach compared to
deterministic mathematics.
• Needed in order to describe the effects
of noise.
Defining our terms
Deterministic mathematics: We can determine with
certainty what the value of n(t) will be at some
future time t.
Probabilistic (stochastic) mathematics: We cannot
determine with certainty what the value of n(t) will
be at some future time t (either because of the
nature of n(t) or because we don’t know enough
about n(t) to be able to do this) but we can describe
the tendencies of n(t) at some future time t.
Defining our terms (continued)
A phenomenon that we cannot describe
deterministically but that we can describe
probabilistically is called random.
Examples:
* Tomorrow’s temperature at noon
* Next week’s stock prices
The consolation prize
Describing a signal using stochastic mathematics
isn’t as good as being able to describe the signal
using deterministic mathematics, but sometimes
it’s the best we can do.
Example: We can’t determine with certainty what the
temperature will be tomorrow at noon, but there is a very high
probability it will be between 50 and 75 degrees. This
information isn’t as good as being able to determine the exact
temperature, but it’s better than nothing.
Relevance to Communication Systems
One of the main types of noise in a
communication system is thermal noise, which is
produced by the random motion of electrons. We
can’t predict the value of thermal noise at any
future time t with certainty, but we can describe
its tendencies.
If we are going to analyze the
effects of noise on communication
systems, we need to develop some
appropriate parameters for
describing the tendencies of a
random signal.
The Random Signal Itself
Let’s use X to represent a measurement of the
random signal (say, a measurement of noise
voltage) at a particular time. X is called a
random variable.
Parameter #1:
Probability Distribution Function
FX (a)  P(X  a)
FX(a) is the probability that the random variable
X is less than or equal to some specified value a.
Relevance to Thermal Noise
The probability that thermal noise is less than or
equal to a (volts) is
a
FX (a) 


1
2  n
e
 ( x n )2

 2 2
n





dx
a
Where n = 0 and n2 is the average normalized
power of the noise at the receiver.
Parameter #2:
Probability Density Function
dFX ( x)
f X ( x) 
dx
Probability density function is good for
determining probability that a random variable
lies between two values, say a and b.
(continued)
Probability Density Function (cont.)
P ( a  X  b)  P ( X  b)  P ( X  a )
 FX (b)  FX (a)
b

f
a
X
( x)dx 

b
  f X ( x)dx
a
f

X
( x)dx
Relevance to Thermal Noise
As established previously, the probability distribution
function for thermal noise is
a
FX (a) 


1
2  n
e
 ( x n )2

 2 2
n





dx
Therefore, the probability density function for thermal noise
 ( xn )2 
is


dFX ( x)
f X ( x) 

dx
1
2  n

e


2 n2


This particular function is called the Gaussian function. It is
the widely-known “bell curve” (more on this subject later).
Parameter #3:
Mean (or expected) value
Mean is symbolized as X

 X   xfX ( x)dx

The mean is just the average value. This is often called
the expected value and is symbolized as E{X}.
The mean value of thermal noise is 0.
Parameter #4:
Variance
Variance is symbolized as X2

   ( x   X ) f X ( x)dx  E{( x   X ) }
2
X
2
2

Variance is a measure of how much values of the random
variable X fluctuate (or vary) about its average. This is a
measure of the unpredictability of X.
The variance of thermal noise is related to its average
normalized power.