2. Contents
• Probability and Random Variables‐brief
Probability and Random Variables brief
introduction/motivation
• Mathematical Models as tools in Analysis and
Mathematical Models as tools in Analysis and
Design
–D t
Deterministic Models
i i ti M d l
– Probability Models
• Examples
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 2
3. A Typical Communication System
A Typical Communication System
• Probabilistic methods in making decisions
/ g
about the transmitted/received message
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 3
4. Digital Communication with
Probability of Error
b bl f
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 4
5. Probability/ Uncertainty Element in
Systems
• Wireless communication networks provide voice
e ess co u cat o et o s p o de o ce
and data communications to mobile users in
severe interference environments.
• The vast majority of media signals, voice, audio,
images, and video are processed digitally.
• The systems the designers build are
unprecedented in scale and the chaotic
environments in which they must operate are
environments in which they must operate are
untrodden terrritory.
• So there is “Uncertainty”
So there is Uncertainty
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 5
6. Probability Models
Probability Models
• Probability models are one of the tools that
Probability models are one of the tools that
enable the designer to make sense out of the
chaos and to successfully build systems that
chaos and to successfully build systems that
are efficient, reliable, and cost effective
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 6
7. MATHEMATICAL MODELS AS TOOLS IN ANALYSIS
AND DESIGN
AND DESIGN
• Experiments: Costly way of testing a design or solve a
problem.
• Model: Approximate representation of a physical situation.
• Useful Model: Able to explain all relevant aspects of a given
Useful Model: Able to explain all relevant aspects of a given
phenomenon.
• Mathematical Models: If observational phenomenon has
measurable properties then a mathematical model
measurable properties then a mathematical model
consisting of a set of assumptions about the system is
employed
• Conditions under which an experiment is performed and a
Conditions under which an experiment is performed and a
model is assumed are very critical. Change the assumptions
then a “good” model can be a great failure.
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 7
9. Computer Simulations & Deterministic
Models
d l
• Computer Simulation Models: They mimic or
p y
simulate the dynamics of a system
• Deterministic Models: Lab and textbook cases,
conditions determine outcome
conditions determine outcome
• 1. Circuit Theory
• 2 Ohm’s Law
2. Ohm s Law
• 3. Kirchoffs’ Laws
• 4 Transforms: FFT; Laplace Transforms
4. Transforms: FFT; Laplace Transforms
• 5. Convolution :Input/output behavior of systems
with well‐defined coefficients
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 9
10. Probability Models
Probability Models
• Probabilistic (Stochastic Random) Models:
Probabilistic (Stochastic, Random) Models:
involve phenomena that exhibit unpredictable
variation and randomness.
variation and randomness
Ex: Urn with three balls; (0,1,2
h h b ll (
marked)
‐‐ Outcome: A number from the set
{ , , }
{0,1,2}
‐‐ Sample Space: All possible
outcomes of an experiment: S =
{0,1,2}
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 10
11. Statistical Regularity
Statistical Regularity
• Relative Frequency:
Relative Frequency:
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 11
14. Properties of Relative Frequency
Properties of Relative Frequency
• Suppose a random experiment has K possible
Suppose a random experiment has K possible
outcomes: S = {1,2,…,K}. Then in “n” trials we
have
have
Ex: Consider the 3‐ball urn experiment
A: even = {0,2} then,
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 14
15. • Disjoint (mutually exclusive) events: If A or B
Disjoint (mutually exclusive) events: If A or B
can occur but not both, then
• Relative frequency of two disjoint events is
the sum of their individual relative frequency.
h f h i i di id l l i f
•
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 15
16. Axioms of Probability
Axioms of Probability
• Kolmogorov’s axioms to form a Theory of Probability:
Assumptions:
1. Random experiment has been defined and the sample
space S has been identified.
2. A class of subsets of S has been specified.
3. Each event A has been assigned a number P(A) such that,
– 1 0 ≤ P(A) ≤ 1
1. 0 ≤ P(A) ≤ 1
– 2. P(S) = 1
– 3. If A and B are mutually exclusive events then
• P(A or B) = P(A) + P(B)
P(A or B) = P(A) + P(B)
• Kolmogorov’s axioms are sufficient to build a consistent
Theory of Probability.
Theory of Probability
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 16
17. Example
• Example: Packet Voice Communication
Example: Packet Voice Communication
system Efficiency
• Due to silences voice communication is very
Due to silences voice communication is very
inefficient on dedicated lines. It is observed
that only “1/3” of the time actual speech goes
through. How to increase this rate by using
prob. approaches???
• Solution: Error vs rate trade off in digital
information (BCS) transmission/storage
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 17
19. Packet Voice Communication system
Efficiency (3)
ff ( )
• A is the outcome of a random experiment, determining which
p , g
packets contain active speech
• M<48 packets are transmitted every 10 ms
• If A≤ M, then all packets are active
• If A> M, then A‐M active packets are discarded randomly
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 19
20. Packet Voice Communication system
Efficiency (4)
ff ( )
• E[A]=48*1/3
E[A]=48 1/3
• E[A]=16
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 20
22. Example: Signal Enhancement Using
Filters
l
• Given a signal x(t) corrupted with noise and
Given a signal x(t) corrupted with noise and
has a Signal‐to‐Noise Ratio value SNR. If you
filter this noisy signal with a properly designed
filter this noisy signal with a properly designed
adaptive filter to suppress noise, we obtain an
enhanced signal, (smoothed by the filter.)
enhanced signal (smoothed by the filter )
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 22
24. Resource Sharing
Resource Sharing
• Example: Multi User Systems with Queues:
Example: Multi User Systems with Queues:
Resource sharing
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 24
26. System Reliability: Cascade vs. Parallel
Systems
• Issues: Need of a clock vs the system delay or
Issues: Need of a clock vs. the system delay or
throughput rate.
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 26
28. Reading Assignment
Reading Assignment
• Text Book Chapter 1 pages 17 ‐34
Text Book, Chapter 1, pages 17 34
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 28
29. Summary
• Mathematical Models
– Relate system parameters and variables
– Allow system designers to predict system performance by using
equations
– Experimentation may be too costly
Experimentation may be too costly
– Experimentation may not be feasible
• Computer simulation models predict system performance
• Deterministic models give output of an experiment with an exact
Deterministic models give output of an experiment with an exact
outcome
• Probability models determine probabilities of the possible
outcomes
• Probabilities and averages for a random experiment can be found
experimentally by computing relative frequencies and sample
averages in a large number of trials of experiment
Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 29
