![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](https://image.slidesharecdn.com/mth263lecture1-121004094656-phpapp01/85/Mth263-lecture-1-20-320.jpg)
Mth263 lecture 1
- 1. MTH 263 Probability and Random Variables Lecture 1 Dr. Sobia Baig Electrical Engineering Department COMSATS Institute of Information Technology, Lahore
- 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
- 18. Packet Voice Communication system Efficiency (2) ff ( ) Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 18
- 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
- 23. Example: Signal Enhancement Using Filters l Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 23
- 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
- 25. Multi User Systems with Queues: Resource sharing h Probability and Random Variables, Lecture 1, by Dr. Sobia Baig 25
- 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