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Unformatted text preview: Probability and Random Signals I: Class Notes for the Course ECSE 305 Benoit Champagne Department of Electrical & Computer Engineering McGill University, Montreal, Quebec, Canada Fall 2003 ii c circlecopyrt 2003 Benoˆ ıt Champagne Compiled September 12, 2006 1 Chapter 1 Introduction 1.1 Randomness versus determinism Determinism in science and engineering: • Deterministic view in science: provided sufficient information is available about the initial state and operating conditions of a natural process or a manmade system, its future behavior can be predicted exactly. • This operational viewpoint has been the prevailing one in most of your college and university education (mechanics, circuit theory, etc.) • A typical example is provided by classical mechanics: Consider the motion of a particle under the influence of various forces in threedimensional space. If we know the initial position and velocity vectors of the particle, its mass and the total force field, Newton’s laws can be used to calculate (i.e. predict) the future trajectory of the particle. c circlecopyrt 2003 Benoˆ ıt Champagne Compiled September 12, 2006 1.1 Randomness versus determinism 2 The concept of randomness: • The above view is highly idealistic: In most ”reallife” scientific and engineering problems, as well as many other situations of interest (e.g. games of chance), we cannot do exact predictions about the phenomena or systems under consideration. • Two basic reasons for this may be identified: we do not have sufficient knowledge of the initial state of the system or the operating conditions (e.g. motion of electrons in a microprocessor circuit). due to fundamental physical limitations, it is impossible to make exact predictions (e.g. uncertainty principle in quantum physics) • We refer to such phenomena or systems as random, in the sense that there is uncertainty about their future behavior: a particular result or situation may or may not occur. • The observation of specific quantities derived from such a random system or phenomenon is often referred to as a random experiment. c circlecopyrt 2003 Benoˆ ıt Champagne Compiled September 12, 2006 1.1 Randomness versus determinism 3 Examples: • Consider the following game of chance: We roll an ordinary sixsided die once and observe the number showing up, also called outcome. Possible outcomes are represented by the set of numbers S = { 1 , 2 , 3 , 4 , 5 , 6 } . We cannot predict what number will show up as a result of this experiment. Neither can we predict that a related event A , such as obtaining an even number (represented by A = { 2 , 4 , 6 } ), will occur. • Consider a more sophisticated example from communications engineering: Using an appropriate modulation scheme, we transmit an analog speech signal s ( t ) over a radio channel....
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 Spring '11
 benoit
 Set Theory, Probability, Probability theory, Natural number

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