MIT6_003S10_chap1 - 1 Difference equations 1.1 Rabbits 2...

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Unformatted text preview: 1 Difference equations 1.1 Rabbits 2 1.2 Leaky tank 7 1.3 Fall of a fog droplet 11 1.4 Springs 14 The world is too rich and complex for our minds to grasp it whole, for our minds are but a small part of the richness of the world. To cope with the complexity, we reason hierarchically. We divide the world into small, comprehensible pieces: systems. Systems are ubiquitous: a CPU , a memory chips, a motor, a web server, a jumbo jet, the solar system, the telephone system, or a circulatory system. Systems are a useful abstraction, chosen because their external interactions are weaker than their internal interac- tions. That properties makes independent analysis meaningful. Systems interact with other systems via forces, messages, or in general via information or signals. Signals and systems is the study of systems and their interaction. This book studies only discrete-time systems, where time jumps rather than changes continuously. This restriction is not as severe as its seems. First, digital computers are, by design, discrete-time devices, so discrete- time signals and systems includes digital computers. Second, almost all the important ideas in discrete-time systems apply equally to continuous- time systems. Alas, even discrete-time systems are too diverse for one method of analy- sis. Therefore even the abstraction of systems needs subdivision. The par- ticular class of so-called linear and time-invariant systems admits power- ful tools of analysis and design. The benefit of restricting ourselves to such 2 1.1 Rabbits systems, and the meaning of the restrictions, will become clear in subse- quent chapters. 1.1 Rabbits Here is Fibonaccis problem [6, 10], a famous discrete-time, linear, time- invariant system and signal: A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? 1.1.1 Mathematical representation This system consists of the rabbit pairs and the rules of rabbit reproduction. The signal is the sequence f where f [ n ] is the number of rabbit pairs at month n (the problem asks about n = 12 ). What is f in the first few months? In month , one rabbit pair immigrates into the system: f [ ] = 1 . Lets assume that the immigrants are children. Then they cannot have their own children in month 1 they are too young so f [ 1 ] = 1 . But this pair is an adult pair, so in month 2 the pair has children, making f [ 2 ] = 2 . Finding f [ 3 ] requires considering the adult and child pairs separately (hier- archical reasoning), because each type behaves according to its own repro- duction rule. The child pair from month 2 grows into adulthood in month 3 , and the adult pair from month 2 begets a child pair. So in f [ 3 ] = 3 : two adult and one child pair....
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This note was uploaded on 12/14/2011 for the course EE 6.003 taught by Professor Freeman during the Spring '11 term at MIT.

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MIT6_003S10_chap1 - 1 Difference equations 1.1 Rabbits 2...

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