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Unformatted text preview: 4 Modes 4.1 Growth of the Fibonacci series 52 4.2 Taking out the big part from Fibonacci 55 4.3 Operator interpretation 57 4.4 General method: Partial fractions 59 The goals of this chapter are: to illustrate the experimental way that an engineer studies sys- tems, even abstract, mathematical systems; to illustrate what modes are by finding them for the Fibonacci sys- tem; and to decompose second-order systems into modes, explaining the decomposition using operators and block diagrams. The first question is what a mode is. That question will be answered as we decompose the Fibonacci sequence into simpler sequences. Each simple sequence can be generated by a first-order system like the leaky tank and is called a mode of the system. By decomposing the Fibonacci sequence into modes, we decompose the system into simpler, first-order subsystems. The plan of the chapter is to treat the Fibonacci system first as a black box producing an output signal F and to develop computational probes to examine signals. This experimental approach is how an engineer stud- ies even abstract, mathematical systems. The results from the probes will show us how to decompose the signal into its modes. These modes are then reconciled with what the operator method predicts for decomposing the system. 52 4.1 Growth of the Fibonacci series Why describe the experimental, and perhaps harder, method for finding the modes before giving the shortcuts using operators? We know the op- erator expression for the Fibonacci system, and could just rewrite it using algebra. The answer is that the operator method has meaning only after you feel modes in your fingertips, a feeling developed only as you play with signals. Without first playing, we would be teaching you amazing feats of calculation on meaningless objects. Furthermore, the experimental approach works even when no difference equation is available to generate the sequence. Engineers often character- ize such unknown or partially known systems. The system might be: computational: Imagine debugging someone elses program. You send in test inputs to find out how it works and what makes it fail. electronic: Imagine debugging a CPU that just returned from the fabri- cation run, perhaps in quantities of millions, but that does not correctly divide floating-point numbers . You might give it numbers to di- vide until you find the simplest examples that give wrong answers. From that data you can often deduce the flaw in the wiring. mathematical: Imagine computing primes to investigate the twin-prime conjecture , one of the outstanding unsolved problems of number theory. [The conjecture states that there are an infinite number of prime pairs p , p + 2 , such as ( 3,5 ) , ( 5,7 ) , etc.] The new field of experimental mathematics, which uses computational tools to investigate mathemat- ics problems, is lively, growing, and a fertile field for skilled engineers [4, 14, 8]....
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- Spring '11