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notes4-8

# notes4-8 - SECTION 4.8 DIFFERENCE EQUATIONS Electrical...

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SECTION 4.8 DIFFERENCE EQUATIONS Electrical engineers, control systems engineers, biologists, physicists, economists, demog- raphers, etc etc, often measure or sample a process at discrete time intervals. This generates a signal, or a sequence of numbers { y k } = ( . . . , y - 2 , y - 1 , y o , y 1 , y 2 , . . . ) . Sometimes the terms y k for k < 0 are either omitted or assumed to be 0, since sampling usually begins at a certain time, but we will continue to think about signals that go on forever in both directions. We will study the collection S of all signals. We’ve already seen that this is a vector space with addition and scalar multiplication done naturally. So we have available all the vector space notions of linear independence, subspaces, bases, etc. WHAT DOES LINEAR INDEPENDENCE IN S MEAN?

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EXAMPLE. Are the signals 1 k , 3 k cos 2 , and 3 k sin 2 linearly independent? One way to make new signals { w k } from old ones { y k } is by setting w k = a 0 y k + n + a 1 y k + n - 1 + · · · + a n - 1 y k +1 + a n y k , where the a i ’s are scalars with a 0 6 = 0 and a n 6 = 0. This is often called a
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notes4-8 - SECTION 4.8 DIFFERENCE EQUATIONS Electrical...

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