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Unformatted text preview: ECE 3250 CONTINUOUSTIME CONCEPTS I Fall 2008 Continuoustime Signals and Convoluton 1. Continuoustime signals We think of the real numbers R as a mathematical model for continuous time. Real number t corresponds to time t . Real number 0 corresponds to time 0. If s > t , then time s is later than time t . Having used R to model continuous time, we can think of a continuoustime signal over F as a function with domain R that takes values in F . As usual, F is either R or C . We denote the set of all continuoustime signals by F R . Working with continuoustime signals is a lot touchier than working with discrete time signals. We have all kinds of analytical things to worry about, e.g. continuity and differentiability. Many continuoustime signals are quite nasty. Fortunately, those signals play a limited role in applications, and we will end up essentially wishing them out of the picture by restricting our attention to what Ill be calling decent signals. Definition 1: A decent signal is a signal x F R that has the following properties: (1) x is continuous except possibly for jump discontinuities. Furthermore, x has at most finitely many jumps in any bounded interval [ t 1 , t 2 ] R . (2) x is bounded on any bounded interval [ t 1 , t 2 ] R . I.e., for any such interval, there exists R > 0 such that  x ( t )  R for every t [ t 1 , t 2 ]. Requirement 1 eliminates a number of signals wed rather not deal with. Consider, for example, the signal x 1 R R with specification x 1 ( t ) = if t is rational 1 if t is irrational . The signal x 1 has no points of continuity. In any bounded interval, x 1 has uncountably many jumps, if you even want to call them jumps. Requirement 1 also rules out the slightly less pathological signal x 2 with specification x 2 ( t ) = 8 > > < > > : if t < if 1 / ( n + 1) < t 1 /n and n is even 1 if 1 / ( n + 1) < t 1 /n and n is odd 1 if t 1 . x 2 has countably infinitely many jumps in the interval [0 , 1], but is flat between the jumps. Note that a decent signal can have at most countably infinitely many jumps because of Requirement 1. Can you see why? Requirement 2 in the definition of decent signals eliminates signals that blow up some where other than as t . Consider, for example, the signal x 3 with specification x 3 ( t ) = 1 t 1 if t 6 = 1 if t = 1 . x 3 ( t ) blows up as t 1 from either side, so x is unbounded on any interval that contains time t = 1. Every continuous signal is a decent signal. Many discontinuous signals are decent as well, including some that well be encountering frequently. The continuoustime unit step is the signal u that has specification u ( t ) = 1 if t if t < ....
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This note was uploaded on 08/10/2010 for the course ECE 4370 at Cornell University (Engineering School).
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