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ContinuousTimeSignalsConvolution - ECE 3250 CONTINUOUS-TIME...

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ECE 3250 CONTINUOUS-TIME CONCEPTS I Fall 2008 Continuous-time Signals and Convoluton 1. Continuous-time 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 continuous-time 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 continuous-time signals by F R . Working with continuous-time 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 continuous-time 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 I’ll 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 we’d rather not deal with. Consider, for example, the signal x 1 R R with specification x 1 ( t ) = 0 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 > > < > > : 0 if t < 0 0 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 = 1 0 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 we’ll be encountering frequently. The continuous-time unit step is the signal u that has specification u ( t ) = 1 if t 0 0 if t < 0 . 1
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2 I hope you don’t object to my using the same notation u for both the discrete- and continuous-time unit steps. u is not continuous, but its only discontinuity is a jump at t = 0. One comment on the unit step: I’ve defined it so u (0) = 1. As it happens, in all the manipulations we do that involve decent signals, it will not matter at all how we define the signals’ values at jump-discontinuity points. I could just as well have set
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