LTISystemsI

LTISystemsI - ECE 3250 DISCRETE-TIME LTI SYSTEMS Fall 2008...

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Unformatted text preview: ECE 3250 DISCRETE-TIME LTI SYSTEMS Fall 2008 In this handout I’d like to summarize the material from class on discrete-time LTI systems. Such systems serve as effective models for a variety of real-world processes that arise in applications. The models are useful not only in ECE — particularly in the areas of signal processing, communication, and control — but in other fields of science and engineering including MAE, ORIE, economics, and dynamical systems theory. The ideas are more transparent in the context of discrete-time models, which is why we start there. As in the handout on discrete-time convolution, we think of the integers Z as a math- ematical model for “discrete time.” Integer n corresponds to “time n .” Integer 0 corre- sponds to “time 0.” If m > n , then “time m is later than time n .” It is not really helpful to think of these “integer times” as being embedded somehow in a familiar “continuous- time axis” or as having standard time units such as seconds or milliseconds or whatever. Integer times are just indices with a natural ordering. Having used Z to model discrete time, we can think of a discrete-time signal over F as a function with domain Z that takes values in F . As usual, F is either R or C . Alternatively and equivalently, a discrete-time signal is a doubly infinite sequence of elements of F . So any such signal x is a sequence ...x (- 2) ,x (- 1) ,x (0) ,x (1) ,x (2) ,x (3) ,... where x ( n ) ∈ F for all n ∈ Z . Think of x ( n ) as the value of the signal x at time n . As before, we denote the set of all discrete-time signals by F Z . The real-world processes we’re interested in modeling are processes that take discrete- time input signals and generate discrete-time output signals. An appealing way to model such a process is as a mapping S : X-→ F Z where X is a subset of F Z that represents the set of possible input signals for the system. The idea of the mapping S is that when x ∈ X is the input signal to the system, S ( x ) ∈ F Z is the output signal that arises. It’s worth formalizing all this with a definition. Definition 1: A discrete-time input-output system consists of • A subset X of F Z representing the system’s set of possible input signals; and • A mapping S : X → F Z that maps each input signal x ∈ X to its corresponding output signal S ( x ) ∈ F Z . One usually assumes that the input space X is “rich enough” to include a lot of signals of interest. We’ll always require that X contain at least all the finite-duration signals. Recall that a signal x ∈ F Z has finite duration if and only if there exist integers N 1 and N 2 so that x ( n ) = 0 when n < N 1 and x ( n ) = 0 when n > N 2 . Recall also the (linear) mappings Shift k o : F Z → F Z defined by Shift k o ( x )( n ) = x ( n- k o ) for all x ∈ F Z and n ∈ Z ....
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LTISystemsI - ECE 3250 DISCRETE-TIME LTI SYSTEMS Fall 2008...

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