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Unformatted text preview: ECE 3250 DISCRETE-TIME LTI SYSTEMS Fall 2008 In this handout Id 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 were 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. Its 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 systems 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. Well 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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