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Unformatted text preview: Ve451 Lecture Notes Dianguang Ma Summer 2010 Chapter 2 DiscreteTime Signals and Systems Representation of DiscreteTime Signals • Functional representation • Tabular representation • Sequence representation Sequence Representations of Signals { } An infiniteduration signal or sequence with the time origin ( 0) indicated by the symbol is represented as ( ) 0,0,1,4,1,0,0, A finiteduration sequence can be represented as ( ) 3, 1, 2,5,0,4, n x n x n ↑ ↑ = ↑ = = K K { } 1 FIR and IIR Systems • It is convenient to subdivide the class of LTI systems into two types, those that have a finiteduration impulse response (FIR) and those that have an infinite duration impulse response (IIR). • We can think of the terms FIR and IIR as general characteristics that distinguish a type of LTI system. Causal FIR LTI System • An FIR LTI system has an impulse response that is zero outside some finite time interval. 1 ( ) 0, 0 and ( ) ( ) ( ) M k h n n n M y n h k x n k = = < ≥ = ∑ Causal IIR LTI System • An IIR LTI system has an infiniteduration impulse response. ( ) ( ) ( ) k y n h k x n k ∞ = = ∑ Recursive and Nonrecursive Systems • We can think of the terms recursive and nonrecursive as descriptions of the structures for realizing or implementing the system. • The fundamental difference between recursive and nonrecursive systems is the feedback loop in the recursive system. Recursive Systems • The output of a recursive system is a function of several past output values and present and past inputs. ( ) [ ( 1), , ( ), ( ), , ( )] y n F y n y n N x n x n M = K K Nonrecursive Systems • The output of a nonrecursive system depends only on the present and past inputs. ( ) [ ( ), ( 1), , ( )] y n F x n x n x n M = K Recursive and Nonrecursive Realizations of FIR Systems • Every FIR system can be realized nonrecursively. On the other hand, any FIR system can also be realized recursively. Let us take the moving averager as an example. The nonrecursive and recursive equations are 1 ( ) ( ) and ( ) ( 1) 1 1 [ ( ) ( 1 )], respectively. 1 M k y n x n k y n y n M x n x n M M = = = + +  + ∑ Recursive and Nonrecursive Realization of an IIR Systems Let us take the accumulator as an example. The nonrecursive realization ( ) ( ) requires the storage of all the input samples ( ) for . Since is increasing, our memory requirements grow...
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 Summer '10
 DianguangMa
 Digital Signal Processing, Signal Processing, LTI system theory, Impulse response, Rxx

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