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Unformatted text preview: Lecture 5: DT Convolution Tie Liu September 13, 2011 1 LTI systems Recall that a CT system is TI if x ( t ) → y ( t ) = ⇒ x ( t − t ) → y ( t − t ) , ∀ t ∈ R and that it is linear x 1 ( t ) → y 1 ( t ) and x 2 ( t ) → y 2 ( t ) = ⇒ ax 1 ( t ) + bx 2 ( t ) → ay 1 ( t ) + by 2 ( t ) , ∀ a, b ∈ R For ECEN 314, we shall focus on systems that are both linear and time invariant (LTI): • Practical importance. • The powerful analysis tools associated with LTI systems. A basic fact: If we know the response of an LTI system to some inputs, we actually know the response to many inputs. Question: What is the smallest set of inputs for which, if we know their outputs, we can determine the output of any input signal? 2 DT Convolution For DT systems, the answer is surprisingly simple: All we need to know is the response (denoted by h [ n ]) to the unit impulse function δ [ n ] = braceleftbigg 1 , n = 0 , n negationslash = 0 Signal decomposition using the unit impulse function and its shifts:...
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 Fall '09

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