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ecen314-lec5

# ecen314-lec5 - Lecture 5 DT Convolution Tie Liu 1 LTI...

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Lecture 5: DTConvolution Tie Liu September 13, 2011 1 LTI systems Recall that a CT system is TI if x ( t ) y ( t ) = x ( t t 0 ) y ( t t 0 ) , t 0 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 0 , n negationslash = 0 Signal decomposition using the unit impulse function and its shifts: x [ n ] = summationdisplay k = -∞ x [ k ] δ [ n k ] 1

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-1 -2 0 1 2 x [ n ] n -2 n -1 n 0 n 1 n 2 n x [ -
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