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Lec_02

Lec_02 - Lecture 2 Discrete-Time Systems and z-Transform...

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1 Lecture 2: Discrete-Time Systems and z -Transform Discrete-time signals v.s. continuous-time signals Discrete-time systems v.s. continuous-time systems z -Transform Inverse z -Transform z -Transform for solving linear difference equations Continuous-Time Signals A signal that changes continuously in time: Defined at all times t (no gap) Signal can take arbitrary values • Example: temperature, position, velocity,… e ( t ) ; ¡1 < t < 1 ; or t ¸ 0 e ( t ) t

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2 A signal (sequence, or series) whose values are defined only at discrete times: Signal defined only at integer times k Signal can still take arbitrary values Discrete-Time Signals 4 ¢¢¢ ¡ 1 k 0 1 3 ¢¢¢ e ( k ) 2 e ( k ) ; k = :::; ¡ 1 ; 0 ; 1 ;::: or k = 0 ; 1 ; 2 ;::: Origin of Discrete-Time Signals Some come naturally Population of a species in different generations Annual growth percentage of GDP Results of a numerical algorithm in different rounds of iteration Some arise by sampling continuous-time signals at regular time intervals, say, every T seconds e ( t ) t e ( kT ) 0 T 2 T 3 T 4 T ¡ T ¢¢¢ ¢¢¢ e ( t ) ; ¡1 < t < 1 ) e ( kT ) ; k = :::; ¡ 1 ; 0 ; 1 ;::: ) e ( k ) ; k = ¢ ¢ ¢ ; ¡ 1 ; 0 ; 1 ;:::; (if T is known from the context)