t d ut dt also A system can be characterized by its unit step response or unit

T d ut dt also a system can be characterized by its

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(t) = d u(t) / dt also * A system can be characterized by its unit step response or unit impulse response. 1 t 1 t ' dt ) (t' u(t) ³ f ² t G

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Slide 34 CUHK(SZ) EIE 3001, Spring 2018/19 How to interpret unit impulse function
Slide 35 CUHK(SZ) EIE 3001, Spring 2018/19 How to interpret unit impulse function

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Slide 36 CUHK(SZ) EIE 3001, Spring 2018/19 Discrete Time Unit Step and Unit Impulse Sequence a Unit Step function: u[n] = 0, n<0 1, n t 0 Note : u[n] at n=0 is defined. a Unit Impulse function: G> n @ = 0, n z 0 1, n=0 x[ n ] n -3 -2 -1 0 1 2 3 1 ... ... x[ n ] n -3 -2 -1 0 1 2 3 1 ... ...
Slide 37 CUHK(SZ) EIE 3001, Spring 2018/19 Discrete Time Unit Step and Unit Impulse Sequence (cont.) a G> n @ = u[n]-u[n-1] : first difference of unit step a u[n]= : running sum of unit sample or equivalently u[n]= ¦ ²f n m m ] [ G ¦ f ² 0 ] [ k k n G u[ n ] n -3 -2 -1 0 1 2 3 1 ... ... G [ n ] n -3 -2 -1 0 1 2 3 G [ n- 1] n -3 -2 -1 0 1 2 3 = + n G [ n- 2] -3 -2 -1 0 1 2 3 + + ...

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Slide 38 CUHK(SZ) EIE 3001, Spring 2018/19 Periodic Properties of Discrete-time Complex Exponential a For continuous-time complex exponential e j Z o t (1) the larger the Z o , the high the rate of oscillation (2) e j Z o t is periodic for any value of Z o . a Are the above two statements still valid for the discrete case e j : o n ? (1) e j( : o +2 S )n = e j : o n e j 2 S n = e j : o n ? Signal with frequency : o = Signal with frequency : o +2 or : o +4 S , … (Note it is not saying that e j : o n is periodic with period 2 S . Why?) o To discuss discrete exponential, only need to consider any interval of 2 S
Slide 39 CUHK(SZ) EIE 3001, Spring 2018/19 Periodic Properties of Discrete-time Complex Exponential a Q: Does e j : o n have a continuously increasing rate of oscillation as : o increase? a A: No. For : o within an interval 0 d : o d 2 the frequency n as : o n for 0 d : o d S the frequency p as : o n for Sd : o d 2 S See Next Figure

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Slide 40 CUHK(SZ) EIE 3001, Spring 2018/19 Periodic Properties of Discrete-time Complex Exponential
Slide 41 CUHK(SZ) EIE 3001, Spring 2018/19 Periodic Properties of Discrete-time Complex Exponential (2) Is e j : o n always periodic? For a signal e j : o n to be periodic with period N (>0), we must have e j : o (n+N) = e j : o n ? e j : o N =1 o : o N is multiple of 2 S o : o N = 2 S m where m is an integer, or : o /2 S m/N e j : o N is periodic if : o /2 S is a rational number

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Slide 42 CUHK(SZ) EIE 3001, Spring 2018/19 [Exercise] 1. What is the condition that the sum of two periodic signals is also periodic? E.g Is x(t)=cos2t+cos(2 0.5 t) periodic? 2. Determine whether the following signal is periodic. If it is, find its period. _ x(t)=sin (2 S /3)t _ x[n]=cos(n/4) _ x[n]=cos 2 ( S n/8)
Slide 43 CUHK(SZ) EIE 3001, Spring 2018/19 A Review

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Slide 44 CUHK(SZ) EIE 3001, Spring 2018/19 Overview of System a System: any process that results in the transformation of signal. x(t) y(t) Continuous-time x[n] y[n] Discrete-time a A system is continuous-time if both input and output are continuous-time. a A system is discrete-time if both input and output are discrete-time.
Slide 45 CUHK(SZ) EIE 3001, Spring 2018/19 Example: Filtering System

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Slide 46 CUHK(SZ) EIE 3001, Spring 2018/19 Interaction of Multiple Systems a (a) Cascade (Series) x(t) y(t) z(t) System #1 System #2 #1 #2 a (b) Parallel
Slide 47 CUHK(SZ) EIE 3001, Spring 2018/19 Interaction of Multiple Systems (cont.) a (c) Series/Parallel #2 #3 #1 x2 ( ) 2 ( ) 2 + - Ex. y[n]=(2x[n]-x[n] 2 ) 2

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