Departmet of Electrical Egieerig Uiversity of Arasas ELEG 4603/5173L Digital Sigal Processig Ch. 1 Discrete-Time Sigals ad Systems Dr. Jigxia Wu wuj@uar.edu
OUTLINE 2 Classificatios of discrete-time sigals Elemetary discrete-time sigals Liear time-ivariat LTI discrete-time systems Causality ad stability Differece equatio represetatio of LTI systems
SIGNAL CLASSICIATION 3 Discrete-time sigal A sigal that is defied oly at discrete istats of time. Represeted as x,, 3, 2, 1,0,1,2,3, x cos 4 x 1 2 exp 4
SIGNAL CLASSICIATION 4 Review: Aalog v.s. digital Cotiuous-time sigal xt, cotiuous-time, cotiuous amplitude aalog sigal Example: speech sigal Cotiuous-time, discrete amplitude Example: traffic light 1 0 2 3 0 2 1 Discrete-time sigal x, Discrete-time, discrete-amplitude digital sigal Example: Telegraph, text, roll a dice 1 0 2 3 0 2 1 Discrete-time, cotiuous-amplitude Example: samples of aalog sigal, average mothly temperature
SIGNAL CLASSIFICATION 5 Periodic sigal v.s. aperiodic sigal x x N Periodic sigal The smallest value of N that satisfies this relatio is the fudametal periods. cos Is periodic? cos 2 is periodic if N is a iteger, ad the smallest N is the fudametal period. Example: cos 3 cos 3 cos 4
SIGNAL CLASSIFICATION 6 Sum of two periodic sigals x x 1 2 : fudametal period : fudametal period N 1 N 2 x1 x2 x1 x2 is periodic if both x ad x 1 2 are periodic. Assume N1 p N2 q N pn 2 qn where p ad q are ot divisible of each other. The period is 1
SIGNAL CLASSIFICATION 7 Example: Is the sigal periodic? If it is, what is the fudametal period? 3 1 cos si 9 7 2
SIGNAL CLASSIFICATION 8 Eergy sigal Eergy: E lim N N N x 2 Review: eergy of cotiuous-time sigal x 2 E x t 2 dt Eergy sigal: 0 E
SIGNAL CLASSIFICATION 9 Power sigal Power of discrete-time sigal P lim N 1 2N 1 N N Review: power of cotiuous-time sigal x 2 P lim T 1 2T T T x t 2 dt Power sigal: 0 P
SIGNAL CLASSIFICATION 10 Example Determie if the discrete-time expoetial sigal is a eergy sigal or power sigal x 2 0. 5 0
OUTLINE 11 Classificatios of discrete-time sigals Elemetary discrete-time sigals Liear time-ivariat LTI discrete-time systems Causality ad stability Differece equatio represetatio of LTI systems
ELEMENTARY SIGNALS 12 Basic sigal operatios Time shiftig x shift the sigal to the right by samples Reflectig x x 3 x 3 x Reflectig x with respect to = 0. x x
ELEMENTARY SIGNALS 13 Basic sigal operatios Time scalig Example: Let 1, x 1, x 2 x 2 1 Fid, eve odd
ELEMENTARY SIGNALS 14 Basic sigal operatios Time scalig Example: x [ 1,2,1,0, 2] the always poits to x0 fid x3, x, x 3 3 2 3
ELEMENTARY SIGNALS 15 Uit impulse fuctio 1, 0, 0, 0. time shiftig 1, 0, Uit step fuctio 0, 0, u 1, 0.,. Relatio betwee uit impulse fuctio ad uit step fuctio u u 1 u
ELEMENTARY SIGNALS 16 Expoetial fuctio Complex expoetial fuctio x is periodic if N is a iteger, ad the smallest iteger N is the 0 fudametal period. Example x exp x exp j cos 0 jsi 0 2 0 Are the followig sigals periodic? If periodic, fid fudametal period. 7 x1 exp j 9 x 2 exp j 7 9
OUTLINE 17 Classificatios of discrete-time sigals Elemetary discrete-time sigals Liear time-ivariat LTI discrete-time systems Causality ad stability Differece equatio represetatio of LTI systems
DISCRETE-TIME SYSTEMS 18 Liear system Cosider a system with the followig iput-output relatioship x 1 System y x 2 1 System y 2 The system is liear if it meets the superpositio priciple x1 x2 y1 y2 System Time-ivariat system Cosider a system with the followig iput-output relatioship x y System The system is time-ivariat if a time-shift at the iput leads to the same time-shift at the output x y System
DISCRETE-TIME SYSTEM 19 Impulse respose of a LTI system The respose of the system whe the iput is System h Ay arbitrary discrete-time sigal ca be decomposed as weighted summatio of the uit impulse fuctios Why? E.g. x x x 3 x 3 Recall: x t x t d
DISCRETE-TIME SYSTEM 20 LTI respose to arbitrary iput Ay arbitrary sigal ca be writte as x x Time-ivariat Liear x LTI LTI h x h x LTI y x h
DISCRETE-TIME SYSTEM 21 Covolutio sum x h The covolutio sum of two sigals ad is Respose of LTI system x h x h The output of a LTI system is the covolutio sum of the iput ad the impulse respose of the system. x h x h
DISCRETE-TIME SYSTEM 22 Examples 1. x m 2. x u, h u x h
DISCRETE-TIME SYSTEM 23 Example 3. Fid the step respose of the system with impulse respose 1 2 h 2 cos u 2 3
DISCRETE-TIME SYSTEM 24 Example: x [0,1,2] h [ 1, 2, 3, 4], Let be two sequeces, fid x h
DISCRETE-TIME SYSTEM 25 Properties: commutativity x h h x x h x h h x h x
DISCRETE-TIME SYSTEM 26 Properties: associativity x h h x h h x h h2 1 2 1 2 1 x h y h y 1 1 2 x h h h2 1 y
DISCRETE-TIME SYSTEM 27 Distributivity h h x h x h x 2 1 1 1 x h 1 + y x h1 h2 y h 2
DISCRETE-TIME SYSTSEMS 28 Example Cosider a system show i the figure. Fid the overall impulse respose. h 2 1 h 1 u h 2 u 1 2 3
OUTLINE 29 Classificatios of discrete-time sigals Elemetary discrete-time sigals Liear time-ivariat LTI discrete-time systems Causality ad stability Differece equatio represetatio of LTI systems
CAUSALITY AND STABILITY 30 Causal system A discrete-time system is causal if the output y 0 depeds oly o values of iput for 0 The output does ot deped o future iput. Example: determie whether the followig systems are causal. 2 y x 3x 2 y x 1 1 y x 1 x x 1 3 y x y 1 x
Causality of LTI system A LTI system is causal if ad oly if h=0 for < 0 Why? A sigal x is causal if x=0 for < 0. Example: For LTI systems with impulse resposes give as follows. Fid if the systems are casual. 31 CAUSALITY AND STABILITY 1 h x h x h x y cos2 h cos2 u h 1 u b u a h 1 u b u a h
CAUSALITY AND STABILITY 32 Bouded-iput bouded-output BIBO stable a system is BIBO stable if, for ay bouded iput x, the respose y is also bouded. BIBO stability of LTI system A LTI discrete-time system is BIBO stable if Why? h
CAUSALITY AND STABILITY 33 Example For a LTI system with impulse resposes as follows. Are they BIBO stable? h 0.5 u h 0.5 u h 0.5 h 2 u h 2 u
OUTLINE 34 Classificatios of discrete-time sigals Elemetary discrete-time sigals Liear time-ivariat LTI discrete-time systems Causality ad stability Differece equatio represetatio of LTI systems
DIFFERENCE EQUATION Differece equatio represetatio of LTI discrete-time system Ay LTI discrete-time system ca be represeted as Review: ay LTI cotiuous-time system ca be represeted as 35 M N x b y a 0 0 M N dt t x d b dt t y d a 0 0
DIFFERENCE EQUATION Simulatio diagram 36 1 1 1 0 1 N x b b x x b N y a y a y N N
DIFFERENCE EQUATION 37 Example Draw the simulatio diagram of the LTI system described by the followig differece equatio 2y 3y 1 0.5x x 1 5x 2
DIFFERENCE EQUATION 38 Example The impulse respose of a LTI system is Fid the differece equatio represetatio Draw the simulatio diagram. h [2,3,0,5]