ECE4270 Fundamentals of DSP. Lecture 2 Discrete-Time Signals and Systems & Difference Equations. Overview of Lecture 2. More Discrete-Time Systems

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1 ECE4270 Fudametals of DSP Lecture 2 Discrete-Time Sigals ad Systems & Differece Equatios School of ECE Ceter for Sigal ad Iformatio Processig Georgia Istitute of Techology Overview of Lecture 2 Aoucemet Discrete-time systems (from Lecture #1) Properties Testig for properties Liear time-ivariat systems (LTI) (from Lecture #1) Covolutio sum Example of evaluatio of discrete covolutio Stability of LTI systems Causality of LTI systems Cascade ad parallel coectios of LTI systems Differece equatios Iitial rest coditios ad LTI Aoucemet Homewor will be assiged weely o Thursdays ad will be due the followig Thursdays. Li to the class web page: More Discrete-Time Systems x[] D-T System L-poit movig average system: L1 y[] 1 L 0 x[ ] y[] x[] 1 x[] x[ 1] x[ L 1] L Accumulator system: y[] x[]

2 Properties of D-T Systems A system is liear if ad oly if ax 1 [] bx 2 [] ax 1 [] bx 2 [] A system is time-ivariat if ad oly if x 1 [] x[ d ] y 1 [] y[ d ] A system is causal if ad oly if y[] depeds oly o x[] for A system is BIBO stable if every bouded iput produces a bouded output; i.e., whe x[] B x for all, the y[] B y for all L1 Movig Averager: y[] (1 / L) x[ ] 0 Liearity: yes 1 L1 ax 1 [ ] bx 2 [ ] a 1 L1 x 1 [ ] b 1 L1 x L 0 L 0 2 [ ] L 0 Time-ivariace: yes 1 L1 x[ L d ] 1 L1 x[( 0 L d ) ] y[ d ] 0 Causality: yes y[] 1 x[] x[ 1] x[ L 1] L Stability: yes y[] 1 L1 x[ ] 1 L1 x[ ] B x L 0 L 0 Dow Sampler: y[] x[m] Liearity yes ax 1 [M] bx 2 [M] ay 1 [] by 2 [] Time-ivariace o y 1 [] x 1 [M] x[m d ] y[ d ] x[m( d )] Causality o y[1] x[m], but y[1] y[m] Stability yes y[] x[m] B x LTI Discrete-Time Systems Liearity (superpositio): x[] [] LTI System ax 1 [] bx 2 [] a x 1 [] b x 2 [] Time-Ivariace (shift-ivariace): x 1 [] x[ d ] y 1 [] y[ d ] LTI implies discrete covolutio: y[] x[]h[ ] x[] h[] h[] x[] y[] h[]

3 Impulse Represetatio of Sequeces x[] x[][ ] a 1 [ 1] a 3 [ 3] a 2 [ 2] a 7 [ 7] p[] a 3 [ 3] a 1 [ 1] a 2 [ 2] a 7 [ 7] impulse LTI Discrete-Time Systems x[] [] LTI System y[] h[] x[][ ] x[]h[ ] x[][ ] x[]h[ ] x[] y[] covolutio sum y[] x[m]h[ m] x[] h[] h[] x[] m impulse respose Discrete Covolutio Two ways to loo at it: As the represetatio of the output as a sum of delayed ad scaled impulse resposes. y[] x[]h[ ] x[0]h[] x[1]h[ 1] x[1]h[ 1] As a computatioal formula for computig y[] ( y at time ) from the etire sequeces x ad h. Form x[]h[ ] for for fixed. Sum over all to produce y[]. Repeat for all. Flippig ad Shiftig 3 h[] h[0 ] h[] 6 3 h[ ] h[( )] 6 3 6

4 Defiitio Discrete Covolutio - I y[] x[]h[ ] h[]x[ ] Example h[] 1, 0 5 0, otherwise y[] x[] x[ ] y[] x[] x[ 1] x[ 5] Discrete Covolutio - II Case 1 y[] 0 for 0 Case 2 y[] a 0 1 a1 1 a (N 1) 0 ad 0 or 0 (N 1) Case 3 Discrete Covolutio - III Stability of LTI Systems Stability: Every bouded iput produces a bouded output. y[] a (N1) an1 a 1 1 a (N 1) 0 or (N 1) y[] h[]x[ ] h[] x[ ] y[] h[]b x Therefore, y[] if h[] Case 1 Case 2 Case 3 This coditio ca also be show to be ecessary as well as sufficiet for BIBO stability.

5 Causality of LTI Systems A system is causal if y[] depeds oly o x[] for less tha or equal to. y[] x[]h[ ] h[ ] 0 for Cascade Coectio of LTI Systems [] [] h 1 [] h[] h 1 [] h 2 [] h 2 [] h[] h 2 [] h 1 [] Sice h 1 []h 2 [] h 2 []h 1 [], it follows that Causality requires h[] 0 for 0 is equivalet to the cascade i either order. Parallel Combiatio of LTI Systems h 1 [] [] h[] h 1 [] h 2 [] h 2 [] The followig system is equivalet to the parallel combiatio. Delay: Accumulator: Examples y[] x[ d ] h[] [ d ] x[][ d ] x[ d ] h[] y[] x[] First differece: y[] x[] x[ 1] [] 0 0 u[] 1 0 h[] [][ 1]

6 Differece Equatios For all computatioally realizable LTI systems, the iput ad output satisfy a differece equatio of the form N a y[ ] b x[ ] 0 0 This leads to the recurrece formula M First-Order Example Cosider the differece equatio y[] ay[ 1] x[] We ca represet this system by the followig bloc diagram: y[ ] N a a 1 0 y[ ] M b a 0 0 x[ ] which ca be used to compute the preset output from the preset ad M past values of the iput ad N past values of the output a Recursive Computatio of Output Let x[] K[] ad y[1] c x[] y[] ay[ 1] x[] 0 c K ac K 0 a(ac K) a 2 c Ka 0 a(a 2 c Ka) a 3 c Ka 2 0 a(a 3 c Ka 2 ) a 4 c Ka 3 Geeral Solutio By iductio, we see that if y[] ay[ 1] x[] with x[] K[] ad y[1] c. the the solutio is If x[] K[ d ], the the output will be y[] ca 1 Ka d for 1 If y[] ca 1 Ka for 1 x[] [], the the output will be y[] ca 1 a for 1 Implies Not TI Implies Not liear

7 LTI Recursive Implemetatio We say that a iput is suddely applied at time d if x[]=0 for all < d. If the iput is suddely applied ad we assume that y[]=0 for all < d, the the iterative computatio will be both liear ad time-ivariat. This assumptio provides the required set of auxiliary coditios {y[ d -1], y[ d -2],,, y[ d -N]} that is required to get the recursio goig. Zero auxiliary coditios are called the iitial rest coditios. For the first-order case of our example, the impulse respose of the LTI system is h[] a u[] Expoetial Impulse Respose With iitial rest coditios, the differece equatio y[] ay[ 1] x[] has impulse respose h[] a u[] LTI Recursio of First-Order DE If we assume iitial rest coditios, the the differece equatio y[] ay[ 1] x[] has impulse respose h[] a u[] I other words, y[] is also give by the covolutio y[] x[]a u[ ] x[]a