Discrete-ime Sigals ad Systems Discrete-ime Sigals ad Systems Dr. Deepa Kudur Uiversity of oroto Referece: Sectios. -.5 of Joh G. Proakis ad Dimitris G. Maolakis, Digital Sigal Processig: Priciples, Algorithms, ad Applicatios, 4th editio, 007. Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 Elemetary Discrete-ime Sigals Sigal Symmetry. uit sample sequece (a.k.a. Kroecker delta fuctio): δ() = {, for = 0 0, for 0 Eve sigal: x( ) =. uit step sigal:. uit ramp sigal: Note: u() = u r () = {, for 0 0, for < 0 {, for 0 0, for < 0 Odd sigal: x( ) = 0-6 -5-4 - - -7 4 5 6 7-6 -5-4 - 0 - -7 4 5 6 7-6 -5-4 - 0 - -7 4 5 6 7 δ() = u() u( ) = u r ( ) u r () u r ( ) u() = u r ( ) u r () Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 4 / 6
Sigal Symmetry Sigal Symmetry Eve sigal compoet: Odd sigal compoet: x e () = [ x( )] x o () = [ x( )] -6-5 -4 0.5 - - -0.5-0 4 5 6 7 8-6 -5-4 (x(-))/ 0.5 - - -0.5 - eve part 0 4 5 6 7 8 Note: = x e () x o () -6-5 -4 x(-) 0.5 - - -0.5 0 4 5 6 7 8-6 -5-4 (-x(-))/ 0.5 - - odd part 0 4 5 6 7 8-0.5 - - Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 5 / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 6 / 6 Simple Maipulatio of Discrete-ime Sigals Simple Maipulatio of Discrete-ime Sigals I Fid x( ). rasformatio of idepedet variable: time shift: k, k Z Questio: what if k Z? time scale: α, α Z Questio: what if α Z? Additioal, multiplicatio ad scalig: amplitude scalig: y() = A, < < sum: y() = x () x (), < < product: y() = x ()x (), < < - - - - 0 4 5 6 7 8 9 0 5 0 - - Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 7 / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 8 / 6
Simple Maipulatio of Discrete-ime Sigals II Simple Maipulatio of Discrete-ime Sigals I Fid x( ). - - - - - 0 4 5 6 7 8 9-0 - 5 0 - - - x( ) < < 0 if is a iteger; udefied otherwise - udefied 0 x() = 5 udefied 4 x(4) = udefied 4 7 x(7) = 7 5 udefied 6 0 x(0) = 7 udefied 8 x() = 9 9 udefied 0 6 x(6) = 5 udefied 9 x(9) = > > 9 0 if is a iteger; udefied otherwise Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 9 / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 0 / 6 Simple Maipulatio of Discrete-ime Sigals Graph of x( ). - - 0 - - 0 4 5 6 7 8 9 4 his sigal is udefied for values of that are ot eve itegers ad zero for eve itegers ot show o this sketch. - - Iput-Output Descriptio of Dst-ime Systems iput/ excitatio Discrete-time sigal Discrete-time System y() Discrete-time sigal output/ respose Iput-output descriptio (exact structure of system is ukow or igored): y() = [] black box represetatio: y() Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6
Classificatio of Discrete-ime Systems ime-ivariat vs. ime-variat Systems Why is this so importat? mathematical techiques developed to aalyze systems are ofte cotiget upo the geeral characteristics of the systems beig cosidered for a system to possess a give property, the property must hold for every possible iput to the system to disprove a property, eed a sigle couter-example to prove a property, eed to prove for the geeral case ime-ivariat system: iput-output characteristics do ot chage with time a system is time-ivarariat iff y() = x( k) y( k) for every iput ad every time shift k. Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 4 / 6 ime-ivariat vs. ime-variat Systems Liear vs. Noliear Systems Examples: time-ivariat or ot? y() = A y() = y() = x ( ) y() = x( ) y() = x( ) y() = y() = e As: Y, N, Y, N, Y, Y, Y Liear system: obeys superpositio priciple a system is liear iff [a x () a x ()] = a [x ()] a [x ()] for ay arbitrary iput sequeces x () ad x (), ad ay arbitrary costats a ad a Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 5 / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 6 / 6
Liear vs. Noliear Systems Causal vs. Nocausal Systems Examples: liear or ot? y() = A y() = y() = x ( ) y() = x( ) y() = x( ) y() = y() = e As: Y, Y, N, Y, Y, N, N Causal system: output of system at ay time depeds oly o preset ad past iputs a system is causal iff for all y() = F [, x( ), x( ),...] Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 7 / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 8 / 6 Causal vs. Nocausal Systems Stable vs. Ustable Systems Examples: causal or ot? y() = A y() = y() = x ( ) y() = x( ) y() = x( ) y() = y() = e As: Y, Y, Y, N, N, N, Y Bouded Iput-Bouded output (BIBO) Stable: every bouded iput produces a bouded output a system is BIBO stable iff for all. M x < = y() M y < Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 9 / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 0 / 6
Stable vs. Ustable Systems he Covolutio Sum Examples: stable or ot? y() = A y() = y() = x ( ) y() = x( ) y() = x( ) y() = y() = e As: Y, N, Y, Y, Y, N, Y Recall: = x(k)δ( k) Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 he Covolutio Sum Let the respose of a liear time-ivariat (LI) system to the uit sample iput δ() be h(). he Covolutio Sum δ() δ( k) α δ( k) x(k) δ( k) x(k)δ( k) h() h( k) α h( k) x(k) h( k) x(k)h( k) y() herefore, y() = x(k)h( k) = h() for ay LI system. Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 4 / 6
Properties of Covolutio Properties of Covolutio Associative ad Commutative Laws: h() = h() [ h ()] h () = [h () h ()] Distributive Law: [h () h ()] = h () h () h () h () h () h () h () * h () = h () * h () h () h () y() y() h () h () y() Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 5 / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 6 / 6 Causality ad Covolutio For a causal system, y() oly depeds o preset ad past iputs values. herefore, for a causal system, we have: Fiite-Impulse Repose vs. Ifiite Impulse Respose Fiite impulse respose (FIR): y() = = = h(k)x( k) h(k)x( k) h(k)x( k) h(k)x( k) y() = Ifiite impulse respose (IIR): y() = M h(k)x( k) h(k)x( k) where h() = 0 for < 0 to esure causality. How would oe realize these systems? wo classes: recursive ad orecursive. Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 7 / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 8 / 6
Ifiite Impulse Respose System Realizatio Ifiite Impulse Respose System Realizatio here is a practical ad computatioally efficiet meas of implemetig a family of IIR systems that makes use of...... differece equatios. Cosider a accumulator: y() = x(k) = 0,,,... Memory requiremets grow with icreasig! Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 9 / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 0 / 6 Ifiite Impulse Respose System Realizatio Liear Costat-Coefficiet Differece Equatios y() = = x(k) x(k) = y( ) y() = y( ) recursive implemetatio Example for a liear costat-coefficiet differece equatio (LCCDE): y() = y( ) Iitial coditios: at rest for < 0; i.e., y( ) = 0 Geeral expressio for Nth-order LCCDE: N a k y( k) = M b k x( k) a 0 Iitial coditios: y( ), y( ), y( ),..., y( N) Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6
Direct Form I vs. Direct Form II Realizatios Direct Form I IIR Filter Implemetatio y() = N a k y( k) k= M b k x( k) is equivalet to the cascade of the followig systems: v() }{{} output y() }{{} output M = b k x( k) }{{} k= iput N = k= a k y( k) v() }{{} iput orecursive recursive LI All-zero system LI All-pole system Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems / 6 Requires: M N multiplicatios, M N additios, M N memory locatios Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 4 / 6 Direct Form II IIR Filter Implemetatio Direct Form II IIR Filter Implemetatio Adder: Costat multiplier: Sigal multiplier: Uit delay: Uit advace: LI All-pole system LI All-zero system Requires: M N multiplicatios, M N additios, M N memory locatios Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 5 / 6 For N>M Requires: M N multiplicatios, M N additios, max(m, N) memory locatios Dr. Deepa Kudur (Uiversity of oroto) Discrete-ime Sigals ad Systems 6 / 6