Discrete-Time Signals and Systems - University of Toronto

1 Discrete-Time Signals and SystemsDr. Deepa KundurUniversity of TorontoDr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems1 / 36 Chapter 2: Discrete-Time Signals and SystemsDiscrete- time Signals and SystemsReference:Sections - ofJohn G. Proakis and Dimitris G. Manolakis,Digital signal Processing:Principles, Algorithms, and Applications, 4th edition, Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems2 / 36 Chapter 2: Discrete-Time Signals and SystemsElementary Discrete-Time Signals1. unit sample sequence ( Kronecker delta function): (n) ={1,forn= 00,forn6= 02.}

2 Unit step signal :u(n) ={1,forn 00,forn<03. unit ramp signal :ur(n) ={n,forn 00,forn<0 Note: (n) =u(n) u(n 1)=ur(n+ 1) 2ur(n) +ur(n 1)u(n) =ur(n+ 1) ur(n)Dr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems3 / 36 Chapter 2: Discrete-Time Signals and SystemsSignal SymmetryEven signal :x( n) =x(n)Odd signal :x( n) = x(n)-110nx(n)-2-3-4-5-6-723456712n-110-2 -3-4-5-6-723456712n-110-2-3-4-5-6-723456 712x(n)x(n)Dr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems4 / 36 Chapter 2: Discrete-Time Signals and SystemsSignal SymmetryEven signal component:xe(n) =12[x(n) +x( n)]Odd signal component:xo(n) =12[x(n) x( n)]Note:x(n) =xe(n) +xo(n)Dr.}}

3 Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems5 / 36 Chapter 2: Discrete-Time Signals and SystemsSignal Symmetryx(n)n -1 1 1 2 3 4 0 6 7 8 5 -1 -2 -3 -4 -5 -6 x(-n)n -1 1 1 2 3 4 0 6 7 8 5 -1 -2 -3 -4 -5 -6 (x(n)+x(-n))/2n -1 1 1 2 3 4 0 6 7 8 5 -1 -2 -3 -4 -5 -6 (x(n)-x(-n))/2n -1 1 1 2 3 4 0 6 7 8 5 -1 -2 -3 -4 -5 -6 even partodd partDr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems6 / 36 Chapter 2: Discrete-Time Signals and SystemsSimple Manipulation of Discrete-Time SignalsITransformation of independent variable:Itime shift:n n k,k ZIQuestion: what ifk6 Z?

4 Itime scale:n n, ZIQuestion: what if 6 Z?IAdditional, multiplication and scaling:Iamplitude scaling:y(n) =Ax(n), <n< Isum:y(n) =x1(n) +x2(n), <n< Iproduct:y(n) =x1(n)x2(n), <n< Dr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems7 / 36 Chapter 2: Discrete-Time Signals and SystemsSimple Manipulation of Discrete-Time Signals IFindx(n) x(n+ 1).x(n)n-12-23-3122-23-111 2 3 406 7 8 95152010-1-2-3Dr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems8 / 36 Chapter 2: Discrete-Time Signals and SystemsSimple Manipulation of Discrete-Time Signals IIx(n)n -1 2 -2 3 -3 1 2 -1 1 1 2 3 4 0 6 7 8 9 5 15 20 10 -1 -2 -3 -x(n+1)-1 -1 -1 1 1 x(n)-x(n+1)Dr.

5 Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems9 / 36 Chapter 2: Discrete-Time Signals and SystemsSimple Manipulation of Discrete-Time Signals IFindx(32n+ 1).n3n2+ 1x(3n2+ 1)< 1< 120 if3n2+ 1 is an integer; undefined otherwise-1 12undefined01x(1) = 1152undefined24x(4) = 23112undefined47x(7) = 35172undefined610x(10) = 27232undefined813x(13) = 19292undefined1016x(16) = 111352undefined1219x(19) = 2>12>190 if3n2+ 1 is an integer; undefined otherwiseDr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems10 / 36 Chapter 2: Discrete-Time Signals and SystemsSimple Manipulation of Discrete-Time SignalsGraph ofx(32n+ 1).

6 N-12-23-3131234067895121110-1-2-31314122 -2-1 This signal is undefined for values of n that are not even integers and zero for even integers not shown on this Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems11 / 36 Chapter 2: Discrete-Time Signals and SystemsInput-Output Description of Dst- time SystemsDiscrete-timeSystemx(n) discrete -t imesignaly(n) discrete -timesignalinput/ex citationoutput/responseIInput-output description (exact structure of system is unknownor ignored):y(n) =T[x(n)]I black box representation:x(n)T y(n)Dr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems12 / 36 Chapter 2: Discrete-Time Signals and SystemsClassification of Discrete-Time SystemsWhy is this so important?

7 Imathematical techniques developed to analyze Systems are oftencontingent upon the general characteristics of the Systems beingconsideredIfor a system to possess a given property, the property must holdfor everypossible input to the systemIto disprove a property, need a single counter-exampleIto prove a property, need to prove for the general caseDr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems13 / 36 Chapter 2: Discrete-Time Signals and SystemsTime-invariant vs. time -variant SystemsITime-invariant system : input-output characteristics do notchange with timeIa system is time -invarariant iffx(n)T y(n) = x(n k)T y(n k)for everyinputx(n) and everytime Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems14 / 36 Chapter 2: Discrete-Time Signals and SystemsTime-invariant vs.

8 time -variant SystemsExamples: time -invariant or not?Iy(n) =A x(n)Iy(n) =n x(n)Iy(n) = x(n) +x2(n 2)Iy(n) =x( n)Iy(n) =x(n+ 1)Iy(n) =11 x(n+2)Iy(n) =e3x(n)Ans: Y, N, Y, N, Y, Y, YDr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems15 / 36 Chapter 2: Discrete-Time Signals and SystemsLinear vs. Nonlinear SystemsILinear system : obeys superposition principleIa system is linear iffT[a1x1(n) +a2x2(n)] =a1T[x1(n)] +a2T[x2(n)]for anyarbitrary input sequencesx1(n) andx2(n), and anyarbitrary constantsa1anda2Dr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems16 / 36 Chapter 2: Discrete-Time Signals and SystemsLinear vs.

9 Nonlinear SystemsExamples: linear or not?Iy(n) =A x(n)Iy(n) =n x(n)Iy(n) = x(n) +x2(n 2)Iy(n) =x( n)Iy(n) =x(n+ 1)Iy(n) =11 x(n+2)Iy(n) =e3x(n)Ans: Y, Y, N, Y, Y, N, NDr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems17 / 36 Chapter 2: Discrete-Time Signals and SystemsCausal vs. Noncausal SystemsICausal system : output of system at any timendepends only onpresent and past inputsIa system is causal iffy(n) =F[x(n),x(n 1),x(n 2),..]for allnDr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems18 / 36 Chapter 2: Discrete-Time Signals and SystemsCausal vs.

10 Noncausal SystemsExamples: causal or not?Iy(n) =A x(n)Iy(n) =n x(n)Iy(n) = x(n) +x2(n 2)Iy(n) =x( n)Iy(n) =x(n+ 1)Iy(n) =11 x(n+2)Iy(n) =e3x(n)Ans: Y, Y, Y, N, N, N, YDr. Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems19 / 36 Chapter 2: Discrete-Time Signals and SystemsStable vs. Unstable SystemsIBounded Input-Bounded output (BIBO) Stable: every boundedinput produces a bounded outputIa system is BIBO stable iff|x(n)| Mx< = |y(n)| My< for Deepa Kundur ( University of Toronto ) Discrete-Time Signals and Systems20 / 36 Chapter 2: Discrete-Time Signals and SystemsStable vs.