Sigal Processig Lecture 02: Discrete Time Sigals ad Systems Ahmet Taha Koru, Ph. D. Yildiz Techical Uiversity 2017-2018 Fall ATK (YTU) Sigal Processig 2017-2018 Fall 1 / 51
Discrete Time Sigals Discrete Time Sigals ATK (YTU) Sigal Processig 2017-2018 Fall 2 / 51
Discrete Time Sigals Impulse Impulse (uit-sample) is the sequece defied as { 0, 0 δ[] = 1, = 0 1 δ[] Figure: Impulse fuctio. ATK (YTU) Sigal Processig 2017-2018 Fall 3 / 51
Discrete Time Sigals A arbitrary sigal as sum of impulses (1/3) Let us cosider a sigal with followig values: x[ 3] = 2, x[0] = 4 x[1] = 1, x[4] = 3 2 4 x[] 3-1 Figure: A arbitrary sequece. ATK (YTU) Sigal Processig 2017-2018 Fall 4 / 51
Discrete Time Sigals A arbitrary sigal as sum of impulses (2/3) The sequece i the figure ca be represeted as x[] = 2δ[ + 3] + 4δ[] δ[ 1] + 3δ[ 4]. 2 4 x[] 3-1 Figure: A arbitrary sequece. ATK (YTU) Sigal Processig 2017-2018 Fall 5 / 51
Discrete Time Sigals A arbitrary sigal as sum of impulses (3/3) More geerally, ay sequece ca be expressed as x[] = x[k]δ[ k] k= ATK (YTU) Sigal Processig 2017-2018 Fall 6 / 51
Discrete Time Sigals Uit step Uit step is the sequece defied as { 0, < 0 u[] = 1, 0 u[] 1... Figure: Uit step fuctio. ATK (YTU) Sigal Processig 2017-2018 Fall 7 / 51
Discrete Time Sigals Uit step as sum of impulses Uit step ca be represeted as sum of impulses as follows: u[] = δ[] + δ[ 1] + δ[ 2] + δ[ 3] +... = δ[ k] k=0 u[] 1... Figure: Uit step fuctio. ATK (YTU) Sigal Processig 2017-2018 Fall 8 / 51
Discrete Time Sigals Uit step as sum of impulses Uit step ca be represeted as sum of impulses as follows: u[] = δ[] + δ[ 1] + δ[ 2] + δ[ 3] +... = δ[ k] k=0 u[] 1... Figure: Uit step fuctio. ATK (YTU) Sigal Processig 2017-2018 Fall 9 / 51
Discrete Time Sigals Differet represetatios of uit step ad impulse δ[] Impulse: δ[] = u[] u[ 1] Step: u[] = δ[k] k= u[]... u[ 1]... ATK (YTU) Sigal Processig 2017-2018 Fall 10 / 51
Discrete Time Sigals Expoetial sequece The sequece is i the form x[] = Aα If α < 0 alteratig (+/-) If α > 0 ot alteratig If α < 1 decreasig If α > 1 icreasig x 1 [] x 2 [] α = 1.15 α = 0.85 ATK (YTU) Sigal Processig 2017-2018 Fall 11 / 51
Discrete Time Sigals Discrete-time siusoids Discrete-time siusoid is described as x[] = A cos(ω + θ) = A cos(2πf + θ) cos(π/8) ATK (YTU) Sigal Processig 2017-2018 Fall 12 / 51
Discrete Time Sigals Discrete-time siusoids: Period The sigal periodic oly if f is a ratioal umber. f = k N, ω = 2π k N, k, N Z If f = k N where commo factors are caceled, the the period is N. cos(2π 3 4 ) ATK (YTU) Sigal Processig 2017-2018 Fall 13 / 51
Discrete Time Sigals Discrete-time siusoids: Aliases Discrete-time siusoids whose frequecies are separated by a iteger of 2π are idetical. cos(ω) = cos([ω + 2πr] ), r Z Ay siusoid with ω > π is idetical to a siusoid with ω π. Siusoids whose ω > π are called aliases (copies). cos( 3 4 π) cos( 5 4 π) ATK (YTU) Sigal Processig 2017-2018 Fall 14 / 51
Discrete Time Sigals Discrete-time siusoids: Highest rate of oscillatio Highest rate of oscillatio is attaied whe ω = π (or ω = π). cos(π) 8 4 4 8 ATK (YTU) Sigal Processig 2017-2018 Fall 15 / 51
Discrete Time Sigals Complex expoetials Complex expoetial sequece is described by x[2] Im x[] = A e jω = A e j2π k N whose elemets are complex umbers. x[3] x[4] x[5] x[1] x[0] x[7] Re The period ad aliases are similar to siusoids. x[6] ATK (YTU) Sigal Processig 2017-2018 Fall 16 / 51
Discrete Time Systems ATK (YTU) Sigal Processig 2017-2018 Fall 17 / 51
Examples Discrete Time Systems: Examples ATK (YTU) Sigal Processig 2017-2018 Fall 18 / 51
Examples What is a discrete-time system? A discrete-time system is defied as a trasformatio (operator) that maps a iput sequece x[] ito a output sequece y[]. y[] = T (x[]) x[] T ( ) y[] Figure: Represetatio of a discrete-system i a block diagram ATK (YTU) Sigal Processig 2017-2018 Fall 19 / 51
Examples Discrete-time system example I: The Ideal Delay The ideal delay system is defied by the equatio y[] = x[ d ], < <. x[] x[ 1] ATK (YTU) Sigal Processig 2017-2018 Fall 20 / 51
Examples Discrete-time system example II: Movig Average (1/3) The geeral movig-average system is defied by the equatio y[] = 1 M 1 + M 2 + 1 M 2 k= M 1 x[ k] = 1 M 1 + M 2 + 1 (x[ + M 1] + x[ + M 1 1] + + x[] + + x[ M 2 1] + x[ M 2 ]) th sample of the output sequece is average of (M 1 + M 2 + 1) samples aroud th sample of iput sequece. ATK (YTU) Sigal Processig 2017-2018 Fall 21 / 51
Examples Discrete-time system example II: Movig Average (2/3) What is the output of a zero-mea oise sigal w[]? w[] y[] ATK (YTU) Sigal Processig 2017-2018 Fall 22 / 51
Examples Discrete-time system example II: Movig Average (2/3) What is the output of a zero-mea oise sigal w[]? w[] y[] ATK (YTU) Sigal Processig 2017-2018 Fall 23 / 51
Examples Discrete-time system example II: Movig Average (3/3) Output of oisy sesor sigals x 1 [] x 2 [] y 1 [] y 2 [] ATK (YTU) Sigal Processig 2017-2018 Fall 24 / 51
Classificatio Discrete Time Systems: Classificatio ATK (YTU) Sigal Processig 2017-2018 Fall 25 / 51
Classificatio Why classificatio matters? To determie which special mathematical tools ca be used to desig/aalyze the system Ex: Liear Time-ivariat (LTI) System Fourier/z-domai aalysis To reach required iformatio from academic literature or olie resources. Example Google searches: Stability aalysis Too geeral, may ot be applicable Stability aalysis for LTI Time-Delay Systems Applicatio specific ATK (YTU) Sigal Processig 2017-2018 Fall 26 / 51
Classificatio Memoryless Systems (1/2) A system is referred to as memoryless if the output y[] at every value of depeds oly o the iput x[] at the same value of. ONLY curret value of the iput x[] NO past values NO future values Example The discrete-time system described as y[] = (x[]) 2 is a memoryless system. ATK (YTU) Sigal Processig 2017-2018 Fall 27 / 51
Classificatio Memoryless System (2/2) System Expressio Memoryless Ideal Delay y[] = x[ d ] Movig Average y[] = 1 M2 x[ k] M 1 + M 2 + 1 k= M1 ATK (YTU) Sigal Processig 2017-2018 Fall 28 / 51
Classificatio Memoryless System (2/2) System Expressio Memoryless Ideal Delay y[] = x[ d ] NO Movig Average y[] = 1 M2 x[ k] NO M 1 + M 2 + 1 k= M1 ATK (YTU) Sigal Processig 2017-2018 Fall 29 / 51
Classificatio Liear Systems Defiitio A system T ( ) is liear if ad oly if followig properties hold: Additivity: T (x 1 [] + x 2 []) = T (x 1 []) + T (x 2 []) = y 1 [] + y 2 [] Homogeeity: T (αx[]) = αt (x[]) = αy[] Combiig two properties yields super-positio: T (αx 1 [] + βx 2 []) = αt (x 1 []) + βt (x 2 []) ATK (YTU) Sigal Processig 2017-2018 Fall 30 / 51
Classificatio Liear System Example: The Accumulator The system, with followig iput-output relatioship, is liear. y[] = x[k] k= Proof. For x 3 [k] = αx 1 [k] + βx 2 [k] super-positio property holds: y 3 [k] = (αx 1 [k] + βx 2 [k]) k= = k= (αx 1 [k]) + k= (βx 2 [k]) = α k= x 1 [k] + β k= x 2 [k] = αy 1 [k] + βy 2 [k] ATK (YTU) Sigal Processig 2017-2018 Fall 31 / 51
Classificatio No-liear System Example: Logarithm The system, with followig iput-output relatioship, is o-liear. y[] = log 10 ( x[] ) Proof. A couter-example violatig super-positio property is x 1 [] = 1 ad x 2 [] = 10. For x 3 [] = x 1 [] + x 2 [], log 10 (10 + 1) = log 10 (11) log 10 (10) + log 10 (1) = 1 ATK (YTU) Sigal Processig 2017-2018 Fall 32 / 51
Classificatio Other Liearity Examples System Expressio Liearity Ideal Delay y[] = x[ d ] Movig Average y[] = 1 M2 x[ k] M 1 + M 2 + 1 k= M1 ATK (YTU) Sigal Processig 2017-2018 Fall 33 / 51
Classificatio Other Liearity Examples System Expressio Liearity Ideal Delay y[] = x[ d ] YES Movig Average y[] = 1 M2 x[ k] YES M 1 + M 2 + 1 k= M1 ATK (YTU) Sigal Processig 2017-2018 Fall 34 / 51
Classificatio Time-ivariat Systems Defiitio The systems, for which a time shift or delay of the iput sequece causes a correspodig shift i the output sequece, are called time-ivariat systems. If the system is time-ivariat, the output is y 2 [] = y 1 [ d ] for the iput x 2 [] = x 1 [ d ] for arbitrary d. ATK (YTU) Sigal Processig 2017-2018 Fall 35 / 51
Classificatio A time-ivariat system example: Accumulator (1/2) The accumulator system is time-ivariat. Proof. For x 2 [] = x 1 [ d ] y 2 [] = k= x 2 [k] = Chage the variable k as k 1 = k d y 2 [] = d k 1 = k= x 1 [k d ] x 1 [k 1 ] = y 1 [ d ] ATK (YTU) Sigal Processig 2017-2018 Fall 36 / 51
Classificatio A time-ivariat system example: Accumulator (2/2) x 1 [] x 2 [] y 1 [] y 2 [] Figure: The simulatio of the system illustrates the time-ivariat property ATK (YTU) Sigal Processig 2017-2018 Fall 37 / 51
Classificatio Not a time-ivariat system example: Compressor (1/2) The system is defied by the relatio y[] = x[m], < <, M Z + The system is ot time-ivariat. Proof. Cosider y 1 [] = x 1 [M]. For x 2 [] = x 1 [ d ], y 2 [] = x 2 [M] = x 1 [M d ] which is ot equal to delayed versio of output y 1 y 2 [] y 1 [ d ] = x 1 [M( d )] ATK (YTU) Sigal Processig 2017-2018 Fall 38 / 51
Classificatio Not a time-ivariat system example: Compressor (2/2) x 1 [] x 2 [] = x 1 [ 1] y 1 [] y 2 [] y 1 [ 1] Figure: The simulatio for M = 3. System is ot time-ivariat. ATK (YTU) Sigal Processig 2017-2018 Fall 39 / 51
Classificatio Causal Systems Defiitio A system is causal if, for every choice of 0, the output sequece value at the idex = 0 depeds oly o the iput sequece values for 0. The output depeds ONLY curret ad past values of the iput NOT depeds o the future values of the iput ATK (YTU) Sigal Processig 2017-2018 Fall 40 / 51
Classificatio Causality Examples Example The backward differece system y[] = x[] x[ 1] is causal. Example The forward differece system y[] = x[ + 1] x[] is ot causal sice output depeds o a future value of iput. ATK (YTU) Sigal Processig 2017-2018 Fall 41 / 51
Classificatio Stable Systems Defiitio A system is bouded-iput bouded-output (BIBO) stable if ad oly if every bouded iput sequece produces a bouded output sequece. For ay bouded iput x[] B x <,, the system is BIBO stable if ad oly if y[] B y <,. ATK (YTU) Sigal Processig 2017-2018 Fall 42 / 51
Classificatio Stable system example The system y[] = x[] 2 is BIBO stable sice for ay x[] B x, y[] = x[] 2 Bx 2. ATK (YTU) Sigal Processig 2017-2018 Fall 43 / 51
Classificatio Ustable system example: The Accumulator System (1/2) The accumulator system y[] = k= x[k] is ot BIBO stable. For x[] = u[], { 0, < 0 y[] = + 1, 0 There is o upper boud of the accumulator system output for the step iput. ATK (YTU) Sigal Processig 2017-2018 Fall 44 / 51
Classificatio Ustable system example: The Accumulator System (2/2) w[] 5 10 y[] 5 10 Figure: Ustable behaviour of the accumulator system. ATK (YTU) Sigal Processig 2017-2018 Fall 45 / 51
Quiz Quiz ATK (YTU) Sigal Processig 2017-2018 Fall 46 / 51
Quiz 1 Is the system liear? Prove or disprove. y[] = max k [,] x[k]. 2 Uder what coditios, the movig-average system y[] = M 2 1 x[ k] 1 + M 1 + M 2 k= M 1 is causal? 3 Is the movig-average system is BIBO stable? Prove. ATK (YTU) Sigal Processig 2017-2018 Fall 47 / 51
Quiz Problem 1 Solutio Let x 1 [] = δ[] ad x 2 [] = δ[]. The, y 1 [] = u[] ad y 2 [] = 0. For x 3 [] = x 1 [] + x 2 [] = 0, the output is y 3 [] = 0 y 1 [] + y 2 [] = u[]. Superpositio does ot hold for the system, hece it is ot liear. δ[] δ[] x 3 [] y 1 [] y 2 [] y 3 [] ATK (YTU) Sigal Processig 2017-2018 Fall 48 / 51
Quiz Problem 2: Solutio The movig average system is M 2 1 y[] = x[ k] 1 + M 1 + M 2 k= M 1 1 = (x[ + M 1 ] + + x[] +... x[ M 2 ]). 1 + M 1 + M 2 System is causal whe M 2 0 ad M 1 0. ATK (YTU) Sigal Processig 2017-2018 Fall 49 / 51
Quiz Problem 3: Solutio (1/2) For iputs satisfyig x[] B x, (1) M y[] = 1 2 x[ k] 1 + M 1 + M 2 k= M 1 M 1 2 = 1 + M 1 + M 2 x[ k] k= M 1 From triagle iequality property of absolute value ( a + b a + b ), y[] M 2 1 x[ k], 1 + M 1 + M 2 k= M 1 ATK (YTU) Sigal Processig 2017-2018 Fall 50 / 51
Quiz Problem 3: Solutio (1/2) ad from (1) y[] 1 1 + M 1 + M 2 (1 + M 1 + M 2 )B x As a result, if x[] B x, the y[] B x. System is BIBO stable. ATK (YTU) Sigal Processig 2017-2018 Fall 51 / 51