Discrete-Time System Properties. Discrete-Time System Properties. Terminology: Implication. Terminology: Equivalence. Reference: Section 2.

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1 Professor Deepa Kudur Uiversity of oroto Referece: Sectio 2.2 Joh G. Proakis ad Dimitris G. Maolakis, Digital Sigal Processig: Priciples, Algorithms, ad Applicatios, 4th editio, Professor Deepa Kudur (Uiversity of oroto) / 23 Professor Deepa Kudur (Uiversity of oroto) 2 / 23 ermiology: Implicatio If A the B Shorthad: A = B Example : it is sowig = it is at or below freezig temperature Example 2: α 5.2 = α is positive Note: For both examples above, B A ermiology: Equivalece If A the B Shorthad: A = B ad If B the A Shorthad: B = A ca be rewritte as A if ad oly if B Shorthad: A B We ca also say: A is EQUIVALEN to B A = B = Professor Deepa Kudur (Uiversity of oroto) 3 / 23 Professor Deepa Kudur (Uiversity of oroto) 4 / 23

2 ermiology: Iput-Output Descriptio Classificatio of Discrete-ime Systems iput/ excitatio x() Discrete-time sigal Discrete-time System y() Discrete-time sigal output/ respose Why is this so importat? mathematical techiques developed to aalyze systems are ofte cotiget upo the geeral characteristics of the systems beig cosidered Iput-output descriptio (exact structure of system is ukow or igored): y() = [x()] black box represetatio: 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 x() y() Professor Deepa Kudur (Uiversity of oroto) 5 / 23 Professor Deepa Kudur (Uiversity of oroto) 6 / 23 Classificatio of Discrete-ime Systems Commo System Properties: static vs. dyamic time-ivariat vs. time-variat liear vs. oliear causal vs. o-causal stable vs. ustable systems Static vs. Dyamic Static system (a.k.a. memoryless): the output at time depeds oly o the iput sample at time ; otherwise the system is said to be dyamic a system is static iff (if ad oly if) for every time istat. y() = [x(), ].. Professor Deepa Kudur (Uiversity of oroto) 7 / 23 Professor Deepa Kudur (Uiversity of oroto) 8 / 23

3 Static vs. Dyamic Cosider the geeral system: y() = [x( N), x( N + ),, x( ), x(), x( + ),, x( + M ), x( + M)], N, M > 0 For N = M = 0, y() = [x()], the system is static. For 0 < N, M <, the system is said to be dyamic with fiite memory of duratio N + M +. For either N ad/or M equal to ifiite, the system is said to have ifiite memory. Static vs. Dyamic Examples: memoryless or ot? y() = A x(), A 0 y() = A x() + B, A, B, 0 y() = x() cos( π ( 5)) 25 y() = x( ) y() = x( + ) y() = x(+2) y() = e 3x() y() = As: Y, Y, Y, N, N, N, Y, N Professor Deepa Kudur (Uiversity of oroto) 9 / 23 Professor Deepa Kudur (Uiversity of oroto) 0 / 23 ime-ivariat vs. ime-variat Systems ime-ivariat vs. ime-variat Systems Examples: time-ivariat or ot? ime-ivariat system: iput-output characteristics do ot chage with time a system is time-ivariat iff x() y() = x( k) y( k) for every iput x() ad every time shift k. y() = A x(), A 0 y() = A x() + B, A, B, 0 y() = x() cos( π 25 ) y() = x( ) y() = x( + ) y() = x(+2) y() = e 3x() y() = As: Y, Y, N, N, Y, Y, Y, Y Professor Deepa Kudur (Uiversity of oroto) / 23 Professor Deepa Kudur (Uiversity of oroto) 2 / 23

4 Liear vs. Noliear Systems Liear system: obeys superpositio priciple Liear Systems: Homogeeity A system is liear iff [a x () + a 2 x 2 ()] = a [x ()] + a 2 [x 2 ()] a system is liear iff [a x () + a 2 x 2 ()] = a [x ()] + a 2 [x 2 ()] for ay arbitrary iput sequeces x () ad x 2 (), ad ay arbitrary costats a ad a 2 Homogeeity: Let a 2 = 0. x() [a x ()] = a [x ()] y() = a x() a y() for ay costat a. Professor Deepa Kudur (Uiversity of oroto) 3 / 23 Professor Deepa Kudur (Uiversity of oroto) 4 / 23 Liear Systems: Additivity A system is liear iff [a x () + a 2 x 2 ()] = a [x ()] + a 2 [x 2 ()] Liear Systems: Additivity herefore: Additivity: Let a = a 2 =. x () y () x 2 () y 2 () [x () + x 2 ()] = [x ()] + [x 2 ()] = x () + x 2 () y () + y 2 () Liearity = Homogeeity + Additivity Need both! If a system is ot homogeeous, it is ot liear. If a system is ot additive, it is ot liear. for ay iput sequeces x () ad x 2 (). Professor Deepa Kudur (Uiversity of oroto) 5 / 23 Professor Deepa Kudur (Uiversity of oroto) 6 / 23

5 Liear vs. Noliear Systems Causal vs. Nocausal Systems Examples: liear or ot? y() = A x(), A 0 y() = A x() + B, A, B, 0 y() = x() cos( π 25 ) y() = x( ) y() = x( + ) y() = x(+2) y() = e 3x() y() = Causal system: output of system at ay time depeds oly o preset ad past iputs a system is causal iff for all y() = [x(), x( ), x( 2),...] As: Y, N, Y, Y, Y, N, N, Y Professor Deepa Kudur (Uiversity of oroto) 7 / 23 Professor Deepa Kudur (Uiversity of oroto) 8 / 23 Causal vs. Nocausal Systems Stable vs. Ustable Systems Examples: causal or ot? y() = A x(), A 0 y() = A x() + B, A, B, 0 y() = x() cos( π ( + )) 25 y() = x( ) y() = x( + ) y() = x(+2) y() = e 3x() y() = Bouded Iput-Bouded output (BIBO) Stable: every bouded iput produces a bouded output a system is BIBO stable iff for all. x() M x < = y() M y < As: Y, Y, Y, N, N, N, Y, Y Professor Deepa Kudur (Uiversity of oroto) 9 / 23 Professor Deepa Kudur (Uiversity of oroto) 20 / 23

6 Discrete-ime Bouded Sigals Stable vs. Ustable Systems x[] x[] Examples: stable or ot? y() = A x(), A 0 y() = A x() + B, A, B, 0 x[] x[] y() = x() cos( π 25 ) y() = x( ) y() = x( + ) y() = x(+2) x[] x[] y() = e 3x() BOUNDED SIGNAL UNBOUNDED SIGNAL y() = As: Y, Y, Y, Y, Y, N, Y, N Professor Deepa Kudur (Uiversity of oroto) 2 / 23 Professor Deepa Kudur (Uiversity of oroto) 22 / 23 Fial Remarks For a system to possess a give property, the property must hold for every possible iput ad parameter of the system. to disprove a property, eed a sigle couter-example to prove a property, eed to prove for the geeral case Professor Deepa Kudur (Uiversity of oroto) 23 / 23