Discrete-time Systems in the Time Domain Domain Chapter 4 Chapter 4 Sections 4.1 - 4.7 Dr. Iyad Jafar
Outline Outline � Definition � Examples � Classification of Discrete-time Systems � Impulse and Step Responses � Linear Time-Invariant Systems (LTI) � Finite-Dimensional LTI Discrete-Time F D l LTI D T Systems y 2
Def Definition nition � A discrete-time system is a process that transforms a given input sequence x[n] into another output sequence g p q p q y[n] with desirable properties. � y[n] T(x[n]) � Discrete-time systems may have more then one input y y p and/or output 3
Exam Examples ples � Accumulator The output y[n] at any instant of time n is the sum of all p y[ ] y previous samples up to instant n. or, the output at instant n is the h h sum of the current f h sample and the previous output y[n-1] 4
Exam Examples ples � The M-point Moving Average (MA) Filter The output y[n] at any instant n is the average of the p y[ ] y g current sample x[n] and the previous M-1 samples. I It is used for smoothing rapid variations in the input d f h d h sequence. 1.5 x[n] y[n] y[ ] 1 0.5 [ n ] x[ 0 M=3 -0.5 -1 0 5 10 15 20 5 n
Exam Examples ples � The M-point Moving Average (MA) Filter Example 3.1: Show that y[n] can be written as p y[ ] y[n] = y[n-1] + 1/M (x[n] – x[n-M]) 6
Exam Examples ples � The Linear Interpolator Used to estimate sample values between pairs of adjacent p p j sample values of a discrete-time sequence. Factor-of-2 Interpolator Factor of 2 Interpolator Factor-of-3 Interpolator 7
Exam Examples ples � The Linear Interpolator 4 4 3 3 U Up-sampling li ] x[ n ] y[ n ] ] 2 2 1 1 0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 n n 4 Interpolation 3 z[ n ] 2 1 0 0 0 2 4 6 8 10 12 n 8
Classification of Discre Classification of Discrete-time Syst -time Systems ems � Linearity In a linear system, if y 1 [n] is the output due to x 1 [n] and y , y 1 [ ] p 1 [ ] y 2 [n] is the output due to x 2 [n], then for an input the output is given by for all possible values of α , β and n Example 3.2: a) y[n] = n x[n] ) y[ ] [ ] b) [ ] = ( [ ]) 2 b) y[n] = (x[n]) 2 9
Classification of Discre Classification of Discrete-time Syst -time Systems ems � Shift-invariant For a shift-invariant system, if y 1 [n] is the output due to y , y 1 [ ] p the input x 1 [n], then the output to an input sequence is basically where n o is an integer. If the independent variable is time, the system is called time-invariant system. y Generally, in time-invariant systems, the output of the system does not depend on the time at which the input is y p p applied. 10
Classification of Discre Classification of Discrete-time Syst -time Systems ems � Shift-invariant Example 3.3: p Determine whether the following g systems are time-invariant. a) y[n] = x[n] – x[n-3] a) y[n] x[n] x[n 3] b) y[n] = n x[n] � � � ⎧ x[n / L] , n = 0, L, 2L, 3L ... c) � ⎨ y[n] 0 , otherwise ⎩ 11
Classification of Discre Classification of Discrete-time Syst -time Systems ems � Causality A system is said to be causal if the computation of the y p output sample n o depends on the input samples in x[n] for n ≤ n o . o Example Example 3 4: Determine 3.4: Determine whether whether the the following following systems are causal. a) y[n] = x[n] – x[n-5] b) y[n] = x[n] - 4 x[n+5] c) y[n] = x[3 – n] ) y[ ] [ ] 12
Classification of Discre Classification of Discrete-time Syst -time Systems ems � Stability There are many definitions for stability. In this course, a y y , system is said to be bounded-input bounded-output ( (BIBO) ) if for every y bounded-input p sequence q the output is also bounded. E ample 3 5: Th Example 3.5: The accumulator is not a BIBO system. It l t i t BIBO t It is not stable when the input is x[n] = u[n]. 13
Classification of Discre Classification of Discrete-time Syst -time Systems ems � Passive and Lossless Systems A discrete-time system is said to be passive if y p � � ∑ ∑ 2 2 � � � y[n] x[n] ��� ��� n n and it is said to be lossless if � � ∑ ∑ 2 2 � � � y[n] x[n] ��� ��� n n Example 3.6: Consider the system y[n] = a x[n – N], N > 0. 14
Classification of Discre Classification of Discrete-time Syst -time Systems ems � Memory Systems A system is a memory (dynamic) system if the y y ( y ) y computation of the output requires storing samples from the input. Otherwise, it is a static system. p , y y[n] = 2 x[n] y[n] = x[n] + x[n 2] y[n] = x[n] + x[n-2] � Recursive Systems A system is said to be recursive when the computation of the output sample requires the knowledge of the previous output sample(s). y [n] = y[n-3] + 6 x[n] y y y[n] = x[n] - x[n-3] 15
Im Impulse pulse and St and Step R ep Responses sponses � Systems are usually characterized by stimulating them with typical signals. The response of the system reflects yp g p y the properties of the system for similar sequences. � The response of a discrete-time system to a unit sample sequence δ [n] is called the unit sample response or δ [ ] i ll d h i l simply, the impulse response , and is denoted by h[n]. � The response of a discrete-time system to a unit step p y p sequence μ [n] is called the unit step response or simply, the step response , and is denoted by s[n]. p p , y [ ] 16
Im Impulse pulse and St and Step R ep Responses sponses Example 3.7: a ) y[n] = α x[n] + α x[n+1] + α x[n-1] + α x[n-2] a ) y[n] α 1 x[n] + α 2 x[n+1] + α 3 x[n-1] + α 4 x[n-2] The impulse response of the system is obtained by setting x[n] = δ [n] x[n] = δ [n] � h[n] = α δ [n] + α δ [n+1] + α δ [n 1] + α δ [n 2] = h[n] � h[n] = α 1 δ [n] + α 2 δ [n+1] + α 3 δ [n-1] + α 4 δ [n-2] = h[n] � h[n] = { α 2 , α 1 , α 3 , α 4 } b) y[n] = x[n] + (x[n-1] + x[n+1]) /2 � h[n] = δ [n] + 0.5 δ [n - 1] + 0.5 δ [n + 1] � h[n] = {0.5, 1 , 0.5} , -1 ≤ n ≤ 1 17
Linear Time-In Linear Time-Invariant Syst riant Systems ems � Linear Time-Invariant (LTI) System - A system satisfying both the linearity and the time-invariance properties both the linearity and the time invariance properties � LTI systems are mathematically easy to analyze and characterize, and consequently, easy to design h i d l d i � There are highly useful signal processing algorithms have g y g p g g been developed utilizing this class of systems over the last several decades � Linear time-invariant systems can be completely characterized by their impulse response h[n]. In other words, characterized by their impulse response h[n] In other words the output of a LTI system for any sequence can be computed by knowing its impulse response! We show this in the y g p p following slides. 18
Linear Time-In Linear Time-Invariant Syst riant Systems ems � Let h[n] denote the impulse response of a LTI discrete- time system time system � We compute its output y[n] for the input x[n] = 0.5 δ [n + 2] +1.5 δ [n − 1] − δ [n − 2] + 0.75 δ [n − 5] 0 δ + 2 +1 δ δ 2 + 0 7 δ 1 � As the system is linear , we can compute its outputs for each member of the input separately and add the individual outputs to determine y[n]. � Since the system is time-invariant δ [ n + 2] → h[n + 2] [ ] [ ] δ [ n − 1] → h[n − 1] δ [ n − 2] → h[n − 2] δ [ n 2] → h[n 2] δ [ n − 5] → h[n − 5] 19
Linear Time-In Linear Time-Invariant Syst riant Systems ems � Likewise, as the system is linear 0 5δ[ n + 2] → 0 5h[n + 2] 0.5δ[ n + 2] → 0.5h[n + 2] − δ[ n − 2] → − h[n − 2] 1.5δ[ n − 1] → 1.5h[n − 1] 1 5δ[ h 0.75δ[ n − 5] → 0.75h[n − 5] � Hence because of the linearity property we get y[n] = 0.5h[n + 2] +1.5h[n − 1] − h[n − 2] + 0.75h[n − 5] � Now, any arbitrary input sequence x[n] can be expressed as a linear combination of delayed and advanced unit sample y p sequences in the form 20
Linear Time-In Linear Time-Invariant Syst riant Systems ems � The response of the LTI system to an input x[k] δ[ n − k] will be x[k]h[n − k]. Hence, the response y[n] to an input be x[k]h[n k]. Hence, the response y[n] to an input will be which can be alternately written as � As we discussed in Chapter 2 this summation is called the � As we discussed in Chapter 2, this summation is called the convolution sum of the sequences x[n] and h[n] and represented compactly as represented compactly as 21
Linear Time-In Linear Time-Invariant Syst riant Systems ems Properties of Convolution 1. 1. The convolution between a length-M and length-N The convolution between a length M and length N sequence has a length of M + N – 1 2. 2 The convolution between a sequence defined over N 1 ≤ n ≤ The convolution between a sequence defined over N 1 ≤ n ≤ M 1 and a sequence defined over N 2 ≤ n ≤ M 2 extends over N 1 + N 2 ≤ n ≤ M 1 + M 2 1 2 1 2 3. Convolution is commutative 4. Convolution is distributive 5. Convolution is associative 22
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