Basic concepts in DT systems. Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 1

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1 Basic concepts in DT systems Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 1

2 Readings and homework For DT systems: Textbook: sections 1.5, 1.6 Suggested homework: pp : Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 2

3 Course outline DT signals (cont d) a brief note on combining time-shift and time reversal Elementary DT signals Periodic signals (cont d) Unit step function Unit impulse function DT systems: System properties: Causality Stability Linearity Invariance Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 3

4 On combining time-shift and time reversal y[ n] = x[ n Two possible ways: x[ n] or reversal x[ n] α] α positive integer advance ( shift left) by α x[ n] x[ n delay ( shift right) by α + α] reversal x[ n x[ n + α] + α] Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 4

5 Periodicity properties of DT CT jω t jω x( t) = e x[ n] = e 0 0 n 1) The larger the magnitude of ω 0, the higher is the rate of oscillation in the signal 2) This signal is periodic for any non-zero value of ω 0 The DT signal x[n] is periodic only if ω 0 /2π is a rational number. m ω 2π Fundamental frequency where m and N have no factors in common. N 0 = Read Table 1.1 for a comparative summary of periodicity properties of CT and DT signals Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 5

6 Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 6 Example Determine the fundamental period of + + = 6 2 2cos 8 sin 4 2cos ] [ π π π π n n n n x

7 Non-periodic elementary signals Building blocks for constructing and representing signals and systems The unit impulse (unit sample): The unit step: δ [] n = 0 1 if if n 0 n = 0 u [] n = 0 1 if if n n < 0 0 Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 7

8 Relationships between DT unit impulse and DT unit step The DT unit impulse is the first difference of the DT unit-step δ [ n] = u[ n] u[ n 1] Conversely, the DT unit step is the running sum of the unit sample n u[ n] u[ n] = = m= δ k= 0 δ [ m] [ n k] Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 8

9 Sampling property of the unit impulse sequence x[ n] δ [ n] = x[0] δ [ n] x[ n] δ [ n n0] = x[ n0] δ [ n n0]; Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 9

10 CT unit impulse and unit step functions CT unit step function: CT unit step function is the running integral of the unit impulse: Conversely, the unit impulse can be thought as the first derivative of the unit step Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 10

11 CT unit step and unit impulse functions (cont d) u(t) is discontinous at t=0 and formally not differentiable; We can imagine a limit process δ Δ ( t) = duδ ( t) dt δ ( t) Δ 0 with DT signals we do not need continuity, derivability etc. Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 11

12 Course outline DT signals (cont d) a brief note on combining time-shift and time reversal Elementary DT signals Periodic signals (cont d) Unit step function Unit impulse function DT systems: example System properties: Causality Stability Linearity Invariance Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 12

13 What is a system? Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 13

14 Example of DT system A rudimentary edge detector Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 14

15 Observations A very rich class of systems (but by no means all systems of interest to us) are described by differential and difference equations Such an equation, by itself, does not completely describe the input-output behaviour of the system: we need auxiliary conditions (initial conditions) In some cases the system of interest has time as the natural independent variable and is causal. However, that is not always the case. Very different physical systems may have very similar mathematical descriptions. Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 15

16 System properties invertibility, causality, linearity, stability, time-invariance etc. Why? Important practical/physical implications They provide us with structure that we can exploit for both system analysis and system design Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 16

17 Invertibility In an invertible system, distinct inputs lead to distinct outputs The original system cascaded with its inverse yield an output w[n] equal to the input x[n] of the original system. example Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 17

18 Invertibility analysis Determine if the following systems are invertible or not. If yes, construct their corresponding inverse systems y[n]=nx[n] y[n]=x[1-n] Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 18

19 Causality A system is causal if the output does not anticipate future values of the input, i.e. if the output at any time depends only of the values of the input up to that time All real-time physical systems are causal. Time only moves forward, and effect occurs after the cause. Causality does notapply to systems processing spatially varying signals (we can move both left and right, up and down) Causality does notapply to systems processing recorded signals (e.g. taped sports games versus live broadcast) Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 19

20 Causality (cont d) Mathematically: A system x[n] y[n] is causal if when and Then x 1 x 1 1 [ n] [ n] = y [ n] = x 2 y y 2 1 [ n] [n] and [ n] for all x 2 n [ n] n for all n 0 n 0 y 2 [n] Causal or non-causal? y[n]=x[-n] y[n]=(1/2) n+1 x 3 [n-1] Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 20

21 Stability a stable system is one in which bounded inputs lead to responses that do not diverge Ex: model for the balance of a bank account from month to month y[n]=1.01y[n-1]+x[n], x[n]>0 for all n>=0. Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 21

22 Stable or unstable? y[n]=nx[n] y[n]=x[4n+1] Strategy: to prove that a system is unstable, we need to find a specific bounded input that leads to an unbounded output. If such an example does not exist, then the system is stable Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 22

23 Time invariance Conceptually: A system is time invariant (TI) if its behaviour does not depend on a particular moment in time. Mathematically: for a DT time-invariant system: Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 23

24 Time invariant or time-varying? y[n]=nx[n] y[n]=sin(x[n]) Strategy: we must determine whether the time invariance property holds for any input and any time shift When a system is suspected of being time-varying, we can seek a counterexample (a specific input signal for which the condition of time-invariance would be violated) Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 24

25 Important If the input to a TI system is periodic, then the output is periodic with the same period. Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 25

26 Linear and non-linear systems Many systems are non-linear (ex: circuit elements: diodes, dynamics of aircraft etc.) In ELEC 310 we focus exclusively on linear systems Why? We can often linearize models to examine small signal perturbations around operating points Linear systems are analytically tractable, providing basis for important tools used in DSP. Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 26

27 Linearity A system is called linear if it has two mathematical properties: Let us consider x 1 [n] y 1 [n] and x 2 [n] y 2 [n] Additivity: x 1 [n] + x 2 [n] y 1 [n] + y 2 [n] Homogeneity (scaling): ax 1 [n] a x 1 [n] We can combine these two properties into one: ax 1 [n]+ bx 2 [n] ay 1 [n] + by 2 [n] Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 27

28 Properties of linear systems Superposition For linear systems, zero input zero output A linear system is causal if and only if it satisfies the condition of initial rest x[ n] = 0 for t 0 y[ n] = 0 for t 0 Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 28

29 Linear Time-Invariant (LTI) Systems Our focus for most of this course A basic fact: If we know the response of an LTI system to some inputs, we actually know its response to many inputs Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 29

30 System interconnections Serial Parallel Feedback example Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 30

31 You know now the primary focus in this class is on linear, timeinvariant LTI systems in the DT domain LTI systems are defined in a similar way in both CT and DT domains How to compute the global input-output function of interconnected systems How to determine whether a system is: Causal Stable Time-invariant Linear Invertible Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 31