1. Introduction to System Hany Ferdinando Dept. of Electrical Engineering Petra Christian University

2. Introduction to System - Hany Ferdinando2 General Overview What is system? How to classify systems? What is LTI system? System interconnection Differential and difference equation

3. Introduction to System - Hany Ferdinando3 Definition A system is a part of environment that causes certain signals in that environment to be related System yx System relates input and output. Usually, inputs are associated with causes and output with effects

4. Introduction to System - Hany Ferdinando4 Classification Causal and non-causal system  y(t) = x(t) + 2x(t-1)  y(t) = x(t+1) – x(t) + 3x(t-2) Memory and memoryless system  y(t) = -4x(t-1) + 2x(t)  y(t) = 2x(t) Lumped and distributed system  It is about the number of state…

5. Introduction to System - Hany Ferdinando5 Two General Systems Continuous-time system  It processes continuous-time signal Discrete-time system  It processes discrete-time signal

6. Introduction to System - Hany Ferdinando6 LTI System All systems we discussed here are LTI LTI is Linear Time-Invariant The system has to be linear The system has to be time-invariant Non LTI systems are not discussed here…, sorry!!

7. Introduction to System - Hany Ferdinando7 Linearity A system is linear if and only if it fulfills homogeneity and additivity law One can say that the superposition theorem can be applied A linear system can be processed easier than non-linear system

8. Introduction to System - Hany Ferdinando8 Linearity Test: Homogeneity Law  If input u gives output y, then input  u has to give output  y Additivity law  If input u 1 and u 2 give output y 1 and y 2 respectively, then input (u 1 +u 2 ) has to give output (y 1 +y 2 ) Combined! Examples and Exercises

9. Introduction to System - Hany Ferdinando9 Time-invariant It means the system does not depend on time Delayed input will result delayed output

10. Introduction to System - Hany Ferdinando10 Time-invariant test: System delay u(n) y(n) y(n-m) delay System u(n) u(n-m) y(n-m) = Examples and Exercises

11. Introduction to System - Hany Ferdinando11 System Interconnection It is series or cascade interconnection The output of system 1 is the input to system 2 Shortly, the output of the previous system is the input to the next system System 1System 2 inputoutput

12. Introduction to System - Hany Ferdinando12 System Interconnection It is parallel interconnection The input signal is applied to both system simultaneously ‘+’ symbol means the output is sum of both output of system 1 and 2 ‘. ’ symbol means the signal is duplicated System 1 System 2 inputoutput+

13. Introduction to System - Hany Ferdinando13 System Interconnection We can combine both interconnections to form a system Beside ‘+’ sign, we can also use ‘–’ sign In the box of the system, we can put any process

14. Introduction to System - Hany Ferdinando14 Example… y(n) = (2x(n) – x(n) 2 ) 2 Multiply by 2 Square +x(n) Square y(n) – +

15. Introduction to System - Hany Ferdinando15 Delay Delay is important in the linear system One may need to delay signal before processing Delay usually expresses in unit delay, it means it will delay one unit per block  For discrete-time system, delay is represented by Z -1

16. Introduction to System - Hany Ferdinando16 Differential Equation Continuous-time system is expressed in the form of differential equation The response of the system is the solution of that equation

17. Introduction to System - Hany Ferdinando17 Difference Equation Discrete-time system is expressed in the form of difference equation Delay is used to express the difference We can draw the difference equation in the system interconnection

18. Introduction to System - Hany Ferdinando18 Difference Equation ¼ + Z -1 ½ x(n) y(n) y(n) = ¼ x(n) + ½ y(n-2)

19. Introduction to System - Hany Ferdinando19 Exercise y(n) = 2x(n) – x(n-1) – ½ y(n-1) y(n) = x(n-1) – x(n-2) + y(n-2) y(n) = x(n) + x(n-1) + y(n-2)

20. Introduction to System - Hany Ferdinando20 Next… Signals and Systems by Alan V. Oppehnheim, chapter 3, p 69-94 Signals and Linear Systems by Robert A. Gabel, chapter 2, p46-68, chapter 3, p 129-138 The basic information about system is discussed. Now we will move to the next topic, i.e. operation on the system. Please read: