Introduction to System

Hany FerdinandoDept. of Electrical Engineering

Petra Christian University

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General Overview

What is system? How to classify systems? What is LTI system? System interconnection Differential and difference equation

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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

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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…

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Two General Systems

Continuous-time system It processes continuous-time signal

Discrete-time system It processes discrete-time signal

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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!!

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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

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Linearity Test:

Homogeneity Law If input u gives output y, then input u has to

give output y Additivity law

If input u1 and u2 give output y1 and y2 respectively, then input (u1+u2) has to give output (y1+y2)

Combined!

Examples and Exercises

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Time-invariant

It means the system does not depend on time

Delayed input will result delayed output

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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

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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 1 System 2input output

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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

input output+

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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

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Example…

y(n) = (2x(n) – x(n)2)2

Multiply by 2

Square

+x(n) Square y(n)–

+

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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

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Differential Equation

Continuous-time system is expressed in the form of differential equation

The response of the system is the solution of that equation

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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

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Difference Equation

¼ + Z-1 Z-1

½

x(n)

y(n)

y(n) = ¼ x(n) + ½ y(n-2)

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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)

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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: