ECE 2221 SIGNALS AND SYSTEMS - International …staff.iium.edu.my/sigit/ece2221/pdf/02 Continuous...

E C E 2 2 2 1 S I G N A L S A N D S Y S T E M S

ECE 2221 Signals and Systems, Sem 3 2010/2011, Dr. Sigit Jarot

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Course Objectives

Dr. Sigit PW JarotECE 2221 Signals and Systems 2

To provide an analysis skill of the continuous‐time signals and systems as reflected to their roles in engineering

practice.

To expose students to both the time‐domain and frequency‐domain methods of analyzing signals and

systems.

To illustrate the potential applications of this course as a Pre‐requisite course to communication engineering and principles, digital signal processing and control system.

OBE (Outcome Based Education)Learning Outcomes


Classify, characterize and conduct basic of signals and systems.

Analyze continuous‐time signals and systems in time domain using convolution.

Analyze continuous‐time signals and systems in frequency domain using Laplace transform.

Analyze continuous‐time signals and systems in frequency domain using Fourier series and Fourier transform.

Acquire introductory‐level knowledge of discrete‐time signals and systems, and sampling theory.

Work in group to perform basic simulation of signals and systems analysis.

After completion of this course the students will be able to:

Course Synopsis


Introduction to Signals

Introduction to Systems

Time‐Domain Analysis of Continuous‐Time Systems

Frequency‐Domain System Analysis: the Laplace Transform

MID‐TERM Examination

Signals Analysis using the Fourier Series

Signals Analysis using the Fourier Transform

Introduction to Discrete Time Signals and Systems Analysis

FINAL Examination

Outline


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Concept of system and system classification Linear time‐invariant continuous‐time systems Linearity and time‐invariance Causality and stability

System Representation Systems represented by differential equations Convolution integral

System interconnection

System Concept


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System is a mathematical transformation of an input signal (or signals) into an output signal (or signals)

SIGNAL A set of data or information. A function or sequence of values that represents information. A function of one or more variables (e.g. time, frequency, space,..) that conveys information on the nature of a physical phenomenon.

SYSTEM A system is an entity that processes a set of signals (inputs) to yield another sets of signals (outputs)

SystemInputSignal

OutputSignal

System Classification


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Static or Dynamic Systems A dynamic system has the capability of storing energy, or remembering its

state, while a static system does not. A battery connected to resistors is a static system, while the same battery

connected to resistors, capacitors, and inductors constitutes a dynamic system.

The main difference is the capability of capacitors and inductors to store energy, to remember the state of the device, that resistors do not have.

Continuous‐Time or Discrete‐Time System Whenever the input(s) and output(s) are both continuous time, discrete

time, or digital, the corresponding systems are continuous time, discrete time, or digital, respectively.

It is also possible to have hybrid systems when the input(s) and output(s) are not of the same type.

LTI Continuous Time Systems


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A continuous‐time system is a system in which the signals at its input and output are continuous‐time signals.

Mathematically we represent it as a transformation S that converts an input signal x(t) into an output signal y(t)=S[x(t)]

Characteristics of System Models


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Linearity The linearity between the input and the output, as well as the constancy of

the system parameters, simplify the mathematical model.

Time invariance Causality Causality, or nonanticipatory behavior of the system, relates to the cause–

effect relationship between the input and the output. It is essential when the system is working under real‐time situations—that is, when there is limited time for the system to process signals coming into the system.

Stability Stability is needed in practical systems. A stable system behaves well under

reasonable inputs. Unstable systems are useless.

Linearity


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A system is said to be linear in terms of the system input (excitation) x(t) and the system output (response) y(t) if it satisfies the following two properties of superposition and homogeneity:1. Superposition:

1( ) ( )x t x t 1( ) ( )y t y t

2( ) ( )x t x t 2( ) ( )y t y t1 2( ) ( ) ( )x t x t x t

1 2( ) ( ) ( )y t y t y t 2. Homogeneity:

( )x t ( )y t ( )ax t ( )ay ta = constant

factor


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ECE 2221 Signals and Systems 12

Show that the system described by the following equation is linear.

ECE 2221 Signals and Systems 13

Show that the system described by the equation is linear.

Let x1(t) → y1(t) and x2(t) → y2(t), then and

Multiple 1st equation by k1, and 2nd equation by k2, and adding them yields:

This equation is the system equation with and

Time Invariance16

A system is said to be time invariance if a time delay or time advance of the input signal leads to an identical time shift in the output signal.

A time‐invariant system do not change with time.

The notion of time invariance. (a) Time‐shift operator St0 preceding operator H. (b) Time‐shift operator St0 following operator H. These two situations are equivalent,

provided that H is time invariant.


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Time InvarianceExample : Inductor

x1(t) = v(t)

y1(t) = i(t)

The inductor shown in figure is described by the input‐output relation:

1 11( ) ( )

ty t x d

L

where L is the inductance. Show that the inductor so described is time invariant.<Sol.>1. Let x1(t) x1(t t0) Response y2(t) of the inductor to x1(t t0) is

0

1 0 11( ) ( )

t ty t t x d

L

2 1 01( ) ( )

ty t x t d

L

2. Let y1(t t0) = the original output of the inductor, shifted by t0 seconds:

3. Changing variables: 0' t

(A)

(B)

(A)0

2 11( ) ( ') '

t ty t x d

L

Inductor is time invariant.

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Time InvarianceExample : Inductor

x1(t) = v(t)

y1(t) = i(t)

The inductor shown in figure is described by the input‐output relation:

1 11( ) ( )

ty t x d

L

where L is the inductance. Show that the inductor so described is time invariant.<Sol.>1. Let x1(t) x1(t t0) Response y2(t) of the inductor to x1(t t0) is

0

1 0 11( ) ( )

t ty t t x d

L

2 1 01( ) ( )

ty t x t d

L

2. Let y1(t t0) = the original output of the inductor, shifted by t0 seconds:

3. Changing variables: 0' t

(A)

(B)

(A)0

2 11( ) ( ') '

t ty t x d

L

Inductor is time invariant.

Signals_and_Systems_Simon Haykin & Barry Van Veen 21

Time InvarianceExample Thermistor

Let R(t) denote the resistance of the thermistor, expressed as a function of time. We may express the input‐output relation of the device as

x1(t) = v(t)

y1(t) = i(t)

1 1( ) ( ) / ( )y t x t R tShow that the thermistor so described is time variant.<Sol.>1. Let response y2(t) of the thermistor to x1(t t0) is

1 02

( )( )( )

x t ty tR t

2. Let y1(t t0) = the original output of the thermistor due to x1(t), shifted by t0seconds:

1 01 0

0

( )( )( )

x t ty t tR t t

3. Since R(t) R(t t0) 1 0 2 0( ) ( ) for 0y t t y t t Time variant!

Signals_and_Systems_Simon Haykin & Barry Van Veen 22

Time InvarianceExample Thermistor

Let R(t) denote the resistance of the thermistor, expressed as a function of time. We may express the input‐output relation of the device as

x1(t) = v(t)

y1(t) = i(t)

1 1( ) ( ) / ( )y t x t R tShow that the thermistor so described is time variant.<Sol.>1. Let response y2(t) of the thermistor to x1(t t0) is

1 02

( )( )( )

x t ty tR t

2. Let y1(t t0) = the original output of the thermistor due to x1(t), shifted by t0seconds:

1 01 0

0

( )( )( )

x t ty t tR t t

3. Since R(t) R(t t0) 1 0 2 0( ) ( ) for 0y t t y t t Time variant!


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Causality


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Causality is independent of the linearity and the time‐invariance properties of a system.

Representation of Systems by Differential Equations


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Response of a Linear System

A system’s output for t ≥ 0 is the result of 2 independent causes:1. The initial condition at t = 0 zero‐input response2. The input x(t) for t ≥ 0 zero‐state response

Decomposition property:

Total response = zero‐input response + zero‐state response

x(t) y(t) +y0(t) x(t) ys(t)=

t

C tdxC

tRxvty0

0)(1)()0()(

zero‐input response zero‐state response

Linear Differential Systems

Many systems in electrical and mechanical engineering where input x(t) and output y(t) are related by differential equations.

For example:

Linear Differential Systems

In general, relationship between x(t) and y(t) in a linear time‐invariant (LTI) differential system is given by (where all coefficients ai and bi are constants):

Use compact notation D for operator d/dt, i.e.

We get

or

)(...)(... 11

1

111

1

1 txbdtdxb

dtxdb

dtxdbtya

dtdya

dtyda

dtyd

NNM

M

MNM

M

MNNNN

N

N

N

.)()( 22

2

etctyDdt

ydandtDydtdy

)()...()()...( 11

111

1 txbDbDbDbtyaDaDaD NNM

MNM

MNNNNN

)()()()( txDPtyDQ

NNM

MNM

MN

NNNN

bDbDbDbDP

aDaDaDDQ

11

1

11

1

...)(

...)(

Zero‐input response basics

Remember that for a Linear System

In this lecture, we will focus on a linear system’s zero‐input response, y0(t), which is the solution of the system equation when input x(t)=0.

Recall

For zero‐input response


)()...()()...( 11

111

1 txbDbDbDbtyaDaDaD NNM

MNM

MNNNNN

)()()()( txDPtyDQ

0)()( 0 tyDQ

0)()...( 011

1 tyaDaDaD NN

NN

General solutions to the zero‐input response equation (1)

From math course on differential equations, we may solve the equation:

By letting where c and λ are constant. Then, by substituting into the above equation

0)()...( 011

1 tyaDaDaD NN

NN

tcety )(0

tNN

NN

t

t

ecdt

ydtyD

ecdt

ydtyD

ecdt

dytDy

00

22

02

02

00

)(

)(

)(


We get

This is identical to the polynomial Q(D) with λ replacing D, i.e.

We can now express Q(λ) in factorized form:

Therefore λ has solutions: λ1, λ2,… λN assuming that all λi are distinct.


Therefore,

has N possible solutions:

where are arbitrarily constants.

It can be shown that the general solution is the sum of all these terms:

In order to determine the N arbitrary constant, we need to have N constraints(i.e. initial or boundary or auxiliary conditions).

Characteristic Polynomial of a Systems

Q(λ) is called characteristic polynomial of the system Q(λ)=0 is the characteristic equation of the system The roots to the characteristic equation Q(λ)=0 i.e. are extremely important.

They are called by different names: Characteristic value Eigenvalues Natural frequencies

The exponential eλit (i=1,2,…,n) are the characteristic modes(also known as natural modes) of the system.

Characteristic modes determine the system behavior

ECE 2221 Signals and Systems

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Repeated Characteristic Roots

If there are repeated roots, the form of the solution is modified.

The solution of the equation:

is given by

In general, the characteristic modes for the differential equation:

are

The solution for yo(t) is


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Complex Characteristic Roots

Solution of the characteristic equation may result in complex roots. For real (i.e. physically realizable) systems, all complex roots must occur in

conjugate pairs In other words, the coefficients of the characteristic polynomial Q(λ) are real.

In other words, if α+jβ is a root, the there must exist the root α-jβ . The zero‐input response corresponding to this pair of conjugate roots is:

For a real system, the response yo(t) must also be real. This is possible only if c1 and c2 are conjugates too.

Let

This gives


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The meaning of 0‐ and 0+

There are subtle differences between time t = 0 exactly, time just before t=0, i.e. t = 0‐ and time just AFTER t=0, i.e. t = 0+.

At t = 0‐ the total response y(t) consists SOLELY of the zero‐input component y0(t).

However, applying an input x(t) at t=0, while not affecting y0(t), in general WILL affect y(t) (because input is now no longer zero).


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Insight into Zero‐input Behavior

Assume a system is initially at rest. Now disturb it momentarily, then remove the disturbance (now it is zero‐input), the system will not come back to rest instantaneously.

In generally, it will go back to rest over a period of time, and only through some special type of motion that is characteristic of the system.

Such response must be sustained without any external source (because the disturbance has been removed).

In fact the system uses a linear combination of the characteristic modes to come back to the rest position while satisfying some boundary (or initial) conditions.

Unit impulse defined as the limit of Gaussian pulse

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Figure 2CT.2 – Unit impulse.

)(lim)(0

tgt

Figure 2CT.3 – Gaussian pulse.

2)/(5.0

21)(

tetg

where

Sampling Property of Unit Impulse49

If you multiply an arbitrary waveform x(t) by an impulse occurring at time , you obtain an impulse have area x() occurring at time .

We can view the impulse as “sampling” x(t) at t =.

Figure 2CT.8 – Sampling property of the unit impulse

)()()()( txttx

Impulse Response50

The response of a LTIC system to a unit impulse is called the system’s impulse response and is denoted as h(t) .

)(t )(thUnit Impulse Impulse Response

Impulse Response of Integrator51

h(t) for t < 0 h(t) for t = 0 h(t) for t > 0

Figure 1.9 – Integrator.

Figure 2CT.9 – Region of integration for eq. (8) (shaded).

Unit Step Function52

If we integrate a unit impulse, we get a unit step. It follows that a unit impulse is the derivative of a unit step.

Figure 2CT.10 – Unit step function.

dttdut )()(

01000

)(21

ttt

tu

Nonlinear systems can be approximated by linear system

Almost all systems become nonlinear when large enough signals are applied Nonlinear systems can be approximated by linear systems for small‐signal analysis – greatly

simply the problem Once superposition applies, analyse system by decomposition into zero‐input and zerostate

components Equally important, we can represent x(t) as a sum of simpler functions (pulse or step)


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Outline

(Review of Previous Lecture) Zero‐Input Response

Unit Impulse Response Convolution Integral Total Response Summary

Response of a Linear System

A system’s output for t ≥ 0 is the result of 2 independent causes:1. The initial condition at t = 0 zero‐input response2. The input x(t) for t ≥ 0 zero‐state response

Decomposition property:


x(t) y(t) +y0(t) x(t) ys(t)=

t

C tdxC

tRxvty0

0)(1)()0()(

zero‐input response zero‐state response

The importance of Impulse Response h(t)

Zero‐state response assumes that the system is in “rest” state, i.e. all internal system variables are zero.

Deriving and understanding zero‐state response depends on knowing the impulse response h(t) to a system.

Any input x(t) can be broken into many narrow rectangular pulses. Each pulse produces a system response.

Since the system is linear and time invariant, the system response to x(t) is the sum of its responses to all the impulse components.

h(t) is the system response to the rectangular pulse at t=0 as the pulse width approaches zero.

Zero‐state Response (1)

We now consider how to determine the system response y(t) to an input x(t) when system is in zero state.

Define a pulse p(t) of unit height and width Δτ at t=0 .

Input x(t) can be represented as sum of narrow rectangular pulses.

The pulse at t=nΔτ has a height x(t)=x(nΔτ) . This can be expressed as x(nΔτ)p (t-nΔτ). Therefore x(t) is the sum of all [x(nΔτ)/ Δτ] . such

pulses, i.e,

)()(lim

)()(lim)(

0

0

ntpnx

ntpnxtx


The term [x(nΔτ)/ Δτ]p(t-nΔτ) represents a pulse p(t-nΔτ) with height x(nΔτ).

As Δτ 0, height of strip ∞, but area remain x(nΔτ) and

)()()()(

ntnxntpnx

)()(lim)(

0ntnxtx


x(t) y(t)


Therefore,

Knowing h(t), we can determine the response y(t) to any input x(t). Observe the all‐pervasive nature of the system’s characteristic modes, which

determines the impulse response of the system.

LTI Systemh(t)x(t) y(t)

y(t) = x(t) * h(t)

)()(lim)(

0nthnxty

dthxty )()()(

dthxty )()()(

How to determine the unit impulse response h(t) (1)

Given that a system is specified by the following differential equation, determine its unit impulse response h(t).

Remember the general system equation:

It can be shown that the impulse response h(t) is given by:

where u(t) is the unit step function, and yn(t) is a linear combination of the characteristic modes of the system.

)()()()( txDPtyDQ

)()]()([)( tutyDPth n

tN

ttn

Necececty ......)( 2121

dtdxty

dtdy

dtyd

)(232

2

How to determine the unit impulse response h(t)? (2)

The constants ci are determined by the following initial conditions:

Note yn(k)(0) is the kth derivative of yn(t) at t=0.

The above is true if M, the order of P(D) is less the N, the order of Q(D)(which is generally the case for most stable systems).

0)0(....)0()0()0()2(

N

nnnn yyyy

1)0()1(

N

ny

Example

Find the unit impulse response of a systems specified by the equation :

)()()23( 2 tDxtyDD

Exercise 1

Find the unit impulse response of a systems specified by the equation :

)()5()()34( 2 txDtyDD

Exercise

Find the unit impulse response of a systems specified by the equation : )()5()()34( 2 txDtyDD

Exercise 2

Find the unit impulse response of a systems specified by the equation : )()117()()65( 22 txDDtyDD

Exercise Find the unit impulse response of a systems specified by the equation :

)()117()()65( 22 txDDtyDD

The Convolution Integral

The derived integral equation occurs frequently in physical sciences, engineering, and mathematics.

It is given the name: the convolution integral. The convolution integral of two function x1(t) and x2(t) is denoted symbolically as x1(t)* x2(t)

And is defined as

Note that the convolution operator is linear, i.e. it obeys the principle of superposition.

dtxxtxtx )()()()( 2121

Properties of Convolution (1)

Commutative Property (order of operands does not matter):

Associative Property (order of operator does not matter):

Distributive Property:


Shift Property: if

then

also

Impulse Property Convolution of a function x(t) with a unit impulse results in the function x(t).


Width Property: Duration of x1(t)=T1 , and duration of x2(t)=T2 , then duration of x1(t)*x2(t)=T1 + T2

Causality Property:

ECE 2221 SIGNALS AND SYSTEMS - International …staff.iium.edu.my/sigit/ece2221/pdf/02 Continuous...

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