THE AUSTRALIAN NATIONAL UNIVERSITY Second Semester … · 2002-08-07 · Models help us cope with complexity. The abstractions of Signals and Systems are general princi-ples for building

THE AUSTRALIAN NATIONAL UNIVERSITY

Second Semester Subject 1999

ENGN2223

Signals and Systems

Lecture #1 ENGN2223 (Signals and Systems) Page 1

Lecturer: Bob Williamson, Room 130, Engineering Build-ing, Phone: x0079, email: bob.williamson@anu

Office Hours: To be negotiated

Textbook: Alan V. Oppenheim and Alan S. Willsky, Sig-nals and Systems(2nd Edition) Prentice-Hall, 1997.

An essential companion book utilized in the course is:John R. Buck, Michael M. Daniel and Andrew C. Singer,Computer Explorations in Signals and Systems usingMATLAB, Prentice-Hall, 1997.

Web Presence:http://spigot.anu.edu.au/

˜williams/teaching/engn2223/engn2223.shtml

Assessment: See handout. (Will be discussed)

Laboratories: There will be 9 computer laboratories in thisunit. MATLAB will be used. See the resources forlearning MATLAB on the Web page.

Students with disabilities: If there are any students withdisabilities in this class who wish to talk to me abouttheir studies in relation to their disability, please makean appointment. The university tries to assist such stu-dents as far as possible, but also respects your privacy.


Some Engineering Problems

How do you design and understand a:

� Broadcast Radio System e.g. ABC-FM

� An Engine Management System

� Hot Water Control System

� Variability of Foreign Currency Exchange Rates

� Aircraft Guidance System

� Telephone Answering Machine

� Blood Pressure Monitor

� Traffic Flow Controllers

� Retrofit some Environmental Cleanup technologies onAging Industrial Plants

� Restore old Audio Recordings

� Development Impacts on Ecological Systems

� Theoretically conceptualise the Organization of LargeCorporations


Yes well : : :

All of these tasks are complicated; some extremely so. Howcan we cope?

Models help us cope with complexity.

The abstractions of Signals and Systems are general princi-ples for building models.

There are “non-systems” models:

Physical thermodynamic models of gas behaviour.

A molecular model to predict fatigue failure of a certainmetal.

Unsurprisingly, for systems engineering tasks, systems basedmodels are useful:

The temperature and pressure of a gas in a vessel governedby some differential equations, with certain exogenous con-trols as input signals.

A dynamical models for an aircraft wing.

ENGN2223 concentrates on some analysis techniques forcertain sorts of Systems, which are very widely used asmodels in Systems Engineering.


What are Signals?

Examples:

� the variation of air pressure caused by speaking

� the current flowing in one loop of an R-L-C network

� US-Australian Dollar Exchange Rate

� output of a microcontroller

are all signals

The idea of a signals is an old one (optical telegraphs over200 years old). Smoke signals older still. These are akin towriting.

The notions of signals covered in this course mainly arosefrom applications such as telephony. Here one wants to rep-resent a human voice.

For the sake of developing a mathematical theory, signalsare mathematical functions.


Voice Signals

The fact that speech can be represented as an undulatingfunction of time (of sound pressure) was one of the key stepsin Bell’s development of the telephone.

“At the suggestion of Clarence Black, Bell built a phonau-tograph using an actual human ear; sound waves cominginto the eardrum were traced on smoked glass by means ofa bristle brush depending on movements from the ossicles.

Once again, as with previous phonautographs, it was borneupon him that the sum or resultant of complex sound vi-brations could be conveyed through a single point and ex-pressed as an irregular wavy line. In this case, he wasstruck further by the way sound waves acting on a tiny mem-brane could move relatively heavy bones”

[Michael E. Gorman, Alexander Graham Bell’s Path to the telephone,

http://jefferson.village.virginia.edu/albell/albell.html]


A Phonautograph

Figure 1: One of Bell’s Phonautographs


Bell’s Harp

“In the summer of 1875, Bell sketched a device consistingof a series of steel reeds over a single electromagnet. Likethe strings of a piano, these reeds would reproduce musicaltones. When one spoke a vowel into the transmitting harp,Bell visualized how a combination of reeds representing thefundamental tone and its overtones would vibrate and thisexact combination would be transmitted to the other side, re-producing the vowel sound. This principle had been clearlyestablished by the Helmholtz device which was Bell’s origi-nal mental model; in this case, however, the single interrupt-ing fork and series of separate resonators were replaced bya series of reeds combining to induce a current in a singleelectromagnet.

“Bell knew he could never build such a device, owing inpart to the multiplicity of reeds that would be required, butit served as a new mental model for a universal transceiver– this harp apparatus could transmit and receive speech,musical tones, or any other pattern of sounds.”

[Michael E. Gorman, Alexander Graham Bell’s Path to the telephone,

http://jefferson.village.virginia.edu/albell/albell.html]


Bell’s Harp (2)

Figure 2: The Harp


Other Types of Signal

Signals need not be one-dimensional functions of time.

Figure 3: Who is it?


What are systems?

A system transforms or generates signals.

Consider the following (and ask what are the signals):

� Tape recorder

� R-L-C network (any electrical circuit in fact)

� Coaxial cable (ethernet)

� Ecological systems (e.g. Mangrove Swamp)

� “The Economy”

� A microphone

� A loudspeaker

� Everything between the microphone in a radio stationand your ears


Some Philosophy

Models are pervasive. Who builds them?

A mother playing with her child on a swing?

You adjusting the temperature of your shower?

You deciding how to arrange your finances?

You guiding your car through a corner?

One view of the way people interact with the world is thatwe alwayshave models, usually implicit. So what we do inENGN2223 is not fundamentally new — its just the techni-cailities that are new.

Conjectural knowledge; Model Building.


More Philosophy

Key points about models:

They are almost never “true”. One never knows how truethey are.

But as for certain truth, no man has known it,Nor will he know it; neither of the gods,Nor yet of all the things of which I speak.And even if by chance he were to uterThe perfect truth, he would himself not know it:For all is but a woven web of guesses

— XENOPHANES

They are very useful conceptually, and analytically.

Most of the hard work in this subject is learning the analyti-cal techniques. But it is very important that the power of theanalysis does not blind you to the fact that all that we willbe analysing will be models of real phenomena. And ourmodels can mislead.


Signals and Systems

Input Signals Output Signals

System

Figure 4: The black-box approach to signals and systems. (Why “black”?)

This illustrates the fact that we are building models. Weoften hierarchically construct complex boxes out of simpleones. (Example: Radio system)

We want to encapsulate our knowledge about a real deviceor thing in the box.

We will do this by mathematical relationships between theinput and output signals.

More importantly for the content of this course, we makesignificant assumptions as to the types of mathematical re-lationship between the inputs and outputs.

Otherwise, every system could be a Universal Turing Ma-chine! (So?)

Two key assumptions are linearity and time-invariance —See lecture 3.


A Peek Ahead: The Fourier Transform

For the class of systems we will study in detail, a Mathemat-ical tool called the Fourier transform is extremely useful forboth analysing and conceptualising a wide range of signalsand systems.

Before that we will define and classify certain standard sig-nals, and elaborate on the properties systems can have, andthe implications that has on their mathematical characteri-zation. (Chapter 1)

Chapter 2 will consider a special class of systems (Lineartime-invariant ones) in rather more detail, and will meet thesimplest version of the Fourier transformation technique inweek 3. The Fourier transform is the mathematical tech-nique used to decompose a signal into a sum of sine waves.

Reminder:

� Reading the text

� Knowing how to use MATLAB

� Doing the Lab Exercises

� Doing lots of the recommended Problems

are the keys to learning the material in this course.

The text has many examples, most of which I will not gothrough in lectures — you are encouraged to read them!


Signals

Most signals that we shall deal with are functions of time.

n

t

A Continuous-time Signal

A Discrete-time Signal

Figure 5: Examples of continuous-time and discrete-time signals.

Notation:

In continuous-time this will be written as x(t), x : R ! R .

In discrete-time this will be written as x[n], x : Z ! R .

If the statement applies to both discrete and continuous sig-nals, then x will be used.

A “signal” is a (mathematical) function. (Our master ab-straction.)


Energy and Power

Many signals are voltages across some impedance. For areal impedance (resistance)

p(t) = v(t)i(t) = v2(t)=R

The total energy over time [t1; t2] isZ t2

t1

p(t)dt =1

R

Z t2

t1

v2(t)dt

The average power is

1

t2 � t1

Z t2

t1

p(t)dt:

Energy or power of anysignal x.

E :=

Z t2

t1

jx(t)j2dt orn2X

n=n1

jx[n]j2

E1 := limT!1

Z T

�T

jx(t)j2dt

P1 := limT!1

1

2T

Z T

�T

jx(t)j2dt

(Analogously for discrete time)


Even, and Odd Signals

A signal x(t) or x[n] is said to be EVEN if it is identicalwith its reflection about the origin:

x(t) = x(�t) or x[n] = x[�n]

The even part of a signal x(t) is given by

EV[x(t)] :=1

2[x(t) + x(�t)]

or

EV[x[n]] :=1

2[x[n] + x[�n]] :

A signal x(t) or x[n] is said to be ODD if it is identical withthe negative of its reflection about the origin:

x(t) = �x(�t) or x[n] = �x[�n]

The odd part of a signal x(t) is given by

OD[x(t)] =1

2[x(t)� x(�t)]

or

OD[x[n]] =1

2[x[n]� x[�n]] :

Why am I telling you this?


Periodic Signals

A signal x is said to be PERIODIC with period T (or N ) if

x(t) = x(t + T ) 8t or x[n] = x[n +N ] 8n: (1)

The fundamental period T0 (or N0) is the smallest positivevalue of T (or N ) for which (1) holds.

� If x is a constant, in which case the period is undefined.

� In discrete time, sin(!0n+�) is not necessarily periodic.(Why?)


Signals: Complex Exponentials (1)

These are a very fundamental type of signal. (Why?)

Read pages 15–30.

Continuous-time

x(t) = Ceat

If

a = 0 then x is a constant.

C 2 R and a 2 R then x is a real exponential.

a 62 R and <fag = 0 then x is a complex exponential.

a 62 R and <fag > 0 then x is an undamped complex ex-ponential.

a 62 R and <fag < 0 then x is a damped complex exponen-tial.

Most of the material here should be very familiar. For ex-ample:

ej!0t = cos!0t + j sin!0t

Hence

A cos(!0t + �) = A2ej�ej!0t + A

2e�j�

e�j!0t

A cos(!0t + �) = A<fej(!0t+�)g

A sin(!0t + �) = A=fej(!0t+�)g

Here !0 is the fundamental frequency.



Discrete-time

x[n] = C�n

If

� = 1 then x is a constant.

C 2 R and � 2 R and

1. � > 1 then x is a growing exponential.

2. 0 < � < 1 then x is a decaying exponential.

3. �1 < � < 0 then x is oscillatory and exponentiallydecaying.

4. � < �1 then x is an oscillatory, growing exponen-tial.

C 2 C and � 2 C and if

C = jCjej�0 and � = j�jej0

then

C�n = jCjj�jn cos(0n + �0) + jjCjj�jn sin(0n + �0)

where j :=p�1.



In discrete-time, values of frequency (0 in the above equa-tion) need only be considered to be in the range

�� 0 < �:

Consider

ej(0+2�k)n = e

j2�knej0n = e

j0n

so that the frequencies 0 and 0 + 2�k are indistinguish-able.

This will lead to the practical problem of aliasingwhen wediscuss sampling later on.


Signals: Unit Step and Impulse (1)

Continuous-time The unit step is defined as

u(t) :=

�0; t < 0

1; t > 0

which is discontinuous at t = 0.

The unit impulse Æ(t) is defined implicitly as

u(t) =

Z t

�1

Æ(� )d�:

Unit ImpulsesUnit Steps

Figure 6: Unit step and impulses in discrete and continuous time.


Signals: Unit Step and Impulse (2)

There is a mathematical difficulty with defining the unit im-pulse explicitly. It is usually defined implicitly by its sam-pling property: Z +1

�1

x(� )Æ(� )d� = x(0)

or Z +1

�1

x(� )Æ(t� � )d� = x(t):

Can develop as derivative of unit step: See pages 32–36.

Also see Question 1.38 of the text.

Discrete-time

In discrete-time, the unit step and impulse are more straight-forward:

u[n] :=

�0; n < 0

1; n � 0

and

Æ[n] :=

�0; n 6= 0

1; n = 0:

Note that the “running integral” relationship exists betweenthese two as well:

u[n] =

nXm=�1

Æ[m]


Discrete-time Periodicity

In order for a signal ej0n to be periodic with period N > 0,we must have

ej0(n+N) = e

j0n

or

ej0N = 1:

so that 0N = 2�m with m 2 Z.

Thus, for ej0n to be periodic with period N , 0=2� must berational, i.e.

0

2�=m

N

for some m 2 Z.


SUMMARY: LECTURE 3

You should understand the following concepts:

� discrete-time and continuous-time signals

� even and odd signals

� periodic signals

� the fundamental period of a periodic signal

� unit step, unit impulse ; their properties

� complex exponential signals

� discrete-time frequencies are between �� and �

� discrete-time periodicity.


Systems

Remember a system is a model. Recall some of our exam-ples:

� R-L-C network (any electrical circuit in fact)

� A microphone

� A loudspeaker itself

� Everything between the microphone in a radio stationand your ears

Think about what the associated signals are.

See the examples on pages 39–40 of the text.

Interconnection of systems: one of the key advantages ofthe system as a model building principle is that it admits anice interconnectability. See pages 41–43 of the text.


System Properties

It is extremely helpful to classify systems in terms of for-mal properties they have. Some key properties we will nowdiscuss are:

� Memory

� Invertibility

� Causality

� Stability

� Time-invariance

� Linearity.

These are useful as they can greatly simplify analysis.

Sometimes they are assumed even when it is knownthey arefalse!


Memory

A system is said to be MEMORYLESS if its output valuedepends only on the current input value.

Example: The input-output relationship for a resistor is

v(t) = i(t)R

where i is the input (current), v is the output (voltage) andR is the resistance.

This system is memoryless.

Example: The input-output relationship for a capacitor is

v(t) =1

C

Z t

�1

i(� )d�

where i is the input (current), v is the output (voltage) andC is the system (effect of capacitance).

This system is not memoryless.


Invertibility

A system is said to be INVERTIBLE if distinct inputs leadto distinct outputs. That is, the input can be recovered uniquelyfrom the output (and knowledge of the system).

Example: The system with input x and output y

y[n] =

nXk=�1

x[k]

is invertible and the inverse system is

z[n] = y[n]� y[n� 1]:

Example: The systems with input x and output y

y[n] = 0

y[n] = (x[n])2

are not invertible.


Causality

A system is said to be CAUSAL if the output only dependson values of the input at the present time and in the past.That is, the output does not depend on future inputs.

For real-time applications, causality must be adhered to. (True?)

In some applications (like image processing), the indepen-dent variable axis is not time, so causality is not of majorimportance.


y[n] = x[n +N ]

with N > 0 is not causal.


Stability

A system is said to be BOUNDED-INPUT / BOUNDED-OUTPUT (BIBO) STABLE if any bounded input leads to abounded output. (The text just says “STABLE”.)


y(t) =

Z t

�1

x(� )d�

is not BIBO stable. (Why?)


y[n] =

+MXm=�M

x[n +m]

is BIBO stable.


Time-Invariance

A system is said to be TIME-INVARIANT if a shift in timeof the input signal causes a time shift in the output signal.Time-invariance is sometimes called shift-invariance.

Specifically, if y[n] is the output of a time-invariant systemwhen x[n] is the input, y[n � n0] will be the output whenx[n� n0] is the input.

Example: Is the system

y(t) = sin(x(t))

time-invariant?

To check, set

y1(t) = sin(x1(t))

then set x2(t) = x1(t� t0) so that

y2(t) = sin(x2(t)) = sin(x1(t� t0)):

Comparing y2(t) with y1(t� t0) we see that

y1(t� t0) = sin(x1(t� t0))

so that the system is time-invariant.


Linearity (1)

(The Reductionist’s dream. (Why?))

Let y1 be the system output to x1 and y2 the system outputto x2. The system is said to be LINEAR if the additivity andhomogeneity properties hold.

These sub-properties are:

ADDITIVITY: The response to x1 + x2 is y1 + y2.

HOMOGENEITY: the response to ax1 is ay1 where a is anarbitrary constant.

Additivity and homogeneity can be combined to give thePRINCIPLE OF SUPERPOSITION: with input ax1+bx2

the system response (output) is ay1 + by2 for arbitrary con-stants a and b.


Linearity (2)

Example: The system

y[n] = 2x[n] + 3

is not linear.

Consider:

y1[n] = 2x1[n] + 3

y2[n] = 2x2[n] + 3

y3[n] = 2 (x1[n] + x2[n]) + 3 = 2x1[n] + 2x2[n] + 3:

But

y1[n] + y2[n] = 2x1[n] + 2x2[n] + 6

so that the superposition principle does not hold.

The system

y[n] = 2x[n] + 3

is called an INCREMENTALLY LINEAR system becauseit can be written as a linear system, y 0[n] = 2x[n], plus anoffset, 3.

(Mathematicians call this AFFINE.)


Examples

Example: What properties does the system

y[n] = Ax[n� 1] +Bx[n� 2]

possess?

The system has memory.

The system is not invertible in general.

The system is causal.

The system is stable (for finite A and B).

The system is time-invariant.

The system is linear.


SUMMARY: LECTURE 4

To date, we have considered

� Signals:

– Even, Odd, Periodic, Fundamental Period.

– Continuous-time: exponentials, unit step, unit im-pulse (Dirac delta).

– Discrete-time: exponentials, unit step, unit impulse(Kronecker delta).

– Frequency range of discrete-time sinusoids.

– Discrete-time sinusoids may not be periodic.

� Systems

– System properties: memory, invertibility, causality,stability, time-invariance and linearity.

– Sub-properties of homogeneity and additivity.

– Incrementally linear systems.

End of Chapter 1 Lectures

Start of Chapter 1 Exercises!


Chapter 2Linear Time-Invariant Systems

Systems which possess the properties of linearity and time-invariance are an important class of systems because theyare easily characterised, as we shall see shortly, by their im-pulse response.

They arise in many situations: for example any RLC circuit.

They are also very good and useful approximationsof manysystems. For example small signal models of mechanicalapparatus (car suspension), or radio transmission channels.

Linear, time-invariant system responses to the Kronecker(Æ[n]) and Dirac (Æ(t)) delta functions (unit impulse func-tions) play an important role because of the sampling orsifting property:

Discrete: x[n] =+1X

k=�1

x[k]Æ[n� k]

Continuous: x(t) =Z +1

�1

x(� )Æ(t� � )d�


Discrete : The Convolution Sum

Consider what happens when the signal

x[n] =

+1Xk=�1

x[k]Æ[n� k] (2)

is input to a linear system.

Superposition means

ax1[n] + bx2[n]! ay1[n] + by2[n]:

This can be used to define the output y[n] in terms of a linearcombination of weighted impulses as in (2).

Suppose that hk[n] is the system response to a unit impulseat time k, Æ[n� k]. Then the system output is

y[n] =

+1Xk=�1

x[k]hk[n]:

If we also suppose our linear system to be time-invariant,then

hk[n] = h0[n� k]

where h0[n] is the response of the system to Æ[n].

Our output is then given by

y[n] =

+1Xk=�1

x[k]h0[n� k] =: x[n] � h[n]

which is known as the convolution sum and h0[n] is knownas the system impulse response.


Example

Consider the calculation of the convolution of

h[n] =

8>>><>>>:

1; n = 0

0:5; n = 1

0:25; n = 2

0; otherwise

with the unit step, u[n].

−20 −15 −10 −5 0 5 10 15 20−1

01

The System Impulse Response Time−Reversed, h[−n]

−20 −15 −10 −5 0 5 10 15 20−1

01

The System Impulse Response, h[n]

−20 −15 −10 −5 0 5 10 15 20−1

01

The Unit Step, u[n]

−20 −15 −10 −5 0 5 10 15 20−1

01

Sample Number−−−>

The System Step Response, y[n] = h[n]*u[n]

Figure 7: Discrete Convolution Example.


Continuous : The Convolution Integral

In a very similar way to the convolution sum derivation,starting with

x(t) =

Z +1

�1

x(� )Æ(t� � )d�

being input to a linear system we can obtain the output

y(t) =

Z +1

�1

x(� )h�(t)d�:

If the system is then assumed to be time-invariant we arriveat the convolution integral:

y(t) =

Z +1

�1

x(� )h0(t� � )d� =: x(t) � h0(t)

where h0(t) is the system response to Æ(t). (And we haveused h�(t) = h0(t� � ).)

The convolution operation in both discrete andcontinuous time is:

COMMUTATIVE: x � h = h � x

ASSOCIATIVE: x � (h1 � h2) = (x � h1) � h2

DISTRIBUTIVE: x � (h1 + h2) = x � h1 + x � h2

Read pages 104–108. See problem 2.71


Example

Consider the convolution of

x(t) =

�1; 0 < t < T

0; otherwise

and

h(t) =

�t; 0 < t < 2T

0; otherwise

s

s s

x(s)

s

s

s

s

s

s

s

s

h(t-s)

0<t<T

T<t<2T

2T<t<3T

t>3T

t<0

h(t-s)

h(t-s)

h(t-s)

h(t-s)

x(s)h(t-s)

x(s)h(t-s)

x(s)h(t-s)

x(s)h(t-s)

x(s)h(t-s)

t

y(t)

Figure 8: Continuous Convolution Example.

See Example 2.7, pp.99–102 of the text.


SUMMARY: LECTURE 5

Lecture 5 introduced linear time-invariant systems.

� We considered the sampling or sifting property of deltafunctions in both discrete and continuous time.

� And thus derived the convolution sum and its graphicalinterpretation.

� Along with the convolution integral and its graphicalinterpretation.

� The impulse response of a linear, time-invariant systemwas seen to characterizesuch systems.

The significance of the convolution representation of LTIsystems is that it is a relatively simple way to determine theresponse of the system to any input. This is of enormouspractical importance.

Compare with general non-LTI systems — have to solve adifferentDE from scratch each time in general.

Fourier transform techniques exploit the convolution repre-sentation, thus providing extremely useful analytical meth-ods and intuition.


Properties of LTI Systems

We previously defined six system properties which I claimedwe should be interested in. What can one say about the re-maining properties when linearity and time-invariance areassumed?

As we have seen, LTI systems are completely characterisedby the system’s impulse response (and initial conditions).[That does not hold in general for non-LTI systems: seeexample 2.9, page 103.]

Therefore, the system properties of

� Memorylessness

� Invertibility

� Causality

� Stability

must be ascertainable from knowledge of the impulse re-sponse.

So let us have an ascertaining lecture...


Memorylessness

A memoryless system is one that only depends on the inputat the present time. Thus

h[n] = KÆ[n]

substituting this into the convolution sum yields

y[n] =

+1Xk=�1

x[k]h[n� k]

=

+1Xk=�1

x[k]KÆ[n� k]

= Kx[n]:

Similarly, for continuous-time memoryless systems,

h(t) = KÆ(t):

This illustrates that memorylessness is indeed a strong as-sumption!

Thus memoryless LTI systems are easy to checkfor: the impulse response must be a simple scal-ing of the appropriate unit impulse.


Invertibility

x y z

H Hinv

Figure 9: A system and its inverse.

For invertibility, we require x = z. Or

z = x � h � hinv = x

that is

h � hinv = Æ (3)

where Æ is the appropriate unit impulse.

See example 2.12 (page 111).

What are applications where invertibility would be nice?

Construct an example of an LTI system that is not invertible.

Invertibility or lack of it has many practical implications(telecommunication channels; control system design).

Practically, one often does not try to invert a system “ex-actly.”

The system is invertible if there exists a functionhinv which satisfies (3).


Causality

Causality means that the output of a system y[n] must notdepend on x[k] for k > n.

A natural assumption for many real-time systems.

From the convolution sum:

y[n] =

+1Xk=�1

h[k]x[n� k]

this means that we require h[n] = 0 for n < 0.

Thus, for causal systems, we can write

y[n] =

+1Xk=0

h[k]x[n� k] =

nXk=�1

x[k]h[n� k]

or (with h(t) = 0 for t < 0),

y(t) =

Z t

�1

x(� )h(t� � )d� =

Z +1

0

h(� )x(t� � )d�:


Stability

A system is said to be BIBO stable if every bounded inputcauses a bounded output. (The text just says “stable”.)

Consider a bounded input

jx[n]j � Bx <1 for all n

and suppose it is applied to a system with impulse responseh[n]:

jy[n]j =

��+1X

k=�1

h[k]x[n� k]

��

+1Xk=�1

jh[k]j jx[n� k]j

� Bx

+1Xk=�1

jh[k]j

Thus if h[�] is absolutely summable, then the system is BIBOstable. In fact “the impulse response of the system is abso-lutely summable” if and only if “the LTI system is BIBO sta-ble” (although stable) absolutely summablehas not beenproved here); see problem 2.49.


Unit Step Response

The unit step response s[n] or s(t) is the output of a systemwhen a unit step u[n] or u(t) is applied.

Key idea:

s[n] = u[n] � h[n]

Thus

s[n] =

nXk=�1

h[k]

Hence

h[n] = s[n]� s[n� 1]

In continuous time

s(t) = u(t) � h(t)

And

s(t) =

Z t

�1

h(� )d�

So

h(t) =ds(t)

dt

= s0(t)


SUMMARY: LECTURE 6

This lecture examined how other system properties are char-acterised if it is known that the system is linear and time-invariant:

MEMORYLESSNESS: Memoryless LTI systems are easyto check for: the impulse response must be a simplescaling of the appropriate unit impulse.

INVERTIBILITY: The system is invertible if there exists afunction hinv which satisfies

h � hinv = Æ:

CAUSALITY: For an LTI system to be causal we requireh[n] = 0 for n < 0.

STABILITY: Thus the output y[n] is bounded (i.e. the sys-tem is BIBO stable) provided

+1Xk=�1

jh[k]j <1:

UNIT STEP RESPONSE Can be related to unit impulseresponse.

s[n] =

nXk=�1

h[k] or s(t) =

Z t

�1

h(� )d�


Causal LTI Systems, Described byDifferential and Difference Equations

Consider

dy(t)

dt

+ 2y(t) = x(t)

where x(t) and y(t) are the input and output respectively ofa LTI system. Note this is an implicit specification of theoutput of the system.

More generally consider the equation

NXk=0

ak

dky(t)

dtk

=

MXk=0

bk

dkx(t)

dtk:

This describes the input-output relationship for an impor-tant sub-class of the linear, time-invariant continuous-timesystems.

These are often described as lumped parameter or finite di-mensional linear systems. Transmission lines and continu-ous membranes are not (“exactly”) modelled by such equa-tions.

The ak and the bk are constants and this sort of equation iscalled a linear, constant coefficient differential equation.


Solving LCCDEs

Suppose we know the input x to the system

dy(t)

dt

+ 2y(t) = x(t)

and we wish to find the output y.

These equations are explicitly solved by finding a y com-prised of

y = yh + yp

where yp is a particular solution of the equation and yh isthe associated homogeneous solution; the solution of theunforced equation

dy(t)

dt

+ 2y(t) = 0

(i) Homogeneous Solution

Generally, the homogeneous solutions will be of exponen-tial form. In this first order case

yh(t) = Ae�2t

where A is an arbitrary constant.


Solving LCCDEs (cont)

(ii) Particular Solution

If x(t) is of the form

x(t) = [K cos(!0t)]u(t) = <fKej!0tg

then the particular solution will be of the form

yp(t) = <fY ej!0tg

where Y 2 C .

In our case,

yp(t) =Kp4 + !

20

cos(!0t� �)u(t)

where � = arctan(!02).

(iii) Combining Solutions

The solution is then

y(t) = yp(t) + yh(t) = Ae�2t +

Kp4 + !

20

cos(!0t� �)

In order to completely specify y(t) some auxiliary condi-tions are required. For example, y(0) = y0 means

A = y0 �K cos(�)p

4 + !20


Auxiliary Conditions and Linearity

Suppose, in the above example, that K = 0 so that the zero-input response is

y(t) = y0e�2t

:

In this case, the system is not linear.

The system is only linear if the auxiliary condition is zero,i.e.

y0 = 0

With non-zero auxiliary conditions, the system is incremen-tally linear.

Linear System

x(t) y(t)

system with zero auxilliary conditions.

response to auxilliary conditions

Figure 10: An incrementally linear system.

In general there will be a set of auxiliary conditions — thevalues of

y(t);dy(t)

dt

;

d2y(t)

dt2; : : : ;

dN�1

y(t)

dtN�1

at some specific point in time, say t = t0.


Difference Equations

Most of the discrete-time systems (LTI) with which we shallbe dealing may be described in the following general differ-ence equation form:

NXk=0

aky[n� k] =

MXk=0

bkx[n� k]

where y is the output and x is the input and the ak and bk areconstant coefficients.

Because the ak and bk are constant, and because of the sim-ple form in which the input and output occur, this sort ofequation is called a linear, constant coefficient differenceequation.


Recursive and Non-recursive

Systems which satisfy

NXk=0

aky[n� k] =

MXk=0

bkx[n� k]

for which at least one of the ak (for k > 0) is non-zero aregenerally recursive systems — that is, the current outputdepends on past (or future) values of the output.

Another term for recursive systems is infinite (duration)impulse response (IIR) systems.

Systems which may be written as

y[n] =1

a0

MXk=0

bkx[n� k]

are called non-recursive.

Because the impulse response of a non-recursive system is

h[n] =

8<:

bn

a0

; 0 � n �M

0; otherwise

such systems are also called finite (duration) impulse re-sponse systems (FIR) systems.


Block Diagram Representations

A very common and useful notation for LCCDEs is that ofthe block diagram. The basic elements are shown below.

b

x1[n]

x2[n]

x1[n]+x2[n]+

x[n] a ax[n]

Dx[n] x[n-1]

+x[n] y[n]

D

-ay[n-1]

Figure 11: Elementary block diagram components (discrete time)

The bottom figure illustrates the LCCDE

y[n] + ay[n� 1] = bx[n]

which we have rewritten as follows

y[n] = �ay[n� 1] + bx[n]


Continuous Time Block Diagrams

In continuous time, we have similar elemental blocks exceptthe delay becomes a differentiation:

Dx(t)

dx(t)____dt

+x(t)

b/a

y(t)

D

dy(t)

dt___

-1/a

Figure 12: Elementary block diagram components (continuous time)

In order to implement the continuous time LCCDE

dy(t)

dt

+ ay(t) = bx(t)

we first rewrite it as

y(t) = �1

a

dy(t)

dt

+b

a

x(t)

This is illustrated in the lower figure.

There are disadvantages in this form of implementation: dif-ferentiators, whilst buildable, are very prone to noise (Why?).


Implementation of LCCDEs using Integrators

Instead, the differential equation

dy(t)

dt

+ ay(t) = bx(t)

can be rewritten as

dy(t)

dt

= bx(t)� ay(t)

and then integrated from �1 to t. If we assume the systemis initially at rest (y(�1) = 0), then

R t

�1

dy(t)

dtdt = y(t) and

we obtain

y(t) =

Z t

�1

[bx(� )� ay(� )]d�:

This can be implemented as shown below:

x(t) x(s)ds

+x(t)

-

t

infinity

y(t)

-a

b

Figure 13: Implementation of DE using an Integrator


Singularity Functions

Read section 2.5 of the text (pages 127–136)

Key points to pick up are:

� The continuous time unit impulse can be considered asan idealised short pulse;

� There are implicit definitions:

g(0) =

Z1

�1

g(� )Æ(� )d�

or Æ(t) is the signal such that

x(t) = x(t) � Æ(t):

� There are “higher-order” impulses (doublets etc) whichare the derivatives of Æ(t).


SUMMARY: LECTURE 7

Topics covered or introduced here:

� difference and differential equations

� recursive, non-recursive, IIR and FIR systems

� solving LCCDEs, particular solutions, homogeneous so-lutions

� auxiliary conditions and linearity

� block diagram notation and implementation of LCCDEs

� singularity functions

End of Chapter 2

You should have started on the text and MATLAB exer-cises for Chapter 2 by now. See the Web page for recom-mendations.


Chapter 3Fourier Series Representation of

Periodic Signals

We now meet the first of the Fourier ideas in the course. Ifyou follow everything covered in this lecture, and can dothe relevant problems, you should have no problems withthe rest of the course!

Read section 3.1 It provides a nice history and motivation.


Response of LTI Systems to Exponentials

This is important. It is the motivation for the use of Fourierseries as opposed to any old orthogonal series.

In studying LTI systems, we should exploitthe linearity prop-erty.

Idea: Represent general signals as linear combinations ofbasic signals where

1. The set of basic signals can be used to construct a broadclass of signals;

2. The response of LTI sytstems to one of the basic signalsis of a simple form, and hence the response to a generalsignal can be easily understood (by exploiting linearity).

It turns out that complex exponentials are the “right” basicsignals to use for analysing LTI systems.


Complex Exponentials and LTI Systems

In continuous time, consider

x(t) = est:

Let h(t) be the impulse response of an LTI system. Theoutput is

y(t) =

Z1

�1

h(� )x(t� � )d�

=

Z1

�1

h(� )es(t��)d�:

But es(t��) = este�s� so

y(t) = est

Z1

�1

h(� )e�s�d�

which we will write as

y(t) = H(s)est

where

H(s) :=

Z1

�1

h(� )e�s�d�

and we assume the integral exists.

[Likewise for discrete time: see pages 183–184.]

Hence if

x(t) =

nXi=1

aiesit

and y(t) = h(t) � x(t), then

y(t) =

nXi=1

aiH(si)esit


Fourier Series Representationof Continuous Time Periodic Signals

Recall that x is said to be periodic with period T if thereexists a T > 0 such that

x(t) = x(t + T ); 8t:

The functions

�k(t) = ejk!0t

; k 2 Z

are periodic with period T = 2�!0

.

(Here we are using s = jk!0.)

The functions

ej!0t

; e�j!0t are called the fundamental components

ej2!0t

; e�j2!0t are called the second harmonics (of ej!0t),

and

ej3!0t

; e�j3!0t are called the third harmonics.

The fundamental period of each of the �k is 2�k!0

= Tk

.

Because each of the �k is also periodic with period T , anysum of �ks will also be periodic with the same period.


Fourier Series (continued)

The representation

x(t) =

+1Xk=�1

ak�k(t) =

+1Xk=�1

akejk!0t

is called a Fourier series representation of the periodic sig-nal x.

Examples (see pages 187–188) show a wide variety of peri-odic waveforms can be generated in this manner.

Suppose x(t) is real. (When isn’t it?) Then can show ak =

a�

�k (see next lecture).

Let ak = Bk + jCk (where Bk and Ck are real). Then onecan show (see page 189) that

x(t) = a0 + 2

1Xk=1

[Bk cos k!0t� Ck sin k!0t]

This is known as the sin-cosine Fourier series.


Finding the ak Coefficients

Suppose we are given an x which is periodic with period T .How can we determine the coefficients fakg of its Fourierseries representation? Indeed, do such fakg exist for all pe-riodic x(t)?

Suppose

x(t) =

1Xk=�1

akejk!0t

:

Multiply both sides by e�jn!0t:

x(t)e�jn!0t =

1Xk=�1

akejk!0t

e�jn!0t

and integrate from 0 to T = 2�=!0:Z T

0

x(t)e�jn!0t =

Z T

0

1Xk=�1

akejk!0t

e�jn!0t

dt

=

1Xk=�1

ak

�Z T

0

ej(k�n)!0t

dt

�(4)

Now

I :=

Z T

0

ej(k�n)!0t

dt =

Z T

0

cos[(k � n)!0t]dt + j

Z T

0

sin[(k � n)!0t]dt:

For k 6= n, both of these integrals are zero (geometricallyobvious: integrating a whole number of periods). And fork = n, I = T . ThusZ T

0

ej(k�n)!0t

dt =

�T; k = n

0; k 6= n

Therefore (4) becomes = Tan.


Finding the ak Coefficients (continued)

Thus we obtain

an =1

T

Z T

0

x(t)e�jn!0tdt:

In fact, we can integrate over anyinterval of length T . (Seetext.) We write

RT

to denote such integration.

We arrive finally at the pair of equations:

x(t) =

1Xk=�1

akejk!0t

(!0 = 2�=T )

ak =1

T

ZT

x(t)e�jk!0tdt

called the synthesis and analysis equations respectively.

Note a0 = 1T

RTx(t)dt, the average value of x(t).

Note that in determining the analysis equation we have shownthat �k(t) are an orthonormal basison [0; T ].

Study the examples on pages 192–195.


Dirichlet Conditions

Suppose we wish to approximate a periodic x(t) with

xN(t) =

+NXk=�N

akejk!0t

:

Consider the error criterion

EN =

ZT

jx(t)� xN(t)j2 dt

If ZT

jx(t)j2dt <1

then

limN!1

EN = 0:

A stronger requirement is that the Fourier series convergesin the sense that

ak =1

T

ZT

x(t)e�jk!0tdt <1

and+1X

k=�1

akejk!0t = x(t);

for all t except possibly for a finite number (per period) ofdiscontinuities ofx.


Dirichlet Conditions (cont)

Dirichlet formulated the following conditions on x whichensure that the Fourier series converges in this stronger sense.

(i)ZT

jx(t)jdt <1

(ii) In any finite time interval, x(t) is of bounded variation(there are a finite number of minima or maxima duringany single period of x).

(iii) In any finite time interval, there are only a finite numberof discontinuities in x.

See the examples on pages 198–199.


Gibbs Phenomenon

Consider the periodic square-wave x(t) with period T0 andwidth 2T1, where 2T1 < T0.

1

+T-T-T +T0 01 1

+T0

Figure 14: Periodic square wave.

The Fourier series coefficients ak are

a0 =1

T0

Z T0=2

�T0=2

x(t)dt =1

T0

Z +T1

�T1

dt =2T1T0

and for k 6= 0

ak =1

T0

Z T0=2

�T0=2

x(t)e�jk!0tdt (!0 =2�

T

)

=1

T0

Z +T1

�T1

e�jk!0t

dt =�1

jk!0T0

e�jk!0t

��+T1

�T1

=sin k!0T1

k�

(!0 =2�

T0

)

Note how we chose the interval of integration (�T0=2; T0=2)

in order to make the evaluation of the integral easy. (Wecould have chosen any interval of length T0.)


Gibbs Phenomenon (continued)

The figure shows the result of truncating the Fourier seriesof the periodic square wave.

0 20 40 60 80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1The building of a square wave: Gibbs’ effect

Figure 15: Example of Gibb’s phenomena.

For those familiar with Lp norms, we have

limN!1

kx� xNk2 = 0

but

limN!1

kx� xNk1 6= 0


SUMMARY: Lecture 8

We have

� Motivated the use of sums of complex exponentials instudying LTI systems;

� Shown how to determine the coefficients of such repre-sentations (which are called Fourier series);

� Briefly considered the convergence of the partial sumsof the series.

You should study the examples in the relevant parts ofthe text very closely.

The Fourier series and transform are the key tools developedin this course. It is vital you understand them now, as wewill be building on these foundations.


Properties of Continuous-Time Fourier Series

Fourier series have lots of properties which provide muchinsight and help us solve problems.

We will use the notation

x(t)FS ! ak

To indicate that the periodic signal x(t) with period T andfundamental frequency !0 = 2�=T has a Fourier series withcoefficients ak.

Table 3.1, p. 206 summarizes the properties. They all fol-low from the more general Fourier transform developed inChapter 4, and thus we will not spend a lot of time derivingthem.

The most important thing about these properties is theiruse: see the examples on pages 206–211.


Linearity, Time Shifting and Time Reversal

Suppose x(t) and y(t) are periodic with period T . If

x(t)FS ! ak

y(t)FS ! bk

then

z(t) := Ax(t) +By(t)FS ! ck := Aak +Bbk

If y(t) = x(t� t0) then

bk =1

T

ZT

x(t� t0)e�jk!0t

dt

which after a change of variable can be shown to equal to

e�jk(2�=T )t0

ak. Thus if x(t)FS ! ak then

x(t� t0)FS ! e

�jk!0t0ak = e

�jk(2�=T )t0ak

A simple reindexing of the synthesis equation (see page 203)

shows that if x(t)FS ! ak then

x(�t) FS ! a�k


Time Scaling, and Multiplication

Suppose x(t) is periodic with period T .

For some � > 0 consider x(�t). This will be periodic withperiod T=�, and fundamental frequency �!0.

If x(t)FS ! ak then

x(�t) =

1Xk=�1

akejk(�!0)t

Thus

x(�t)FS ! ak but with period T=�

If

x(t)FS ! ak

y(t)FS ! bk

then

x(t)y(t)FS ! hk =

1Xl=�1

albk�l

This is the discrete convolution of the coefficients ak and bk.

See later (section 4.4) where this relationship will be care-fully derived from the more general Fourier transform.


Conjugate Symmetry and Parseval’s Relation

If

x(t)FS ! ak

then

x�(t)

FS ! a�

�k

(Follows from definition readily.)

Observe that if x(t) is real, then x�(t) = x(t) and so

a�k = a�

k

Hence

ja�kj = jakj

Problem 3.46 asks you to show that Parseval’s relationshipholds:

1

T

ZT

jx(t)j2dt =1X

k=�1

jakj2

Observing too that

1

T

ZT

��ake

jk!0t��2dt =

1

T

ZT

jakj2dt = jakj2

we conclude that the total average power of a periodic sig-nal is equal to the sum of the average powers of all of itsharmonic components.

This is extremely useful.

Now look at some examples from pages 208–211


Fourier Series Representation ofDiscrete-Time Periodic Signals

Suppose x is periodic with period N :

x[n] = x[n +N ] for all n:

With !0 = 2�=N , have the set of discrete time complexexponentials:

�k[n] = ejk!0n = e

jk(2�=N)n; k = 0;�1;�2; : : :

Note that for any integer r

�k[n] = �k+rN [n] for all n:

Thus if we want a representation of the form

x[n] =Xk

ak�k[n]

it only makes sense to sum over a range of N successivevalues of k, for example k = 0; 1; : : : ; N � 1. We indicatethis by the notation k = hNi:

x[n] =Xk=hNi

ak�k[n]

This is referred to as the discrete time Fourier Series


Determination of Fourier SeriesRepresentation of a Periodic Signal

Using the fact (problem 3.54) that

Xn=hNi

ejk(2�=N)n =

�N; k = 0;�N;�2N; : : :0; otherwise

Can show (see page 213) that if x[n] is an N -periodic signal,then its Fourier series coefficients are given by

ar =1

N

Xn=hNi

x[n]e�jr(2�=N)n

and hence we have the discrete-time Fourier series pair:

x[n] =Xk=hNi

akejk!0n =

Xk=hNi

akejk(2�=N)n

ak =1

N

Xn=hNi

x[n]e�jk!0n =1

N

Xn=hNi

x[n]e�jk(2�=N)n

Can check that ak = ak+N .

See examples on pages 214–221.


Properties of Discrete-Time Fourier Series

Very similar to continuous time Fourier series. See table onpage 221.

Some differences:

If

x[n]FS ! ak

and

y[n]FS ! bk

then

x[n]y[n]FS ! dk =

Xl=hNi

albk�l

where the expression on the right is the periodic convolutionof ak and bk.

Analogously, Parseval’s relation becomes

1

N

Xn=hNi

jx[n]j2 =Xk=hNi

jakj2

See examples pages 224–226.


Fourier Series and LTI Systems

We previously saw that if x(t) = est is the input to an LTI

system, then the output is of the form y(t) = H(s)est where

H(s) =

Z1

�1

h(� )e�s�d�

and h(t) is the impulse response of the LTI system.

Analogously in discrete time, you should have seen (by thereading the pages I suggested) that if x[n] = z

n is the in-put to a discrete-time LTI system, then the output is y[n] =H(z)zn where

H(z) =

1Xk=�1

h[k]z�k

and h[n] is the impulse response of the LTI system.


Restricting s and z

When s or z are arbitrary complex numbers, H(s) and H(z)

are called the system functions of the corresponding sys-tems.

Much can be learned about systems even when we restrictour attention to the case where the real part of s is zero: thuss = j! and hence est = e

j!t. The system function in thisspecial case is

H(j!) =

Z1

�1

h(t)e�j!tdt

and is called the frequency response of the system.

Analogously, restricting z such that jzj = 1, which we willparametrize by z = e

j!, we have as the special case

H(ej!) =

1Xn=�1

h[n]e�j!n

Clearly we will be able to express the output of an LTI sys-tem when the input is a linear combination of signals of theform e

j!t (or ej!n in discrete time) easily in terms of thefrequency response of the system.


The Output in terms of Frequency Response

Suppose x(t) is a periodic signal with the Fourier series rep-resentation

x(t) =

1Xk=�1

akejk!0t

If we apply x(t) to an LTI system with impulse responseh(t), then by considering sk = jk!0, we can conclude thatthe output is

y(t) =

1Xk=�1

akH(jk!0)ejk!0t

Thus from

x(t)FS ! ak

we conclude

y(t)FS ! akH(jk!0)

Analogously (see page 230) in discrete time, if x[n] is theinput, and y[n] the output of an LTI system with impulseresponse h[n], and

x[n]FS ! ak

then

y[n]FS ! akH(ej2�k=N) (!0 =

2�

N

)

where H(ej!) is the frequency response.

See examples 228–231.


Filtering

Filtering of signals is extremely widespread. The analysisand design of filters is one of the things the material in thiscourse is useful for.

We will concentrate on the analysis of filters mainly (logi-cally prior to the design of filters).

Some examples:

� Tone controls (or graphic equalizer) on a HiFi

� Artificial reverberation

� Approximate inverse filtering in a Modem

� Trend estimates of economic indices

� Image processing/enhancement (blurring=convolution)

� Telecommunications (channelization and Frequency Di-vision Multiplexing)

� Removal of noise in transducer measurements

� Removal of mains hum


Examples of Filters

The text discusses frequency shaping filters, and includesplots of the log-magnitude of the frequency response of somepractical systems.

Also mentions a differentiator. That is y(t) = dx(t)

dt. Can

show H(j!) = j!. (Try to show this! Recall ddtsin at =

a cos at)

Another example is an early telephone microphone (fairlylow-fi!)


Some Questions

Some questions you might ponder are

� Can one design a filter with an arbitrary magnitude re-sponse?

� Does the phase of the response matter (rememberH(j!)

is in general complex valued).

� How can one roughly preserve the “shape” of a signalwhilst filtering off unwanted noise?

� What sort of filters can one make with a finite numberof components (capacitors, inductors and resistors)

� Can one always use discrete time filters (implementedon a computer) rather than analog (continuous time fil-ters) implemented in special hardware.

� Given a desired frequency response, how do I design animplementable filter than (approximately) obtains thatfrequency response?


Frequency-Selective Filters

There are a variety of ideal filters that it is useful to thinkabout.

These are ideal since it can be shown that they can not beimplemented exactly in a practical sense (in continuous timethey would need an infinite number of components; in dis-crete time they would need an infinitely long impulse re-sponse and hence and infinite number of computations peroutput sample).

The book illustratea few of them. the simplest one is theideal low pass filter with cutoff frequency !c. This has afrequency response

H(j!) =

�1; j!j � !c

0; j!j > !c

Note the terminology “passband” and “stopband”.

Read sections 3.10 and 3.11


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THE AUSTRALIAN NATIONAL UNIVERSITY Second Semester … · 2002-08-07 · Models help us cope with complexity. The abstractions of Signals and Systems are general princi-ples for building