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DEE2363DEE2363Signals And SystemsSignals And Systems

by Mr. Koay Fong Thaikoayft@psdc.org.my ;

ftkoay@gmail.com

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Announcement (1)

Schedule Feedback to Lecturer

(First two weeks of the semester) The availability of Room, Students, Lecturer.

Plagiarism 7 cases in August’08 Final/Resit Exams. 1 expelled, 6 with Warning Letters and Re-

take.

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Announcement (2)

Badge Warning Letter.

Absent 3 times – Warning Letter and inform parents. Less than 75% attendance to be barred

from final exam.

Use of Handphones and Laptops are STRICTLY PROHIBITED in Class and Lab.

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Dos

Feel free to ask questions during the class (without disturbance), or

my consultation hours, every Wednesday and Friday, 9am – 11am.

Be sensitive to find my errors. Work hard to get a good grade. Be on time in the class. Be on time to submit coursework (take note

of the format).

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Don’ts

Use of cell phones and computers in the class.

Play games!! Sleep and/or talk in the class. Copy tutorials/assignments (you will get zero

if I catch you). …etc, to be defined by me….

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Assessment

Tutorial : 10% Quiz : 10% Assignment : 10% Mid-Term Tests : 20%

50% **** Final Exam : 50% .

100%

Less than 25% to be barred from final exam!!

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Lecture and Tutorial ScheduleLecture

WeekWed

8am-10amB19 EEA

Tues, 10am-12pmB19 EEB

Tues, 8am-10amB19 CE

Thurs, 10am-12pm1 6 --2 13 --3 20 Tut 19 19 214 27 Tut 26 26 285 4 --6 11 Tut 10 10 127 18 Tut 17 17 198 25 Tut 24 24 269

10 8 Tut 7 7 911 15 Tut 14 14 1612 22 Tut 21 21 2313 29 --14 6 Tut 5 5 715 13 Tut 12 12 1416 ----- Tut 19 19 21

25 July - 31 July : Study Week

1 August - 14 August : Exam Weeks

June

July

Mid-Term Break (30 May - 5 June)

May

Tutorial (Lab??)

April

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Lecture Schedule with Outcome

WeekWed

8am-10amOutcome Contents

1 6 --

2 13 --

3 20 Tut

4 27 Tut

5 4 --

6 11 Tut

7 18 Tut

8 25 Tut9

10 8 Tut

11 15 Tut

12 22 Tut

13 29 --

14 6 Tut

15 13 Tut

16 ----- – Tut Revision (Tutorials)

Lecture

April1 Introduction

Continuous/discrete time signals & signal operations

2Signals and

Systems in Time

Impulse response, Convolution Integral/Sum, Differential/Difference Equations, Linear Constant-Coefficient Differential/Difference Equations.

25 July - 31 July : Study Week1 August - 14 August : Exam Weeks

May

June

Discrete-time Fourier Transform, the properties and applications in signals and systems, Inverse

4Laplace Transform

Its properties and the inverse transform;Laplace Transform analysis of signals and

July5

z-TransformIts properties and the inverse transform;z-Transform analysis of signals and systems.

3Fourier Transform

Signals in frequency domain, Continuous/discrete Fourier Series; Continuous-time Fourier Transform,

Mid-Term Break

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Broad Aims To introduce the students to the idea of signal and

system along with the analysis and characterization.

To introduce the students the transformation methods for both continuous-time and discrete-time signals and systems.

To provide a foundation to numerous other courses that deal with signal and system concepts directly or indirectly, for instance, communication, control, instrumentation, and so on; as well as to students of disciplines such as, mechanical, chemical and aerospace engineering.

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Objectives

By the end of the course, you would have understood: Basic signal analysis (mostly continuous-time) Basic system analysis (also mostly continuous

systems) Time-domain analysis (including convolution) Laplace Transform and transfer functions Fourier Series and Fourier Transform Sampling Theorem and Signal Reconstructions Basic z-transform

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Topics Signals and Systems in the time domain

Impulse response, Convolution Integral/Sum, Differential/Difference Equations, Linear Constant-Coefficient Differential/Difference Equations.

Fourier Transform Continuous/discrete Fourier Series and Transform, the

properties and applications in signals and systems, Inverse Fourier Transform.

Laplace Transform Properties, inverse transform; analysis of signals and

systems. z-Transform

Properties, inverse transform; analysis of signals and systems.

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Reference Books B.P. Lathi, “Signal Processing and Linear Systems”, 1998,

Oxford University Press. PSDC Library : TK5102.9 Lat 1998

M.J. Roberts, “Signals and Systems: Analysis Using Transform Methods and MATLAB”, 2004, McGraw Hill. PSDC Library : TK5102.9 Rob 2004

Alan V. Oppenheim, Alan S. Willsky, S. Hamid Nawab, “Signals and Systems”, 2nd Edition 1997, Pearson Education,. PSDC Library : QA402 Opp 1997

Simon S. Haykin, B. Van Veen, “Signals and Systems”, 2nd Edition 2005, John Wiley & Sons.

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Introduction to Signals Introduction to Signals (CT and DT)(CT and DT)

By Koay Fong Thai

koayft@psdc.org.my

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Topics

Introduction Size of a Signal Classification of Signals Some Useful Signal Operations Some Useful Signal Models Even and Odd Functions

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Introduction

The concepts of signals and systems arise in a wide variety of areas:communications, circuit design, biomedical engineering, power systems, speech processing, etc.

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What is a Signal?

SIGNAL A set of information or data. Function of one or more

independent variables. Contains information about the

behavior or nature of some phenomenon.

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Examples of Signals

Electroencephalogram (EEG) signal (or brainwave)

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Examples of Signals (2)

Stock Market data as signal (time series)

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What is a System?

SYSTEMSignals may be processed further

by systems, which may modify them or extract additional from them.

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

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What is a System? (2)

A system may be made up of physical components, as in electrical or mechanical systems (hardware realization).

A system may be an algorithm that computes an outputs from an inputs signal (software realization).

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Examples of signals and systems

Voltages (x1) and currents (x2) as functions of time in an electrical circuit are examples of signals.

A circuit is itself an example of a system (T), which responds to applied voltages and currents.

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Signals Classification

Signals may be classified into: 1. Continuous-time and discrete-time signals 2. Analogue and digital signals 3. Periodic and aperiodic signals 4. Causal, noncausal and anticausal signals 5. Even and Odd signals 6. Energy and power signals

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Signals Classification (2) – Continuous versus Discrete

Continuous-timeA signal that is specified for everyvalue of time t.

Discrete-timeA signal that is specified only at discrete valuesof time t.

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Signals Classification (3) – Analogue versus Digital (1)

Analogue, continuous

Analogue, discrete

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Signals Classification (4) – Analogue versus Digital (2)

Digital, continuous

Digital, discrete

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Signals Classification (5) – Periodic versus Aperiodic

A CT signal x(t) is said to be periodic if for some positive constant To,

The smallest value of To that satisfies the periodicity condition is the fundamental period of x(t).

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Signals Classification (6) – Periodic versus Aperiodic (2)

Fundamental period = To.Then, fundamental frequency is fo = 1/To in Hz or cycles per second.

or Angular frequency, ωo = 2/To, radian per

second. Example, f(t) = C cos (2fot+)

C : amplitude; fo : frequency; : phase Rewriting f(t) = C cos (ωot+)

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Signals Classification (7) – Periodic versus Aperiodic (3)

A DT signal x[n] is said to be periodic if for all positive integer N,

The smallest value of N is the fundamental period of x[n].

Fundamental angular frequency, is defined by = 2/N.

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Signals Classification (6) – Periodic versus Aperiodic (4)

For the signal,

Find the period and the fundamental frequency of the signal.

Solution (hint: sin = cos ( -/2)):

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Signals Classification (6) – Periodic versus Aperiodic (5)

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Signals Classification (8) – Periodic versus Aperiodic (6)

Aperiodic (Nonperiodic) signals?

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Signals Classification (9) – Causal vs. Noncausal vs. Anticausal

Causal (因果 ) signal: A signal that does not start before t =0. f(t) = 0; t <0

Noncausal signal: A signal that starts before t =0, such as charge in

capacitor before switch is turned on. Anticausal signal:

A signal that is zero for all t 0.

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Signals Classification (10) – Even versus Odd

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Signals Classification (11) – Even versus Odd (2)

A signal x(t) or x[n] is referred to as an even signal if CT: DT:

A signal x(t) or x[n] is referred to as an odd signal if CT: DT:

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Signal Classification (12) – Energy versus Power

Signal with finite energy (zero power)

Signal with finite power (infinite energy)

Signals that satisfy neither property are referred as neither energy nor power signals

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Size of a Signal (1)

A number indicates the largeness or strength of the signal.

Such a measure must consider both amplitude and duration of the signal.

Measurement of the size of a human being, V with variable radius, r and height, h with assumption of cylindrical shape given by

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Size of a Signal (2)

Assuming f(t) = sin t, f(t) could be a large signal, yet its positive and negative areas cancel each

other. Then, indicates a signal of small size. This can be solved by defining the signal size

as the area under f2(t) (f2(t) always > 0).

tdtsin

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Size of a Signal, Energy (Joules)

Measured by signal energy Ex:

Generalize for a complex valued signal to: CT: DT:

Energy must be finite, which means

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Size of a Signal, Power (Watts)

If amplitude of x(t) does not → 0 when t → ∞, need to measure power Px instead:

Again, generalize for a complex valued signal to: CT:

DT:

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Example

Determine the suitable measures of the signals in the figure below:

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Example

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Summary

By the end of the class, you would have understood: Examples of signals Signals classification

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Signal Operations for Signal Operations for CT SignalsCT Signals

by Koay Fong Thai

koayft@psdc.org.my

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Signal Operations

Signal operations are operations on the time variable of the signal, involve simple modification of the independent variable. Time Shifting Time Scaling Time Inversion (Reversal) Combined operations

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Signal Operations: Time Shifting

Shifting of a signal in time adding or subtracting the amount of the

shift to the time variable in the function. x(t) x(t–to)

to > 0 (to is positive value),signal is shifted to the right (delay).

to < 0 (to is negative value),signal is shifted to the left (advance).

x(t–2)? x(t) is delayed by 2 seconds. x(t+2)? x(t) is advanced by 2 seconds.

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Signal Operations: Time Shifting (2)

Subtracting a fixed amount from the time variable will shift the signal to the right that amount.

Adding to the time variable will shift the signal to the left.

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Signal Operations: Time Shifting (3)

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Signal Operations: Time Scaling Compresses (压缩 ) and dilates (膨胀 ) a

signal by multiplying the time variable by some amount.

x(t) x(t) If >1, the signal becomes narrower

compression. If <1, the signal becomes wider dilation.

Play audio recorded, f(t) in mp3 player at twice the normal recording speed? f(2t) or f(t/2)? f(2t)

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Signal Operations: Time Scaling (2)

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Signal Operations: Time Scaling (3)

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Signal Operations: Time Scaling (4)

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Signal Operations: Time Inversion (Reversal)

Reversal of the time axis, or folding/flipping the signal (mirror image) over the y-axis.

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Signal Operations: Time Inversion (Reversal) (2)

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Signal Operations: Combined Operations

Signal f(at–b) can be realized in TWO ways:1. Time-shift f(t) by b f(t–b),

then, time-scaled f(t–b) by a f(at–b)

2. Time-scale f(t) by a f(at),then, time-shift f(at) by b/a f[a(t – b/a)]

If a<0, it’s time inversion operation.

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Example

Given the signal as shown in figure below. Plot i. x (t + 1) ii. x (1-t)

iii. iv.

tx

2

3

1

2

3tx

x (t)

0 1 2t

x (t)

0 1 2t

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Example (i)

Shift to the left by one unit along the t axis.

x (t + 1)

-1 0 1t

-2

1

x (t + 1)

-1 0 1t

-2

1

x (t)

0 1 2t

x (t)

0 1 2t

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Example (ii)

Replace t with –t in x(t + 1) x(-t + 1) x(1-t).

It is obtained graphically by reflecting x(t + 1) about the y axis.

x (-t + 1)

-1 0 1t

-2

1

x (-t + 1)

-1 0 1t

-2

1

x (t + 1)

-1 0 1t

-2

1

x (t + 1)

-1 0 1t

-2

1

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Example (iii)

The signal x(3/2 t) compression of x(t) by a factor of 2/3.

x (t)

0 1 2t

x (t)

0 1 2t

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Example (iv)

First, advance or shift to the left x(t) by 1 as shown in figure below.

Then, compress this shifted signal by a factor of 2/3.

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Signal Operations

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Some Useful Signal Some Useful Signal ModelsModels

by Koay Fong Thai

koayft@psdc.org.my

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Signal Models: Unit Step Function

Continuous-Time unit step function, u(t):

u(t) is used to start a signal, f(t) at t=0 f(t) has a value of ZERO for t <0

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Signal Models: Unit Step Function (2)

f(t) = e-at x u(t) a causal form of e-at.

x =

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Signal Models: Unit Step Function (3)

Realize the rectangular pulse below:

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Signal Models: Unit Impulse Function Continuous-Time unit impulse function, (t) is

defined by P.A.M. Diarc:

We can visualize an impulse as a tall and narrow rectangular pulse of unit area.

When 0, the height is very large, 1/.

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Signal Models: Unit Impulse Function (2)

A possible approximation to a unit impulse:An overall area that has been maintained at unity.

Multiplication of a function by an Impulse?

b(t) = 0; for all t0is an impulse function which the area is b.

Graphically, it is represented by an arrow "pointing to infinity" at t=0 with its length equal to its area.

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Signal Models: Unit Impulse Function (3)

May use functions other than a rectangular pulse. Here are three example functions:

Note that the area under the pulse function must be unity.

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Signal Models: Unit Ramp Function

Unit ramp function is defined by: r(t) = tu(t)

Where can it be used?

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Signal Models: Example

Describe the signal below:

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Signal Models: Example (2)

x =t[u(t) – u(t-2)]

x = -2(t-3)[u(t-3) – u(t-2)]

t[u(t) – u(t-2)] - 2(t-3)[u(t-3) – u(t-2)]

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Signal Models: Exponential Function est

Most important function in SNS where s is complex in general, s = +j

Therefore,est = e(+j)t = etejt = et(cost + jsint)(Euler’s formula: ejt = cost + jsint)

If s = -j, est = e(-j)t = ete-jt = et(cost - jsint)

From the above, etcost = ½(est +e-st )

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Signal Models: Exponential Function est (2)

Variable s is complex frequency. est = e(+j)t = etejt = et(cost + jsint)

est = e(-j)t = ete-jt = et(cost - jsint)etcost = ½(est +e-st )

There are special cases of est :1. A constant k = ke0t (s=0 =0,=0)

2. A monotonic exponential et (=0, s=)

3. A sinusoid cost (=0, s=j)

4. An exponentially varying sinusoid etcost (s= j)

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Signal Models: Exponential Function est (3)

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Signal Models: Exponential Function est (4)

In complex frequency plane:

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Even and Odd Functions

A function fe(t) is said to be an even function of t if

fe(t) = fe(-t)

A function fo(t) is said to be an odd function of t if

fo(t) = -fo(-t)

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Even and Odd Functions: Properties

Property:

Area: Even signal:

Odd signal:

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Even and Odd Components of a Signal (1)

Every signal f(t) can be expressed as a sum of even and odd components because

Example, f(t) = e-atu(t)

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Even and Odd Components of a Signal (2)

Example, f(t) = e-atu(t) casual?

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Signal Models: Summary

Unit step function, u(t) Unit impulse function, (t) Unit ramp function, r(t) Exponential function, est

Even and off function All these functions are used for

CONTINUOUS-TIME signals!!!!

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SamplingSampling

By Koay Fong Thai

koayft@psdc.org.my

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Sampling Theorem

SamplingSampling is the process of is the process of converting a converting a continuous signal into a discrete signalcontinuous signal into a discrete signal..

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Sampling Theorem (2)

The The sampling frequencysampling frequency or sampling or sampling rate, rate, ((ffss))

the number of samples per second the number of samples per second taken from a continuous signal to make taken from a continuous signal to make a discrete signal.a discrete signal.

It is measured in It is measured in hertz (Hz)hertz (Hz). .

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Sampling Theorem (3)

Sampling periodSampling period or sampling time ( or sampling time (TT) ) The inverse of the sampling frequency The inverse of the sampling frequency which is the which is the time between samplestime between samples..

Given the sampling period T, the sampling Given the sampling period T, the sampling

frequency is given byfrequency is given by

)( 1Hz

Tf s

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Sampling Theorem (4)

The discrete-time signal The discrete-time signal x[n]x[n] is obtained is obtained by “taking-samples” of the analog signal by “taking-samples” of the analog signal xc(T) every T second.every T second.

x[n] = xc(nT) It is measured in It is measured in hertz (Hz)hertz (Hz). . The relationship between the variable t of

analog signal and the variable n of discrete-time signal is

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Sampling Theorem (5)

We refer to a system that implements the We refer to a system that implements the operation of the above equation as an ideal operation of the above equation as an ideal continuous-to-discrete-time (C/D) convertercontinuous-to-discrete-time (C/D) converter::

Block diagram representation of an ideal C/D Block diagram representation of an ideal C/D Converter Converter

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Sampling Theorem (6)

Sampling is represented as an Sampling is represented as an impulse train impulse train modulation followed by the conversion into a modulation followed by the conversion into a sequencesequence..

Figure below illustrates a mathematical representation Figure below illustrates a mathematical representation of sampling with a periodic impulse train followed by a of sampling with a periodic impulse train followed by a conversion to a discrete-time sequence.conversion to a discrete-time sequence.

(a) Overall system;

(b) xs(t) for two sampling rates. The dashed envelope represents xc(t);

(c) The output sequence for the two different sampling rates.

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Sampling Theorem (7)

Figure below is the DT sequences of two Figure below is the DT sequences of two

CT signals at sampling frequency of 5 CT signals at sampling frequency of 5

Hertz (samples/second).Hertz (samples/second).

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Signal Operations for Signal Operations for DT SignalsDT Signals

By Koay Fong Thai

koayft@psdc.org.my

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Signal Operations

Signal operations are operations on the time variable of the signal, involve simple modification of the independent variable. Time Shifting Time Scaling Time Inversion (Reversal) Combined operations

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Signal Operations: Time Shifting

Shifting of a signal in time

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Signal Operations: Time Scaling Compresses and expanses a signal by

multiplying the time variable by some integers.

x(k) x(k) If >1, the signal becomes narrower

compression data losses decimation (抽取 ) or downsampling (下降抽样 ).

If <1, the signal becomes wider expansion. Insert missing samples using an interpolation

formula interpolation (插值 ) or upsampling (后续采样 )

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Signal Operations: Time Scaling (2)

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Signal Operations: Time Inversion (Reversal)

Reversal of the time axis, or folding/flipping the signal (mirror image) over the y-axis.

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Some Useful Signal Some Useful Signal ModelsModels

By Koay Fong Thai

koayft@psdc.org.my

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Signal Models: Unit Step Function

Discrete-Time unit step function/sequence, u(k):

The unit step is the running sum of an impulse: u n kk

n[ ] [ ]

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Signal Models: Unit Impulse Function

Discrete-Time unit impulse function/sequence, (k):

The unit impulse is the first-difference of a unit step:

[ ] [ ] [ ]n u n u n 1

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Signal Models: DT Exponential Function k

In CT SNS, CT exponential est can be expressed in another form, t where t = est.

However, in DT SNS, it is proven that k is more convenient than ek (k is the integer).

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Signal Models: DT Exponential Function k (2)

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Signal Models : DT Exponential Function k (3)

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Questions and Answer

Any questions?