chapter1-mg1

TEL252E Signals and Systems

Prof.Dr. Muhittin GkmenDept. Of Computer Eng.

MG2011 1TEL252E Signals and Systems

Lecture 11) General Informations about course

2) Signals

3) Systems

4) Examples


General Information

Course Name: TEL252E Signals and Systems

Course Staff:Instructors: Prof.Dr.Muhittin Gkmen

Room #: EEB 4316e-mail:[email protected]

Do.Dr. Sabih AtadanRoom #:EEB 5109e-mail: [email protected]

Teaching Assistant: Muzaffer Ege Alper Room #: Research Lab3


General Information (Cntd)

Course Material:Lecture NotesOppenheim, A. V., and A. S. Willsky, with

S. H. Nawab. Signals and Systems. 2nd ed. New Jersey: Prentice-Hall, 1997. ISBN: 0138147574.


General Information

Evaluation:4 Homeworks%102 Quizes %15Mid-Term %35Final %40

Detailed information can be found in the course web-page:

http://www.cs.itu.edu.tr/~canberk/tel252e.htm


Calendar

Week 1 IntroductionWeek 2 Continuous-Time and Discrete-Time Signals and Systems. System Properties.

Singular functions.Week 3 Convolution. Periodic Signals. Week 4 Continuous- and Discrete-Time Fourier Series.Week 5 Continuous-Time Fourier Transform.Week 6 Continuous-Time Fourier Transform (cont.). Discrete-Time Fourier Transform.Week 7 Discrete-Time Fourier Transform (cont.).Week 8 First and Second Order Continuous- and Discrete-Time Systems. Ideal and Non-Ideal

Filters.Week 9 Midterm ExamWeek 10 Sampling. Impulse-Train Sampling. Sampling Theorem and Aliasing. Zero and First

Order Hold. Analog-to-Digital and Digital-to-Analog Conversions.Week 11 Laplace Transforms, Unilateral and Bilateral z-Transforms, Region of Convergence

(ROC). The relationships between Laplace Transform, (Continuous and Discrete) Fourier Transforms and z-Transform.

Week 12 Transfer Functions using the Laplace- and z-Transforms, Pole-Zero Plot in s- and z-planes, Stability.

Week 13 Constant Coefficient Linear Differential and Difference Equations. Week 14 Block Diagram Representation of Continuous- and Discrete-Time Systems. Direct

Form, Series and Cascade Filter Realizations. Feedback Structure in s-Domain.


MG2011 TEL252E Signals and Systems 7

Course Outline (Tentative) Fundamental Concepts of Signals and Systems

Signals Systems

Linear Time-Invariant (LTI) Systems Convolution integral and sum Properties of LTI Systems

Fourier Series Response to complex exponentials Harmonically related complex exponentials

Fourier Integral Fourier Transform & Properties Modulation (An application example)

Discrete-Time Frequency Domain Methods DT Fourier Series DT Fourier Transform Sampling Theorem

Z Transform Stability analysis in z domain

Chapter ISignals and Systems


SIGNALSSignals are functions of independent variables

that carry information about the behavior or nature of some phenomenon

For example: Electrical signals --- voltages and currents in a circuit Acoustic signals --- audio or speech signals (analog or digital) Video signals --- intensity variations in an image (e.g. a CAT scan) Biological signals --- sequence of bases in a gene


TEL252E Signals and Systems 10

What is Signal?

Signal is the variation of a physical phenomenon / quantity with respect to one or more independent variable

A signal is a function.

Example 1: Voltage on a capacitor as a function of time.

RC circuit

isV I C

R

+

-cV

MG2011


What is Signal?

Example 2 : Closing value of the stock exchange index as a function of days

M T W FT S S

Index

Fig. Stock exchange

Example 3:Image as a function of x-y coordinates (e.g. 256 X 256 pixel image)

THE INDEPENDENT VARIABLES

Can be continuous Trajectory of a space shuttle Mass density in a cross-section of a brain

Can be discrete DNA base sequence Digital image pixels

Can be 1-D, 2-D, N-D For this course: Focus on a single (1-D) independent variable

which we call time.

Continuous-Time (CT) signals: x(t), t continuous valuesDiscrete-Time (DT) signals: x[n], n integer values only


CT Signals

Most of the signals in the physical world are CTsignalsE.g. voltage & current, pressure,temperature, velocity, etc.


DT Signals

x[n], n integer, time varies discretely

Examples of DT signals in nature: DNA base sequence Population of the nth generation of certainspecies


Many human-made DT SignalsEx.#1 Weekly Dow-Jonesindustrial average

Ex.#2 digital image

Courtesy of Jason Oppenheim.Used with permission.

Why DT? Can be processed by modern digital computersand digital signal processors (DSPs).



Continuous-Time vs. Discrete Time

Signals are classified as continuous-time (CT) signals and discrete-time (DT) signals based on the continuity of the independent variable!

In CT signals, the independent variable is continuous (See Example 1 (Time))

In DT signals, the independent variable is discrete (See Ex 2 (Days), Example 3 (x-y coordinates, also a 2-D signal)) DT signal is defined only for specified time instants! also referred as DT sequence!


Continuous-Time vs. Discrete Time

The postfix (-time) is accepted as a convention, although some independent variables are not time

To distinguish CT and DT signals, t is used to denote CT independent variable in (.), and n is used to denote DT independent variable in [.] Discrete x[n], n is integer Continuous x(t), t is real

Signals can be represented in mathematical form: x(t) = et, x[n] = n/2

y(t) =

Discrete signals can also be represented as sequences: {y[n]} = {,1,0,1,0,1,0,1,0,1,0,}

0 x[n-n0] is the delayed version of x[n] (Each point in x[n] occurs later in x[n-n0])

x[n]

x[n-n0]

. . . . . .

. . . . . . . . . . .

n

nn0

Time shift


Examples of Transformations

t0 < 0 x(t-t0) is an advanced version of x(t)

x(t)

x(t-t0)

t

tt0

Time shift



Reflection about t=0

x(t)

x(-t)

t

t

Time reversal



x(t)

t

x(2t)

t

x(t/2)

t

compressed!

stretched!

Time scaling



Given the signal x(t):

Let us find x(t+1):

Let us find x(-t+1):

x(t)

t

1

1 20

(Time reversal of x(t+1)) t

1

10-1

(It is a time shift to the left)

x(t+1)

t

x(-t+1)10-1

1

It is possible to transform the independent variable with a general nonlinear function h(t) ( we can find x(h(t)) )

However, we are interested in 1st order polynomial transforms of t, i.e., x(t+)



For the general case, i.e., x(t+),

1. first apply the shift (),

2. and then perform time scaling (or reversal) based on .x(t+1)

t10-1

1

x((3/2)t+1)

t2/30-2/3

1

Example: Find x(3t/2+1)


Periodic Signals A periodic signal satisfies:

Example: A CT periodic signal

If x(t) is periodic with T then

Thus, x(t) is also periodic with 2T, 3T, 4T, ...

The fundamental period T0 of x(t) is the smallest value of T for whichholds

0, )()( >+= TtTtxtx

)(tx

0 TTT2 T2

++= ZmmTtxtx for )()(

0, )()( >+= TtTtxtx


Periodic Signals A non-periodic signal is called aperiodic. For DT we must have

Here the smallest N can be 1,

The smallest positive value N0 of N is the fundamental period

0, ][][ >=+ NnnxNnx

Period must beinteger!

a constant signal


Even and Odd Signals

If even signal (symmetric wrt y-axis)

If odd signal (symmetric wrt origin)

Decomposition of signals to even and odd parts:

][][or )()( nxnxtxtx ==

][][or )()( nxnxtxtx ==

odd

t

x(t)

t

evenx(t)

{ } [ ])()(21)( txtxtxEV +=

{ } [ ]1( ) ( ) ( )2

OD x t x t x t=

{ } { })()()( txODtxEVtx +=


Exponential and Sinusoidal Signals

Occur frequently and serve as building blocks to construct many other signals

CT Complex Exponential:

where a and C are in general complex.

Depending on the values of these parameters, the complex exponential can exhibit several different characteristics

atCetx =)(

x(t) x(t)

C Ctt

a < 0a > 0 Real Exponential (C and a are real)



Periodic Complex Exponential (C real, a purely imaginary)

Is this function periodic?

The fundamental period is

Thus, the signals ej0t and e-j0t have the same fundamental period

tjwetx 0)( =

for periodicity = 1

00

2

=T

TjwtjwTtjwtjw eeeetx 0000 .)( )( === + += ZnnT

0

2



+= t0

sincossincosje

jej

j

=

+=

By using the Eulers relations:

We can express: (put

( ) ( )

( ) ( )tjjtjjtjtj

tjjtjjtjtj

tj

eeeej

Aeej

AtA

eeeeAeeAtA

tjte

0000

0000

0

22)sin(

22)cos(

sincos

)()(0

)()(0

00

++

++

==+

+=+=+

+=

Sinusoidals in terms of complex exponentials



Alternatively,( )( ))(0

)(0

0

0

Im)sin(

Re)cos(

+

+

=+

=+tj

tj

eAtAeAtA

cosA

)(tx 00

2

=T

t

)cos()( 0 += tAtx

CT sinusoidal signal

)cos( 0 +tA



Complex periodic exponential and sinusoidal signals are of infinite total energy but finite average power

As the upper limit of integrand is increased as

However, always Thus,

1)(

1

)(1

0

00

2 000

=+

=

=+=== ++

periodperiod

TT

T

TT

T

tjperiod

ETTT

P

TTTTdtdteE

++ periodETTTT ,...3,2 00

1=periodP

==

T

T

tj

Tdte

TP IMPORTANT 1

21lim

20

Finite average power!


Harmonically Related Complex Exponentials

Set of periodic exponentials with fundamental frequencies that are multiplies of a single positive frequency 0

,...2,1,0for )( 0 == ketx tjkk

0kfrequency lfundamenta with periodic is )(0constant a is )(0

txktxk

k

k

=

00

0

0

2 where,2 period lfundamenta and

== TkT

k


Harmonically Related Complex Exponentials kth harmonic xk(t) is still periodic with T0 as well Harmonic (from music): tones resulting from

variations in acoustic pressures that are integer multiples of a fundamental frequency

Used to build very rich class of periodic signals


General Complex Exponential Signals

Here, C and a are general complex numbers

Say,

(Real and imaginary parts) Growing and damping sinusoids for r>0 and rrt

)cos()( 0 += tCetxrt

0,


DT Complex Exponential and Sinusoidal Signals

[ ]

)0for n alternatioSign (1,01,10,1

real are and C :signals lexponentia Real of instead use tocustomary and convenient more isIt

numberscomplex generalin are and C where

)1 decaying and ,1 (growing ssinusoidal are expcomplex general DT of partsimaginary and Real

1


Periodicity Properties of DT Signals

Zkk

n

+

+

==

+

,2 valuesfreq identical has exp DT valuesfreqdistinct has exp CT

exp.complex CT fromdifferent very is This2 WITHFN THE AS SAME THE IS FREQ WITHFN THE SO

eee find sLet'

e :expcomplex DT heConsider t

0

0

00

1

n2jnj)n2j(

j

00

0

It is sufficient to consider an interval from 0 to 0 +2 to completelycharacterize the DT complex exponential!

Result:



exp?complex DT ofy periodicit about theWhat around is expcomplex DT arying)(rapidly v freqHigh

2 and 0 around is expcomplex DT varying)(slow freq low Hence,zero. ton oscillatio of rate the,2 until more as

,n oscillatio of rate the, to0 from as exp DTFor

lyindefinite n oscillatio of rate the as exp CTFor

-or 20 kesusually ta One

0

00

0

0

0

00

=

==



!!otherwise! periodicnot

number, rational a is 2

when periodic is exp DT So

integers. bemust and that (**) and (*) from conditions thehave We

*)*(* 2

ly equivalentOr

2 havemust weminteger some sother wordIn (**) .2 of multipleinteger an is if holds This

(*) eee :conditiony Periodicit

0

0

0

0

unity bemust

jj)(j 000

Nm

NmNm

mNN

NnNn

=

=

=

=+



!signals.)! sinusoidal DTfor validalso ist developmen same (The

*)*(*in as 2

express toneed weexpcomplex an of freq fund thefind toTherefore

**)*(* 2 then is period lfundamenta The

2 then isfrequency lfundamenta The

outfactor common theTake

0

0

0

=

=

mN

mN


Periodicity Properties of DT SignalsExamples

121

121N , **)*(* usingby so

common,in factors no 121

2

122 n)cos()12

n2cos( x[n]

12. period fund with periodic is )12n2cos( x[n]:Ex

000

=

=

====

=

62

121N , **)*(* usingthen

, )6()1(

122

2

124 n)cos()12

n4cos( x[n]

6. period lfundamenta with periodic is )12n4cos( x[n]:Ex

000

=

=

==

=====

=

Nn



OBSERVATION: With no common factors between N and m, N in (***) is the

fundamental period of the signal Hence, if we take common factors out

Comparison of Periodicity of CT and DT Signals:

Consider x(t) and x[n]

6 61

20 == N

x(t) is periodic with T=12, x[n] is periodic with N=12.

122cos 12

2cos )n(x[n])t(x(t) ==



But, if and

x(t) is periodic with 31/4. In DT there can be no fractional periods, for x[n] we have

then N=31.

If and

x(t) is periodic with 12, but x[n] is not periodic, because there is no way to express it as in (***)

Study Fig.1.27 page 27, Table 1.1 in Opp. Example 1.6 as well

( )31t8cos)( =tx ( )318cos][ nnx =

314

20 =

( )6cos)( ttx = ( )6cos][ nnx =

12

12

0 =


Harmonically Related Complex Exponentials (Discrete Time)

Set of periodic exponentials with a common period N

Signals at frequencies multiples of (from 0N=2m)

In CT, all of the HRCE, are distinct

Different in DT case!

,...2,1,0for ][2

==

kenn

Njk

k

N2

,...2,1,0for 0 =ke tjk


Harmonically Related Complex Exponentials (Discrete Time)

Lets look at (k+N)th harmonic:

Only N distinct periodic exponentials in k[n] !! That is,

][.][1

222)(

neeen knj

nN

jknN

Nkj

Nk

====

+

+

nN

Nj

N

nN

jnN

jenenenn

2)1(

1

4

2

2

10 ][,,][,][,1][

====

][][,][][ 110 nnnn NN ==


Unit Impulse and Unit Step Functions

Basic signals used to construct and represent other signals

DT unit impulse:

DT unit step:

Relation between DT unit impulse and unit step (?):

=

=

0,10,0

][nn

n

0

[n-k]- - -- - -

n k0

Interval of summation

n


Unit Impulse and Unit Step Functions (Continuous-Time)

CT unit step:

CT impulse:dt

tdut )()( =

dtut

= )()(

> 2fmax Attention: assumes 0 quantization error, unrealistic quantization noise Music typically extends from 20 Hz to 20 kHz Speech 100 Hz to 10 kHz, major energy in band from 200Hz to 4kHz

Quantization depth determined by desired sound quality (quant. noise): Typically 8 (256 levels) or 16 (65,536 levels)Samples always "per channel": (e.g., 2x for stereo)Logarithmic Quantization:compensates for fact that quantizationerror much more audible around 0 amplitude

cont


Aud

io3 Audio Quality of Common Appliances


Aud

io3 Sampling Rates


Coding = representation of sampled values by integers / bitsPCM (Pulse Code Modulation)

integer value = (quantized) sampled value simple but requires high number of bits

DPCM (Differential PCM) integer value = difference between current value and predicted value prediction based on previous values requires less bits than PCM for same qualityDM (Delta Modulation)

as DPCM but only differences of 1 and -1 allowed requires minimal number of bits but quality can be poor

ADPCM (Adaptive Differential PCM): as DPCM but adapts predictor to signal characteristics simplest prediction: "difference remains" also adapts width of quantization steps to signal characteristics better quality than DPCM with same storage requirements

Aud

io3 Common Coding Methods


Aud

io3 Physics Of Acoustics cont

Frequency Pitch

Amplitude Loudness


ExampleA

udio

3

Phonem: Stairs

Diphon

Half syllable

Syllable


Speech Synthesis in Frequency DomainA

udio

3


Aud

io3 Speech Input


Aud

io3 Speech Recognition

Sound patternWord model


Aud

io3 Sound Enhancement

Declicking Noise Reduction Echo suppression


Aud

io3 Declicking

Removing impulse distortions, also called declicking ofrecorded sound, is performed in two steps. In a first stepimpulse distortions - clicks - are detected within a signal,which are going to be removed from the signal in the secondstep.


Aud

io3 Declicking cont


Aud

io3 Noise Reduction


Aud

io3 Echo Suppression

sonogram


Baseline JPEG Encoder Block Diagram


Baseline JPEG Decoder Block Diagram


Baseline JPEG Pros and Cons

Advantages Memory Efficient Low complexity Compression efficiency Visual model utilization Robustness

Disadvantages Single resolution Single quality No target bit rate No lossless capability No tiling No ROI Blocking artifacts Poor error resilience


Noise reduction Edge Enhancement

Zooming


*1 1 11 1 1 1 1 1

1 1 11 1 1 1 1 1

1 1 11 1 1 1 1 1

1 1 11 1 1 1 1 1

1 1 11 1 1 1 1 1

1 1 11 1 1 1 1 1

1 1 11 1 1 1 1 1

1 1 11 1 1 1 1 1

1 1 11 1 1 1 1 1

=


61 62

57 60

59 65

63 56

59 55 58 57

49 53 55 45

C1,2=

Median Operation

1 1

1 1

1

1

1 1 1

62

60

59

63

65

56

55 58 57

62

59

65

60

63

56

57

58

55

98765432159

rank


9x9 Median


Edge Detection

What is an edge A large change in image brightness of a short

spatial distance Edge strength = (I(x,y)-I(x+dx,y))/dx


Roberts Operator

Does not return any information about the orientation of the edge

[ ] [ ]22 ),1()1,()1,1(),( yxIyxIyxIyxI +++++

),1()1,()1,1(),( yxIyxIyxIyxI +++++

or

1 00 -1

0 1-1 0+


Prewitt Operator

-1 -1 -10 0 0 1 1 1

-1 0 1-1 0 1 -1 0 1

P1= P2=

Edge Magnitude =

Edge Direction =

22

21 PP +

2

11tanPP


Robinson Compass Masks

-1 0 1-2 0 2 -1 0 1

0 1 2-1 0 1 -2 -1 0

1 2 10 0 0 -1 -2 -1

2 1 01 0 -1 0 -1 -2

1 0 -12 0 -2 1 1 -1

0 -1 -2-1 0 -1 2 1 0

-1 -2 -10 0 0 1 2 1

-2 -1 0-1 0 1 0 1 2


0 1 2-1 0 1 -2 -1 0

1 2 10 0 0 -1 -2 -1


2D Laplacian Operator

( ) ( )2

2

2

22 ,,),(

yyxf

xyxfyxf

+

=

0 -1 0-1 4 -1 0 -1 0

1 -2 1-2 4 -2 1 -2 1

-1 -1 -1-1 8 -1-1 -1 -1

Convolution masks approximating a Laplacian


0 -1 0-1 4 -1 0 -1 0

Input Mask Output


Chapter 4Image Enhancement in the

Frequency Domain



Frequency Domain



Spatial Domain



Spatial Domain


TEL252E Signals and SystemsProf.Dr. Muhittin GkmenDept. Of Computer Eng.Slide Number 2General InformationGeneral Information (Cntd)General InformationCalendarCourse Outline (Tentative)Chapter ISignals and SystemsSIGNALSWhat is Signal?What is Signal?THE INDEPENDENT VARIABLESCT SignalsDT SignalsMany human-made DT SignalsContinuous-Time vs. Discrete TimeContinuous-Time vs. Discrete TimeContinuous-Time vs. Discrete TimeSignal Energy and PowerSignal Energy and PowerSignal Energy and PowerTransformation of Independent VariableExamples of TransformationsExamples of TransformationsExamples of TransformationsExamples of TransformationsExamples of TransformationsExamples of TransformationsPeriodic SignalsPeriodic SignalsEven and Odd SignalsExponential and Sinusoidal SignalsExponential and Sinusoidal SignalsExponential and Sinusoidal SignalsExponential and Sinusoidal SignalsExponential and Sinusoidal SignalsHarmonically Related Complex ExponentialsHarmonically Related Complex ExponentialsGeneral Complex Exponential SignalsDT Complex Exponential and Sinusoidal SignalsDT Sinusoidal SignalsPeriodicity Properties of DT SignalsPeriodicity Properties of DT SignalsPeriodicity Properties of DT SignalsPeriodicity Properties of DT SignalsPeriodicity Properties of DT SignalsExamplesPeriodicity Properties of DT SignalsExamplesPeriodicity Properties of DT SignalsExamplesHarmonically Related Complex Exponentials (Discrete Time)Harmonically Related Complex Exponentials (Discrete Time)Unit Impulse and Unit Step FunctionsUnit Impulse and Unit Step FunctionsUnit Impulse and Unit Step Functions (Continuous-Time)Continuous-Time ImpulseContinuous-Time ImpulseCT and DT SystemsWhat is a system?CT and DT SystemsExamplesInterconnection of SystemsInterconnection of SystemsSystem PropertiesMemory vs. Memoryless SystemsSystem PropertiesInvertibilitySystem PropertiesCausalitySystem PropertiesStabilitySystem PropertiesTime-InvarianceSystem PropertiesLinearitySystem PropertiesLinearitySuperposition in LTI SystemsSuperposition in LTI SystemsSlide Number 69Slide Number 70Slide Number 71Slide Number 72Slide Number 73Slide Number 74Slide Number 75Slide Number 76Slide Number 77Slide Number 78Slide Number 79Slide Number 80Slide Number 81Slide Number 82Slide Number 83Slide Number 84Slide Number 85Slide Number 86Slide Number 87Slide Number 88Slide Number 89Slide Number 90Slide Number 91Slide Number 92Slide Number 93Median OperationSlide Number 95Edge DetectionRoberts OperatorPrewitt OperatorRobinson Compass MasksSlide Number 1002D Laplacian OperatorSlide Number 102Slide Number 103Slide Number 104Slide Number 105Slide Number 106

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