Analy/cal Methods for Electrical Engineering Tamara Bonaci

EE590BPMP:Analy/calMethodsforElectricalEngineering

Autumn2016

Tamara Bonaci [email protected]

Thank you to Professor Maryam Fazel for sample slides and materials.

Signals

10/19/16

Signalsaresetsofinforma/onordatathatcanbemodeledasfunc/onsofoneormoreindependentvariables(oHen/meorspace)

EE 590 B, Autumn 2016 – Lecture 4

Signals• Signals are sets of information or data that can be

modeled as functions of one or more independent variables (often time or space)

1d 2d 3dWe will focus on the single variable case…

2

System

10/19/16

Asystemdescribesarela/onshipbetweeninputandoutputsignals

EE 590 B, Autumn 2016 – Lecture 4

More examples of systemsComplex, dynamical, interconnected systems, lots

of inputs and outputs…

3

10/19/16 EE 590 B, Autumn 2016 – Lecture 4

5

System Properties

1 1{ ( )} ( )T x t y t

2) Time-invariance: A System is Time-Invariant if it meets this criterion:

3) Stability: A System is BIBO Stable if it meets this criterion:

“System Response to a linear combination of inputs is the linear combination of

the outputs.”

“System Response is the same no

matter when you run the system.”

“The system doesn’t blow up if given reasonable

inputs.”

1) Linearity: A System is Linear if it meets the following two criteria:

Additivity:

Scaling:

{ ( )} ( )T x t y t 0 0{ ( )} ( )T x t t y t t� �If then

BIBO = “Bounded input, bounded output”

| ( ) |x t M td � f � | { ( )} | | ( ) |T x t y t L t d � f �If then

2 2{ ( )} ( )T x t y t

1 2 1 2{ ( ) ( )} { ( )} { ( )}T x t x t T x t T x t� �

If and

Then

{ ( )} ( )T x t y t If { ( )} { ( )}T ax t aT x t then

SystemProper/es

4

SimpleExamplesofSystem


6

System Properties

4) Invertibility:A System is Invertible if it meets this criterion:

5) Causality:A System is Causal if it meets this criterion:

“If you know the output signal, then you know exactly what the input

signal was.”

“The system does not anticipate the input.”

(It does not laugh before it’s tickled!)

6) Memory: A System is Memoryless if it meets this criterion:

The output depends only on current or past values of the input.

You can undo the effects of the system.

{ ( )} ( ) . . { ( )} { { ( )}} ( )i i iT x t y t T s t T y t T T x t x t � � If

If T{x(t)}=y(t) then y(t+a) depends only on x(t+b) where b<=a

If T{x(t)}=y(t) then y(t+a) depends only on x(t+a)(If a system is memoryless, it is also causal.)

“The output depends only on the current value

of the input.”

SystemProper/es

5

BuildingSignalsfromImpulses


6

8

Another View: Building Signals from Impulsesx(t)

t

G'(t)

t

'

1/'

( ) ( ) ( )

lim 0 : ,

( ) ( ) ( )

k

x t x k t k

k d

x t x t d

G

W W

W G W W

''

f

�f

' ' � '

'o 'o 'o

�

¦

³

10/19/16 EE 590 B, Autumn 2016 – Lecture 4 7

ImpulseResponseofanLTISystem

18

Another View

{ ( )} { ( ) ( ) }T x t T x t dW G W Wf

�f

�³

An LTI system can be completely described by its impulse response!

LINEARITY( ) { ( )}x T t dW G W W

f

�f

�³TIMEINVARIANCE( ) ( )x h t dW W W

f

�f

�³


TheConvolu/onIntegral

9

The Convolution Integral

( ) ( )* ( ) ( ) ( )y t x t h t x h t dW W Wf

�f

�³function of W

=h(-(W-t))function of W, flipped and shifted by t

W, not t


StepResponse

46

Convolution of the impulse response of an LTI system with a unit step

( )* ( ) ( ) ( ) ( )h t u t h u t d s tW W Wf

�f

� ³produces its step response, s(t)

Step Response


Tes/ngSystemProper/esGivenanImpulseResponseh(t)


Tes/ngSystemProper/esGivenanImpulseResponseh(t)

Sinceimpulseresponseh(t)fullyspecifiesanLTIsystem,wecanuseitinpropertytes/ng


CausalityandImpulseResponseh(t)

4

Causality of LTI SystemsA system is causal if the output does not depend on future times of the input.

( ) ( ) ( )y t h x t dW W Wf

�f

�³

For an LTI system: general convolution formula

If LTI system is causal, we have:

0

( ) ( ) ( )y t h x t dW W Wf

�³

An LTI system is causal if h(t)=0 for t<0


CausalityandImpulseResponseh(t)

5

Causality of LTI systems

An LTI system is causal if and only if its impulse response h(t) is a “causal signal” (that is, h(t)=0 for all t<0)

If h(t) is causal, h(t-W)=0 for all (t-W )<0 or all t < W

Only Integrate to t for causal

systems

( ) ( ) ( )t

y t x h t dW W W�f

�³

( ) ( ) ( )y t x h t dW W W�f

�f

�³


StepResponseandCausality

7

Step Response and Causality

Step Response: T u(t)*h(t)u(t)

Impulse Response: T h(t)G(t)

The step response is a running integrator of the impulse response.

( ) ( ) * ( ) ( ) ( )s t u t h t h u t dW W Wf

�f

�³�

For a causal system,

0

( ) ( )* ( ) ( )t

s t u t h t h dW W ³


BIBOStabilityandImpulseResponseh(t)

Bounded inputsystem

Bounded output

B1 , B2, B3 are constants

An LTI system is BIBO stable if

Integral of abs value of impulse response must be finite

Stability of LTI Systems

3( )h d BW W�f

�f

d � f³


BIBOStabilityandImpulseResponseh(t)

Bounded input

system

Bounded output

B1 , B2, B3 are constants

Because system is LTI.

abs(sum) <= sum(abs)

Show that

is sufficient for BIBO stability for an LTI system.

abs(prod) = prod(abs)

by assumption

if


Exponen/alResponseofanLTISystem


Exponen/alResponseofanLTISystemExponential response of LTI system

T h(t) (Impulse response)

T

T Exponential resp.

∗

∗

s in general is complex (s= )


Exponen/alResponseofanLTISystemExponential response of LTI system

For a given s, output is just a constant times the input!

Input

Output


FrequencyViewofSignals


TheViolin The violin

G3

D4

A4

E5

196 Hz

293.66 Hz

440 Hz

659.26 Hz

Fundamental frequencies


SignalsintheFrequencyDomainSignals in the frequency domain

196

293.66

440

659.26

Each signal can be represented by its frequency content

G(f )

D (f )

A(f )

E (f )

t (seconds)f (Hz)

g(t)

d(t)

a(t)

e(t)


AnExamplePeriodicSignalPeriodic Examples

TIME

FREQ

flute ‘ae’ as in ‘bat’

fundamental frequency

harmonics

fundamental period


FourierSeries


FourierSeries:Intui/onBasicidea:•  Youcanbuildsignalsoutofsinusoids,orequivalently,

outofcomplexexponen/als(CEs)Forperiodicsignals:•  YouoHendon’tneedtomanysinusoids/CEs•  Theonesthatyouneedallhavefrequenciesthatrelate

totheoriginalperiodFornon-periodicsignals:•  Youneedacon/nuum(hint:FourierTransform)


FourierSeries:Intui/onThe Basic Idea

+


FourierSeries:BasicIdeabasic idea:

Hz440 880

time

time

time

•  Sinusoidsaredescribedintermsoffrequency,whichlinkstoarepresenta/onthatourearsuse

•  Harmonicallyrelatedsinusoidsare‘orthogonal’

– Orthogonalitysimplifiesmathema/calanalysis

•  Sinusoidsarebuiltfromcomplexexponen/als– PassingthroughLTIsystemsjustscalescomponents– Convolu/onin/mebecomesscalingforeachcosine


WhySinusoids?

•  Vectorscanberepresentedassumsoforthogonalvectors

•  Signalscanberepresentedassumsoforthogonalsignals


SignalRepresenta/on

Signal Representation

x

y a = 2x + y

Vectors can be represented as a sum of orthogonal vectors.

Signals can be represented as a sum of orthogonal signals.


VectorOrthogonalityVector Orthogonality

Dot product between two vectors f and x

Angle betweenthe vectors

Vectors and signals are orthogonal if their dot product is zero.

f

x

aka inner productaka scalar product


SignalOrthogonalitySignal Orthogonality

then real signals f and x are orthogonal from time to time

For d-dimensional vectors a and b, dot product is

Similarly for signals f and x, dot product is

·

· 0

· ∗ 0 for complex signals

If


FourierSerieswithComplexExponen/als


FourierSerieswithComplexExponen/alsFourier Series with Complex Exps

00 0( ) 2 /jk t

kk

x t c e TZ S Zf

�f

¦

Building periodic signals with complex exponentials

We’ll come back to the cosine version later

^ `0 :jk te k ZZ �The complex exponential orthogonal basis:


BuildingPeriodicSignalswithComplexExponen/als–Example1Easy Example 1

0( ) 1 cos(5 .6) 5, 2 / 5x t t TZ S � �

0 0

(.6) (.6)0 0 0

( .6) ( .6)

0 (1) ( 1)

1( ) 1 ( )

21 12 2

j j

j t j t

t j t j t

x t e e

e e e e e

Z Z

Z Z Z�

� � �

�

� �

� �

0 1c 0.61 0.5 jc e

0.61 0.5 jc e��

ck = 0 for all other k


BuildingPeriodicSignalswithComplexExponen/als–Example2Easy Example 2

( ) cos( ) 3sin(2 ) cos(5 .6)x t t t t � � �

0 (2 , , 2 / 5) 2T LCM S S S S 0 1Z

0 0

0 0

0 0

2 2

(5 .6) (5 .6)

1( ) ( )

23( )

21( )2

j t j t

j t j t

j t j t

x t e e

e ej

e e

Z Z

Z Z

Z Z

�

�

� � �

�

� �

� �

1 1 0.5c c�

(.6) (0.6) *5 5 51 1

;2 2

j jc e c e c��

*2 2 2

3 3;

2 2c c c

j j�

�

ck = 0 for all other k


TheComplexExponen/alBasis:KeyQues/onsKey Questions

1. Is this an orthogonal basis?

2. How can we find a signal’s representation in this basis?

3. What signals can be represented in this basis?

4. What other bases are equivalent?

^ `0 :jk te k ZZ �The complex exponential basis:


TheComplexExponen/alBasis:KeyQues/onsThe Complex Exponential Basis

Is this an orthogonal basis? ^ `0 :jk te k ZZ �

⋅

Denote the kth basis signal by

cos j sin

cos sin

cos sin 0Let m k n

Ifm 0

Ifm 0 ⋅ cos 0 sin 0Orthogonal!


TheComplexExponen/alBasis:KeyQues/onsThe Complex Exponential Basis

00 0( ) 2 /jk t

kk

x t c e TZ S Zf

�f

¦

How can we find a signal’s representation in this basis?That is, how do we find for all k?

Easy case: sums of sines and cosines1. Find fundamental frequency2. Expand sinusoids into complex exponentials3. Write CEs in terms of k times the fundamental frequency4. Read off coefficients ck


TheComplexExponen/alBasis:KeyQues/ons

Inner product is the length of a projected ontovector x/|x|

of x

of y

For a 2Dvector and2D orthogonalbasis

For a signal f(t)from t1 to t2and an orthogonalset of signals x1(t), x2(t), x3(t), … xN(t)

of

of

⋅⋅⋅⋅

⋅⋅

⋅⋅

⋅⋅⋅⋅

⋅⋅

Signal RepresentationHowmuchofeachbasis?


TheComplexExponen/alBasis:KeyQues/onsHow much of each basis signal?

of

of

⋅⋅

⋅⋅

⋅⋅


Because are orthogonal, we need

of⋅⋅

Howmuchofeachbasis?



Howmuchofeachbasis?Fourier Series with Complex Exps

0

0

0 0

0

2 /

Synthesis: ( )

( ) ( )Analysis:

jk tk k k

k k

jk tk

kk k

T

x t c c e

x t x t ec

T

Z

Z

S Z

M

MM M

f f

�f �f

� �

�

¦ ¦

0

00

1 ( ) jk tk

T

c x t e dtT

Z�� ³

⋅⋅

Denotes integrating over any length interval


TheComplexExponen/alBasis:KeyQues/onsFinding in general

0

0

0

0 0

0

Synthesis: ( ) 2 /

1Analysis: ( )

jk tk

k

jk tk

T

x t c e T

c x t e dtT

Z

Z

S Zf

�f

�

¦

³



TheComplexExponen/alBasis:KeyQues/onsKey Questions







TheComplexExponen/alBasis:KeyQues/onsWhat Signals Can Be Represented?

Aside: First, lets measure how good a representation is


⋅⋅

Error of an approximation using only first N harmonics.

| |

Energy of the error

Result:

If | | ∞, then

→ 0 as → ∞May not be equal everywhere


ConvergenceIllustra/on:SquareWaveIllustrating Convergence: Square Wave

−4 −3 −2 −1 0 1 2 3 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Goal f(t)

Firstbasis signal is just a constant.5(1)

−4 −3 −2 −1 0 1 2 3 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Onebasissignal

Three basis signals: ଵଵ ଵଵ


ConvergenceIllustra/on:SquareWave

−4 −3 −2 −1 0 1 2 3 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−4 −3 −2 −1 0 1 2 3 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Making a square wave using cosines

Three basis signals: ଵଵ ଵଵ

Five basis signals: ଶଶ ଵଵ ଵଵ ଶଶ


ConvergenceIllustra/on:SquareWaveMaking a square wave using cosines

−4 −3 −2 −1 0 1 2 3 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−4 −3 −2 −1 0 1 2 3 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Five basis signals:

Seven basis signals: ⋯


GibbsEars Gibbs ears

−4 −3 −2 −1 0 1 2 3 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

undershoot

Gibbs phenomenon: at discontinuous jumps –overshoot and undershoot of about 9%. Never really goes away as you add terms…

Does that keep our approximation error from getting small? No


DirichletCondi/onsDirichlet Conditions

The Dirichlet conditions are:

1. | | ∞ (absolutely integrable over period)

2. Over any finite interval, has a finite number of maxima and minima

3. In any finite interval, has a finite number of discontinuities, and each discontinuity is finite


If the Dirichlet conditions hold, then:• The FS representation converges to except at

points of discontinuity, where it converges to the average value on either side

All practical, real-world periodic signals satisfy these conditions


Key Questions








Addi/onalBasesAdditional Bases

0 0 01

0 0

( cos( ) cos( )

sin( ) sin( ))

k kk

k k

c c k t c k t

jc k t jc k t

Z Z

Z Z

f

�

�

� � �

� � �

¦

00 0( ) (cos( ) sin( ))jk t

k kk k

x t c e c k t j k tZ Z Zf f

�f �f

�¦ ¦

0 0 01

(( ) cos( ) ( )sin( )k k k kk

c c c k t j c c k tZ Zf

� �

� � � �¦

0 0 01

( cos( ) sin( ))k kk

a a k t b k tZ Zf

� �¦


Sin/CosBasisSin/Cos Basis

0 0 01

( ) ( cos( ) sin( ))k kk

x t a a k t b k tZ Zf

� �¦

Basis Signals:

1cossincos 2sin 2cos 3sin 3

⋮

If is real, then , realChallenge: show this

Provides a real valued representation of real valued periodic signals

Basis set: harmonically related sinusoids

,


CosBasis Cos Basis

0 01

( ) 2 cos( )k kk

x t d d k tZ Tf

� �¦Basis Signals:

1coscos 2cos 3

⋮

Basis set: harmonically related shifted cosines

,

For real …

Provides a real valued representation of real valued periodic signals


FourierSeriesProper/esFourier Series Properties

0

( )( ) ( )( ) ( )( ) ( )

( ) ( )

x ty t ax t by t x aty t x t

y t x t t

l � l

l � l

� l 0 0

0 0

0 0

0,

,

k

y yk k

y yk k

yk k

jk tyk k

c

c ac k c ac b

c c a

c c

c c e Z

Z Z

�

�

z �

Added constant only affects constant term

Shift in time –t0 Add linear phase term –jkZ�t0

Time scale,a > 0 Same ck,

scale Z0reverse reverse

Linear ops


Sped-upSignalsSped-up signals

Fourier spectra(FS coefficients shown vs index k)0

Double the speed of the time signal, what happens to the FS coeff’s?

and

0 0

0 0

21 1( ) (2 ) let 2

/ 2

1 1( ) / 2 ( )

/ 2

gjn t jn tgn

g

jn jnn

d g t e dt f t e dt tT T

f e d f e d dT T

Z Z

Z W Z W

W

W W W W

f f� �

�f �f

f f� �

�f �f

³ ³

³ ³ SAME!!

k


Sped-upSignals

02( ) jn tn

ng t d e Z

f

�f

¦

Sped-up signals

Fourier spectra

0

and

The period of g(t) is T/2

Component frequencies are twice as far apart as f(t)’s

Double the speed of the time signal, what happens to the FS coeff’s?


Sped-upSignalsspeed changes: same coefficients but need faster

or slower sinusoids…

MagnitudeFourier spectraof f(t)

0

0

0

MagnitudeFourier spectraof f(2t)

MagnitudeFourier spectraof f(t/2)

1

1

1

Note inverse time-frequency changes: more compact in time, more spread out in frequency


ProblemProblem

Fourier series represents a periodic time signalby the sum of discrete frequency exponentials.

Many signals of interest are NOT periodic in time.

Is there any way to deal with them?


FourierTransform


FourierTransform:BasicIdeaFourier Transform: Basic IdeaStretch a periodic signal so that its period T goes to infinity.

Fourier SeriesIf we make T twice as longthen we need frequency components that are twice as slow (or half as fast),

0

Fourier Spectrafor T

Fourier Spectrafor 2T


FourierFormulasFourier Formulas

Fourier series

Periodic functions w/ period

Fourier transform

Arbitrary practical signal

In general ck and F(jZ) are complex.

(memorize these formulas)

0( ) jn tn

nf t d e Z

f

�f

¦Synthesis: 12

Analysis: 1

FT:

Inverse FT:


Nota/on/TerminologyNotation / Terminology Note

is really a function of – some people use instead

sometimes called the “spectrum” of

If is the impulse response of an LTI system, thenis called the “frequency response” of the system

will cover this more later…

usually referred to as the “Fourier transform” of

Short-hand notation:


Nota/on/Terminology

Fourier transform:Decompose any “practical” signal into an integral (sum) of complex exponentials

Makes it easy to get the response of LTI systems

Finding the Fourier transform: Evaluate the integral.Or use cool tricks.


FourierSymmetriesforRealValuedSignalsFourier Symmetries for Real-Valued Signals

Magnitude is even, phase is odd.

Fourier series

Periodic functions w/ period

Fourier transform

Arbitrary practical signal

0( ) jn tn

nf t d e Z

f

�f

¦Synthesis: 12

Analysis: 1

FT:

Inverse FT:

Real part is even, imaginary part is odd.Challenge: show this


WhatSignalsHaveFourierTransform?

• Some signals blow up when you take the Fourier transform integral.• Those signals “don’t have” a Fourier transform.• Example: where 0

• Like the FS, we have sufficient conditions for the existence of a FT

What Signals Have a Fourier Transform?


DirichletCondi/ons,Part2Dirichlet Conditions, Part 2The Dirichlet conditions for the Fourier Transform are:

1. | | ∞ (absolutely integrable)

2. Over any finite interval, has a finite number of maxima and minima

3. In any finite interval, has a finite number of discontinuities, and each discontinuity is finite

If the Dirichlet conditions hold, then:• The FT representation converges to except at

points of discontinuity, where it converges to the average value on either side

Most practical, real-world signals satisfy these conditions,(but some signals of theoretical importance do not )


Example:FourierTransformofaRectFourier transform of a rect

Let

t-T/2 0 T/2

1

Find the Fourier transform:

Define

12


Example:FourierTransformofaRectFourier transform of a rect: Plotted

t1

0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0-1

-0 . 5

0

0 . 5

1

1 . 5

2

2 . 5

3

0 w

-3/2 3/2


SomeProper/esofFourierTransformSome Fourier transform properties

Scaling?

Additivity?

Convolution?

Time shift?

time domain Fourier transform


SomeKnownFourierTransformsFourier Transforms You’ve Seen

0cos( )tZ 0 0( ) ( )SG Z Z SG Z Z� � �

0j te Z02 ( )SG Z Z�

0( )t tG � 0j te Z�

1 2 ( )SG Z

0sin( )tZ 0 0( ) ( )j jSG Z Z SG Z Z� � �

, 0

Resources


71

Materialfortoday’slecturefrom:•  A.V.Oppenheim,A.S.Willsky,SignalsandSystems,p.177–

205.•  H.P.Hsu,Schaum’sOutlinesofTheoryandProblemsofSignals

andSystems,p.211–231.

Analy/cal Methods for Electrical Engineering Tamara Bonaci

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