Frequency Response - University of Queensland2 ELEC 3004: Systems 10 April 2013 - 3 Cool Signal...

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Frequency Response

ELEC 3004: Systems: Signals & ControlsDr. Surya Singh

Lecture 10

[email protected]://robotics.itee.uq.edu.au/~elec3004/

© 2013 School of Information Technology and Electrical Engineering at The University of Queensland

April 10, 2013

ELEC 3004: Systems 10 April 2013 - 2

Today:Week Date Lecture Title27-Feb Introduction1-Mar Systems Overview6-Mar Signals & Signal Models8-Mar System Models

13-Mar Linear Dynamical Systems15-Mar Sampling & Data Acquisition20-Mar Time Domain Analysis of Continuous Time Systems22-Mar System Behaviour & Stability27-Mar Signal Representation29-Mar Holiday

10-Apr Frequency Response & Fourier Transform12-Apr Analog Filters17-Apr IIR Systems19-Apr FIR Systems24-Apr z-Transform26-Apr Discrete-Time Signals1-May Discrete-Time Systems3-May Digital Filters8-May State-Space

10-May Controllability & Observability15-May Introduction to Digital Control17-May Stability of Digital Systems22-May PID & Computer Control24-May Information Theory & Communications29-May Applications in Industry31-May Summary and Course Review

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Cool Signal Share: Eulerian Video Magnification for Revealing Subtle Changes in the World

• Assignment 1 Solutions:– Posted– Please try to get the peer review marks in

by April 16 at 11:59pm


Announcements:

!

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Signals Processing: Seeing The Light

White light(all wavelengths)

Colour spectrum

f= 1/

Think of a Signal Processing like a prism:“Destructs a source signal into its constituent frequencies”

10 April 2013 -ELEC 3004: Systems 5

• Any finite power, periodic, signal x(t)– period T

• can be represented as () summation of– sine and cosine waves

• Called: Trigonometrical Fourier Series

Fourier Series

)sin()cos(2

)( 001

0 tnwBtnwAA

tx nn

n

• Fundamental frequency w0=2/T rad/s or 1/T Hz• DC (average) value A0 /2


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• Signal measured (or known) as a function of an independent variable– e.g., time: y = f(t)

• However, this independent variable may not be the most appropriate/informative– e.g., frequency: Y = f(w)

• Therefore, need to transform from one domain to the other– e.g., time frequency– As used by the human ear (and eye)

Transform Analysis

Signal processing uses Fourier, Laplace, & z transforms etc


0 1 2 3 4 5 6-2

-1

0

1

2

time (t)

y =

f(t)

0 1 2 3 4 5 6 7 80

0.5

1

1.5

frequency (f)

Am

plitu

de

Frequency representation (spectrum) shows signal contains:• 2Hz and 5Hz components (sinewaves) of equal amplitude


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• An & Bn calculated from the signal, x(t)– called: Fourier coefficients

Fourier Series Coefficients

,3,2,1)sin()(2

,2,1,0)cos()(2

2/

2/

0

2/

2/

0

ndttnwtxT

B

ndttnwtxT

A

T

T

n

T

T

n

Note: Limits of integration can vary, provided they cover one period


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

time (s)

signal (black), approximation (red) and harmonic (green)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

time (s)

harm

onic


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

time (s)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

time (s)

harm

onic


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

time (s)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

time (s)

harm

onic


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

time (s)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

time (s)

harm

onic



Fourier Series Coefficients

• Approximation with 1st, 3rd, 5th, & 7th Harmonics added, note:– ‘Ringing’ on edges due to series truncation – Often referred to as Gibb’s phenomenon

• Fourier series converges to original signal if– Dirichlet conditions satisfied– Closer approximation with more harmonics

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• For Fourier series to converge, f(t) must be• defined & single valued• continuous and have a finite number of finite discontinuities

within a periodic interval, and• piecewise continuous in periodic interval, as must f’(t)

Dirichlet Conditions


Example: Square wave

))cos(1(2

cos11coscoscos

)sin()sin()sin()(

0sinsin

)cos()cos()cos()(

).2(

;21,1

;10,1

)(

2

1

1

0

2

1

1

0

2

0

2

1

1

0

2

1

1

0

2

0

nn

B

n

n

nnn

n

n

tn

n

tnB

dttndttndttntxB

n

tn

n

tnA

dttndttndttntxA

tx

t

t

tx

n

n

n

n

n

periodic! i.e., x(t + 2) = x(t)

No cos terms as sin(n) = 0 nx(t) has odd symmetry

Sin terms only

cos(2n) = 1 n

1610 April 2013 -ELEC 3004: Systems

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Example: Square wave

etc

harmonic)(fifth )5sin(5

4harmonic)(fourth 0

harmonic) (third)3sin(3

4harmonic) (second0

al)(fundament)sin(4

x(t)

gives, terms theExpanding

)sin())cos(1(2

)(

is, seriesFourier ricTrigonomet Therefore,

1

t

t

t

tnnn

txn

• Only odd harmonics;• In proportion

1,1/3,1/5,1/7,…• Higher harmonics

contribute less;• Therefore, converges


Complex Fourier Series (CFS)• Also called Exponential

Fourier series– As it uses Euler’s relation

• FS as a Complex phasor summation

j

tjnwtjnwtnw

tjnwtjnwtnw

twjAtwAtjwA

2

)exp()exp()sin(

2

)exp()exp()cos(

implies,which

)sin()cos()exp(

000

000

000

n

n tjnwXtx )exp()( 0Where Xn are theCFS coefficients


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Complex Phasor Summation


Complex Fourier Coefficients• Again, Xn calculated from

x(t) • Only one set of coefficients,

Xn– but, generally they are

complex

2/

2/

0 )exp()(1 T

T

n dttjnwtxT

X

Remember: fundamental w0 = 2/T !


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Relationships • There is a simple

relationship between – trigonometrical and

– complex Fourier coefficients,

.0,2

;0,2

20

0

njBA

njBA

X

AX

nn

nn

n

Therefore, can calculate simplest form and convert

Constrained to besymmetrical, i.e.,complex conjugate

*nn XX


Example: Complex FS• Consider the pulse train

signal• Has complex Fourier series:

).(

;2

,0

;2

0,

)(

Ttx

Tt

tA

tx

T

2exp

2exp

exp1

exp)(1

00

0

2

2

0

2

2

0

jnjn

Tjn

A

dttjnAT

dttjntxT

XT

T

n

A

Note: n is theind. variable

Note: ×by / …


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Example: Complex FS• Which using Euler’s identity

reduces to:

Tw

nwT

A

nw

nw

T

AX n

2

)2(sa2

)2sin(

0

00

0

sin2sincossincos

sincossincos

expexp

jjj

jj

jj

Note: letting 20 n

Note:cos(-) = cos(): evensin(-) = -sin(): odd


-20 -15 -10 -5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5Duty Cycle /T = 1/2

Am

plitu

de, a

bs(X

n)

Fourier coefficient, n

-20 -15 -10 -5 0 5 10 15 20-4

-2

0

2

4

Pha

se, a

ngle

(Xn)


Xn complex plot mag & phase As e.g. 1.2 but X0 0

2nd, 4th, 6th, 8th,… = 0

Note: complex conj symmetry

± -ve real value


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-20 -15 -10 -5 0 5 10 15 200

0.05

0.1

0.15

0.2


Am

plitu

de, a

bs(X

n)


-20 -15 -10 -5 0 5 10 15 20-4

-2

0

2

4

Pha

se, a

ngle

(Xn)


4th, 8th, 12th,… = 0 Coefficients getting smaller!


-20 -15 -10 -5 0 5 10 15 200

0.05

0.1

0.15


Am

plitu

de, a

bs(X

n)


-20 -15 -10 -5 0 5 10 15 20-4

-2

0

2

4

Pha

se, a

ngle

(Xn)


8th, 16th,… = 0


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-20 -15 -10 -5 0 5 10 15 200

0.02

0.04

0.06


Am

plitu

de, a

bs(X

n)


-20 -15 -10 -5 0 5 10 15 20-4

-2

0

2

4

Pha

se, a

ngle

(Xn)


16th, 32nd,… = 0


• Amplitude spectrum has ‘sinc’ (‘sa’) envelope– Amplitude reduces as duty cycle decreases– DC coefficient X0 (n=0) present

• compare to first example

– Duty cycle /T: XM/T = 0 (M = 1,2,3,…)– Even symmetry: |X-n| = |Xn|

• For real (not complex) signals (all we shall consider)• Often only plot positive frequency, e.g., Matlab

• Phase spectrum– Odd symmetry: X-n = - Xn

Complex FS of Pulse Train

Complex conjugate (Hermitian) symmetry is general property for real x(t)


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• Represents periodic signals (T = 2/w0)– as sum of cosine waves: “cosine series”

• at harmonic frequencies 0, w0, 2w0, 3w0, …• |Xn| is half amplitude of nth harmonic• Xn is phase shift of nth harmonic

• Distribution with harmonic number of – both amplitude & phase– called a frequency spectrum (discrete)

Interpretation of Fourier Series

1

00 cos||2)(n

nn XtnwXXtx

i.e., Harmonics only


• Trigonometrical and Complex FS both– Orthogonal expansions

• Because harmonically related – sines, cosines, and complex phasors are all orthogonal

• Product of f1(t) & f2(t) integrates to zero over 1 period

Orthogonal Expansions

2/

2/

0*

0 . if,0

; if,1)(exp)exp(

1 T

T nm

nmdttjmwtjnw

T

This means we can calculate each Xn independently(instead of solving n simultaneous equations)


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Example: Calculate Power in x(t)

Orthogonal 0

Total = sum of power in 2 sine waves


• Direct consequence of orthogonality• Calculate power of a signal either

– in time domain, i.e., from x(t), or– in frequency domain, i.e., from Xn

Parseval’s Theorem

T

nnXdttx

TP

0

22 )(1

Continuous &periodic

Discrete


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-10 -8 -6 -4 -2 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1Duty Cycle /T = 1/2

Nor

mal

ised

Am

plitu

de, |

Xn|

*T/

Angular frequency (w)

-10 -8 -6 -4 -2 0 2 4 6 8 10-4

-2

0

2

4

Pha

se, a

ngle

(Xn)


Note change of axes!


-10 -8 -6 -4 -2 0 2 4 6 8 100

0.2

0.4

0.6

0.8


Nor

mal

ised

Am

plitu

de, |

Xn|

*T/


-10 -8 -6 -4 -2 0 2 4 6 8 10-4

-2

0

2

4

Pha

se, a

ngle

(Xn)

Angular frequency (w) 10 April 2013 -ELEC 3004: Systems 34

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-10 -8 -6 -4 -2 0 2 4 6 8 100

0.2

0.4

0.6

0.8


Nor

mal

ised

Am

plitu

de, |

Xn|

*T/


-10 -8 -6 -4 -2 0 2 4 6 8 10-4

-2

0

2

4

Pha

se, a

ngle

(Xn)


-10 -8 -6 -4 -2 0 2 4 6 8 100

0.2

0.4

0.6

0.8


Nor

mal

ised

Am

plitu

de, |

Xn|

*T/


-10 -8 -6 -4 -2 0 2 4 6 8 10-4

-2

0

2

4

Pha

se, a

ngle

(Xn)


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• Fourier series– Only applicable to periodic signals

• Real world signals are rarely periodic• Develop Fourier transform by

– Examining a periodic signal– Extending the period to infinity

Fourier Transform


• Problem: as T , Xn 0– i.e., Fourier coefficients vanish!

• Solution: re-define coefficients– Xn’ = T x Xn

• As T – (harmonic frequency) nw0 w (continuous freq.)– (discrete spectrum) Xn’ X(w) (continuous spect.)– w0 (fundamental freq.) reduces dw (differential)

• Summation becomes integration

Fourier Transform


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Fourier Transform

dwjwtwXtx

wtjnwXtx

dtjwttxwX

dttjnwtxwX

dttjnwtxX

nT

T

TT

T

T

n

)exp()(2

1)(

2)exp(lim)(

)exp()()(

)exp()(lim)(

)exp()(

00

'

2/

2/

0

2/

2/

0'

Fourier transformX(w) = F{x(t)}

Inverse Fourier transformx(t) = F

-1{X(w)}

Note missing 1/T

Modified Fourier series

Inverse Fourier series

Note: as T n0

10 April 2013 - 39ELEC 3004: Systems

Note the similarity to Previous FS examples

j

ee jj

2sin

Using Euler’s relation


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Frequency Response - University of Queensland2 ELEC 3004: Systems 10 April 2013 - 3 Cool Signal...

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