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
elec3004@itee.uq.edu.auhttp://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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8
9
10
11
12
6
2
3
4
5
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ELEC 3004: Systems 10 April 2013 - 3
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
ELEC 3004: Systems 10 April 2013 - 4
Announcements:
!
3
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
10 April 2013 -ELEC 3004: Systems 6
4
• 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
10 April 2013 -ELEC 3004: Systems 7
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
10 April 2013 -ELEC 3004: Systems 8
5
• 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
10 April 2013 -ELEC 3004: Systems 9
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
10 April 2013 -ELEC 3004: Systems 10
6
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
10 April 2013 -ELEC 3004: Systems 11
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
10 April 2013 -ELEC 3004: Systems 12
7
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
10 April 2013 -ELEC 3004: Systems 13
ELEC 3004: Systems 10 April 2013 - 14
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
8
• 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
10 April 2013 -ELEC 3004: Systems 15
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
9
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
10 April 2013 -ELEC 3004: Systems 17
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
10 April 2013 -ELEC 3004: Systems 18
10
Complex Phasor Summation
10 April 2013 -ELEC 3004: Systems 19
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 !
10 April 2013 -ELEC 3004: Systems 20
11
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
10 April 2013 -ELEC 3004: Systems 21
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 / …
10 April 2013 -ELEC 3004: Systems 22
12
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
10 April 2013 -ELEC 3004: Systems 23
-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)
Fourier coefficient, n
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
10 April 2013 -ELEC 3004: Systems 24
13
-20 -15 -10 -5 0 5 10 15 200
0.05
0.1
0.15
0.2
0.25Duty Cycle /T = 1/4
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)
Fourier coefficient, n
4th, 8th, 12th,… = 0 Coefficients getting smaller!
10 April 2013 -ELEC 3004: Systems 25
-20 -15 -10 -5 0 5 10 15 200
0.05
0.1
0.15
0.2Duty Cycle /T = 1/8
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)
Fourier coefficient, n
8th, 16th,… = 0
10 April 2013 -ELEC 3004: Systems 26
14
-20 -15 -10 -5 0 5 10 15 200
0.02
0.04
0.06
0.08Duty Cycle /T = 1/16
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)
Fourier coefficient, n
16th, 32nd,… = 0
10 April 2013 -ELEC 3004: Systems 27
• 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)
10 April 2013 -ELEC 3004: Systems 28
15
• 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
10 April 2013 -ELEC 3004: Systems 29
• 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)
10 April 2013 -ELEC 3004: Systems 30
16
Example: Calculate Power in x(t)
Orthogonal 0
Total = sum of power in 2 sine waves
10 April 2013 -ELEC 3004: Systems 31
• 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
10 April 2013 -ELEC 3004: Systems 32
17
-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)
Angular frequency (w)
Note change of axes!
10 April 2013 -ELEC 3004: Systems 33
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.2
0.4
0.6
0.8
1Duty Cycle /T = 1/4
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)
Angular frequency (w) 10 April 2013 -ELEC 3004: Systems 34
18
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.2
0.4
0.6
0.8
1Duty Cycle /T = 1/8
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)
Angular frequency (w) 10 April 2013 -ELEC 3004: Systems 35
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.2
0.4
0.6
0.8
1Duty Cycle /T = 1/16
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)
Angular frequency (w) 10 April 2013 -ELEC 3004: Systems 36
19
• 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
10 April 2013 -ELEC 3004: Systems 37
• 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
10 April 2013 -ELEC 3004: Systems 38
20
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
10 April 2013 -ELEC 3004: Systems 40
21
ELEC 3004: Systems 10 April 2013 - 41
Summary!
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