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CHEE825 Fall 2005 J. McLellan 1
Spectral Analysis and Input Signal Design
CHEE825 Fall 2005 J. McLellan 2
Outline
• Fourier series• Fourier transforms
» continuous» discrete
• frequency spectrum• spectra and dynamic elements• input signal design in the frequency domain
CHEE825 Fall 2005 J. McLellan 3
Projection of Vectors
Suppose we have a vector u, and we want to approximate it as the sum of two orthonormal vectors v and w -
- optimal solution in least squares sense
where < , > is an inner product (dot product)• residual is orthogonal to space spanned by v and w
wwuvvuu ><+>≈< ,,
CHEE825 Fall 2005 J. McLellan 4
Projection of Vectors
Pictureu
v
w
space spanned by v and w
CHEE825 Fall 2005 J. McLellan 5
Periodic Functions and Inner Products
Defn: A function f(t) is periodic if f(t+T) = f(t).
Suppose the period of the function is T, and define the following inner product for functions defined on [-T/2,T/2]:
• functions on this interval are now the vectors • this inner product maps vectors (functions on the
interval) into real numbers
∫>=<−
2/
2/)()(
1,
T
Tdttgtf
Tgf
CHEE825 Fall 2005 J. McLellan 6
Orthogonal Bases of Periodic Functions
Example
{sin(kt), cos(kt)} form an orthogonal basis with respect to this inner product - with T= 2
Example
{ } form an orthonormal basis with respect to the following inner product
where overbar denotes complex conjugate, and
tjke ω
∫>=<−
2/
2/
___)()(
1,
T
Tdttggf
Tgf
ω2
=T
CHEE825 Fall 2005 J. McLellan 7
Approximation of Periodic Functions
If we have a function with period T, choose sinusoids (or complex sinusoids) with frequencies that are multiples of
in order align basis with period of function.
The orthogonal basis is now { }
Fourier series can be developed in terms of sines and cosines, or in terms of exponentials
» exponentials are more succinct, and I will use these.
Tω 2
0 =
tjke 0ω
CHEE825 Fall 2005 J. McLellan 8
Fourier Series
Given the orthogonal basis { }, we can approximate the periodic function f(t) with period T as
tjke 0ω
∫>==<=
∑≈
−
−
∞
−∞=
2/
2/0
00
0
)(1
),(,2
)(
T
T
tjkk
k
tjkk
tjketf
Tetfcand
T
where
ectf
ωω
ω
πω
CHEE825 Fall 2005 J. McLellan 9
Fourier Series
Conceptual analogy
wwuvvuu ><+>≈< ,,
v
w
space spanned by v and w
∫>==<
+≈
−
−2/
2/
221
00
00
)(1
),(
)(
T
T
tjkk
tjtj
tjketf
Tetfc
whereecectf
ωω
ωω
CHEE825 Fall 2005 J. McLellan 10
Fourier Series
• Issues– convergence– functions that can be approximated– definition of integral
CHEE825 Fall 2005 J. McLellan 11
What if f(t) is non-periodic?
Suppose f(t) is defined on an interval [-T/2, T/2] but is not periodic. To create a periodic function, define the periodic extension of f(t):
“cut and paste” to create periodic function defined on real line
⎩⎨⎧
±±=++−∈+−∈
=K,2,1],2/,2/[)(
]2/,2/[)()(*
llTTlTTtlTtf
TTttftf
CHEE825 Fall 2005 J. McLellan 12
Fourier Series of Non-Periodic Functions
Work with periodic extension:
∫>==<
>=<=
∑≈
−
−
∞
−∞=
2/
2/
*0
*
00
0
0
)(1
),(
),(,2
)(
T
T
tjk
tjkk
k
tjkk
tjketf
Tetf
etfcandT
where
ectf
ωω
ω
ω
πω
CHEE825 Fall 2005 J. McLellan 13
Towards the Fourier Transform
Note that if the fundamental frequency is
then and we can write the Fourier series for f*(t) as
Tω 2
0 =
0
2ω=T
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛∫=
∑≈
∞
−∞= −
−
∞
−∞=
k
tjkT
T
k
tjkk
eetf
ectf
tjk00
0
2/
2/
0
*
)(2
)(
ω
ω
ω
πω
CHEE825 Fall 2005 J. McLellan 14
Towards the Fourier Transform
What if we let ?
Now
∞→T
00 →⇒ ω
which is in the form of a Riemann sum
CHEE825 Fall 2005 J. McLellan 15
Towards the Fourier Transform
Riemann sum
Now0ωδ =
∫ ⎟⎟⎠
⎞⎜⎜⎝
⎛∫=
∫=∑=
∑=
∞
∞−
∞
∞−
−
∞
∞−
∞
−∞=→
∞
−∞=∞→
ω
ωω
ω
ωω
ωωω
ω
dedtetf
dee
etf
tjtj
tj
k
tjk
k
tjk
T
)(21
lim
lim)(
00
0
0
0
0
CHEE825 Fall 2005 J. McLellan 16
Fourier Transform
Given function f(t), define F(w) as the Fourier transform of f(t):
∫=ℑ=
∫=ℑ=
∞
∞−
−
∞
∞−
−
ωωω
ω
ω
ω
deFFtf
dtetftfF
tj
tj
)()(()(
)(21
))(()(
1
CHEE825 Fall 2005 J. McLellan 17
Fourier Transforms of Sampled Signals
Suppose that f(t) is sampled at a sampling period Ts, i.e., we have { f(kTs) }. In order to take the Fourier transform of the sampled signal, we can paste it back together using the impulse sampling representation:
∑ −=∞
−∞=kss kTtkTftf )()()(
~δ
paste together using“stick” functions - impulsefunction
CHEE825 Fall 2005 J. McLellan 18
Towards the Discrete Fourier Transform
Now take the Fourier transform of the impulse-sampled function:
∑=
∫ ⎟⎟⎠
⎞⎜⎜⎝
⎛∑ −=
∫=
∞
−∞=
−
∞
∞−
−∞
−∞=
∞
∞−
−
k
kTjs
tj
kss
tj
sekTf
dtekTtkTf
dtetfF
ω
ω
ω
π
δπ
πω
)(21
)()(21
)(~
21
)(
CHEE825 Fall 2005 J. McLellan 19
Discrete Fourier Transform
Given a sampled signal { f(kTs) },
Note that because of sampling, we have periodicity on the interval so we need only consider F(w) on this interval.
∫=
∑=
∞
∞−
∞
−∞=
−
ωω
ω
ω
ω
deFkTf
ekTfF
s
s
kTjs
k
kTjs
)()(
)(21
)(
],[ −
CHEE825 Fall 2005 J. McLellan 20
Energy in a Signal
Given a continuous signal f(t) defined on [-T/2,T/2], the squared magnitude (norm) of the signal can be defined in terms of the inner product:
In terms of the Fourier series representation of f(t), this yields
dttftfT
tftftfT
T∫>==<
− 2/
____2 )()(1)(),()(
∑=
>∑∑=<∞
−∞=kk
ktjk
ktjk
c
ecectf
2
2 00 ,)( ωω
where__
2kkk ccc =
Parseval’s Relation
CHEE825 Fall 2005 J. McLellan 21
Energy in a Signal
The Fourier coefficients describe the breakout of energy in the signal by frequency. We can plot them as a function of frequency.
E.g., for the sawtooth wave expansion -
Energy is distributedat low frequencies
CHEE825 Fall 2005 J. McLellan 22
Energy in a Signal - via Fourier Transform
We can examine the frequency content in a signal via the Fourier transform by plotting versus frequency.
We can follow a similar approach using the Discrete Fourier Transform (DFT) to analyze the energy distribution by frequency in a sampled signal.
2)(ωF
CHEE825 Fall 2005 J. McLellan 23
The Spectrum of a Stochastic Signal
The spectrum (spectral density) of a stochastic signal is defined as the discrete Fourier transform of its autocovariance function:
∑=
=Φ
∞
−∞=
−
k
kjy
yy
ek
kDFT
ωγπ
γω
)(21
))(()(
CHEE825 Fall 2005 J. McLellan 24
Properties of the Spectrum
1. Since the autocovariance is an even function
the spectrum is a real-valued function.
Why? - next slide
2. Since cos is an even function, the spectrum is symmetric about the origin.
)()( kk yy −=γγ
CHEE825 Fall 2005 J. McLellan 25
Why?
)}0())(cos((2{21
)}0())sin())(cos(())sin())(cos(({21
)}0())sin())(cos(())sin())(cos(({21
)(21
)(
1
1
1
1
1
yk
y
yk
yk
y
yk
yk
y
k
kjyy
ktk
ktjktkktjktk
ktjktkktjktk
ek
γωγ
γωωγωωγ
γωωγωωγ
γ
ω ω
+∑=
+−∑++∑=
++∑++∑=
∑=Φ
∞
=
−∞=
∞
=
−∞=
∞
=
∞
−∞=
−
CHEE825 Fall 2005 J. McLellan 26
Example - Spectrum of an MA(1) Disturbance
The MA(1) model is:
with autocovariance function:
)1()()( −−= tataty θ
⎪⎩
⎪⎨
⎧
>=−
=+=
10
1
0)1(
)( 2
22
kfor
kfor
kfor
k a
a
y θσ
σθ
γ
CHEE825 Fall 2005 J. McLellan 27
Example - Spectrum of MA(1) Disturbance
Taking the DFT:
Maple demo
CHEE825 Fall 2005 J. McLellan 28
Variance and the Spectrum
We can recover the variance of the signal from the spectrum.
Since we have the Fourier transform pair,
then
∫Φ=
∑==Φ
∞
∞−
∞
−∞=
−
ωωγ
γ
γω
ω
ω
dek
ekkDFT
kjyy
k
kjyyy
)()(
)(21
))(()(
∫Φ=∫Φ==∞
∞−
∞
∞−ωωωωσγ ω dde y
jyyy )()()0( 02
The variance is the area under the spectrum.
CHEE825 Fall 2005 J. McLellan 29
Spectra and Transfer Functions
For single-input single-output transfer functions, we have the following:
where
)()()(
)()()(2
ωω ωu
jy eG
tuqGty
Φ=Φ
=
)()()()()(______
2 ωωωωω jjjjj eGeGeGeGeG −==
CHEE825 Fall 2005 J. McLellan 30
Check - MA(1) Disturbance
Starting from the transfer function, we have
Since
we have
)cos(21
)1)(1()(
2
2
ωθθ
θθ ωωω
−+=
−−= −jjj eeeG
σωθθω2
))cos(21()(2
2 ay −+=Φ
σω2
)(2a
a =Φ
CHEE825 Fall 2005 J. McLellan 31
Sample Spectrum
The spectrum can be estimated from sample autocovariance functions - “sample spectrum”.
Note that the Fourier transform is taken over a finite record length– sample spectrum is an asymptotically unbiased estimator for
the spectrum, in part because the sample autocovariance is an asymptotically unbiased estimator for the autocovariance
Smoothing can be applied to the sample spectrum– involves weighting sample spectra values
∑=−
−−=
−1
)1()(ˆ
21
)(ˆn
nk
kjekf ωγπ
ω
CHEE825 Fall 2005 J. McLellan 32
Periodograms
… are direct analyses of the frequency content in a signal
» compute directly from data, instead of via the sample autocovariance function
• compute Fourier series over the finite data record length – requires inner product over the finite data record length, and
different complex exponential basis functions that are orthogonal over the data record
– basis functions are t
nk
je
2
CHEE825 Fall 2005 J. McLellan 33
Periodograms
Fourier series coefficients are computed for the series in terms these basis functions
• squared magnitudes of coefficients vs. frequency (term number)
• indicates frequency breakdown in data
CHEE825 Fall 2005 J. McLellan 34
The Cross-Spectrum
… is defined in a manner similar to the (auto)spectrum. Given the cross-covariance function between stochastic signals y and u,
Transfer function relationship:
kj
kyuyuyu ekkDFT ωγ
γω −∞
−∞=∑==Φ )(
21))(()(
)()()(
)()()(
ωω ωu
jyu eG
tuqGty
Φ=Φ
=
CHEE825 Fall 2005 J. McLellan 35
The Cross-Spectrum
Note - Since the cross-covariance function is not necessarily even (symmetric about the origin), in general the cross-spectrum will be complex-valued
Consider - • Co-spectrum - real component• Quadrature spectrum - imaginary component
OR…• Amplitude spectrum - magnitude• Phase spectrum - phase angle
CHEE825 Fall 2005 J. McLellan 36
Input Signal Design
Recall our general process+disturbance model:
and the prediction error formulation:
If our model structure is correct, then with perfect knowledge of the parameters, {e(t)} would be the random shocks driving the disturbance.
What happens if there are differences in model structure?
)()()()()( teqHtuqGty +=
))()()(()(
1)( tuqGty
qHte −=
CHEE825 Fall 2005 J. McLellan 37
Input Signal Design
Model Bias Case:
True plant -
where {a(t)} is an iid random shock sequence.
Examining the prediction errors under this scenario,
)()()()()( 00 taqHtuqGty +=
CHEE825 Fall 2005 J. McLellan 38
Input Signal Design
The spectrum of the residuals {e(t)} is related to that of the input and random shocks:
We can express the variance of the residuals as:
)()(
)()(
)(
))()(()(
20
20 ωωω ω
ω
ω
ωωaj
j
uj
jj
eeH
eH
eH
eGeG Φ+Φ−=Φ
ωωω ω
ω
ω
ωωd
eH
eH
eH
eGeGteVar aj
j
uj
jj
∫⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛Φ+Φ
−=
∞
∞−)(
)(
)()(
)(
))()(())((
20
20
CHEE825 Fall 2005 J. McLellan 39
Input Signal Design
Expanding
The impact of the input signal design on residual variance due to model bias is
∫+∫ Φ−=
∫ Φ+∫ Φ−=
∞
∞−
∞
∞−
∞
∞−
∞
∞−
ω
σωω
ωωωω
ω
ω
ω
ωω
ω
ω
ω
ωω
deH
eHd
eH
eGeG
deH
eHd
eH
eGeGteVar
j
ja
uj
jj
aj
j
uj
jj
20
220
20
20
)(
)(2
)()(
))()((
)()(
)()(
)(
))()(())((
ωωω
ωωd
eH
eGeGteVar uj
jj
∫ Φ−
=∞
∞−)(
)(
))()(())((
20
CHEE825 Fall 2005 J. McLellan 40
Bias-Oriented Designs
If we choose a low frequency test signal, we are penalizing the prediction of low frequency process behaviour the greatest
» bias will be reduced in low frequency range
Recall our design cost function discussion earlier ...
CHEE825 Fall 2005 J. McLellan 41
Experimental Design Objective
Design input sequence to minimize the following:
design
cost
error in
predicted frequency response
importance
function
our designobjectives
difference in predicted vs.true behaviour- function of frequency, andthe input signal used
CHEE825 Fall 2005 J. McLellan 42
Accounting for Model Error - Interpretation
Optimal solution in terms of frequency content:
spectral density
frequencyerror in model vs.true process
spectral density
frequency
importance to ourapplication
low
highvery important
not important*J=
CHEE825 Fall 2005 J. McLellan 43
Accounting for Model Error
Consider frequency content matching
Goal - best model for final application is obtained by minimizing J
J G e G e C j dj T j T
frequencyrange
∫ $( ) ( ) ( )ω ω ω ω2}
bias in frequencycontent modeling
}
importanceof matching- weightingfunction
CHEE825 Fall 2005 J. McLellan 44
Example - Importance Function for Model Predictive Control
spectral density
frequency
high frequency disturbance rejectionperformed by base-levelcontrollers- > accuracy not importantin this range
require good estimateof steady state gain,slower dynamics
CHEE825 Fall 2005 J. McLellan 45
Bias-Oriented Designs in Closed-Loop ID
Recall from earlier that in the case of a dither signal w(t) introduced in closed-loop experimentation, our input is effectively
The model is estimated between u(t) and y(t) pretending that it is open-loop:
and the prediction error expression is again
)()()(1
1)( tw
qGqGtu
c+=
)()()()()( teqHtuqGty +=
))()()(()(
1)( tuqGty
qHte −=
CHEE825 Fall 2005 J. McLellan 46
Bias-Oriented Designs in Closed-Loop ID
Now the output in this instance is
and this leads to
which can be simplified to:
)()()(1
)()(
)()(1)(
)(0
0
0
0 taqGqG
qHtw
qGqGqG
tycc +
++
=
)()()(1
)()(
1)()()(
)()(1)(
)(1
)(0
0
0
0 taqGqG
qHqH
tuqGtwqGqG
qGqH
tecc
⎟⎟⎠
⎞⎜⎜⎝
⎛+
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
+=
)()()(1
)()(
1)(
)()(1)()(
)(1
)(0
0
0
0 taqGqG
qHqH
twqGqG
qGqGqH
tecc
⎟⎟⎠
⎞⎜⎜⎝
⎛+
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
CHEE825 Fall 2005 J. McLellan 47
Bias-Oriented Designs in Closed-Loop ID
The variance of the residuals is:
The imposition of closed-loop experimentation essentially introduces additional weighting by the “sensitivity function” of the true plant:
∫+
+
∫ Φ+
−=
∞
∞−
∞
∞−
ω
σ
ωω
ωωω
ω
ωωω
ωω
deGeGeH
eH
deGeGeH
eGeGteVar
jc
jj
ja
wjc
jj
jj
2
0
02
2
0
0
))()(1)((
)(2
)())()(1)((
))()(())((
)()(11
0 qGqG c+
CHEE825 Fall 2005 J. McLellan 48
Bias-Oriented Designs in Closed-Loop ID
Running the experiment in closed-loop introduces a component in the “importance function” that weights frequency prediction in the range associated with disturbance rejection / setpoint tracking.