-
Time domain identification, frequency domain
identification.Equivalencies! Differences?
J. Schoukens, R. Pintelon, and Y. RolainVrije Universiteit
Brussel, Department ELEC, Pleinlaan 2, B1050 Brussels, Belgium
email: [email protected]
Abstract
- In the first part of this paper, the full equiva-lence between
time and frequency domain identification isestablished. Next the
differences that show up in the practi-cal applications are
discussed. Finally, an illustration on theidentification of a
servo-system in feedback is given.
I. INTRODUCTION
For a long time, frequency domain identification and timedomain
identification were considered as competingmethods to solve the
same problem: building a model for alinear time-invariant dynamic
system. In the end, thefrequency domain approach got a bad
reputation because thetransformation of the data from the time
domain to thefrequency domain is prone to leakage errors: noiseless
datain the time domain resulted in noisy frequency responsefunction
(FRF) measurements. This is illustrated in thesimulation below. A
system is excited with a random input.The input and the output are
sampled in 256points with . No disturbingnoise is added. Starting
from these measurements, the FRF
is measured and compared to the true FRF (FIG.1.). It can be
seen that is strongly disturbed. This was themajor reason to drop
the frequency domain approach. Thestatement: Why would we move from
the time to thefrequency domain? The only thing we buy for it
areproblems! expresses very well the feeling that lived in
theidentification society.
This problem is further analysed in Section III where itis shown
that i) exactly the same problem is present in thetime domain; ii)
by extending the models, a full equivalenceexists between both
domains.
Once this equivalence between both domains is estab-lished, one
can wonder why to bother about it? Are thereany differences at all?
The answer is definitely yes.Although both domains carry exactly
the same information,it may be more easy to access this information
in one
u0 t( ) y0 t( )u0 k( ) y0 k( ), k 1 2 … 256, , ,=
Ĝ jωl( ) G0
-40
-20
0
20
0 0.1 0.2 0.3 0.4 0.5
Am
plitu
de (
dB)
Frequency
FIG. 1. Comparison of the true FRF (__)with the estimatedFRF
(+)
G0Ĝ
Ĝ
domain than in the other because the same information
isrepresented differently.
When discussing the differences between time- and fre-quency
domain methods, a clear distinction should be madebetween the
aspects that are intrinsically due to the fre-quency domain
formulation, and on the other hand the addi-tional signal
processing possibilities that are opened byusing periodic
excitations. The latter might also be usefulfor time domain
identification methods.
The practical aspects that take advantage of the fre-quency
domain formulation are: arbitrary selection of theactive
frequencies where the model is matched to the meas-urements;
continuous-time modelling; and identification ofunstable models. By
including also periodic excitations, wewill be able to address in
addition: the use of nonparametricnoise models; separation of plant
and noise model estima-tion; errors-in-variables identification and
identificationunder feedback; separation of nonlinear distortions
and(process) noise. These aspects are discussed in Section IVand V.
A comprehensive discussion of other aspects can befound in
(Pintelon and Schoukens, 2001; Ljung, 2004).
In Section VI, an extensive case study on the closedloop
identification of a compact disc servo-system is pre-sented. A
frequency domain approach using periodic excita-tions is made. Many
of the aspects mentioned before areillustrated.
II. SETUP
The discussion in this paper is completely focused
onsingle-input-single-output linear systems (SISO), but thereader
should be aware that many results can be directlyextended and
generalized to multiple-input-multiple-output(MIMO) systems, and
even a nonlinear behaviour can beincluded in the framework
(Schoukens
et al.
, 2003). Consider the discrete time system
, (1)
with , and . The aim ofsystem identification is to extract the
best model for the plant , and at times the disturbing noise
powerspectrum .
III. TRANSIENTS: THE KEY TO THE EQUIVALENCE OF TIME AND
FREQUENCY DOMAIN
In practice the identification should be done from a finite
setof measurements
, . (2)
The description (1) should be extended to include theimpact of
reducing the observation window from
y t( ) G0 q( )u0 t( ) H0 q( )e t( )+ y0 t( ) v t( )+= =
t : ∞– ∞→ x t 1–( ) q 1– x t( )=G q θ,( )
G q( )H q θ,( ) 2
u0 t( ) y t( ), t 0 1 … N 1–, , ,=
-
to . This is illustrated in FIG.2. The records are split in
three parts: the observation win-dow, the preceding and following
unobserved signals. Twoeffects can be seen:
i) The past excitation (before the start of the observation)
contributes to the output in the observation window:
a tail is added to the observed output (FIG. 2.a). Theseeffects
are well known in the time domain, a transient term
has to be added to describe the contributions of theinitial
conditions to the output.
ii) When the data are processed in the frequency domain,it is
implicitly assumed that the observed signals are periodi-cally
repeated. At that moment not only the tail (a) is addedto the
output, also the tail (b) will be missing. Although thislooks more
complicated than the previous situation it leadseventually to
expressions that are completely equivalent tothe time domain
description. Both extended models aregiven below.
A. Extended time domain description
The reduction from to addsinitial conditions effects on the
output of the dynamicsystems (plant and noise filter). These are
described by theplant and the noise filter transients . Both
decayexponentially to zero.
, . (3)
B. Extended frequency domain description
The finite data records aretransferred to the frequency domain
using the discreteFourier transform (DFT):
, (4)
with , and . Although the spectra are affected by leakage, their
relation remains
remarkably simple: it is again sufficient to add a transientterm
to the system equations, completely similar to what is
done in the time domain (Pintelon
et al.
, 1997; Pintelon andSchoukens, 2001). This term describes the
impact of the tails(a) and (b) on the input-output relation.
. (5)
The relation between and is:
,
with . (1)
The reader should be aware that this expression is valid
forarbitrary signals, no periodic excitation is required.
Loosely spoken, the impact of the transients disappearsat a rate
of or faster. In practice the noise transient
is always omitted. For simplicity we do the same in
thispaper.
C. Equivalence between time- and frequency domain
During the identification step, parametric plant- and
noisemodels are identified by minimizing the squared
predictionerrors. This can be done in the time- or in the
frequencydomain (Ljung, 1999; S derstr m, and Stoica, 1989):
( stands for the data). (6)
Both expressions result in exactly the same value for thecost
function, and this for arbitrary excitations. This estab-lishes the
full equivalence between the time- and frequencydomain formulation
of the prediction error framework.
Remark: The transfer function model and the transientmodel have
very similar expressions:
, (7)
Hence the inclusion of the transient term does not increasethe
complexity of the cost function because both rationalforms have the
same denominator.
D. Differences?
Although we established a full equivalence between thetime- and
frequency domain formulation, there are someremarkable differences
in the practical use of theseexpressions.
1) Unstable plant models: If an unstable plant model
isidentified, the time domain calculations are tedious and
oftenimpossible because the calculation noise explodes. FromForsell
and Ljung (2000) it is known that the predictorshould be stable .
In the frequency domain, this prob-lem is not a problem because
only multiplications of thefinite DFT-spectra are made, and the
stability of the models
t : ∞– ∞→ t 0 … N 1–, ,=
-3
0
3
0 200 400-3
0
3
0 200 400
-3
0
3
0 200 400-3
0
3
0 200 400
-3
0
3
0 200 400-3
0
3
0 200 400
-3
0
3
0 200 400-3
0
3
0 200 400
u(t)
y(t)
u(t)
u(t)
u(t)
y(t)
y(t)
y(t)
observation window
FIG. 2. : Illustration of the effect of the finite observation
window.
(a)
(b)
t 0=
TG t( )
t : ∞– ∞→ t 0 … N 1–, ,=
TG t( ) TH t( ),
y t( ) G0 q( )u0 t( ) H0 q( )e t( ) TG t( ) TH t( )+ + += t
0≥
{u0 t( ) y t( ), : t 0 … N 1}–, ,=
X k( ) 1
N-------- x t( )e
j2πN------ tk–
t 0=N 1–
∑= k 0 1 … N 1–, , ,=
x u y e, ,= X U Y E, ,=U0 Y0,
Y0 k( ) G0 zk1–( )U0 k( ) TG k( )+=
U0 E, Y
Y k( ) G0 zk1–( )U0 k( ) H0 zk
1–( )E k( ) TG k( ) TH k( )+ + +=
zk ej2πk N⁄=
N 1 2/–
TH
VPE θ Z,( ) H1– q θ,( ) y t( ) G q θ,( )u0 t( )– TG q θ,( )–(
)
2
t 0=
N 1–
∑=
Y k( ) G zk1– θ,( )U0 k( )– TG zk
1– θ,( )– 2
H zk1– θ,( ) 2
-----------------------------------------------------------------------------------------k
0=
N 1–
∑=
Z
G zk1– θ,( )
bnzn–
n 0=
nb∑
anzn–
n 0=
na∑------------------------------= TG zk
1– θ,( )inz
n–n 0=
max na nb,( )∑
anzn–
n 0=
na∑----------------------------------------=
H 1– G
-
is not at the order, even not for the intermediate models
dur-ing the iteration process (The zeros of should be kept inthe
unite circle!).
2) It is very easy in the frequency domain to restrict thesum to
a selected set of frequencies, for example the fre-quency band of
interest; to those frequencies with a goodsignal-to-noise ratio; or
to eliminate spurious components.Although this should not affect
the prediction error result (itcan be interpreted as a maximum
likelihood estimator, andthrowing away information does never
improve the result), ithas a strong impact on the calculation of
initial estimateswith sub optimal (linear) methods like ARX or
subspaceidentification.
IV. PERIODIC SIGNALS: A FREE ACCESS TO THE NONPARAMETRIC NOISE
MODEL
A. Introduction
The prediction error method can be considered as thesolution of
a weighted least squares problem:
(8)
with , , the covariance matrix of . The major problem is
that is a large dense matrix that should be inverted whichis
impractical. This is nicely circumvented in the predictionerror
method by whitening the residuals with a parametricnoise model ,
leading to
. (9)
This is exactly the time domain form of in (5). Inthe frequency
domain the matrix inversion does not comeinto play. The covariance
matrix of the frequencydomain noise is asymptotically ( ) a
diagonal matrix:
, (10)
with . This gives an alternative fre-quency domain
representation of the cost function:
.(11)
The full details and the proof of the equivalence of (5) and(11)
are given in Schoukens et al. (1999).
B. Taking advantage of periodic excitations
In the classical prediction error framework, the plantmodel and
the noise model are identifiedsimultaneously because this is the
only possibility to sepa-rate the signal and the noise . However,
if the exci-tation is periodical, , it is possible to collect
successive periods, and to average the measurementsover these
repeated periods. This process is exemplified forthe output
measurement in FIG. 3.:
,
. (12)
The sample variance is a nonparametric estimate of. Substituting
it in the frequency domain expression of
(11) gives:
. (13)
This results in a two step procedure: i) The nonparametricnoise
model is determined in the pre-processing step, ii) the plant model
is estimated inthe 2nd step, keeping the noise model fixed.
C. Discussion
This approach has many advantages: i) It is no longerrequired to
estimate plant and noise model simultaneously;ii) Even before
starting the identification process, it ispossible to verify the
quality of the raw data as is illustratedin the example of Section
VI; iii) The estimated noise modelis no longer influenced by the
plant model errors; iv) Thismethod can be extended to the
errors-in-variables problem.This includes identification in
feedback as a special case.
The price to be paid for all these advantages is therestriction
to periodic signals and the need for multiple peri-ods to be
measured, resulting in a frequency resolution loss.However, the
required number of periods can be small,for example is enough for
consistency, and reduces the efficiency loss to less than 33% in
variance(Schoukens et al., 1997).
D. Extension: the errors-in-variables problem
In some applications, there is not only process noise on
theoutput. Also the input measurements can be disturbed bynoise, as
is for example the case for identification in
feedback. The nonparametric noise model is extended withthe
input noise variance and the input-outputcovariance ( denotes the
complex conjugate):
H
V v̂T
Cv1– v̂=
v̂T
v̂ 1( ) … v̂ N( )= v̂ k( ) y t( ) G q θG,( )u0 t( )–=Cv N N×
v
Cv
ε t( ) H 1– q θ,( )v t( )=
VPE θ Z,( ) εTε=
VPE θ Z,( )
CVN ∞→
CV diag σY2 1( ) … σY
2 N( )( )=
σY2 k( ) H ejωk( ) 2=
VPE θ Z,( )Y k( ) G zk
1– θ,( )U0 k( )– TG zk1– θ,( )– 2
σY2 k( )
-----------------------------------------------------------------------------------------k
0=
N 1–
∑=
G q θ,( ) H q θ,( )
y0 t( ) v t( )u t T+( ) u t( )=
M
Ŷ k( ) 1M----- Y l[ ] k( )
l 1=M
∑=
σ̂Ŷ2 k( ) 1
M----- 1
M 1–-------------- Y l[ ] k( ) Ŷ k( )–
2
l 1=M
∑=
u t( )
y 1[ ] t( ) y 2[ ] t( ) y l[ ] t( )t
FIG. 3. Processing periodic excitations: is the period.y l[ ] t(
) lthFIG. 3. Processing periodic excitations: is the period.y l[ ]
t( ) lth
… …
σ̂Y2 k( )
σY2 k( )
VPE
VPE θ Z,( )Ŷ k( ) G zk
1– θ,( )Û k( )– TG zk1– θ,( )–
2
σ̂Y2 k( )
--------------------------------------------------------------------------------------k
0=
N 1–
∑=
σ̂Y2 k( ) var Y k( )( )=
G zk1– θ,( )
MM 4= M 6=
u0 t( ) y0 t( ) y t( )
u0 t( ) y0 t( ) y t( )
ny t( )
v t( )
u t( )
nu t( )
FIG. 4. : Standard and alternative noise assumption.
+
+
+
plant
plantAlternative
Standard
FIG. 4. : Standard and alternative noise assumption.
σ̂U2 k( )
σ̂YU2 k( ) x
-
(14)
The corresponding cost function to be minimized is thesample
maximum likelihood (Pintelon and Schoukens,2001):
(15)
The use of nonparametric noise models combined with
iden-tification in feedback is illustrated in SectionVI.
V. FREQUENCY DOMAIN IDENTIFICATION: A HIGHWAY TO CONTINUOUS
TIME
IDENTIFICATION
A. Arbitrary excitation signals
In the last theory section of the paper, we combine
theadvantages of the time- and frequency domain
identificationapproaches. A continuous-time modelling procedure
isproposed using arbitrary excitations. The relation betweenthe
spectra of band limited measurements is the continuous-time
model(see FIG. 5):
. (16)
By combining this with a discrete time noise model, a
mixedBox-Jenkins continuous time identification method is foundthat
is based on the minimization of the cost function (Pinte-lon et
al., 2000):
. (17)
The major advantages of this method compared to clas-sical
continuous time identification techniques are that: i)The need for
approximate differentiation or integration iscompletely removed;
ii) There is no need for huge oversam-pling rates, the full
bandwidth can be used; iii) A noisemodel is included which
increases the efficiency; iv) It is alogical extension of the well
known Box-Jenkins identifica-tion method.
The major drawback compared with the discrete timeBox-Jenkins
method is the loss of consistency in feedbackwhich is due to the
mixture of a discrete- and continuous-time model.
B. Periodic excitation signals
For periodic excitation signals the sample maximumlikelihood
method (15) can be used without any modifica-tion to identify a
continuous-time model.
VI. CASE STUDY: IDENTIFICATION OF A SERVO- SYSTEM IN CLOSED LOOP
OPERATION
In this example we illustrate all the aspects that werediscussed
before:
- use of periodic signals - extraction of a non parametric noise
model - use of a selected set of frequencies - removal of large
spurious components- separation of the signals, the noise, and the
nonlinear
distortions- identification of a continuous or discrete time
model in
feedback using the errors-in-variables frame work-
identification of an unstable model (due to nonlinear
distortions);
A. Experimental set-up
The open loop transfer function of the radial posi-tion
servo-system of a CD player is identified. Figure 6shows a
simplified block diagram of the compact disc (CD)
player measurement setup. The block stands for the cas-cade of a
power amplifier, a lowpass filter, the actuator sys-tem and,
finally, the optical position detection system. Theblock stands for
the parallel implementation of a lead-lagcontroller with some
additional integrating action, that stabi-lizes the unstable
actuator characteristics and takes care forthe position control. In
order to excite and to measure theopen loop transfer function, two
operational amplifiers havebeen inserted in between the lead-lag
controller and thepower amplifier at the input of the process .
B. Need for closed loop identification
The actuator transfer function represents the dynamics of thearm
moving over the compact disc, and is, in a firstapproximation,
proportional to . In practice, due to thefriction, the double pole
at the origin moves into the left halfplane to a highly underdamped
position. This explains whythe characteristics of the position
mechanism of a CD-playeris very hard to measure in open loop.
C. Use of the errors-in-variables framework
An external reference signal is injected in the loop,
theresulting signals are measured (the input is not exactlyknown).
Moreover, the loop is also disturbed by the processnoise , mainly
induced by tracking irregularities due topotato shaped spirals; non
eccentric spinning of the disc;
σ̂YU2 k( ) 1
M----- 1
M 1–-------------- Y l[ ] k( ) Ŷ k( )–( ) U l[ ] k( ) Û k( )–(
)
l 1=M
∑=
VML θ Z,( )
Ŷ k( ) G Ωk θ,( )Û k( )–2
σ̂Y2 k( ) σ̂
U2 k( ) G Ωk θ,( )
2 2Re σ̂YU2 k( )G Ωk θ,( )( )–+
------------------------------------------------------------------------------------------------------------------------k
1=
F
∑=
ZOH plant plant
AAAA
ZOH- BL-
u t( )
u t( )
y t( )
y t( )
ucyc yc
FIG. 5. : Ideal ZOH- and BL-setup. AA: an ideal anti-alias
filter.
G s θ,( ) bnsn
n 0=
nb∑ ansnn 0=na∑⁄=
VPE θ Z,( )Y k( ) G s θ,( )U k( )– TG zk
1– θ,( )– 2
H zk1– θ,( ) 2
--------------------------------------------------------------------------------k
0=
N 1–
∑=
GC
FIG. 6. Setup of the CD measurements.
r u G C y-
++
d
G
C
CG
1 s2⁄
ru y,
d
-
dirt, stains and scratches on the disc surface. The
followingrelations exist between the Fourier spectra (assuming
thatthey all exist):
(18)
In the absence of disturbances, the open loop transfer func-tion
between and is and it is the aim of thissection to provide a
parametric model for it. A periodic refer-ence signal will be
applied, the contributions
are considered as noise.Hence the input and output measurements
are disturbed bymutually correlated noise which fits perfectly in
the errors-in-variables approach.
D. Design of a periodic excitation, use of a well selected
set
of frequencies
For the sake of control, a 582.5 Hz sinusoidal wobble signalis
internally injected in the feedback loop. It is measured
atdifferent points in the electronic circuit, and serves as aninput
signal for an automatic gain controller (AGC), tocompensate,
amongst other things, for the effect that thedisplacement of the
arm is not perpendicular to the trackover the whole disc, resulting
in a variable gain of theprocess. The wobble signal complicates the
measurementprocess significantly, since it is more than 20 dB above
thenormal signal levels in the loop. For this reason, we had tomake
a careful experiment design to eliminate its impact onthe
measurements.
As an external reference signal a multisine with ,
, , and, is used (Schoukens and Pintelon, 2001).
The frequencies are selected to allow for the detection
ofnonlinear distortions at the non-excited frequencies.
Themultisine is generated with a clock frequency of
and points in one period.
E. Preprocessing: extraction of the signals and noise levels
In the first experiment, 256 K points are measured. The
longrecord is broken in 16 blocks of 4 basic periods each( ). This
is done in order to reduce the leakage effectof the wobble signal
on the rest of the spectrum. Themeasurement window does not contain
an integer number ofperiods of the wobble signal since its
frequency is notsynchronized to the measurement system (Figure 7).
In thisfigure, it can be seen that the contribution of the
referencesignal is clearly above the disturbances level. Also
thewobble signal (with its leakage) is clearly visible,
itsamplitude is more than 20 dB above the signals of interest.
Starting from the 16 repeated spectra, are esti-mated, together
with the (co-)variances andshown in Figure 8. It turns out that
there is an extremely highcorrelation ( ) between the noise on and
. From (18), it is seen that this indicates that the proc-ess noise
dominates completely the measurement noise
F. Quantifying the nonlinear distortions
Checking the non-excited odd frequencies in Figure 7 seemsto
indicate the presence of odd nonlinear distortions(Schoukens et
al., 1998; Pintelon and Schoukens, 2001;Schoukens et al. 2003), but
they are almost completelyhidden under the noise level of the test.
Hence, a secondexperiment with a reduced set of frequencies:( with
) is made. This does not affectthe relative level of the
nonlinearities with respect to thelinear contributions, but it
increases the SNR (Pintelon andSchoukens, 2001). From the results
in Figure 9 it is seen thatthe odd distortions are now well above
the noise level of thetest. Especially at the lower frequencies a
very highdistortion can be seen, indicating that the linearised
models
U jω( ) 11 G jω( )C jω(
)+--------------------------------------R jω( ) C jω( )
1 G jω( )C jω( )+--------------------------------------D jω(
)–=
Y jω( ) G jω( )C jω( )1 G jω( )C jω(
)+--------------------------------------R jω( ) C jω( )
1 G jω( )C jω( )+--------------------------------------D jω(
)+=
u y G jω( )C jω( )
r t( )C jω( )D jω( )( ) 1 G jω( )C jω( )+( )⁄
r t( ) Ak 2πf0lkt ϕk+( )sink 1=F∑= F 305=
f0 2.3842 Hz= lk 1 3 9 11 17 19 25 27 …, , , , , , , ,=Ak
constant=
10 MHz 210⁄ N 4096=
M 16=
Û Ŷ,σ̂U
2 σ̂y2 σ̂YU
2, ,
σYU2 σU
2 σy2⁄ 1–≈ U
Y
Input Output
-100
-80
-60
-40
-20
0 2500 5000
Am
plitu
de (
dB)
Frequency (Hz)
-100
-80
-60
-40
-20
0 2500 5000
Am
plitu
de (
dB)
Frequency (Hz)
Signal
Signal
Disturbances Disturbances
Wobble Wobble
FIG. 7. Pilot test with a special odd multisine signal.
-100
-80
-60
-40
0 1000 2000 3000
Am
plitu
de (
dB)
Frequency (Hz)
signal
noise
FIG. 8. SNR of the output measurements after processing the
rawdata.
Y
σ̂Y
f0 19.07 Hz= F 39=
-100
-80
-60
-40
-20
0 1000 2000 3000A
mpl
itude
(dB
)Frequency (Hz)
Linear
Cubic
FIG. 9. nonlinearity test: power on the FRF
measurementfrequencies (linear), detected distortions after
compensationfor the linear feed through (cubic), --- .σY
-
will have poor value in this frequency band.
G. Identification of a continuous or discrete time model
After these non-parametric tests, we have already a goodidea
about the limiting quality of the model. For the giveninput power,
the nonlinearities are for sure less than 30 dBbelow the linear
output. Next a linear model is identified thatapproximates the
system as good as possible. A 24th orderdiscrete-time or continuous
time model ( )was identified (both give very similar results). The
measuredFRF is compared to the parametric model in Figure 10. Ascan
be seen, a very good fit is obtained. The residuals arebelow the
noise level. Only at the low frequencies, where thenonlinearities
detected in the nonlinearity test are very large,the fit is poor.
Because we knew in advance that in this bandthe data are of poor
quality, the frequencies below 230 Hzwere not considered during the
fit. The cost function of thefit is 842.6, while a theoretical
value of 256.5 is expected.This points to model errors. However,
the auto-correlation ofthe residuals is white, therefore we can
conclude that the bestlinear approximation is extracted. The
remaining errors aredue to the nonlinear behaviour of the
process.
A stability analysis showed that two poles of the modelwere
unstable ( ), but the correspond-ing closed loop model is stable
and hence the model is valu-able for a closed loop analysis. This
instability is due to thefact that the system has 2 poles, almost
equal to one (doubleintegration in -domain), that are very
difficult to estimatedue to the presence of the nonlinearities in
this band.
VII. CONCLUSIONS
In this paper the equivalencies and differences of time-
andfrequency domain identification are discussed. Thisdiscussion
should not be mixed up with the use of periodicexcitations.
The major conclusion is that there exist a full equiva-lence
between both approaches from theoretic and informa-tion point of
view. Transforming data from the time to the
frequency domain does neither create or delete
information.Hence, what can be done in one domain can also be done
inthe other domain. However, in practice it might be easier
toaccess the information in one of both domains. Arbitraryselection
of active frequency bands, continuous time model-ling, and
identification of unstable models are typical exam-ples.
It is also shown that the restriction to periodic inputsopens a
number of practical possibilities and this for time-and frequency
domain identification. It does not only allowto extract
nonparametric noise models, it also simplifies sig-nificantly the
identification under feedback conditions andgives access to a
nonlinear distortion analysis.
VIII. ACKNOWLEDGEMENT
This work was supported by the Flemish government (GOA-ILiNos),
and the Belgian government as a part of the Belgianprogram on
Interunivrsity Poles of Attraction (IUAP V/22).
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[11] S derstr m, T. and P. Stoica (1989). System
Identification.Prentice-Hall, Englewood Cliffs
na nb 24= =
-60
-20
20
0 1000 2000 3000
Am
plitu
de (
dB)
Frequency (Hz)
0
90
180
270
0 1000 2000 3000
Phas
e (d
eg.)
Frequency (Hz)
FIG. 10. : Comparison of the estimated transfer function (full
line)with the measured FRF (dots). The residuals (+) are compared
tothe 95% noise level (thin full line).
z 1.021 j0.00344±=
z
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