Optimal Settings for Measuring Frequency Response ...

3276 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 58, NO. 9, SEPTEMBER 2009

Optimal Settings for Measuring FrequencyResponse Functions With Weighted

Overlapped Segment AveragingJérôme Antoni and Johan Schoukens, Fellow, IEEE

Abstract—This paper investigates the measurement errors in-volved in estimating frequency response functions (FRFs)—andrelated quantities such as the coherence function—from weightedoverlapped segment averaging, a technique that has become astandard in modern data analyzers due to its computational ad-vantages. Particular attention is paid to leakage errors, for whichthis technique has frequently been criticized. Our main result isthat a half-sine or diff window with about 2/3 overlap achieves thebest compromise to reduce leakage errors in the case of stationaryrandom excitations, and this is independently of the system FRF.This conclusion is to be contrasted with the customary habit ofusing a Hanning window with 1/2 overlap. The same reasoningconfirms that a rectangular window without overlap minimizesmeasurement errors in the special case of multisine excitations.Moreover, practical formulas are provided to the reader forcomputing the bias and variance of the frequency response andcoherence functions in the general case.

Index Terms—Frequency response function (FRF), optimalwindow, spectral leakage, transient effects, weighted overlappedsegment averaging (WOSA).

I. INTRODUCTION

THE MEASUREMENT of frequency response functions(FRFs) from experimental data is of continuous interest

in numerous engineering fields and, more specifically, in anyapplication where one is concerned with inferring the structureof an unknown linear system from its measured inputs andoutputs. Although the parametric identification of the systemis usually the ultimate objective—on which there exists avast literature—the nonparametric estimation of FRFs offers asimple and yet very useful preliminary description of the systemin the frequency domain. For instance, measured FRFs—orrelated quantities such as the coherence function—can be usedto better understand a system order or to check its (non)-linearity. Modern FRF measurement techniques are extensivelybased on the discrete Fourier transform (DFT), owing to its veryefficient computation by means of the fast Fourier transform(FFT) algorithm. This flexibility, plus the constantly increasingcapacity of computational devices, has led to the proposal ofseveral estimators whose relative optimality strongly dependson the statistical nature of the excitation signals (i.e., inputs; see

Manuscript received May 6, 2008; revised April 9, 2009. Current versionpublished August 12, 2009.

J. Antoni is with the Laboratory Roberval of Mechanics, University ofTechnology of Compiègne, 60205 Compiègne, France (e-mail: [email protected]).

J. Schoukens is with the Department of Electrical Engineering, Vrije Univer-siteit Brussel, 1050 Brussels, Belgium.

Digital Object Identifier 10.1109/TIM.2009.2022376

[1] for a comprehensive overview on the subject). Surely, thebest possible choice is to use the maximum-likelihood estimatortogether with periodic excitation signals, since only in that casecan the DFT be exactly used without finite-length effects. Theremay be, however, some technical or psychological reasonsthat lend the experimenter to prefer other types of excitations.Among those, a popular subclass is encompassed by stationaryrandom signals (e.g., binary signal, random uniform, and ran-dom Gaussian). Different FRF estimators have been proposedover the years in this context, which essentially find their rootsin experimental spectral analysis. They all rely on variousfrequency smoothing or averaging strategies, with the funda-mental assumption (hope) that the estimator variability can sobe stabilized without distorting too much the smooth structureof the FRF. In any case, this obviously leads to a difficultbias/variance tradeoff, which nonparametric FRF measurementtechniques have always been criticized for.

In this paper, we focus on the so-called weighted overlappedsegment averaging (WOSA) or Welch’s procedure. The WOSAprocedure consists of segmenting the available records intoseveral short-length segments, computing their DFTs, and thenaveraging their products to get a stable estimator of the systemFRF. Although the idea of the WOSA procedure is quite old[2], it has surprisingly received very little attention in thescientific literature. When it comes to the FRF measurement,its exact statistical performance and optimal settings remainvirtually unknown. It is the objective of this communication topartially fill in these gaps. In particular, we address importantand practical questions.

1) Which window shape should we use to taper the data?2) How much can we gain from overlapping adjacent

segments?3) Which optimal amount of overlap should we set between

adjacent segments?

The answers to these questions will obviously provide theexperimenter with some useful guidelines to make optimalusage of the WOSA technique in FRF measurements.

This paper is organized as follows. In Section II the principleand limitations of the WOSA procedure are reminded.The statistical errors involved by the WOSA procedure arequantified in Section III, in terms of systematic and stochasticerrors due to leakage and measurement noise. From theseresults, the optimal settings of the WOSA parameters are thendeduced in Section IV. Section V shows that the conclusionsremain unchanged for the WOSA estimation of other spectral

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ANTONI AND SCHOUKENS: OPTIMAL SETTINGS FOR MEASURING FRFs WITH WOSA 3277

Fig. 1. Plant model.

quantities such as the coherence function. Section VI provesthat the same reasoning can be applied to the special case ofmultisine excitations, although the optimal settings are thenfound to be very different. Finally, Section VII presents somenumerical experiments. The proofs of our main results aresketched in the Appendix.

II. FRF MEASUREMENT WITH THE WOSA PROCEDURE

A. Classical Approach

Let us consider a pair of input/output stationary random sig-nals {u(n); y(n)}, n ∈ Z, and an assumed linear time-invariantrelationship, i.e.,

y(n) =∞∑

k=0

g(k)u(n − k) + νy(n) (1)

where g(k) stands for the (causal) system impulse response,and νy(n) for the additive measurement noise at the output(see Fig. 1). The measurement of the system FRF then con-sists of estimating the values of G(ω) =

∑∞k=0 g(k)e−jωk =

F{g(k)} at some specific frequencies ωk in a finite set Ω fromthe finite-length records {u(n)}L−1

n=0 and {y(n)}L−1n=0. The

classical approach to this problem is to estimate G(ω) asthe ratio of the smoothed cross-periodogram to the smoothedperiodogram, i.e.,

G(ωk) =∑M

i=−M giYL

(ωk − 2πi

L

)U ∗

L

(ωk − 2πi

L

)∑Mi=−M gi|UL

(ωk − 2πi

L

) |2 (2)

where

YL(ωk) =L−1∑n=0

y(n)e−jωkn = DFT{y(n)}, ωk =2πk

L

(3)

is the DFT of {y(n)}L−1n=0—and similarly for UL(ωk)—and

where the gi’s, i = −M, . . . ,M , are the taps of a suitablydesigned low-pass filter. The FRF estimator (2) is directlyinherited from the early works of Blackman and Tukey in powerspectrum estimation [3] (it is also known as the Blackmanand Tukey procedure). Note that the so-defined G(ωk) is theweighted mean-square estimator of G(ωk) given the set of(noisy) observations {YL(ωk − 2πi/L)}M

i=−M and the set of re-gressors {UL(ωk − 2πi/L)}M

i=−M . The statistical performanceof this approach has been thoroughly studied in references suchas [4]–[6].

B. WOSA Procedure

Whereas the Blackman and Tukey estimator (2) controlsstability by smoothing over adjacent frequency lines, the basic

Fig. 2. Principle of the segmenting and windowing process. The signalof length L is divided into M (weighted) segments of length N that areoverlapping by N − R samples.

idea of the WOSA estimator (5) is to arrive at the sameresult by averaging over segments of data. Namely, let usdenote

Ywi(ωk) =

iR+N−1∑n=iR

w(n − iR)y(n)e−jωkn, ωk =2πk

N

(4)

the short-time DFT of the segment of data {y(n)}iR+N−1n=iR an-

chored at time iR and tapered by the smoothed N -long window{w(n)}N−1

n=0 ; the WOSA FRF estimator is then obtained asthe ratio of the averaged cross-periodogram to the averagedperiodogram, i.e.,

G(ωk) =∑M−1

i=0 Ywi(ωk)U ∗

wi(ωk)∑M−1

i=0 |Uwi(ωk)|2 (5)

with M = �(L − N)/R� + 1 as the total number of data seg-ments (�x� standing for the greatest integer less than or equalto x) and 1 − R/N as the fraction of overlap between adjacentsegments (see Fig. 2).

The WOSA procedure seems to have become a standard incommercial data analyzers in virtue of its simplicity and manycomputational advantages.

1) It can be very efficiently computed by means of severalshort FFTs (which is always faster than computing onevery long FFT).

2) It requires few memory allocation, and it can actuallybe recursively computed in almost real time (this isthe solution implemented in current commercial dataanalyzers).

3) It is robust against nonstationary, outlying, or missingdata since the process of segmenting allows for easydetrending and removal of corrupted segments.

The performance of WOSA has been investigated in someearly papers within the context of spectrum estimation. In a se-ries of papers [7]–[9], Nutall showed that the WOSA procedurecan be made nearly as efficient (in the sense of achieving thesame statistical stability given a similar frequency resolution)as the Blackman and Tukey procedure provided that sufficient



Fig. 3. Transient effects with the (left) rectangular window and (right) Hanning window.

overlap between adjacent segments is applied. From numericalsimulations, Nutall reported that ideally 62.5% overlap shouldbe imposed, but that 50% is more practical and almost justas efficient. This recommendation seems to have become thestandard nowadays. In another series of papers, the key role oftapering the segments with a smooth data window (as opposedto the rectangular window) was clearly demonstrated (see [10]and the references therein). Nevertheless, there has been nounique answer so far to the question of an optimal windowshape; instead, many “good” windows have been proposed overthe years, each with its own peculiarities [11]–[15].

Surprisingly, there are very few related results when it comesto applying WOSA for the measurement of FRFs [16], [17].Indeed, this issue happens to be more involved than that ofspectral analysis. One major peculiarity of the WOSA estimatorwhen used for FRF measurements is that the use of severalsegments of reduced length makes it prone to severe leakageerrors. If the mechanism of leakage due to Fourier transforminga truncated time series is well known in spectral analysis, itsimplication in the WOSA technique is more subtle and requiressome explanations. It stems from the inherent but erroneousassumption that the tapered output segments are identical to thesystem responses of the tapered input segments, despite of thenonlinearity of the windowing process. This produces transientartifacts in the signals at the onset and end of each segment,which will perturb the FRF estimation process. As illustratedin Fig. 3, a typical solution to reduce the transient magnitudeis to use a smooth data window—e.g., the Hanning windowas opposed to the rectangular window—but this is generally atthe price of an increase in the transient duration so that somecompromise has to be found. Another intuitive solution is toincrease overlap between adjacent segments so that transientartifacts got averaged out to a further extent. We now addresssuch questions as whether there is an optimal window shape totaper the data with, as well as how much can be gained fromoverlapping adjacent segments.

The answers to these questions are nontrivial, and whenit comes to FRF measurements, the approach to obtain themis quite different from that used in classical spectral analy-sis. The proposed approach proceeds by first analyzing themeasurement errors involved in the WOSA procedure. Theseare shown to stem from three sources, such as the following:1) systematic errors; 2) stochastic errors due to spectral leakage;and 3) stochastic errors due to additive measurement noise.

This can be summarized by the following remarkably simpleformula (see the proof in the Appendix)

G(ωk) = G(ωk) + βL(ωk) + βN (ωk) (6)

where βL(ωk) is a nonzero-mean stochastic error accountingfor spectral leakage, and βN (ωk) is a zero-mean stochasticerror accounting for measurement noise. We then pay particularattention to the leakage error βL(ωk), the statistical analysisof which provides closed-form (asymptotic) expressions for thebias and the variance of the WOSA FRF as a function of thedata-window shape and length, the number of segments, andthe amount of overlap between adjacent segments.

Incidentally, (6) proves that leakage affects both the bias andthe variability of G(ωk) since

E{

G(ωk)}

= G(ωk) + E {βL(ωk)} (7)

Var{

G(ωk)}

= Var {βL(ωk)} + Var {βN (ωk)} . (8)

It seems that only the deterministic nature of leakage (i.e., thefirst term on the right-hand side of (7) has been recognizedin classical references on the subject) and that its randomcontribution has been ignored from variance analyses [2], [5].However, there are many instances where the latter dominatesthe former [17].

III. STATISTICAL ERRORS

A. Assumptions and Notations

In this section we assume, for simplicity, that

A1) the input signal u(n) is real, zero mean, and stationary;A2) its correlation length is much smaller than the system

time constant1

τc =(∑∞

k=0 |g(k)|k2∑∞k=0 |g(k)|

) 12

(9)

as expected from a well-designed input signal [1];A3) the measurement noise is independent of u(n);

1Note that this definition of the time-constant encompasses both notions ofpure delay and of group delay of the impulse response of the system.



Fig. 4. (Left) Hanning and (right) half-sine windows v(τ) and their corre-sponding (normalized) autocorrelation functions κ(τ).

A4) the system time constant τc is much smaller than thedata-window length (as classically recommended withWOSA and more generally with any nonparametricmeasurement of FRFs).

Furthermore, to propose concise expressions, let us introducethe continuous window shape v(τ) defined over the support set[0; 1[. The discrete N -long data window w(n) used in WOSA[e.g., in (5)] is obtained by sampling v(τ) with N samples perunit length, i.e.,

w(n) = v(n/N), n = 0, . . . , N − 1. (10)

Let us also denote

R2v(τ) =

1−|τ |∫0

v (|τ | + x) v(x)dx (11)

the autocorrelation function of v(τ), and

κ(τ) =R2v(τ)R2v(0)

(12)

its normalized version (see Fig. 4). Finally, let 100(1 − θ)be the percentage of overlap between two adjacent segments,where θ = R/N (see Fig. 2).

With these notations, we are now in position to define the twocorrection factors

C2v (θ) = − κ′′(0) − 2

M(θ)−1∑i=1

(1 − i

M(θ)

)κ′′(iθ)κ(iθ)

(13)

D2v(θ) = 1 + 2

M(θ)−1∑i=1

(1 − i

M(θ)

)κ(iθ)2 (14)

which are explicit functions of the percentage of overlap, thenumber

M(θ) = �(L − N)/Nθ� + 1 (15)

TABLE ICORRECTION FACTORS FOR THE HANNING WINDOW

TABLE IICORRECTION FACTORS FOR THE HALF-SINE WINDOW

of segments, and the window shape v(τ). Note that C2v (θ) exists

only for windows having a bounded derivative, a requirementthat precludes, in particular, the rectangular window.

B. Systematic Error

Under Assumptions A1–A4 and provided that v(τ) has abounded derivative, the systematic error Bias{G(ωk)} of (7)can be shown to have the asymptotic expression2 (see theAppendix)

E {βL(ωk)}−−−−→Nτc

C2v (1)

2N2G′′(ωk) ∼ O(N−2) (16)

where the correction factor C2v (1) = −κ′′(0) is as given in

(13) with θ = 1 and may be interpreted as the curvature of theautocorrelation function of the window at lag 0—see the firstrow in Tables I and II for some particular values—and G′′(ωk)is the second derivative of G(ωk). Expression (16) is instructivein two instances: First, since the bias is an O(N−2), long datawindows should be used. Second, since it is proportional to−κ′′(0), data windows implying minimum curvatures shouldbe used. This issue will be addressed in the next section.

C. Stochastic Errors

From (8), the total variance of the WOSA FRF estimatordecomposes into one contribution due to leakage and one con-tribution due to measurement noise. We now provide the gen-eral expressions for these two contributions given an arbitrary

2The notation−−−−−−−−−→(·)

N�τc means “converges to (·) as N becomes muchlarger than τc.”



percentage of overlap and an arbitrary window shape. Thesketch of the proofs is provided in the Appendix.

1) Stochastic Errors Due to Leakage: UnderAssumptions A1–A4 and provided that v(τ) has a boundedderivative, the variance of G(ωk) due to leakage is (see theAppendix)

Var {βL(ωk)}−−−−→Nτc

C2v (θ)

M(θ)N2|G′(ωk)|2, ωk �= 0 mod (π)

∼ θC2v (θ) · O(L−1N−1) (17)

with the correction factor C2v (θ) as given in (13). Equation (17)

can be somehow simplified when particular cases are con-sidered: we deemed it useful to provide the reader with theequivalent formula reported in Tables I and II for the typicalcases of a Hanning and a Half-sine window, respectively, with0, 1/2, and 2/3 overlap.

Since the variance is shown to be an O(L−1N−1), onesolution to reduce stochastic errors due to leakage (given a fixedsignal length L) is to use long data windows. Another solutionis to set the fraction of overlap θ = R/N that minimizes C2

v (θ).This issue will be addressed in the next section.

2) Stochastic Errors Due to Measurement Noise: UnderAssumptions A1–A3, the variance of G(ωk) due to measure-ment noise is (see the Appendix)

Var {βN (ωk)} ≈ D2v(θ)

M(θ)· NSRout(ωk)

∼ θD2v(θ) · O(L−1N) (18)

with the correction factor D2v(θ) as given in (14), and where

NSRout(ωk) = S2νy(ωk)/S2u(ωk) stands for the output noise-

to-signal ratio (NSR) at frequency ωk. Again, the reader mayfind it convenient to use the equivalent formulas reported inTables I and II for the typical cases of a Hanning and a half-sine window, respectively, with 0, 1/2, and 2/3 overlap.

Expression (18) is similar to that reported in [2]. However,we emphasize again the fact that it cannot alone explain thevariability of the WOSA FRF estimator.

IV. OPTIMAL SETTINGS OF THE WOSA PARAMETERS

We now make use of the preceding results (16)–(18) to findthe optimal parameters to be used with the WOSA procedure.Our first result concerns the optimal window shape that isfound to minimize the errors due to leakage (both systematicand stochastic). Our second result specifies how much overlapshould be used to minimize the overall stochastic errors (bothleakage and measurement noise).

A. Optimal Window Shape

The optimal window shape should minimize the followingmean-square error (MSE):

MSE(ωk)= |E {βL(ωk)}|2+Var {βL(ωk)}+Var {βN (ωk)} .

For the sake of simplicity and to avoid having the optimumdepending on the unknown signal-to-noise ratio (SNR) and

Fig. 5. C2v(θ)/M(θ) as a function of the amount of overlap (1 − θ) for

the (dotted line) half-sine and (continuous line) Hanning windows, with L =10 000 and N = 100. Note that M(θ) = �(L − N)/Nθ� + 1.

actual FRF, we consider only the minimization of the errorsdue to leakage (i.e., the first two terms on the right-hand sideof the preceding equation). This amounts to finding the optimalwindow in the noise-free case.

Turning back to (16) and (17), it is shown that, in theparticular case of no overlap (θ = 1), such a window shouldminimize the curvature −κ′′(0) of its autocorrelation function.In the family of real symmetric windows, it can be shown(e.g., by standard variational analysis) that such a minimum isachieved by the half-sine window shape (see Fig. 4), i.e.,

v(τ)−−−−→Nτc

sin(πτ), 0 ≤ τ ≤ 1. (19)

The half-sine window has several appealing properties.

1) It has the same minimum-bias performance as the re-cently proposed (complex and asymmetric) “diff” win-dow [17].

2) Its optimality is asymptotically (when N τc) indepen-dent of the system FRF.

3) It is incidentally very similar to the parabolic windowhistorically used in Welch’s original paper [2].

Moreover, it can be shown that the optimality of the half-sinewindow still approximatively holds when overlap is increased(θ < 1). This is because the value of Var{βL(ωk)} becomesalmost independent of the window shape as θ → 0, as shownin Fig. 5.

In fact, the superiority of the half-sine window over theHanning window can be checked in Fig. 4, wherein the largercurvature of κ(τ) at τ = 0 implies a smaller bias accordingto (16), and in Fig. 5, wherein the smaller correction factorC2

v (θ) implies a smaller variance according to (17). Precisely,Tables I and II) indicate a reduction of 1.33 on the bias and,more modestly, of 1.33, 1.39, and 1.09 on the variance with0, 1/2, and 2/3 overlap respectively. This superiority may be



Fig. 6. D2v(θ)/M(θ) as a function of the amount of overlap (1 − θ) for

the (dotted line) half-sine, (continuous line) Hanning, and (gray line) rectan-gular windows, with L = 10 000 and N = 100. Note that M(θ) = �(L −N)/Nθ� + 1.

explained by the fact that, in the frequency domain, the half-sinewindow achieves a better balance between local leakage (widthof the main lobe) and global leakage (roll-off of the secondarylobes).

Finally, it should be emphasized that the half-sine win-dow is no longer optimal—in the sense of minimizing theoverall MSE—in the presence of (significant) measurementnoise. It can be seen from (14) and (18) that the reductionof Var{βN (ωk)} requires a window shape with small inertia(a Dirac impulse in the limit), which is conflicting with the re-duction of the leakage errors. Therefore, the optimum windowwill be a compromise in the general case, depending on boththe unknown SNR and FRF. Addressing this difficult issue isoutside the scope of this paper and is also unlikely to providethe end user with any simple recommendation.

B. Optimal Amount of Overlap

Having set the window shape that minimizes the effect ofleakage errors, it is still possible to significantly reduce theMSE by acting on the overlap. As mentioned in the precedingsection, the intuition is that overlapping will average out sto-chastic errors to a further extent by providing a larger numberM(θ) of segments. Indeed, turning back to (13) and (14) andtheir illustration in Figs. 5 and 6, it is shown that C2

v (θ)/M(θ)and D2

v(θ)/M(θ) are globally decreasing functions of theamount of overlap (1 − θ). More precisely, Tables I and II showthat from 0 to 2/3 overlap, the number of windows is roughlytripled, whereas the correction factors C2

v (θ) and D2v(θ) are

only slightly changed. For instance, with a Hanning window,this yields a reduction by 3.8 for the variance due to leakageand by 2 for the variance due to measurement noise. With ahalf-sine window, the reduction is by 3 for the variance due toleakage and by 1.7 for the variance due to measurement noise.

Moreover, Figs. 5 and 6 show that not much more can begained by forcing adjacent segments to overlap more than about2/3 and 1/2, respectively, because the added information thenbecomes extremely redundant. This means that, depending onthe window shape and whether leakage noise or measurementnoise predominates, the optimal fraction of overlap is some-where between 1/2 and 2/3. Since the actual SNR will rarelybe known, a conservative choice is about 2/3 overlap, a limitabove which the extra computational cost involved by forcingmore overlap is surely not justified. More accurate settings maybe formulated on a case-by-case basis, depending on both theused window and the leakage- or noise-free case, but we believethat the simple and conservative “2/3 rule” is just good enoughin practice—it can actually be shown to conservatively hold forall smooth (i.e., with a bounded derivative) data windows, justas observed here above on the Hanning and half-sine windows.3

V. EXTENSION TO OTHER SPECTRAL FUNCTIONS

It is a natural question to investigate whether the optimalsettings found hitherto for measuring FRFs also apply to themeasurement of other spectral functions. Of particular interestis the WOSA estimator of the (squared-magnitude) coherencefunction and the alternative WOSA measurement of FRFs—theso-called “H2” function.

A. Coherence Function

The coherence function

γ2(ωk) =

∣∣∣∑M−1i=0 Ywi

(ωk)U ∗wi

(ωk)∣∣∣2∑M−1

i=0 |Ywi(ωk)|2∑M−1

i=0 |Uwi(ωk)|2 (20)

is a useful quantity for checking the linearity of the systemunder study. For the sake of simplicity, we only address herethe impact of leakage noise on its measurement (i.e., the noise-free case), which, to our knowledge, has never been investigatedbefore in the literature. Investigation of the combined effect ofleakage and measurement noise is left for future research.

Under Assumptions A1–A4 and provided that v(τ) has abounded derivative, the bias and variance of γ2(ωk) due toleakage are (see the Appendix)

Bias{γ2(ωk)

}−−−−→Nτc

C2v (1)

2N2

|G′(ωk)|2|G(ωk)|2 (21)

Var{γ2(ωk)

}−−−−→Nτc

E2v(θ)

M(θ)N4

|G′(ωk)|4|G(ωk)|4 , ωk �= 0 mod (π)

∼ θE2v(θ) · O(L−1N−3) (22)

where

E2v(θ) = κ′′(0)2 + 2

M(θ)−1∑i=1

(1 − i

M(θ)

)κ′′(iθ)2. (23)

3The formal proof of this assertion is only of technical interest and will bepublished elsewhere; it can actually easily be verified by computing (13) and(14) for different window candidates.



Fig. 7. Noise assumption for the H2 estimator.

Interestingly, the variance due to leakage is now an O(N−3)for the coherence function instead of an O(N−1), as previouslyfound for the FRF.

B. H2 Function

The H2 function is the mean-square estimator of a FRF in thepresence of input noise only, as illustrated in Fig. 7. Denotingby x(n) = u(n) + νx(n) the noisy measurement of the inputsignal, it is given by

G2(ωk) =∑M−1

i=0 |Ywi(ωk)|2∑M−1

i=0 Xwi(ωk)Y ∗

wi(ωk)

. (24)

Under Assumptions A1–A4 and provided that v(τ) has abounded derivative, the bias and variance of G2(ωk) due toleakage are (see the Appendix)

Bias{

G2(ωk)}−−−−→Nτc

C2v (1)

2N2

{G′′(ωk)+2

|G′(ωk)|2G(ωk)∗

)

}(25)

Var{

G2(ωk)}

=C2

v (θ)M(θ)N2

|G′(ωk)|2+D2

v(θ)M(θ)

· NSRin(ωk),

ωk �= mod(π) (26)

where NSRin(ωk) = S2νx(ωk)/S2u(ωk) stands for the input

NSR at frequency ωk, and where the correction factors C2v (θ)

and D2v(θ) are identical to those found for G(ωk) (output-noise-

only case).

C. Conclusion

It is noteworthy that (21), (22), (25), and (26) all involve thesame structure as (16) and (17). In conclusion, those settingsthat were found to be optimal for G(ωk)—i.e., a half-sine ora diff window with about 2/3 overlap—are also optimal formeasuring the coherence function and the H2 function.

VI. MULTISINE EXCITATIONS

So far, our analysis has focused on the case where theexcitation signal is random stationary. A related question ofinterest is whether the same reasoning can be followed to findthe optimal settings in the case of a multisine excitation.

For that purpose, let us write the multisine excitation signalof period N as

u(n) = u(n + N) =∑

ωk∈Ω

Akejωkn+φk (27)

where Ak are real deterministic amplitudes, φk are inde-pendent random phases uniformly distributed over [0; 2π[(whose outcomes are fixed for a given experiment), and Ω ={2πk/N, k = 0, . . . , N − 1} is a set of frequency lines.

Assuming the model plant of Fig. 1, the systematic error isthen found to have the expression

Bias{

G(ωk)}≈

N−1∑l=0

|Al|2 |W (ωl − ωk)|2 [G(ωl) − G(ωk)]

N−1∑l=0

|Al|2 |W (ωl − ωk)|2

(28)

with W (ω) as the DTF of the data window w(n). Clearly, thebias is exactly zero provided that W (ωl − ωk) = 0 for any ωl �=ωk, that is, provided that W (ω) has zeros at all frequenciesω = 2πk/N , k = 1, . . . , N − 1. For an N -long window, thiscan only be achieved by the rectangular shape.

Similarly, the expression of the variance due to leakage noisecan be found to depend on factors involving W (ωl − ωk), sothat those windows that cancel the systematic error also cancelthe stochastic leakage errors.

Finally, the variance due to measurement noise is foundto be

Var{

G(ωk)}≈ F 2

v (θ)M(θ)

· Bw · NSRout(ωk)

∼ θF 2v (θ) · O(L−1) (29)

where NSRout(ωk) = S2νy(ωk)/|Ak|2 stands for the output

NSR, Bw = NR2v(0)/|W (0)|2 stands for the bandwidth ofthe data window (e.g., Bw = 1/N for the rectangular win-dow), and

F 2v (θ) = 1 + 2

M(θ)−1∑i=1

(1 − i

M(θ)

)κ(iθ) (30)

is a positive correction factor very similar to (14) but withoutthe power 2 on κ(iθ).

Contrary to the previous cases, it can be checked that F 2v (θ)

is now minimum at θ = 1, that is, when there is no overlap.Moreover, the window shape that minimizes (29) is that withthe minimum bandwidth Bw, which is again the rectangularwindow. This is illustrated in Fig. 8. It is also shown inthis figure that overlapping the rectangular window has someunexpected consequences due to the “bouncing” behavior ofF 2

v (θ): the worst choice is 1/4 overlap; suboptimal choices are1/2, 2/3, 3/4, . . . overlap, but with an increased computationalcost. This is summarized in Table III.

In conclusion, the WOSA optimal settings for measuringFRFs in the case of multisine excitations with period N isa rectangular window of length N without overlap. Such achoice completely cancels the impact of leakage noise and min-imizes that of measurement noise (in that case, Var{G(ωk)} ≈NSRout/L). As expected, this conclusion is fully coherent withthat of the maximum-likelihood approach presented in [1].



Fig. 8. BwF 2v (θ)/M(θ) as a function of the amount of overlap (1 − θ) for

the (continuous line) rectangular and (dotted line) half-sine windows, with L =10 000 and N = 100. Note that M(θ) = �(L − N)/Nθ� + 1.

TABLE IIICORRECTION FACTORS FOR THE RECTANGULAR WINDOW

VII. NUMERICAL VERIFICATIONS

The results established in the preceding sections are nowverified on simulations. The simulated system is a one-degree-of-freedom oscillator with the transfer function

G(z) =1 − 2z−1 + z−2

1 − 1.6079z−1 + 0.9875z−2(31)

which shows a sharp resonance at ω0 = 0.2π. According to(9), its corresponding time constant τc is about 223 samples.The system was excited by a random Gaussian input of unitvariance and of length L = 214, as illustrated in Fig. 9. Notethat significant bias and variance are expected at the resonancefrequency ω0 according to (16) and (17), which involve the firstand second derivatives of G(ω) (Table IV).

To make the computation of experimental bias and variancepossible, the same experiment was repeated 1000 times butwith different (independent) inputs, so that, concerning themeasurement of the FRF, for instance, its experimental bias andvariance could be computed as follows:

Bias{

G(ωk)}

=1

1000

1000∑i=1

G(ωk; i) − G(ωk)

Var{

G(ωk)}

=1

1000

1000∑i=1

∣∣∣∣∣G(ωk; i) − 11000

1000∑i=1

G(ωk; i)

∣∣∣∣∣2

Fig. 9. Simulated system.

TABLE IVMEASUREMENT ERRORS AT THE RESONANCE FREQUENCY OF A

ONE-DEGREE-OF-FREEDOM OSCILLATOR

with i as the index of the experiment. In each case, the windowlength was set to N = 210, which is about four times as greatas the system time constant τc.

A. Validation of Bias and Variance Formulas

Our primary concern is to check the validity of the mainresults (16), (17), (21), (22), (25), and (26). No measurementnoise was added to better emphasize the impact of leakage noiseonly on the systematic and stochastic errors. The parameters forthe WOSA measurements were set with a half-sine window and2/3 overlap. Fig. 10 compares the experimental bias and stan-dard deviation with the corresponding formulas provided in thispaper, for the coherence function, the classical FRF estimate,and the “H2” function. A very good match is generally notice-able. As expected from our formulas, the maximum errors occurin the vicinity of the resonance frequency where the leakageeffect is the strongest. Concerning the FRFs, the bias error be-comes so considerable in that frequency region that it overrunsthe stochastic errors (i.e., standard deviation; see Fig. 10(b) and(c)]. As for the coherence function [see Fig. 10(a)], its biaserror largely dominates over the whole frequency range; thisis because its variance was found to be very small: an O(N−3)instead of an O(N−1) for the FRFs [see (17), (22), and (26)].Finally, it is noteworthy that the bias of the “H2” functionsuddenly drops at exactly ω0. Again, this behavior can be wellpredicted by plugging (31) into (25), and it suggests that “H2”may be an excellent estimator of the resonance amplitude (thisis true, in general, for any resonant system).

B. Validation of the Optimal Settings

Our second concern is to experimentally check the impact ofoverlap and of the window shape on the WOSA measurement.



Fig. 10. Comparison of (continuous lines) experimental and (dotted lines)theoretical biases and standard deviations in the case of a half-sine windowwith 2/3 overlap. (a) γ2(ωk). (b) G(ω0). (c) G2(ω0). Bold lines depict theactual quantities to be measured.

A first experiment was conducted to compare the estimationerrors in the cases without overlap and with 2/3 overlap. Nomeasurement noise was added, and a half-sine window wasused. The experimental biases and standard deviations areshown in Fig. 11. It is obvious from these results that 2/3overlap can reduce the standard deviation of stochastic errors byalmost a factor of 2. Precisely, the case without overlap yieldsM(θ = 1) = 16 segments and, according to Table I, a cor-rection factor C2

v (1) = π2, whereas the case with 2/3 overlapyields M(θ = 1/3) = 46 and C2

v (1/3) = 9.61(1 + 0.13/46);thus, the reduction in standard deviation is like the squareroot of C2

v (1)/M(1) over C2v (1/3)/M(1/3), that is, 1.7. In

addition, it is checked that overlap has no effect on the biaserror, as predicted from (16), (21), and (25).

A second experiment was conducted to obtain more quantita-tive results with respect to both overlap and the window shape.To summarize each simulation by a single couple of points, theexperimental bias and variance results were integrated over thewhole positive frequency axis, i.e.,

Bias2 =L/2−1∑k=1

∣∣∣Bias{

G(ωk)}∣∣∣2

Var =L/2−1∑k=1

Var{

G(ωk)}

.

Fig. 11. Comparison of experimental biases and standard deviations (dottedlines) without overlap and (continuous lines) with 2/3 overlap. (a) γ2(ωk).(b) G(ω0). (c) G2(ω0). Bold lines depict the actual quantities to be measured.

Note that, according to these definitions, the total MSE issimply obtained as Bias2 + Var. The corresponding results aredisplayed in Fig. 12. As expected, it is shown that the squaredbias is (statistically) constant against overlap, whereas thevariance is a monotonically decreasing curve, with its minimumnearly reached at 2/3 overlap. This is independently of thewindow shape. These experimental curves fairly well reproducethe trends of Fig. 5, notwithstanding a different scaling factorrelating to the system characteristics. Indeed, from (17), ittheoretically holds that

Var � C2v (θ)

M(θ)N2

L/2−1∑k=1

|G′(ωk)|2

where C2v (θ)/M(θ) is the quantity displayed in Fig. 5. This

again validates (17) and its implication about the optimalityof the 2/3 rule. Finally, Fig. 12 demonstrates that the half-sinewindow is much better than the Hanning window as long asleakage errors (bias and variance) are of concern. A reductionof about 1.8 is observed on the squared bias, which is in perfectaccordance with what can be predicted from (16) and Tables Iand II, i.e.,

Bias2(Hanning)

Bias2(half-sine)=∣∣∣∣C2

v (1)Hanning

C2v (1)half-sine

∣∣∣∣2 =(

43

)2

� 1.778.

The reduction on the variance term is less significant, particu-larly at high levels of overlap. Equation (17) and Tables I and II,



Fig. 12. Integrated squared bias and variance due to leakage with respect tothe amount of overlap for the Hanning and half-sine windows.

in the case of no overlap, give

Var(Hanning)Var(half-sine)

=C2

v (1)Hanning

C2v (1)half-sine

=43� 1.33

which fairly well compares with the reduction of about 1.4, asshown in Fig. 12, and in the case of 2/3 overlap, i.e.,

Var(Hanning)Var(half-sine)

=C2

v (1/3)Hanning

C2v (1/3)half-sine

=10.50

(1 + 0.27

M(1/3)

)9.61

(1 + 0.13

M(1/3)

)�1.10, which again is coherent with Fig. 12.

Very similar results were obtained for the measurement ofthe coherence and the H2 functions, which are not displayedhere due to the lack of space.

C. Investigation of the Effect of Measurement Noise

Our last concern is to check the scope of validity of ourconclusions when the SNR becomes low. To do so, the totalMSE Bias2 + Var was computed for different output SNRs (seeFig. 1), and the values were reported for both the Hanningand half-sine windows at various amounts of overlap. Theexperimental results are displayed in Fig. 13. It is shown thatthe half-sine window still outperforms the Hanning window,until the SNR reaches the unrealistically low value of −40 dB.At this stage, measurement noise completely overwhelms leak-age noise, and the curves of Fig. 13(d) actually reproduce thetrends of Fig. 6. This demonstrates that, at least for the veryresonant system simulated in this section, leakage errors arevery predominant (actually mainly the bias) that the half-sinewindow remains a very good choice, even in the presence of(realistic levels of) additive noise.

VIII. CONCLUSION

The measurement of FRFs by means of the WOSA proceduresuffers from three types of errors, namely, a systematic error

Fig. 13. Overall MSE as a function of the SNR for the (· · · • · · ·) Hanningand (· · · ◦ · · ·) half-sine windows. (a) SNR = ∞ dB. (b) SNR = 0 dB.(c) SNR = −20 dB. (d) SNR = −40 dB.

due to leakage, a stochastic error due to leakage noise, and astochastic error due to measurement noise. We have proven thatthe optimal real symmetric data-window shape that globallyminimizes the leakage errors is the half-sine window, whichincidentally achieves all the same properties as the (complex-valued and nonsymmetric) diff window recently proposed in[17]. Moreover, the stochastic errors can significantly be re-duced by increasing the percentage of overlap between adjacentsegments. The ultimate variance reduction that can achievedthis way is about 3 on the leakage errors and 1.7 on the mea-surement noise errors with the half-sine window. An importantpractical result is that, in all cases, this is nearly achieved bysetting about 2/3 overlap independently of the system FRF andof the data window. The “2/3 law” should replace the customary1/2 law used in commercial data analyzers without increasingtoo much the computational demand.

Moreover, it has been shown that the same recommendationsapply as well to the measurement of other spectral quantitiesand, in particular, to the coherence function, which is often usedin conjunction with FRF measurements.

As a final note, we emphasize again that the optimality ofthese settings applies provided that the data-window lengthis set larger than the system time constant—which is in anycase the usual recommendation with WOSA—and only withstationary random excitations. The special case of multisine ex-citations was verified to yield completely different conclusions.

APPENDIX

The sketch of our proofs starts from the demonstra-tion of relation (6). From Fig. 3, the leakage error dueto segment i is twi

(n) = y(n)w(n − iR) −∑∞k=0 g(k)u(n −

k)w(n − k − iR), i.e.,

Twi(ωk) =

∫Wi(ωk − ν) (G(ν) − G(ωk)) dU(ν) (32)



where Wi(ωk − ν) = W (ωk − ν)e−jiR(ωk−ν), and dU(ν) isthe spectral increment of process u(n) at frequency ν [6].Therefore, Ywi

(ωk) = G(ωk)Uwi(ωk) + Twi

(ωk) + Nwi(ωk),

from which (6) immediately follows with

βL(ωk) =∑M−1

i=0 Twi(ωk)Uwi

(ωk)∗∑M−1i=0 Uwi

(ωk)Uwi(ωk)∗

(33)

βN (ωk) =∑M−1

i=0 Nwi(ωk)Uwi

(ωk)∗∑M−1i=0 Uwi

(ωk)Uwi(ωk)∗

. (34)

Now, from Assumption A4, Twi(ωk) may be approximat-

ed as∫Wi(ωk − ν)

((ν − ωk)G′(ν) − 1

2(ωk − ν)2G′′(ν)

)dU(ν).

(35)

All proofs proceed by making use of (33) and (34) and a set ofproperties:

(P1): E{dU(ν1)dU(ν2)} = S2u(ν1)δ(ν1 + ν2)dν1dν2.(P2): E{dU(ν1)dN(ν2)} = 0.(P3):

∫ |W (ν)|2νp ≈ (−j)p2πN (1−p)R(p)2v (0), p = 0, 1, 2.

(P4): The input signal u(n) is assumed to be Gaussian.

Property P1 follows from Assumption A1, Property P2 fromAssumption A3, and Property P3 from the assumed differentia-bility of the window shape v(τ). Property P4 is invoked herefor simplicity, although not fundamentally necessary since theDFT makes (most) signal tend to Gaussianity in virtue of thecentral limit theorem [6].

A. Equation (16)

From Assumption A4, the expected value of the ratio(33) tends to the ratio of the expected values. Namely,from P1, P3 with p = 0, and Assumptions A1 andA2, E{∑M−1

i=0 Uwi(ωk)Uwi

(ωk)∗} = S2u(ωk)2πNR2v(0)M ,and from P3 with p = 2, E{∑M−1

i=0 Twi(ωk)Uwi

(ωk)∗} =−S2u(ωk)πN−1R′′

2v(0)MG′′(ω). Taking the ratio and usingdefinition (12), (16) immediately follows.

B. Equation (17)

From Assumption A4, the variance of the ratio (33) tends tothe ratio of the variance of the numerator to the square of theexpected value of the denominator. Since the latter has just beenevaluated in the preceding discussion, we only need to workout the former. Namely, from P1, P3 with p = 0, 1, P4, andAssumptions A1 and A2, Var{∑M−1

i=0 Twi(ωk)Uwi

(ωk)∗} =S2

2u(ωk)(2π)2MC2v (θ)R2

2v(0)|G′(ω)|2. Taking the ratio andusing definition (12), (17) immediately follows.

C. Equation (18)

From Assumption A4, the variance of the ratio (34) tends tothe ratio of the variance of the numerator to the square of the

expected value of the denominator. Here, again, only the formerquantity needs to be evaluated. From P1, P2, P3 with p = 0, P4,and Assumptions A1–A3, Var{∑M−1

i=0 Nwi(ωk)Uwi

(ωk)∗} =S2u(ωk)S2ν(ωk)(2π)2MD2

v(θ)R22v(0)N2. Taking the ratio

and using definition (12), (18) follows.

D. Equations (21) and (22)

The evaluation of (21) and (22) starts with replacing Ywi(ωk)

in (20) by its expansion G(ωk)Uwi(ωk) + Twi

(ωk). FromAssumption A4, the magnitude of the transient term Twi

(ωk)is much smaller than Uwi

(ωk) on the average. Expanding ratio(20) to the second order with respect to Twi

(ωk) then yields theapproximation

γ2(ωk) � 1 −∑M−1

i=0 |Twi(ωk)|2∑M−1

i=0 |Ywi(ωk)|2 . (36)

The rest of the proof follows as for the bias and variance ofG(ωk) by taking the expected value and the variance of (36),simplifying the statistics of a ratio as the ratio of the statisticsdue to Assumption A4, and using Assumptions A1–A2 andProperties P1–P4.

E. Equations (25) and (26)

The evaluation of (25) and (26) proceeds as for the H2

function; expansion of the ratio (24) to the second order withrespect to Twi

(ωk) then yields the approximation

G2(ωk) � G(ωk) ·(

1 +∑M−1

i=0 Twi(ωk)Y ∗

wi(ωk)∑M−1

i=0 |Ywi(ωk)|2

). (37)

The rest of the proof follows as before, by taking the ex-pected value and the variance of (37), simplifying according toAssumption A1–A4, and using Properties P1–P4.

REFERENCES

[1] R. Pintelon and J. Schoukens, System Identification—A FrequencyDomain Approach. Piscataway, NJ: IEEE Press, 2001.

[2] P. Welch, “The use of the fast Fourier transform for the estimation ofpower spectra: A method based on time averaging over short, modifiedperiodograms,” IEEE Trans. Audio Electroacoust., vol. AU-15, no. 2,pp. 70–73, Jun. 1967.

[3] R. B. Blackman and J. W. Tukey, The Measurement of Power SpectraFrom the Point of View of Communications Engineering. New York:Dover, 1959.

[4] G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications.Boca Raton, FL: Emerson-Adams Press.

[5] J. Bendat and A. Piersol, Random Data: Analysis and MeasurementProcedures, 2nd ed. New York: Wiley-Interscience, 1986.

[6] D. R. Brillinger, Time Series: Data Analysis and Theory. New York:SIAM, 1975.

[7] A. H. Nuttall and G. C. Carter, “A generalized framework for powerspectral estimation,” IEEE Trans. Acoust., Speech, Signal Process.,vol. ASSP-28, no. 3, pp. 334–335, Jun. 1980.

[8] A. H. Nuttall, “Spectral estimation of means of overlapped FFT process-ing of windowed data,” Naval Underwater Syst. Center, New London, CT,Rep. 4169. suppl. TR4169S.



[9] A. H. Nuttall, “On the weighted overlapped segment averaging method forpower spectral estimation,” Proc. IEEE, vol. 68, no. 10, pp. 1352–1354,Oct. 1980.

[10] D. R. Brillinger, “The key role of tapering in spectrum estimation,” IEEETrans. Acoust., Speech, Signal Process., vol. ASSP-29, no. 5, pp. 1075–1076, Oct. 1981.

[11] F. J. Harris, “On the use of windows for harmonic analysis with thediscrete Fourier transform,” Proc. IEEE, vol. 66, no. 1, pp. 51–83,Jan. 1978.

[12] A. Eberhard, “An optimal discrete window for the calculation of powerspectra,” IEEE Trans. Audio Electroacoust., vol. AU-21, no. 1, pp. 37–43,Feb. 1973.

[13] N. Geçkinli and D. Yavuz, “Some novel windows and a concise tutorialcomparison of window families,” IEEE Trans. Acoust., Speech, SignalProcess., vol. ASSP-26, no. 6, pp. 501–507, Dec. 1978.

[14] A. Nuttall, “Some windows with very good sidelobe behavior,” IEEETrans. Acosut., Speech, Signal Process., vol. ASSP-29, no. 6, pp. 84–91,Feb. 1981.

[15] J. Le Roux and J. Ménez, “A cost minimization approach for optimalwindow design in spectral analysis of sampled signals,” IEEE Trans.Signal Process., vol. 40, no. 4, pp. 996–999, Apr. 1992.

[16] J. L. Douce and L. Balmer, “Statistics of frequency-response estimates,”Proc. Inst. Elect. Eng., vol. 137, no. 5, pp. 290–296, Sep. 1990.

[17] J. Schouken, Y. Rolain, and R. Pintelon, “Analysis of windowing/leakageeffects in frequency response function measurements,” in Proc. 16th IFACWorld Congr., Prague, Czech Republic, Jul. 4–8, 2005.

Jérôme Antoni was born in Strasbourg, France, onNovember 11, 1972. He received the Ph.D. degree(with highest honors) in signal processing from thePolytechnic Institute of Grenoble, Grenoble, France,in 2000.

He is currently a Lecturer with the University ofTechnology of Compiègne, Compiègne, France. Hisresearch interests include nonstationary signal analy-sis, system identification, and signal separation. Hisapplications are concerned with noise and vibrationanalysis, modal analysis, and system diagnosis.

Johan Schoukens (M’90–SM’92–F’97) receivedthe degree in engineering and the Ph.D. degree inapplied sciences from the Vrije Universiteit Brussel(VUB), Brussels, Belgium, in 1980 and 1985,respectively.

He is currently a Professor with the VUB. Theprime factors of his research are in the field of systemidentification for linear and nonlinear systems.

Dr. Schoukens was the recipient of the Best PaperAward and the Society Distinguished Service Awardfrom the IEEE Instrumentation and Measurement

Society in 2002 and 2003, respectively.


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