Robust Broadband Periodic Excitation Design

Gyula Simon*, Johan Schoukens**

Department of Measurement and Information SystemsTechnical University of Budapest, H-1521 Budapest, Hungary

e-mail: simon@mit.bme.hu

**

*

Department ELEC, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium

e-mail: Johan.Schoukens@vub.ac.be

Abstract

commonly used, and also effective algorithms ([2], [3]),and handy design tools are available to design andgenerate such signals [4].

The main advantage of such signals is that there is noleakage effect (precise FRF-measurements can be made),the desired band of excitation usually can be selected(especially by multisine), and good crest factor values canbe achieved to provide good signal-to-noise ratio duringthe measurement. However, the good properties hold onlyif the designed signal is properly used. Inexperiencedusers may not adequately use the carefully designedsignals, namely they may tend to use fractional periodsinstead of full ones. (Typical examples are when themeasurement process would be too long and isterminated, or the recorded data is divided into two partsby the user for identification and validation purposes.)The result may be quite unexpected: certain frequencies inthe band of interest are excited very poorly (even 30-40dB loss can happen) thus resulting in bad signal-to-noiseratio during the measurement process. (Note, that whilethe leakage effect can be taken into account in the signalprocessing algorithms, the bad SNR due to loss ofinformation can not be compensated.) This effect canhappen not only in the swept sine, but even in themultisine case. While the swept sine signals do not allowhelping this problem, the multisine excitation signals canbe designed to avoid this phenomenon.

Section 2 will formulate the problem and exampleswill be given on the effect of misused excitation signals.Also a simple and easy-to-use solution is suggested todesign multisine signals which are robust: when not a fullperiod is used the proposed excitation signal has only asmall power loss in the band of interest, and thus satisfac-tory results can be obtained even in case of misuse.

In Section 3 the suggested solution is analyzed, and thetheoretical results are verified through practical examples.Based on the results Section 4 provides guidelines todesign robust multisine excitation signals.

This paper considers a rather practical problem arisingwhen inexperienced users misuse the (otherwise welldesigned) periodic broadband excitation signals duringthe measurement or signal processing phase of an identi-fication process. Using a fractional period of the excita-tion signal instead of full periods may affect not only theprecision because of the well known leakage effect, butmay cause a serious loss of information on the measuredsystem as well. The power of the excitation signal incertain frequency bands may be much lower (30-40 dB)than it would be expected, and thus the measurement in anoisy environment may give poor result. A solution ispresented here to make periodic broadband (multisine)excitation signals more robust against such misuse. Thesuggested solution is analyzed, and the theoretical resultsare verified by practical examples.

1. Introduction

In system identification processes the device under testis excited by an appropriate excitation signal, and theresponse of the system is measured. Given the excitationand the respective system response the desired systemparameters can be derived by different digital signalprocessing methods.

The excitation signal sometimes comes from thenature, but most commonly is artificially generated. In thelatter case the designer of the excitation signal can selectthe appropriate signal type and can often set a lot of signalparameters to achieve optimal result.

In practice the following types of general-purposeexcitation signals are widely used: random noise, pseudo-random binary sequences, swept sine (periodic chirp) andmultisine.

Since the deterministic signals usually have superiorproperties compared to the random noise [1], they are

2. Effect of Fractional Periods on the PowerSpectrum

Periodic excitation signals can be – and usually are –designed to excite only a frequency band of interest andthey have no significant power outside that band. Thishelps to keep excitation power as low as possible and toavoid unnecessary non-linear effects. But it is even moreimportant that the excitation really be present in thosefrequencies (or nearby) where the system is to bemeasured and modeled. If certain bands are not excitedsufficiently then the amount of information gathered tomodel those bands may be low, and the result of theidentification process would be poor.

However, misused excitation signals may result inquite different spectra than it would be necessary: theband of interest may be partly unexcited. Fig. 1 illustratesthe phenomenon using popular excitation signals: sweptsignal, Schroeder multisine, and random phase multisine[1], [5].

In the case of the swept signal the frequency is sweptbetween fmin and fmax in time with period T, where fmin andfmax are the lowest and highest frequencies to excite:)XOO3HULRG󰀀6ZHSVLQH󰀆󰀅󰀀󰀆󰀇󰀀󰀀IV󰀃󰀄IV󰀃󰀅󰀀󰀆󰀅󰀀󰀆󰀇󰀀󰀀IV󰀃󰀄⎛⎛π(fmax−fmin)⎞⎞

⎟x(t)=2Asin⎜tf2(1)+π⎜min⎟ t⎟.⎜T⎝⎠⎝⎠

A multisine signal is a sum of harmonically relatedsinusoids:

x(t)=

∑A

k=1

N

k

cos(2πfkt+φk),

(2)

where fk=lk*f0, lk positive integer, so that fmin ≤ fk ≤ fmax.The phases are often chosen so that the crest factor besmall, like in the Schroeder multisine, where the phasesare calculated by:

k(k−1)(3)φk=−π.N

All the signals in Fig. 1 were designed to excite thefrequency band from 0 to fs/6, where fs is the samplingfrequency. The spectra were calculated by DFT with arectangular window, as it is common in identificationprocesses. Using full periods the required results areobtained, but when only half of the signal length wasused, the resulted spectra seriously differ from therequired: in case of swept sine and Schroeder multisine alarge part of the band of interest is not excited adequately.

󰀅3/+DOI3HULRG󰀀󰀆󰀅󰀀󰀆󰀇󰀀IV󰀃󰀅󰀀IV󰀃󰀄IV󰀃󰀅󰀒+DOI3HULRG󰀀6FKURHGHU󰀆󰀅󰀀󰀆󰀇󰀀󰀀IV󰀃󰀄IV󰀃󰀅󰀀󰀆󰀅󰀀󰀆󰀇󰀀󰀀IV󰀃󰀄IV󰀃󰀅󰀀󰀆󰀅󰀀󰀆󰀇󰀀󰀀IV󰀃󰀄IV󰀃󰀅5DQGRP3KDVH󰀀󰀆󰀅󰀀󰀆󰀇󰀀󰀀IV󰀃󰀄IV󰀃󰀅󰀀󰀆󰀅󰀀󰀆󰀇󰀀󰀀IV󰀃󰀄IV󰀃󰀅󰀀󰀆󰀅󰀀󰀆󰀇󰀀󰀀IV󰀃󰀄IV󰀃󰀅Figure 1. Examples of correctly an incorrectly used excitation signals. 1st column: Spectra ofcorrectly used signals, 2nd and 3rd column: spectra of truncated (half) periods. 1st row: Swept sine,2nd row: Schroeder multisine, 3rd row: random phase multisine. The power spectra are shown in dB.In the random phase case no wide unexcited bands can beseen.

Unfortunately, this kind of misuse is very commonsince a lot of users do not have – and, of course, do notneed to have – solid theoretical background in the field ofdigital signal processing. Instead the excitation signalshould be robust against misuse.

In the next section it will be proven that the randomphase multisine is really robust against “fractional-periodmisuse”. First a framework will be defined to analyze thestatistical properties of the spectral power loss, and then the behavior of the random-phase multisine will beexamined.

3. Statistical Properties of the Power Loss

In an identification process it is crucial, that during themeasurement the frequency band of interest be wellexcited to gain enough information to determine thesystem’s properties. To identify an arbitrary unknownsystem it would be necessary to excite all the frequencylines in the band of interest. However, in the case ofpractical systems with “not too rapidly changing” transferfunctions it is not crucial if some lines are poorly excitedwhen the surrounded lines have enough power. The datagained from the well-excited lines is enough to describethe system’s properties in the close neighborhood. Thismeans that it is usually satisfactory if the total power issufficient in all subbands of the band of interest, where asubband contains more than one line. In practice the sizeof the subband depends on the system to be identified, butusually means a smaller fraction of the band of interest,e.g. 1/10th of it, and thus can contain some tens or evenhundreds of lines.

In this framework the global quantity of “smallestexcitation power” in the band of interest can be defined asthe minimum of the average powers in all subbands:

S=minj

(Pj), j=1...Ns,(4)

P1K

2j=K∑

Ai,j, (5)

i=1

where K is the number of frequency lines in a subband, Nsis the number of subbands in the band of interest, and Ai,jis the ith spectrum line in the jth subband.

It is clear, that if S is sufficiently large even if only afraction of the full period is used, then the excitationsignal is robust against the “fractional period” misuse.To determine the stochastic behavior of S, thefollowing general-purpose excitation signal will beconsidered:

•the length of the full period is N,

•the spectrum is flat in the band of interest between fmin

and fmax, and fmin•the excited frequencies are fk=fs / N*k, for all possible

integer k, so that fmin ≤ fk ≤ fmax,

•the phase of the sinusoid components is random andequally distributed between -π and +π.

The power spectrum of the above excitation signalcomputed by taking the absolute square of an N-point

DFT has equal values ⎛⎜⎝A2=A2

i,j⎞⎟⎠

for all lines in the

band of interest between fmin and fmax, and zeros otherwise,and the phase of the DFT is random. When the signalsequence is truncated to MN⎛⎜2

⎝0,A2⎞⎟⎠

.According to (5), Pj (in the band of interest) is the sumof 2K squared, normally distributed random variables, soPj is also a random variable, and its distribution is chi-square with 2K degree of freedom.

If the cumulative distribution function (c.d.f.) of a chi-square distribution with f degrees of freedom is denotedby Ff(x), then the c.d.f. of Pj will be:

P(Pj⎝A2⎟⎠

,

(6)where K is the number of lines in the subband.

If (after the M-point DFT) the band of interest is di-vided into Ns subbands with K lines in each, then the c.d.f.of S, e.g. the c.d.f. of the smallest Pj value in the band ofinterest can be expressed using order statistics [6]:

Ns

P(S1−FKx2K⎜⎞⎞⎝A2⎟⎠⎟⎟

⎠.(7)It is clear that, apart from the normalizing coefficientA2

, the distribution depends only on the number of lines inthe subbands (K) and the number of subbands (Ns). Basedon (7), Table 1 contains the most probable power lossvalues as well as the upper limits for the maximum power

Values of K1030100300s51.9 / 4.91.0 / 2.60.5 / 1.40.3 / 0.8N fo102.5 / 5.31.3 / 2.80.7 / 1.50.4 / 0.9 seu203.0 / 5.71.6 / 3.00.8 / 1.60.5 / 0.9laV303.3 / 6.01.7 / 3.10.9 / 1.60.5 / 0.9403.5 / 6.11.8 / 3.21.0 / 1.70.6 / 1.0Table 1. Most probable power drop / maximumpower drop (with 99 % confidence level) values indB for different Ns and K values

loss with confidence level of 99%, for different K and Nsvalues.

From data shown in Table 1, it can be seen that for aconstant number of subbands (constant Ns) the powerdrop decreases as the number of lines (K) in the subbandsincreases. For a constant K the power drop increases as Nsincreases. As it is intuitively expected, for a constantnumber of frequency lines in the band of interest (i.e. forconstant K⋅Ns) the power drop increases as the number ofsubbands increases.

In a practical case, where the number of subbands isnot too high and in each subband there is a reasonablyhigh number of frequency lines, the expected power dropof the misused random-phase multisine is around 1...3 dB,which is much less than the power drop in the case of theSchroeder or swept-sine cases (compare with Fig. 1).

To validate the theoretical results, 1000 excitationsignals were designed with the same power spectrum,each with random phase. The band of interest was similarto that of Fig.1, the number of points in the full periodwas N = 2048, and the signal before calculating the DFTwas truncated to M=512. For test purposes 5 subbandswere selected in the band of interest with 10 consecutivelines in each (Ns=5, K=10). The smallest average subbandpower (S) was calculated for each excitation signalaccording to (4). The experimental distribution is shownin Fig. 2.a, with the theoretical distribution functioncalculated from (7), and scaled according to the numberof experiments. The match clearly validates thetheoretical results.

In Fig 2.b and Fig2.c similar plots can be seen for

signals designed starting from random-phase multisinesusing the crest factor optimizer algorithms in [2] and [3].The results verify the conjecture that the algorithmspreserve the random-like behavior of the phasecomponents, and thus the theoretical results for randomphase multisine signals can be applied to determine thebehavior of the modified signals as well.

Based on the results the next section will provideguidelines of robust multisine excitation signal design.

4. Robust Multisine Signal Design

If the designed multisine excitation signal will notsurely be adequately used then the designer or theautomatic designer algorithm should take care of therobustness against fractional-period use. The free designparameters are the phase values. The sophisticatedSchroeder multisine – which provides satisfactory lowcrest factor without any additional optimization, and thusis very popular – is one of the worst solutions from thispoint of view. Even the use of Schroeder multisine as aninitial signal for crest-factor minimization algorithms ([2],[3]) is a bad choice, since the result of the minimizationstill behaves almost as badly as the initial signal,according to experiments.

A better choice is to start from a random phasemultisine and then apply a crest factor minimizationalgorithm. According to experiments the known crest-factor minimization algorithms do not significantlychange the random-like behavior of the spectrum, thus the

󰀆([SHULPHQWDODQG7KHRUHWLFDO'LVWULEXWLRQVRIWKH6PDOOHVW$YHUDJH6XEEDQG3RZHU󰀆󰀆󰀄IUHTXHQF\\󰀉󰀊󰀋󰀄󰀄󰀅󰀅󰀅󰀁󰀁󰀁󰀀󰀁󰀂󰀀󰀄D󰀂󰀀󰀁󰀂󰀀󰀄E󰀂󰀀󰀁󰀂󰀀󰀄F󰀂Figure 2. Experimental (bars) and theoretical (solid line) distributions of the smallest average subbandpower values (horizontal axis in dB, vertical axis in percentage) for a random-phase multisineexcitation signal (Ns=5, K=10, 1000 experiments). a. without crest factor minimization, b. with crestfactor minimization algorithm [2], and c. with crest factor minimization algorithm [3].Schroeder1.66Schroeder with Algorithm [2]1.49Schroeder with Algorithm [3]1.42Random phase3.0 … 3.5 … 4.8Random phase with Algorithm [2]1.53 … 1.63 … 1.76Random phase with Algorithm [3]1.37 … 1.39 … 1.41Table 2. Typical crest factor values of multisineexcitation signals

gained theoretical results hold even if the initial random-phase signal is modified by the algorithms (see thehistograms in Fig 2).

Table 2 contains typical crest factor values of theSchroeder multisine, random-phase multisine, and theoutput of crest-factor minimization algorithms in [2] and[3] starting from Schroeder and random initial phases. Itis clear, that the optimization methods give very goodresults when the initial state is a random phase multisine,and moreover, the gained signal is robust against misuse.The only drawback of the random phase multisinesignals is that an additional iterative optimization processis required to reach the desired low crest factor values.The amount of time necessary to optimize a multisinesignal’s crest factor with the algorithms described in [2]and [3] depends mainly on the desired number of excitedfrequency lines. Table 3 shows typical running times forthe optimization algorithms with typical settings, on aPentium-class PC, in MATLAB environment. From theresults in Table 2 and Table 3 it is clear that Algorithm [3]gives better crest factor values and is faster thanAlgorithm [2] when the number of excited lines is low.For very high number of frequency lines only Algorithm[2] can be used.

Note, that the results shown in Table 3 are for typicalsettings, where the number of iterations is set high toachieve low crest factor values. If the number of iterations(and thus the running time) is decreased by a factor offive then the resulted crest factor values are typically 5-10percent higher.

Number ofRunning time of

excited lines

Algorithm [2]Algorithm [3]

103 sec2 sec304 sec3 sec1009 sec4 sec30045 sec30 sec10002 min50 min300010 min-10000

60 min-

Table 3. Typical running time values of crest-factor minimization algorithms for differenttypes of excitation signals

5. Conclusion

In this paper a solution was suggested to avoid

problems arising from the misuse of periodic broadbandexcitation signals.

Examples illustrated that the power-loss of thespectrum in the band of interest can be serious when afractional period of the popular Schroeder multisinesignals is used. The power-loss causes insufficientexcitation in the band of interest and thus the result of theidentification process may be of lower quality.

It was proven that the random-phase multisine signalshave superior properties, they are much less sensitive tothe improper use. Even if some frequency lines are poorlyexcited, the excitation power is evenly distributed in theband of interest and no larger regions remain withoutexcitation.

The theoretical results were verified by experiments.The theoretical distribution function of the power loss wasshown to match with the distribution gained from 1000experiments.

The theoretical results can also be used to predict theloss of power in the band of interest when the excitationsignal is misused.

The weak crest factor of the random-phase multisinecan be effectively decreased by appropriate optimizationmethods. Two optimization algorithms known in theliterature were tested. The proposed combined method(random-phase multisine with optimization) provides“good quality” multisine with low crest factor values,which is also robust against misuse.

References:

[1]: J. Schoukens, P. Guillaume, R. Pintelon, Design of

Broadband Excitation Signals, Chapter 3 in:“Perturbation Signals for System Identification,” editedby K. Godfrey, Prentice Hall, 1993.

[2]:E. Van der Ouderaa, J. Schoukens, R. Renneboog, “Peak

Factor Minimization Using a Time-Frequency DomainSwapping Algorithm,” IEEE Trans. Instrum. Meas.,Vol.37, pp. 982-9, March 1988.

[3]:P. Guillaume, J. Schoukens, R. Pintelon, I. Kollár, “Crest-Factor Minimization Using Nonlinear ChebishevApproximation Methods,” IEEE Trans. Instrum. Meas.,Vol. 40, pp. 982-9, Dec. 1991.

[4]:I. Kollár, Frequency Domain System Identification

Toolbox User’s Guide, The MathWorks Inc., 1995.

[5]:M. R. Schroeder, “Synthesis of Low-Peak Factor Signals

and Binary Sequences with Low Autocorrelation,” IEEETrans. on Inform. Theory, Vol. IT 16, pp. 85-, Jan.1970.

[6]: A. Stuart and J. K. Ord, Kendall’s Advanced Theory

of Statistics (Vol. 1), Charles Griffin & Co., London,1987.