Multiple coherence method in time domain for an analysis ...past.isma-isaac.be/downloads/isma2010/papers/isma2010_0289.pdf · sense stationary processes, power spectral density and

Multiple coherence method in time domain for the analysisof the transmission paths of noise and vibrations with nonstationary signals

A. Gomez Gonzalez 1, J. Rodrıguez 2, X. Sagartzazu 3, A. Schuhmacher 4, I. Isasa 5

1 USC, Department of Applied Mathematics,Campus Sur S/N, 15782 Santiago de Compostela, A Coruna, Spaine-mail: [email protected]

2 USC, Department of Applied Mathematics,Campus Sur S/N, 15782 Santiago de Compostela, A Coruna, Spain

3 Ikerlan S. Coop., Department of Mechanical Engineering,Po Jose Marıa Arizmendiarrieta, 2, 20500 Arrasate-Mondragon, Guipuzcoa, Spain

4 Bruel & Kjær Sound & Vibration Measurement A/SSkodsborgvej 307, DK-2850, Nærum, Denmark

5 Orona EIC S. Coop., Elevator Innovation Center,Pol. Lastaola s/n, 20120, Hernani, Guipuzcoa, Spain

AbstractIn this work we analyze the transmission paths of noise and vibration through the multiple coherence method.The non-stationariness of the signals in our applications motivates the introduction of the time variable. Theanalysis is performed using Fast Fourier transform-based techniques and auto-regressive modeling. Thesemethods will be compared in terms of computational expenses and reliability. We illustrate their behaviorcomputing the power spectral density and the coherence of synthetic and experimental stationary and non-stationary signals.

1 Introduction

In many industrial applications, it is vitally important to be able to identify the relationships existing betweena certain output signal and a series of input signals. Thus, for example, in the car manufacturing industry, itis important to identify the contribution level of the different sources of the noise reaching the driver’s seat(the engine, the wheels, the aerodynamics, etc.) [10]. This enables us to pinpoint the predominant noisesource, in order to modify the design of the vehicle accordingly, so as to reduce the noise level in the driver’sseat. Other applications can be found in the field of geophysics, radar applications, civil engineering andbiomedicine. In this last field, there is one example in neuroscience in which the technique is used to analyzeencephalograms in order to study cerebral asymmetry or locate epileptic foci [9].

In these situations, spectral coherence [9, 6] is one of the fundamental tools used to analyze the linearrelationships existing between two random signals or processes. This magnitude, defined as the quotientbetween the square of the cross spectrum between two signals and the product of their autospectra, establishesthe correlations between the spectral content of both signals. It is therefore vitally important to obtain anaccurate estimate of the aforementioned spectra.

Currently, there are many different methods which enable us to calculate both the autospectra and the cross

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spectra [3, 4, 2]. The more conventional procedures, including the periodogram and the Blackman-Tukeymethod, are based on techniques which use the Fast Fourier Transform (FFT), making them highly efficientfrom a computational perspective. This, coupled with the fact that they are also very easy to implement,makes them extremely attractive. Nevertheless, the spectral estimators obtained are of poor quality, sincethey have a high level of variance and a low spectral resolution. These limitations stem mainly from thelimited (and therefore finite) nature of the data. The use of the Fourier transform as the main tool obligesyou to extend the data, usually by completing the signal with zeros. This in turn results in a loss of resolutionin the frequency domain, a phenomenon known as leakage. When analyzing a signal which has large peaksin the spectral field, this effect may mask or distort smaller peaks. These problems can be reduced bymultiplying the signal by adequately selected windows, although this often results in a loss of accuracy.

The methods outlined above implicitly assume that the non-observed data, or in other words, the data locatedoutside the window support, are all zero - a highly unrealistic hypothesis. Attempts to overcome this difficultygave rise to modeling-based or parametric methods. Parametric estimation thus becomes a procedure withthree different stages [2, 3, 4]: i) firstly, a model is selected (based on a priori knowledge) which reproducesour data as faithfully as possible; ii) next, the parameters present in the model are calculated; iii) finally, thespectra of the new approximation are calculated; in general, these spectra can be calculated exactly, giventhe simple structure of the approximated signal. Within the field of parametric estimation, the most popularmodels include autoregressive moving average methods [3] (also known as ARMA methods), in which thedata are approximated using random processes with very simple structure (generally white noise) and whichsatisfy a recurrence ratio with a low number of terms. In general (for a wide variety of signals), these methodsoffer a higher level of spectral resolution than that provided by methods based on the FFT, although they alsoinvolve a very high computational cost. Parametric models are of particular interest in situations in whichonly few data are available.

The methods described above assume that the analyzed signal is stationary, which basically means thatthey assume that the spectral composition does not vary over time. However, in many applications, thisassumption is not reasonable, for a number of different reasons [5]. On many occasions, the signals emittedcannot be considered stationary [9]. In other situations of interest, the emitting source of one of the inputsignals may be in movement in relation to the location of the receiver or vice versa, thus generating shifts inthe spectrum over time, due to the Doppler effect. In this case, it is necessary to introduce the time variableinto the spectral study and the calculation of the coherences. An initial idea to tackle this is to divide thesignals into short time blocks, so as to capture the time dependency and, by restricting the signal to each ofthe individual blocks, enable it to be considered stationary, i.e. a pseudo-stationary approach. Then, one ofthe methods valid for stationary signals is applied to each block [9]. This enables you to calculate the timeevolution of the spectrum and the coherences. Note, however, that increasing the time resolution by reducingthe size of the blocks results in a drop in spectral resolution. This is the idea which will be developed in thepresent paper. Another way of tackling the problem is to analyze the signals using transforms which includeboth the frequency and time variables. Two examples of this method are the Wigner-Ville transform andwavelet-based techniques [9, 8].

The present paper is organized as follows: section 2 outlines some basic notations of signal processing, widesense stationary processes, power spectral density and spectral coherence. In section 3 a series of spectralestimation techniques for stationary signals are presented and compared, ending with an explanation of theintroduction of the time variable to deal with non stationary signals. Finally, section 4 analyzes differentsignals, both stationary and non stationary, analytically and experimentally, including an analytic case inwhich a moving source is considered.

2 Spectral analysis and coherence calculation

The use of the multiple coherence method for our purposes, requires to perform a spectral analysis of thesignals involved.

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In this way, we define the power spectral density (PSD) by:

Pxx(f) =+∞∑

k=−∞rxx[k]e−i2πfk, (1)

whererxx [k] = E {x∗ [n]x [n+ k]} , (2)

is the autocorrelation function.

In an analogous way we define the cross power spectral density (CPSD) by:

Pxy(f) =+∞∑

k=−∞rxy[k]e−i2πfk, (3)

where rxy is now the cross correlation function:

rxy[k] = E{x∗[n]y[n+ k]}. (4)

Note at this point, that for these quantities to be well defined it is necessary for the process considered to bewide sense stationary (WSS) (see [3]).

The PSD describes the distribution in frequency of the power of x[n] and, as such, is real and non negative.The CPSD, on the other hand, is in general complex. The magnitude of the CPSD indicates whether fre-quency components in x[n] are associated with large or small amplitudes at the same frequency in y[n], andthe phase indicates the phase lag or lead of x[n] with respect to y[n] for a given frequency component.

A WSS process which is frequently used is that of white noise, which is characterized by having an autocor-relation function given by:

rxx[n] = σ2xδ[n], (5)

where δ[n] is the unit impulse function (which takes value 1 for n = 0 and zero elsewhere) and σ2x is called

the variance of the noise. This tells us that none of the samples are correlated with any of the others. Now,using the relation (1), for this type of process we get:

Pxx(f) = σ2x, (6)

which is completely flat regarding frequency. Alternatively, we could say that white noise is made up byequipotential contributions at all frequencies.

Once we have defined these concepts we can introduce the ordinary coherence between two signals x[n]and y[n] by:

γ2xy(f) =

|Pxy(f)|2Pxx(f)Pyy(f)

∈ [0, 1]. (7)

Ordinary coherence is, then, a statistical indicator which indicates whether or not a given input and outputare correlated (i.e. linearly related). It has values between 0 and 1, with 1 indicating a perfectly linearrelationship between the input and the output, and 0 indicating a complete lack of correlation.

For cases in which there are many inputs and one or more outputs, it is sometimes worth estimating thedegree of correlation existing between one group of selected inputs X[n] = {x1[n], . . . , xm[n]} and oneoutput y[n]. This is the basis of the concept of multiple coherence, defined as:

γ2Xy(f) =

PHXy(f)P−1XX(f)PXy(f)Pyy(f)

, (8)

TRANSFER PATH ANALYSIS AND SOURCE IDENTIFICATION 3929

where γ2Xy is the multiple coherence function between the series of inputs X and the output y, PXy is the m

dimensional vector of the CPSD between inputs X and the output y, PXX is the m×m dimensional matrixof the PSD and CPSD of the series of inputs and Pyy is the PSD of the output.

Multiple coherence also takes values of between 0 and 1. A value of 1 indicates that all the inputs in theseries are linearly related to the output, while a value of 0 indicates that none of the inputs are correlated tothe output.

Coherence is used to estimate the degree of participation of possible sources in the perceived signal. In areal case, first of all the possible sources are identified (using a priori knowledge of the object under study).Then reference sensors are placed in the vicinity of them (the inputs), and some other signals are measured inplaces of interest (the outputs). The autospectra and cross spectra between the output and the contribution dueto each source (by applying an appropriate transfer function to the inputs) acting separately are calculated:PXX , PXy and Pyy. Then the ordinary and multiple coherence can be calculated. For more detail see [7].

We should keep in mind that the method does not work properly if the sources are coherent. Mathematically,this leads to a poor conditioning of the PXX matrix and incorrect results for the inverse. In practice, thismeans that coherent sources cannot be separated. The usual procedure in this case is to initially analyzethe ordinary coherences between each pair of input signals, in order to determine the coherence betweenthem. When they are coherent, they are grouped together and become a group of inputs which are analyzedtogether, with the multiple coherence between the group and the output being considered. If the number ofincoherent sources in the measured sound field is less than the number of inputs the matrix will also becomerank-deficient and care should be taken when inverting, for example by introducing some regularization step.

As we have seen, all these definitions have a meaning when the signal is WSS. This is clearly a severerestriction, and it is doubtful whether any ”real-life” processes possess this property. In this sense we findsome works by Priestley, see for example, [5] in which he presents a theory of evolutionary (i.e. time-dependent) power spectra.

3 Spectral estimation

In this section we will present two different ways to calculate PSD and CPSD of a given signal. The firstapproach is based on the fast Fourier Transform (FFT), whereas the second one is based on auto-regressivemodeling of the signal.

3.1 FFT based techniques

We will just describe in this section the PSD estimator based on FFT techniques known as the Blackman-Tukey estimator (often abbreviated to the BT estimator). Given a signal {x[n]}N−1

n=0 this estimator is givenby:

PBT(f) =N−1∑

k=−(N−1)

rxx[k]e−i2πfk, (9)

where rxx[k] is the autocorrelation function. This estimator is based on equation (1). In practice, we do nothave access to the autocorrelation function. It is therefore necessary to replace rxx[k] with an estimator. Afrequent solution is:

rxx[k] =

1N

N−1−k∑n=0

x∗[n]x[n+ k], for k = 0, 1, . . . , N − 1,

r∗xx[−k], para k = −(N − 1),−(N − 2), . . . ,−1,

(10)


the so-called biased autocorrelation estimator.

Another possible approach to autocorrelation is as follows:

rxx[k] =

1

N − |k|N−k+1∑n=0

x∗[n]x[n+ k], for k = 0, 1, . . . , N − 1,

r∗xx[−k], para k = −(N − 1),−(N − 2), . . . ,−1,

(11)

this is known as the non-biased estimator, which in turn provides another PSD estimator, different from theprevious one.

The BT estimator for the CPSD can also be defined by estimating the cross correlation function rxy[k] in acompletely analogous way.

Once the autocorrelation function (respectively the cross correlation function) has been estimated in one orthe other way, we must calculate its Fourier transform. For this operation we have a very efficient algorithm,the fast Fourier transform having an algorithm complexity of O(N logN).

Note that, with these estimators we are not taking into account the expectation which appears in the definitionof the autocorrelation function. Therefore, various realizations of the process would be required. In practice,however, only one single realization of the process is generally available. It is therefore necessary for a singlerealization of the process: x[n], −∞ < n < +∞, to be enough to determine the autocorrelation function. Arandom process with the above-mentioned property is known as an ergodic process for the autocorrelation(see [2]). In general, a strictly ergodic process allows one to determine ensemble averages by replacing themwith time averages. Since checking that x[n] has this property is generally fairly complicated, it is simplyassumed by hypothesis.

Many of the problems of spectral estimation based on FFT are due to the suppositions made regarding thedata located outside the measurement interval. Implicitly we are multiplying the real infinite signal by arectangular window and as a consequence supposing that the non-measured data have a value of zero. Whenthe real power of the signal is concentrated in a narrow frequency bandwidth, it can be broadened to nearbyfrequencies, due to the phenomenon known as leakage. This multiplication by a rectangular window mayalso affect the spectral resolution. This is why different windows may be considered when calculating theseestimators, with the aim of reducing the effects of leakage, even if it is not completely eliminated. It is alsowell known that these FFT based techniques are inconsistent estimators and that they have a high variability.

3.2 AR based techniques

These techniques are introduced so as to be able to extrapolate the signal outside the measure interval withoutsupposing extension with zeros. In general, the steps to follow are:

1. Choose the parametric model-based method.

2. Calculate the parameters or coefficients of the above-mentioned model.

3. Finally the spectrum or PSD of the modeled signal can be obtained in a simple way due to its specificstructure.

In the first step, related to the choice of model, it may be useful to know, a priori, something about what thespectrum or PSD that we want to calculate is like; to some extent, the idea is to extrapolate the unknownvalues of the signal, so as to avoid having to complete it with zeros or supposing periodicity. Therefore, theneed for windows is eliminated, along with their distorting effect. The level of improvement in the spectralresolution and fidelity (where applicable) is due to the suitability of the selected model and its capacity toadapt to the measured data or the autocorrelation sequence, with few parameters.


The techniques analyzed in this section are a specific kind of parametric model, which possess rational systemfunctions and are conducted by white noise. Concretely we will center in autoregressive (AR) models.

In an autoregressive model the signal is approximated by the linear difference equation with complex coeffi-cients:

x[n] = u[n]−p∑

k=1

a[k]x[n− k], (12)

where x[n] is the output sequence of a filter (of order p) which models the observed data, and u[n] is theconducting noise, which will be supposed to be a white noise with zero mean and variance σ2

u.

To obtain the coefficients a[k], with k = 1, . . . , p and the variance of the white noise σ2u from a finite series

of data we must solve the so called Yule-Walker equations, which are just a linear system. In principle wewould need O(p3) operations to solve it, but its particular structure allows us to solve it in O(p2) operationsby the Levinson algorithm. Note that this order p will be related to the length of the signal, but will not behigher than half this length and in general will be much lower (see [2]).

It can be checked that the PSD of a signal with this structure is described totally in terms of the modelparameters (see [3]) through the equation:

PAR(f) =σ2u∣∣∣∣∣1 +

p∑k=1

a[k]e−i2πfk∣∣∣∣∣2 . (13)

Note that with this approximation we are already taking into account the randomness of the signals, sinceonce the signal has been modeled, the PSD is obtained directly and analytically. That’s why in this case, theneed for time averages disappears.

For the calculation of the CPSD it is necessary to define multivariable models which allow to consider bothsignals together.

One critical aspect when using this type of method is the choice of p, the order of the filter to be used. Theoptimum filter order is not known a priori, which is why in practice, various orders need to be postulated.Based on this, some kind of error criterion is defined to indicate which order to select. If the order is toolow, it produces an excessively softened spectral estimator, while if it is too high spurious peaks may appear.An intuitive approximation would be to construct a series of autoregressive models with increasingly highorders, until the error level reaches a minimum. In [2] the authors outline different criteria for selecting thisorder.

3.3 Introduction of the time variable

The spectral estimators we have analyzed in the two previous sections assume that the signal is WSS; how-ever, in general, the signals we tend to analyze are not necessarily of this type.

As a consequence we shall divide each signal into blocks localized in time. We will assume that the restrictionof the signal to each block is WSS. Each block is analyzed with one of the preious techniques. In the case ofmethods based on the FFT, this is known as the Short Time Fourier Transform, often abbreviated to STFT,and has also been used to obtain coherences, see for example, in [6]. In this paper, in addition to analyzingmethods based on the FFT, we have also calculated the spectra of each of the blocks using techniques basedon parametric models.

We will now distinguish between both techniques:

• In the case of FFT-based techniques, it was already necessary to divide the signal in blocks in orderto take time averages since once the ergodicity has been assumed, this is equivalent to calculating


the expectancies in a random process or signal. Indeed, since we were multiplying each block bya window, it is usual to consider an overlap between the blocks, this makes that we will have moreblocks, but in the end we will be obliged to take time averages, so we will lose some of them, reducingthe time resolution.

• In the case of AR modeling we recall that neither this need for averages nor the overlap was necessary.So in this case we will be able to have better frequency resolution (bigger blocks) and quite a goodtime resolution.

What is important to bear in mind is the fact that the frequency resolution obtained will be directly related tothe size of the blocks considered; in fact, the larger the blocks, the better the frequency resolution obtained.However, there is a tradeoff between the frequency and the time resolution since a reduced number of blocksimplies a very poor time resolution. This is related to the Heisenberg uncertainty principle.

4 Numerical and experimental validation

4.1 Calculation of power spectral densities: FFT vs AR

Before calculating coherences, we will first analyze the basic differences between the two techniques de-scribed so far in this document, applied to the PSD calculation of a short WSS signal.

Let us consider, an ARMA(2, 3) signal, i.e:

x[n] = u[n]−2∑

k=1

a[k]x[n− k] +3∑

k=1

b[k]u[n− k], (14)

where a[1] = −1.4001, a[2] = 0.9801 are the AR coefficients, b[1] = 0.3314 b[2] = −0.2651, b[3] =−0.5120 are the MA coefficients, and u[n] is a white noise with zero mean and varince σ2

u = 1. This signalwill be considered to be measured in a time interval of 1 s with 5000 samples. One realization of this signalis represented in Figure 1.

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Figure 1: Signal of which we are going to calculate the PSD.

For this signal we can calculate explicitly its PSD (see Figure 2(a)), which we will compare with thoseobtained with the FFT-based techniques dividing it in 8 blocks and making averages (see Figure 2(b)) andwith AR based method, where there is no need for averages (see Figure 2(c)). The order of the AR modelconsidered is p = 12 (we see that it is much lower than half the size of the signal which would be 2500).

The first difference we see between both results is that the PSD obtained using autoregressive techniques ismuch smoother than that obtained using FFT-based techniques, even taking into account that in the secondcase we are making time averages. In other words, the PSD calculated using FFT-based techniques presentsa much higher degree of variability.

Qualitatively speaking both results are rather similar. One should keep in mind that the PSD is a density, sothe most important thing is that the integral between two frequencies coincide.


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Figure 2: PSD of an ARMA(2, 3) signal.

Nevertheless, one area in which a major difference can be observed between the two techniques is in thecalculation time. Even when calculating only the PSD of a single block, such as the case in question here,autoregressive techniques are slower than FFT-based ones. And this difference is even more noticeable inthe case of long signals, and when the block sizes are fairly large (which is necessary in order to obtain goodfrequency resolution); in fact, the calculation time is closely related to the filter order used, and its estimationin turn depends directly on the size of the blocks for which we want to calculate the PSD.

Consequently, autoregressive techniques offer a greater range of advantages over FFT-based ones when, forexample, the signal considered is very short, i.e. when few samples are available. In this case, the fact thatautoregressive techniques do not require time average calculations enables us to obtain better time resolution.When the size signal is large, the computational time becomes too expensive with respect to the FFT-basedtechniques. When it can be assumed that the autoregressive model is valid, less biased spectral estimatorsare obtained, with a lower level of variability (i.e. with better properties from the probabilistic perspective ingeneral) than obtained using conventional spectral estimators, based on the FFT.

4.2 Calculation of coherences on a model problem

In this section we will consider two sources emitting different signals. These signals will be received at areceptor located at a distance R1 from one source and R2 from the other one.

We will distinguish between two cases: numerical and experimental with the same configuration, but withsome key differences which we will describe in the next sections. The mentioned configuration is shown inthe sketch included in figure 3(a), where R1 = 1.17 and R2 = 3.13. The corresponding working scenariofor the experimental cases is shown in the photograph in figure 3(b).

M1

M2

M3

0.17 m

0.17 m

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0.83 m

2.32 m

1.86 m

(a) Sketch of the working scenario. (b) Working scenario.

Figure 3: Sketch and photograph of the working scenario.


4.2.1 Numerical validation

For the numerical case we assume that the source is puntual and the emitted signal is known. As a con-sequence we don’t need the microphones we have in the sketch in Figure 3(a). Firstly, it is necessary tocalculate the transfer functions in order to determine, given the signal emitted by the source the contributionassociated to that source. In the time domain, the signal received will have a time lag (the time requiredto reach the position in which the receptor is located), along with an attenuation. The time a wave takes totravel the distance Ri is Ri/c where c is the speed of sound (c = 343 m/s). In the frequency domain, thetransfer function is determined by:

Fi(f) =eRii2πf/c

4πRi. (15)

To determine the transfer function in the time domain, hi(t), we only have to carry out the inverse Fouriertransform of the previous function.

If we use Qi(t) to denote the temporal signal being emitted by the i-th source, and we wish to determine thesignal being received by the receptor (i.e. the contribution of each of the sources at the receptor position),we simply have to calculate its convolution with the corresponding transfer function in the time domain:

Ci(t) = hi(t) ∗Qi(t). (16)

To obtain the total contribution, we simply have to add together both contributions: C(t) = C1(t) + C2(t).

The ultimate aim is to analyze the coherences between:

• The contribution of each of the sources and the output (ordinary coherence).

• The contribution of both inputs and the output (multiple coherence).

In the numerical case, since no noise has been added to the output it is merely the sum of each of thecontributions. Thus we should expect the multiple coherence to be always one.

Stationary case

In order to illustrate the stationary case, each of the sources is assumed to be emitting a white noise. Figure4 shows the signals emitted by both sources.

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Figure 4: Stationary case. Signals emitted by the sources.

Here, we could have made the analysis without the introduction of the time variable, because since it is astationary case, the introduction of the time variable will not provide us with any additional information.Anyway it will be useful to validate, to some extent, the subsequent results.

Our intuition tells us that if both sources are emitting independent and uncorrelated stationary signals withthe same sound power, and one source is nearer the receptor than the other, then that source should contributemore.


(a) Nearer input (b) Farther input (c) Multiple coherence

Figure 5: Coherences obtained in a stationary case.

Figure 5 shows the results obtained. As we expected, we see how the nearer input contributes, for everyfrequency and for every time, more than the farther input. And, indeed, the multiple coherence is alwaysone.

Non stationary case

In this case, a random stationary signal (white noise) emitted by the far source was considered along with arandom non stationary signal emitted by a near source.

As non stationary signal we have selected a random signal, increasing in magnitude over time. Figure 6shows the signals emitted.

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In this case, it is reasonable to assume that there will be a time-related evolution of the coherence: at first,the far source will have more weight, but as the amplitude of the signal emitted by the near source increases,so will its contribution.

Figure 7 shows the results obtained, and illustrates how there is indeed a time-related evolution of the co-herence. We see how around time t = 10 s nearest input starts contributing more than the farther as itsamplitude increases.

Note that the results are almost constant in frequency, since white noises have been considered.


Figure 7: Coherences obtained in a non stationary case.


4.2.2 Experimental validation

In the experimental validation the working scenario consisted of two loudspeakers with two microphoneslocated nearby to measure the signal being emitted. A third reference microphone to record the output wasalso used. This microphone was located at a certain distance from the other two (see Figure 3).

The loudspeakers used were MCM audio select Clear Polypropylene Woofers (8 Ohm loudspeaker, nominalpower: 40 W, frequency response from 44 Hz to 8 kHz, open air resonance frequency: 61 Hz, acousticsuspension of volume: 0.0187 m3)

The microphones used were of type 4189 from Bruel&Kjær.

First of all we will briefly describe some of the difficulties which we encountered when making the measures:

1. Non coherent sources:When defining coherence, the multiple coherence method is not capable ofseparating the contributions of different sources if the signals emitted are coherent; for this reason itis important to check, before carrying out the analysis, that the input signals are not coherent. This, inthe experimental case, is more complicated, because despite the fact that we placed one microphonenear one of the loudspeakers, we are always going to be measuring part of what is being emitted bythe other loudspeaker.

To be sure both signals emitted were non coherent we used two different generators which enabled usto carry out the analysis.

2. Transfer functions: The transfer functions for this case had been obtained experimentally. In general,transfer functions are measured as quotients between sound pressures and volumetric flows, but due tothe difficulty of measuring these volumetric flows in practice, we used transmissibilities instead, i.e.ratios between the sound pressures measured between the reference microphone and each of the sourcemicrophones (near and far). This is the main difference between the numerical and the experimentalcase.

3. Total contribution: In the numerical case, the total contribution was obtained simply by adding to-gether the separate contributions of each of the sources. Here, the aim was to consider the signalmeasured at the reference microphone as the total contribution. This does not coincide exactly withthe sum of the two separate contributions. As we shall see, this gives rise to errors; for example, themultiple coherence between the output and both inputs will not exactly be equal to 1.

4. Frequency range: The frequency range in which we can consider the loudspeakers as ideal sourcesis between 0 Hz and 500 Hz. Consequently, our analysis was restricted to frequencies in this range.

5. Signal to noise ratio (SNR): One aspect that may be particularly important when carrying out anexperimental test is background noise, which can be measured using the signal to noise ratio, explainedbelow.

Since the experiments were carried out inside a semi-anechoic chamber, it was reasonable to assumethat the signal to noise ratio (SNR) would be high enough and that we would not encounter anyproblems as a result of background noise. Nevertheless, the corresponding check was carried out,obtaining a high signal to noise ratio.

6. Analysis of the contributions: Finally, we tried to check whether or not we were really calculatingcorrectly the contributions of each of the loudspeakers, and therefore that each microphone was basi-cally measuring the signal emitted by the nearest loudspeaker, rather than the one located further away.Due to lack of space we will not show here the results, but they guaranteed us we were indeed gettingthe correct contribution.


Stationary case

The signals emitted were, as in the numerical case, white noises. In Figure 8 we see the measured signals ata point close to the emission point.

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Figure 8: Stationary case. Signals measured.

The results obtained in the aforementioned frequency range are shown in Figure 9. Since in this range thebehavior of the source can be considered ideal we see basically the same results as in the numerical case.As expected, the multiple coherence is not exactly one, which is due to reflections within the chamber or,alternatively, to the non-ideal nature of the source considered. We can observe that for very low frequencieswe have a strange behavior, which may be due to reflections in the semi-anechoic chamber.


Figure 9: Stationary case. Coherences obtained.

Non stationary case

In this case, we also emit analogue signals to the ones considered in the numerical validation. The signalsmeasured at a point close to the emission point are shown in Figure 10.

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Figure 10: Nonstationary case. Signals measured.

Representing the coherence in the frequency range from 0 Hz to 500 Hz (see Figure 11) we observe howthe near source does not always contribute more than the far source. However, in the experimental case, theprecise moment in which it starts contributing more not is not as clear as in the numerical case. As in thestationary experimental case for very low frequencies we have a strange behavior.



Figure 11: Non stationary case. Coherences obtained.

4.3 Calculation of coherences with a moving source

In this final section of the validation we will consider a model numerical problem consisting of two sources,the first one static, emitting an ARMA(2,3) signal whose PSD we can see in Figure 12(a), plus a simulatednoise and the second one a moving source emitting an ARMA(4,2) whose PSD is shown in Figure 12(b)plus another simulated noise. Since both signals emitted are WSS we show their PSDs just as a functionof frequency. The evolution in time of the coherences will be due to the movement of one of the sourceswith respect to the receptor. As the final output we will consider the sum of both contributions plus anothernoise. All the noises considered will be of an amplitude of 5% with respect to the signal, and are included tosimulate possible errors in the measures.

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Figure 12: Exact PSD of the signals emitted.

For the academic case considered in this section, we will assume the speed of sound to be c = 1. The movingsource will move at a speed of 0.5 m/s along the line y = −0.5, so the Mach number will be equal to 0.5.We show the configuration in Figure 13. The static source will emit the signal from the point (0, 1.5, 0),while the moving signal will move from (−2.5,−0.5, 0) to (2.5,−0.5, 0). Note that we will suppose thatthe sources do not start emitting and/or moving in t = 0, but that they are emitting and/or moving also inprevious instants, although the experience will center on a specific time interval (from 0 s to 10 s). Thereceptor R will be located at point (0, 0, 0).

In this case we have to take into account the movement of the source in the transfer function, so as to obtainthe signal which will be received by the receptor. Indeed, we have to consider the Doppler effect whichdepending on the case in consideration will play an important role. In Figure 14 we have the signals receivedat the receptor point as a consequence of the static and moving signal considered and the output. Note thatthe instant in which the contribution due to the moving source is maximum does not coincide exactly witht = 5 s, but t = 5.5 s, since the speed of sound is 1 m/s in this experience.

The coherences obtained are shown in Figure 15. If we compare the coherence from the static source withits PSD we see how the peak we should be seeing at 200 Hz, appears a bit masked by the peaks at 100 Hzand 1500 Hz from the moving source, since this values of the frequency are changing over time: when thesource is approaching the receptor the frequency is higher than the emitted, while when it is moving away


x

y

R = (0, 0, 0)

Es = (0, 1.5, 0)

Em(0) = (−2.5,−0.5, 0) Em(5) = (0,−0.5, 0) Em(10) = (2.5,−0.5, 0)

Figure 1: Configuration of the numerical experience.

1

Figure 13: Configuration of the numerical experience.

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Figure 14: Signals received at the receptor point.

from the receptor this frequency is lower due to the Doppler effect. The Doppler effect can also be seen onthe other peak of the PSD of the signal emitted by the moving source. Looking at the PSDs of the signalemitted by the static source we also see that its magnitude increases with the frequency, while the one ofthe moving source decreases. This is also seen in the coherences. Indeed, in the central interval of time, themoving source is nearer the receptor than the static source. For not very high frequencies we see how thisimplies that it is contributing more, but for the high frequencies it is the static input the one that contributesmore.

(a) Static input (b) Moving input (c) Multiple coherence

Figure 15: One moving source. Coherences obtained.

5 Conclusions

We have explored different techniques in particular FFT and AR based for multiple coherence calculations.We have seen that the FFT based method has a high degree of variability, while AR based techniques providesa smoother approach, however in computational time the FFT is a faster algorithm. We have exhibited


the importance of introducing the time variable when considering non stationary signals, or even if thesignals considered are stationary but one of the sources is moving. In the future work we will consider time-frequency (such as Wigner-Ville) or time-scale transforms (such as Wavelets) for the study of non stationarysignals.

Acknowledgements

The first author (A. Gomez) is supported by Spanish Ministry of Education through Grant Number AP2008-01243.The second author (J. Rodrıguez) is supported by the MCINN (Spanish Ministry of Science andInnovation) contract RYC-2008-03543.

References

[1] H. Van der Auweraer, K. Wyckaert, J. Top, Spectral estimation of time-variant signals, in Publicationsat ISMA 17th, Leuven, Belgium, 1992 September, Leuven (1992), pp. 207-224.

[2] S. M. Kay, S. L. Marple Jr, Spectrum analysis - a modern perspective, Proceedings of the IEEE (1981),vol. 69, pp. 1380-1419, 1981.

[3] S. M. Kay, Modern Spectral Estimation: Theory and Application, Prentice Hall Signal Processing Series(1988).

[4] S. L. Marple Jr, Digital Spectral Analysis with Applications, Prentice Hall Signal Processing Series(1987).

[5] M.B. Priestley, Time-dependent spectral analysis and its application in prediction and control, Journalof Sound and Vibration, Vol. 17, No. 4, Academic Press (1971), pp. 517 - 534.

[6] M. W. Trethewey, Temporal noise source separation by short time coherence output power, Proceedingsof ISMA 2008, Leuven, Belgium, 2008 September, Leuven (2008), pp. 3051 - 3060.

[7] R. Potter. Matrix formulation of multiple and partial coherence. Journal of the Acoustical Society ofAmerica, Vol. 61, Issue 3 (1977), pp. 776 - 781.

[8] J. K. Hammond and P. R. White, The analysis of non-stationary signals using time-frequency methods,Journal of Sound and Vibration, Vol. 190, No. 3, Academic Press (1996), pp. 419 - 447.

[9] A. Klein, T. Sauer, A. Jedynak and W. Skrandies, Conventional and Wavelet Coherence Applied toSensory-evoked Electrical Brain Activity, IEEE Transactions on Biomedical Engineering (2006), Vol.53, No. 2, pp. 266 - 272.

[10] N. Møller, M. Batel and F. Henderson, Obtaining Maximum Value from Source/Path Contribution Anal-ysis, Proceedings. JSAE Annual Congress 2005, Japan, Japan (2005), pp. 7 - 10.



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