Coherence length as a figure of merit in multireference near-field acoustical holography^a) Open Access

Near-field acoustical holography (NAH)¹ requires a spatially coherent hologram to give a successful reconstruction of the sound field. This may be obtained from a simultaneous measurement of all field points. However, hologram measurements often consist of hundreds or thousands of measurement grid points, making the number of field microphones required impractical. In these cases, a scan-based measurement with a small, dense, field array, in combination with a fixed-location reference array, is used.

When a sound field is generated by a single, coherent source, only one reference microphone is required to tie the scans together. Hald² developed a multireference procedure called spatial transformation of sound fields (STSF), which is a partial field decomposition (PFD) method to accommodate sound fields of multiple independent (incoherent) subsources. In STSF, a singular value decomposition (SVD) of the cross-spectral matrix of reference signals can be used as the basis for a linear projection of the field measurement, which results in a set of mutually incoherent partial fields. The incoherent sum of these partial fields represents the total field. Each partial field may then be projected individually using NAH and summed to provide the total reconstruction. Because the partial fields in this method are based on an SVD, they do not directly represent independent physical subsources.

Additional PFD methods have been developed to generate partial fields that can be associated with physical subsources.^3,4 These are most successful when each reference is located as closely as possible to each individual subsource. Kim et al.⁵ introduced a method that uses holographic projection to determine the optimal reference locations and then places a set of virtual references at those locations. However, in the STSF, virtual reference, and all PFD methods, it is important to understand that the physical reference array must completely sense all independent sources to begin with—the number of reference microphones must equal or exceed the number of subsources and each subsource must be sensed by at least one reference—if the total signal energy at the hologram is to be represented in the decomposition.⁵ In practice, when the subsources are localized or spatially distinct, this requirement is simple to meet.

What is to be done when independent subsources are not localized, such as in the case of an aeroacoustic source? An aeroacoustic source may be considered a continuum of partially coherent subsources where the number of subsources is ambiguous. Lee and Bolton,⁶ Shah etal.,⁷ Wall etal.,⁸ and others performed scan-based NAH on jet-noise sources. In each of these experiments, the sufficiency of the reference array in completely sensing all subsources was verified after the measurement was taken, using the virtual coherence method.^9–11 However, no quantitative guidelines exist in current literature to predict reference-array sufficiency a priori or guide the design of a reference array when the number and locations of independent subsources are unknown. Although this study is directed toward an application in aeroacoustic measurements, no attempt is made here to model the sound field of an actual aeroacoustic source. Rather, the partial spatial coherence of an aeroacoustic source is represented by a simple array of partially coherent point monopoles.

The purpose of this paper is to present a simple, quantitative figure of merit that yields the inter-microphone spacing necessary, given an array aperture, for the deployment of a sufficient reference array near partially coherent complex sources. This guideline is based on a spatial analysis of the (ordinary) coherence in the region of the source, such as those performed by Wall etal.,¹² that can be performed with relatively few sensors. From such coherence measurements, an average spatial coherence length in the sound field is determined. The spatial density of microphones in the reference array dictates how many references, on average, are located within an average coherence length. This figure of merit is called “references per coherence length” (RPL_C).

An analysis of the spatial coherence measured in the geometric near field of partially spatially coherent sources leads to the definition of RPL_C.¹³ To obtain a spatial coherence measurement, a linear sensor array (collinear with the eventual location of the reference array) is placed near and along the length of an extended source. Coherence values,¹⁴ $γzm,z2$⁠, between a given sensor at a location z_m and all sensors along the array are calculated. From these data, the coherence length, L_C, is determined. Although the term “coherence length” is used in other studies, such as in optical holography applications¹⁵ and underwater acoustics,¹⁶ here it is defined as the spatial distance L_C along the array over which $γzm,z2$ drops from unity to some desired threshold. In this paper, a coherence threshold of $γzm,z2$ = 0.5 is used. The value of L_C is assigned to the location z_m of the given sensor. This process is repeated for all N_R sensor locations, resulting in an array of coherence length values, L_C(z_m). The mean of L_C(z_m) over all locations yields the mean coherence length,

which summarizes the spatial coherence of the sound field into a single quantity.

Note that $γzm,z2$ generally drops off in both directions of increasing and decreasing z away from z_m, so L_C(z_m) may be calculated in either direction. If the source radiates symmetrically, then 〈L_C〉 is the same for either direction chosen. This may not be the case when the sound field is asymmetric.¹² 

With ⟨L_C⟩ established, references per coherence length is defined as

where the distance Δz_R is the physical spacing between sensors in a reference array. The figure of merit RPL_C quantifies the spatial density of sensors in the reference array in terms of spatial field coherence. As spatial source coherence decreases, ⟨L_C〉 decreases and the number of effective independent subsources increases. A greater number of reference sensors are required to completely sense these subsources, so the inter-reference spacing must be more dense (i.e., for a reference array of fixed aperture length, the sensor spacing Δz_R must decrease). The quantity RPL_C takes advantage of the fact that information about the number of independent subsources is contained in ⟨L_C〉. One might expect that the value of RPL_C required for a sufficient reference array is invariant with source coherence because the necessary Δz_R decreases with decreasing ⟨L_C〉. Such a relationship is demonstrated in this paper.

To assess the utility of RPL_C, a quantitative method of determining the sufficiency of a reference array is desired. Therefore the quantity “mean virtual coherence sum,” ⟨$Σγ̃2$〉, is defined. This quantity is calculated after application of the virtual coherence method.⁹ In a PFD based on the SVD of the reference cross-spectral matrix, the strengths of the individual partial fields are determined by the ordered, monotonically decreasing singular values, each of which corresponds to one of the normalized partial fields. The partial fields with the highest singular values correspond to dominant noise sources. Theoretically, when there are S independent subsources and low measurement noise, there will be S large source-related singular values. Then the first S partial fields can be propagated using NAH; the rest can be discarded.

However, in aeroacoustic sources, the number of subsources is ambiguous. This is reflected in a more gradual decrease of the ordered singular values with no clear distinction between source and noise-related singular values.^9,10 Virtual coherence, in conjunction with the STSF method, provides a way to estimate the number of significant partial fields.^9,17 The virtual coherence function,

quantifies the amount of (normalized) coherent energy in the ith partial field at the jth measurement location. The quantity $Cvipj$ is the cross-spectrum between the virtual reference signal v_i and the field signal p_j, $Cvivi$ is the autospectrum of the virtual reference, and $Cpjpj$ is the autospectrum of the field signal. If the sum of the virtual coherence function over all the partial fields, $∑iγ̃i,j2$⁠, approaches unity for each field location j, then the reference array has sufficiently sensed all of the subsources. If a subset of the first K partial fields returns a sum that approaches unity for each j, then these K partial fields fully contain the source information (i.e., K ≥ S), and the remaining partial fields are discarded. Examples of $∑iγ̃i,j2$ plotted over all j are shown by Lee and Bolton (see Ref. 9, their Figs. 15 and 17).

It is useful to collapse the information in the spatial maps of maps of $∑iγ̃i,j2$ to a single number, which has not been done previously. Taking the average over field locations, the “mean virtual coherence sum” is obtained, i.e.,

where J is the total number of field measurements. If $〈∑γ̃2〉$ approaches unity, then it is likely $∑iγ̃i,j2$ is nearly unity for all j. Hence, $〈∑γ̃2〉$ quantifies the sufficiency of a reference array in this paper. A calculation of $〈∑γ̃2〉$ as a function of RPL_C results in an RPL_C criterion for a sufficiently dense reference array regardless of frequency and spatial coherence.

To investigate the relationship between RPL_C and $〈∑γ̃2〉$⁠, a numerical experiment was performed in which an approximately continuous, partially coherent source was generated one frequency at a time. See Fig. 1 for a schematic. An array of N_Q = 1000 point monopoles, marked by red circles, were spaced evenly along the z axis between z = 0.9 and 2.1 m with an inter-source separation of d such that kd $≪$1, k being the acoustic wavenumber. A “primary” signal vector, $ŝ1$⁠, was generated of complex numbers with unit magnitude and random phase. This signal defined the amplitude of the first monopole source, i.e., $q̂1=ŝ1$⁠. Subsequent source amplitudes were defined by

where $q̂n−1$ were the complex amplitudes of the previous source, $ŝn$ were newly generated random complex signals, and the coherence factor b dictated the portion of new, random signal energy that was added to the previous source signal. Division by the L2 norm, ||·||₂, of the total signal ensured that all monopole magnitudes were unity. The result was a line array of monopoles with normalized magnitude, comprising a partially coherent source. The spatial coherence properties of this source, and consequently the radiated field, depend on the coherence factor, b, and on frequency, f. For example, a b value of zero results in a perfectly coherent source, and as b approaches infinity the sources becomes completely incoherent.

A scan-based measurement of the sound field was simulated by propagating sound pressures from the monopoles to the measured hologram using the free-space Green’s function. The measurement grid was 10 cm from the sources and consisted of a linear array of 11 receivers, placed at 43 locations, to generate a hologram of 11 × 43 points with equal 7.62 cm (3 in.) spacing. Its location is marked by the blue dashed line in Fig. 1. An infinite signal-to-noise ratio (SNR) was assumed in the measurement. The holographic projection of these data was not carried out in this work. Rather the hologram data serve as the field signals for which the reference array sufficiency is determined. When a reference array sufficiently senses all sources, then all energy in the field measurements is represented in the partial field decomposition.

The reference array, shown by the green “×” symbols in Fig. 1, was placed at a standoff distance of x_R = 0.3 m from the source with an aperture that spanned from z = 0.3 to 2.7 m (twice the length of the source) and with variable spacing, Δz_R, as illustrated in Fig. 1. With the fixed aperture length, L_R, Δz_R depended on the number of references, N_R.

The simulated scan-based measurement of the partially coherent source was repeated as f was varied from 0.1 to 10 kHz, b was varied from 0.01 to 10, and RPL_C was varied from 0.2 to 2. The analysis for each parameter set proceeded as follows. (1) From the L_C values found in the direction of increasing z, 〈L_C〉 was calculated. (2) Reference sensor spacing Δz_R was determined for a given RPL_C value using Eq. (2). (3) A PFD of the sound field was performed using a reference array with spacing Δz_R. (4) The mean virtual coherence sum, $〈∑γ̃2〉$⁠, was calculated to determine reference-array sufficiency. Results of this experiment are reported in the following section.

Figure 2 shows examples of coherence $γzm,z2$ as a function of z, referenced to two values of z_m, and for 300 and 900 Hz. The coherence factor was b = 0.05. Note the locations where $γzm,z2=0.5$⁠, which value was used to define L_C. The values of L_C tend to be smaller for large f and lower b. This trend also holds true for 〈L_C〉. It is also important to note that 〈L_C〉 depends somewhat on the reference-array aperture; calculated coherence lengths vary over location and can become large far from the source.

The values of $〈∑γ̃2〉$ are plotted against N_R in Fig. 3(a) for two frequencies (300 and 900 Hz), and two b values: 0.05 for the highly coherent case and and 10 for the highly incoherent case. For the purpose of comparison, $〈∑γ̃2〉$ ≥ 0.99 is considered to be “sufficient” in sensing all subsources. The minimum N_R at which this occurs is represented by $ÑR$⁠, the critical number of reference microphones. Figure 3(a) illustrates how $ÑR$ depends on both b and f, although it is far more sensitive to f than to b in this case. In general, it is difficult to predict the sufficiency of a reference array for an arbitrary source configuration based on an analysis of $〈∑γ̃2〉$ as a function of N_R.

The figure of merit RPL_C provides an alternative way to view these data that shows a more consistent trend. The data in Fig. 3(a) are shown again in Fig. 3(b), but this time they are plotted against RPL_C. Note how the curves now more closely collapse, and $〈∑γ̃2〉$ exceeds 0.99 above RPL_C ≈ 1. For all f and b tested in this work, with the definition of 〈L_C〉 given in Sec. II, one reference microphone per coherence length (RPL_C = 1) is considered sufficient to fully decompose the subsources.^13,18 Hence, RPL_C, as defined in Sec. II, is a useful figure of merit when source coherence properties or the number of independent subsources is ambiguous. It should be noted that although the criterion RPL_C = 1 may be intuitive, it is dictated by the definition of the coherence length, L_C, which in turn is dictated by the coherence threshold. For example, a coherence threshold of $γzm,z2=0.8$ resulted in a criterion of RPL_C = 0.2 for the same experiment.

It is important to understand the extent to which RPL_C = 1 is a reasonable guideline. This guideline can become invalid when test parameters are varied to the extreme. As one example, decreasing b to smaller orders of magnitude than those shown here results in a highly coherent source, with long coherence lengths, and the calculation of RPL_C becomes unreliable. However, in this case, only one or two references are needed, eliminating the utility of such an analysis. For all practical realizations of source coherence and frequency, RPL_C = 1 is quite robust.

To test the robustness of RPL_C = 1, the experiment was repeated with b = 0.05, f = 900 Hz and with a varying reference array aperture and standoff distance. The criterion RPL_C = 1 was used to determine Δz_R. First, $〈∑γ̃2〉$ was calculated as a function of x_R for L_R = 2.4 m (twice the source length) and then as a function of L_R for x_R = 0.3 m. The results are shown in Fig. 4. Note that $〈∑γ̃2〉$> 0.99 for a range of x_R = 0.1 to 1.1 m in Fig. 4(a) and for a range of L_R = 1.2 to 4.0 m in Fig. 4(b). This suggests that RPL_C = 1 results in a sufficient reference array for a wide range of apertures and standoff distances. Extreme values of L_R and x_R can cause the RPL_C = 1 guideline to break down, such as moving the reference array far from the source. However, this experiment shows that a reference-array design that satisfies RPL_C = 1 based on a rudimentary knowledge of the source location and extent sufficiently senses all subsources.

A useful figure of merit for predicting a sufficient inter-microphone spacing of a reference array, deployed near a partially spatially coherent, extended source for the application of partial field decomposition (PFD) and near-field acoustical holography (NAH) methods, has been introduced. This quantity, “references per coherence length” (RPL_C) can be calculated from simple coherence measurements near the source. It has been shown that if microphones are placed such that there is one reference per coherence length, according to the definition of RPL_C given here, all independent subsources will be sufficiently sensed. This has been shown to be a robust guideline regardless of frequency, spatial source coherence, reference aperture, and standoff distance, over a broad range of these parameters.

There exist a myriad of additional source and measurement configurations to which such an RPL_C study can be applied to further ensure that RPL_C = 1 is a robust criterion. Future investigations could include the effects of amplitude-weighted sources and the directional radiation due to phase shading, and measurement noise. In addition, the validity of RPL_C = 1 to aeroacoustic or other physical, partially coherent sources needs to be investigated.

Special thanks to Scott D. Sommerfeldt and Jonathan D. Blotter for their insightful contributions.