Chaotic reverberation in a cavity, when coupled with time reversal acoustics, can be harnessed to build a perfect time-reversal mirror for transmitting and receiving highly focused sounds with a small number of transducers. In this article, a virtual receiving array, comprised of a single receiving transducer and a chaotic cavity, is developed based on time reversal processing of the reverberation inside the cavity. A prototype array, having 10 × 10 virtual receiving elements, is built and evaluated against a comparable physical array in terms of its localization and waveform reproduction capabilities. It turns out that the most crucial factor in the success of a virtual array is the ergodicity of its chaotic cavity, the exact mathematical expression for which is also derived. The virtual receiving array presented here may find some niche applications in reverberant environments, where a physical array turns out to be too costly or cumbersome to operate.

## I. INTRODUCTION

Since the concept of time reversal acoustics (TRA) in a chaotic cavity was first introduced in Refs. 1–3, virtual transmitting arrays employing time-reversal chaotic cavities have been developed for generating and steering highly-focused sound beams with only a handful of transducers. A variety of chaotic cavities (e.g., Sinai billiard, cylindrical, and planar-layer reverberators) and time reversal techniques were proposed for use in fluids^{4–8} and solids,^{9–13} respectively. Also, time-reversal focusing performance was compared among the different types of chaotic cavities and time reversal techniques.^{14–17} In contrast, relatively little attention has been paid to virtual receiving arrays based on the same principle. Quieffin *et al.*^{5} were among the first to consider a virtual receiving array within the purview of acoustic source localization in fluids. Their method, employing a solid cavity and a single contact transducer immersed in water, involved a two-step procedure. The first step was to construct an impulse response library by collecting impulse responses between the contact transducer and several points in water. In the second step, the contact transducer recorded the reverberation inside the cavity, of a pulse sent from an acoustic source of unknown location. Then a matched field processing (MFP)^{18} of a kind was performed, where the recorded signal was correlated with each impulse response in the library to locate the source with the maximum correlation. In a similar vein, Sinelnikov *et al.*^{8} and Catheline *et al.*^{19} demonstrated source localization in water and air, respectively, using planar-layer reverberators and a human skull as chaotic cavities. As for virtual receiving arrays in solids, Ing *et al.*^{20} explored the possibility of impact localization in a glass plate, which led to investigations in more complex structures such as aluminum fuselages.^{21–23} It is worth noting that these localization studies, almost invariably employing MFP in their execution, are indeed embodiments of receive-mode TRA thanks to the mathematical equivalence between TRA and MFP. See Sec. II and the Appendix for more details.

In this paper, building upon the idea of time reversal in a chaotic cavity, we present the principle, construction, and evaluation of a general-purpose virtual receiving array for two-dimensional (2-D) acoustic measurements in fluids. Using a single receiving transducer mounted on a chaotic cavity, time reversal processing allows the virtual array to operate as a bona fide receiving array, in which each “virtual element” replaces and performs like an actual receiving transducer (see Fig. 1). In this regard there are two key requirements for each virtual element. First, a virtual element, upon time reversal processing, should be uniquely associated with a position $ ( x , y , z )$ in space. This is the localization problem that has been dealt with extensively in the existing time reversal acoustics literature.^{5,8,19–23} Second, a virtual element has to faithfully reproduce the pressure-time signal $ p i ( t )$ arriving at its *i*th position, where $ p i ( t )$ can be of *any waveform* such as a sinusoid, a pulse, a random noise, or even a signal as complicated as music. At the time of this writing, seemingly no previous studies, with the notable exception of Ref. 14, have addressed in depth the aspect of waveform reproduction, which is the primary focus of our investigation.

The outline of the paper is as follows. The principle of a general-purpose (i.e., capable of doing both source localization and waveform reproduction) virtual receiving array is described in Sec. II. In particular, the mathematical criterion for the ergodicity of a cavity is laid out, which quantifies the level of chaotic reverberation inside a cavity. The construction of a prototype virtual array and its performance judged against an actual 2-D scan of the acoustic field are presented in Sec. III. Possible extensions and applications of virtual receiving arrays in reverberant environments are discussed in Sec. IV, and conclusions are given in Sec. V.

## II. THEORY

The schematic of a virtual receiving array is shown in Fig. 2. Here, a chaotic cavity is formed by a rigid surrounding wall with an opening at one end that comprises the virtual array aperture. The cavity is flooded with the same fluid medium as that of the incident wave for minimizing reflection at the aperture. A receiving transducer, mounted flush with the interior wall of the cavity, records the reverberant sound $ p ( r \u2032 , t )$ at $ r = r \u2032$, as the incident wave enters the cavity through the aperture and bounces off the wall multiple times. The virtual array aperture contains a prescribed set of grid points or virtual elements (labeled with $ i = 1 , \u2009 2 , \u2009 \u2026 , \u2009 N ,$ where $N$ is the number of elements), at which the corresponding pressure-time waveforms $ p i ( t )$ of the incident wave are to be determined via time reversal processing of the reverberant sound $ p ( r \u2032 , t )$.

^{24}defined as $ a ( t ) \u2217 b ( t ) = \u222b \u2212 \u221e \u221e a ( \tau ) b ( t \u2212 \tau ) d \tau $. Although seemingly obvious in light of linear system theory, Eq. (1) needs to be put on a rigorous mathematical footing. Specifically, there are two pertinent questions. First, is there such a function $ g ( r a | r \u2032 ; t )$ that enables one to write Eq. (1)? If so, how is it obtained analytically? To shed light on this, consider a version of the Helmholtz integral equation in the time domain,

^{25}

^{,}

*a priori*, Eq. (6) is a Fredholm equation of the second kind

^{26}with the unknown function $ g ( r a | r w ; t )$ [or $ g ( r a | r \u2032 ; t )$ for that matter]. Thus it can be solved for $ g ( r a | r w ; t )$ numerically at a set of discrete points $ r w , \u2009 k$ $ ( k = 1 , \u2009 2 , \u2009 \cdots , \u2009 K )$ on the wall.

^{26}This proves the existence and solvability of $ g ( r a | r w ; t )$.

*N*virtual elements on the aperture

*i*th virtual element. The reverberation $ p ( r \u2032 , t )$ is picked up by a single receiving transducer at $ r = r \u2032$ and converted to a voltage signal $ r ( t )$, here expressed in a mathematical form similar to Eq. (7) as

*i*th virtual element encapsulates both the acoustic impulse response $ g i ( t ) \Delta S i$ from $ r a , i$ to $ r \u2032$ [see Eq. (7)] and the frequency-dependent receiving voltage sensitivity $ M v$ of the transducer. A collection of impulse responses $ h i ( t )$ for all

*N*virtual elements ought to be obtained from measurements prior to the operation of the virtual array, a process often referred to as training.

^{20,22}

*T*is the duration of $ r ( t )$, ideally long enough to capture the lingering reverberation inside the cavity. Then the time-reversed signal $ r ( T \u2212 t )$ is computationally rebroadcast to the

*j*th element using an appropriate impulse response $ h \u0302 j ( t )$, which is tantamount to the convolution of the two functions,

*j*th virtual impulse response used for waveform reproduction at the

*j*th element, which contains the voltage-to-pressure sensitivity $ M v \u2212 1$ of the transducer as well as the acoustic impulse response from $ r \u2032$ to $ r a , j$. Because it is convenient to model $ h \u0302 j ( t )$ after the physical impulse response $ h j ( t )$, it is assumed that (i) electroacoustic transduction in either direction (pressure-to-voltage or vice versa) incurs negligible loss of energy nor information such that $ M v M v \u2212 1 \u2248 1$ (an assumption mostly valid for piezoelectric transducers but incorrect for moving-coil loudspeakers) and (ii) due to the acoustic reciprocity the acoustic impulse response for the rebroadcast from $ r \u2032$ to $ r a , j$ is also given by $ g j ( t ) \Delta S j$, originally defined from $ r a , j$ to $ r \u2032$.

*i*th element, and provides a litmus test for ergodicity of the cavity that is central to the operation of the virtual array. [Note that the electroacoustic sensitivities $ M v$ and $ M v \u2212 1$ of the receiving transducer are conveniently removed from $ h i ( T \u2212 t ) \u2217 h \u0302 j ( t )$ because of the assumption $ M v M v \u2212 1 \u2248 1$.] If a cavity is truly chaotic, the impulse responses $ h i ( t )$ and $ h \u0302 j ( t )$, representative of the room acoustics of the cavity, are to satisfy the following ergodicity conditions:

^{27}and energy conservation. Ergodicity requires that every single ray spawned by a unit impulse $ \delta ( t )$ at the

*i*th element should eventually reach the receiving transducer at $ r \u2032$ upon multiple reflections, and therefore all the acoustic energy of the unit impulse is accounted for in the received signal $ h i ( t )$ given a sufficiently long recording time (known as the Heisenberg time

^{6,17,28}). When the time-reversed signal $ h i ( T \u2212 t )$ is rebroadcast toward the same element ( $ j = i$) via $ h \u0302 j ( t )$, it has to be restored as a time-reversed unit impulse $ \delta ( T \u2212 t )$ at the

*i*th element according to the conservation of energy. This leads to the first condition in Eq. (11). The second condition between different elements ( $ j \u2260 i$) turns out to be a corollary to the first because all of the acoustic energy is refocused on the

*i*th element, there is no energy to spare for any other elements.

^{24}defined as $ a ( t ) \u2009 \u2605 \u2009 b ( t ) = \u222b \u2212 \u221e \u221e a ( \tau ) b ( t + \tau ) d \tau $. Using Eq. (12) the ergodicity conditions [Eq. (11)] can be rewritten in terms of correlation

*j*th element. Thus, $ p j ( T \u2212 t )$ is time-reversed again to recover the original waveform

^{14}Therefore, we adopt the procedure of inverse filtering,

^{14,29}known to improve the quality of wave energy focusing

^{9,10,13–17}and impact localization,

^{21}to enhance the performance of the virtual receiving array. To apply inverse filtering, time-reversal rebroadcast [Eq. (10)] is carried out in the frequency domain with the virtual impulse response $ h \u0302 j ( t )$ given by

^{14}

## III. EXPERIMENT

### A. Construction of the virtual receiving array

A single-channel virtual receiving array was constructed for use in water. As shown in Fig. 3(a), the chaotic cavity, machined from stainless steel, measured 110 mm × 110 mm × 80 mm in its exterior dimensions and had a square window of 35 mm × 35 mm. The hollow cavity was completely flooded with water so that chaotic reverberations could occur inside the cavity. This particular design allows the incident sound to enter the cavity through the square window with little reflection, thereby minimizing the information loss on waveforms to be extracted. Part of the interior cavity wall was in the shape of a mushroom,^{30} where the top part was a quarter-sphere with a radius of 50 mm, and the lower part was a half-cylinder with a radius of 20 mm and a height of 20 mm [Fig. 3(b)]. A steel ball (radius: 10 mm) and a tetrahedral block (base: 50 mm × 50 mm × 70 mm, height: 20 mm) were attached to increase the ergodicity of the cavity via symmetry breaking. An immersion transducer (model V384-SU, Olympus, Waltham, MA), with an aperture diameter of 0.25 in. (6.35 mm) and a center frequency of 3.5 MHz, was mounted flush with the cavity wall.

Given the hardware of the chaotic cavity and the receiving transducer, the construction of the virtual array amounts to “training,” in which a library of impulse responses $ h i ( t )$ is collected over the prescribed virtual array aperture. Here, the virtual array aperture was set to a square plane of 27 mm × 27 mm, located 50 mm away from the window of the cavity [Fig. 3(a)]. The aperture contains 10 × 10 virtual elements with an inter-element spacing of $ \Delta x = \Delta y = 3 \u2009 mm$ (or dimensionless spacing $ \Delta x / \lambda = \Delta y / \lambda = 7$ with respect to the central wavelength *λ*). It would have been desirable to set up the virtual array aperture coincident with the cavity window as described in Fig. 2, but a number of physical constraints forced the placement of the aperture plane away from the window. Because of the finite size (aperture diameter: 0.25 in.) of the transducer used in training (Olympus, model V384-SU), cramming a large number of virtual elements into the cavity window without the training transducer bumping into the cavity wall was practically impossible. More importantly, the front surface of the training transducer, when in close proximity to the cavity window, could have presented a solid wall reflector rather than a free space required of the virtual array aperture. The 50-mm spacing between the cavity window and the virtual array aperture introduces an additional acoustic impulse response, arising from the free-field propagation between the two planes, to $ h i ( t )$ of Eq. (8). However, the overall form and integrity of Eq. (8) and the subsequent theory would remain the same regardless.

*i*th virtual element, and $ R ( \omega )$ is the Fourier transform of the reverberation signal $ r ( t )$. One should exercise caution with the impulse response obtained using a band-limited training signal such as a tone burst. Notice that the spectrum for the training signal [Fig. 4(d)] has most of its energy between 2.2 and 5.0 MHz corresponding to the 20 dB bandwidth. Because there is relatively little energy beyond the 20-dB bandwidth, the frequency-domain division according to Eq. (17) could amount to division by near-zero and amplify the noise outside the effective frequency band. Thus, a bandpass (2.2–5.0 MHz) filter is applied following the frequency-domain division in order to suppress this unwanted artifact.

The impulse responses so obtained were checked for the ergodicity conditions [Eq. (13)]. For example, Fig. 5 illustrates the ergodicity check applied to $ h 45 ( t )$, the impulse response associated with the 45th virtual element located at $ ( x , y ) = ( 12 \u2009 mm , 12 \u2009 mm )$ (shown with circle in the aperture). The first row of Fig. 5 shows the autocorrelation $ h 45 ( t + T ) \u2009 \u2605 \u2009 h \u0302 45 ( t )$, where the virtual impulse response $ h \u0302 45 ( t )$[Fig. 5(b)] was obtained by processing $ h 45 ( t )$ [Fig. 5(a)] with inverse filtering [Eq. (16)] and then scaling its amplitude so as to yield the autocorrelation peak of unity [Fig. 5(c)]. The autocorrelation, exhibiting a single dominant peak, is reasonably close to the Dirac delta function, thus in conformance with the first ergodicity condition that dictates the fidelity of waveform reproduction. The deviation from the exact delta function is mainly due to the finite bandwidth (2.2–5.0 MHz) of the impulse response [recall Fig. 4(e)]. In fact, the characteristic shape of the autocorrelation—a “Mexican hat” with decaying oscillations—is that of a bandpass filtered delta function. The cross correlation of $ h 45 ( t )$ with any other impulse response, on the other hand, was found to be almost identically zero, where a criterion was used that any cross correlation peak less than 10% was deemed small enough to be ignored against the autocorrelation peak of unity. Note that the criterion for cross correlation peak depends mostly on the cavity shape, hence the ergodicity of the cavity. With increasing ergodicity via improved symmetry breaking the threshold value for cross correlation peak could be made smaller. The second row of Fig. 5, for instance, shows the cross correlation $ h 45 ( t + T ) \u2009 \u2605 \u2009 h \u0302 44 ( t )$ between the impulse responses for the 45th element and the adjacent 44th element (shown with square). Despite the close proximity of the two elements (only 3 mm apart), the corresponding impulse responses are remarkably *un*correlated [Fig. 5(f)], satisfying the second ergodicity condition pertaining to the localization performance. All the 100 impulse responses were inspected and passed the ergodicity check. With that, the construction of the virtual receiving array was complete.

### B. Evaluation of the array performance

The performance of the virtual receiving array was judged against that of a comparable physical array. Because it was too costly to build an actual array of 100 receiving transducers, it was replaced with a 2-D raster scan of acoustic pressure over the same measurement plane as the virtual array aperture. The receiving transducer (Olympus, model V384-SU) used for the 2-D raster scan was identical to that of the virtual receiving array, and was attached to an axial positioner for automated scanning across 10 × 10 points in 3 mm increments. By employing the identical receiving transducer for both the virtual array and the 2-D raster scan, the comparison could clearly bring out the effectiveness of the time-reversal chaotic cavity as an element multiplier. A 19-cycle Gaussian tone burst with a center frequency of 3.5 MHz was emitted by an underwater sound source (Olympus, model V384-SU), situated on the axis $ [ ( x , y ) = ( 13.5 \u2009 mm , 13.5 \u2009 mm ) ]$ and 100 mm away from the measurement plane (or the virtual array aperture).

Comparison of the pressure waveforms/spectra obtained from the virtual array (solid line) and the 2-D raster scan (dotted line) are given in Fig. 6. The pressure waveforms from the 2-D raster scan were computed using the nominal sensitivity of the receiving transducer at 3.5 MHz ( $ M \xaf v = 6.06 \xd7 10 \u2212 6 \u2009 V / Pa$), previously obtained via comparison calibration with a needle hydrophone (model NH1000, Precision Acoustics, Dorchester, UK). On the contrary, the pressure waveforms from the virtual array were calculated according to Eqs. (9), (10), and (15), which did not require any explicit use of the receiving transducer sensitivity, hence no need for calibration of the receiving transducer. (However, calibration of the training transducer was needed prior to the training of a virtual receiving array.) Figures 6(a) and 6(b) show the waveforms observed at two different locations respectively; the former close to the axis (shown with circle in the aperture) and the latter further away from the axis (indicated with square). It is clear from the comparison that the virtual receiving array is capable of faithfully reproducing the pressure-time waveform at each location. This is rather remarkable given the fact that no adjustment of waveforms or fine-tuning of parameters was made to obtain the match between the waveforms. Perhaps one downside of the virtual array is the lingering temporal oscillations in the extracted waveform [see Fig. 6(a)], which is believed to be a limitation of one-channel time reversal, and can be reduced by increasing the number of cavity transducers and the duration of the time window.^{1–3,31} Figures 6(c) and 6(d) show the corresponding spectra at the two locations, respectively. As each wavelet bounces around with multiple reflections inside the chaotic cavity, it is attenuated by thermoviscous absorption in the medium, which gets larger with increasing frequency according to the frequency-square absorption law. Thus, the frequency components above 3.5 MHz for the virtual array (solid line) are underrepresented compared with those for the 2-D raster scan (dotted line).

Comparison of the measured spatial distributions of acoustic pressure amplitude, taken at some representative time [*t* = 103.1 *μ*s; see Fig. 6(a)], is shown in Fig. 7, which is reflective of the localization performance of the virtual array. Each pressure distribution is normalized by its own spatial peak amplitude across the measurement plane. Contours are drawn in 3 dB decrements with respect to the peak amplitude (0 dB) such that the contour closest to the center is indicative of the half-power beam width. Note that Fig. 7(a) is a single snapshot of the pressure distribution taken by the virtual array, whereas Fig. 7(b) has to be assembled over 100 successive measurements as the 2-D raster scan progresses. Apart from some noise beyond the fourth contour ( –12 dB) in Fig. 7(a), the pressure distribution obtained from the virtual array agrees favorably with that from the 2-D raster scan. However, because the virtual elements have nonuniform ergodicity properties (i.e., autocorrelation and cross correlation waveforms) among themselves, the contour plot of the virtual receiving array is not as symmetrical as that of the raster scan. The half-power beam widths, obtained by the virtual receiving array and the 2-D raster scan, are 6.1 and 5.7 mm, respectively.

## IV. DISCUSSION

^{32}Eq. (18) along with all the virtual array mathematics carry over to the case of Fig. 8(c). [Note that in Fig. 8(c) the integral over the narrow tube connecting the outer wall and the virtual array aperture is negligibly small and thus ignored.]

This particular realization of a virtual receiving array—hereafter dubbed an “inner aperture virtual array”—may find niche applications in some naturally reverberant surroundings for which a physical array is not suitable. Figure 9(a), for example, posits a scenario in which a virtual hydrophone array of relatively large aperture is easily built and operated in a flooded dry dock (a reverberant enclosure) for the acoustic holography of ship radiated noise. The feasibility of an inner aperture virtual array as illustrated in Fig. 9(a) was examined via finite element simulation using COMSOL Multiphysics. The aim was to check whether the virtual receiving array exploiting solely the reverberation in the dry dock could meet the ergodicity conditions of Eq. (13). Figure 9(b) shows the three-dimensional (3-D) finite element model of a hypothetical 1800-ton diesel-electric submarine (length: 60 m, diameter: 6.3 m) in a dry dock (width × length × depth: 100 m × 150 m × 15 m) flooded with sea water. A 7 × 7 virtual array with an inter-element spacing of $ \Delta x = \Delta y = 0.3 \u2009 m$ (or $ \Delta x / \lambda = \Delta y / \lambda = 0.3$ with respect to the central wavelength *λ*) was assumed to be measuring the machinery noise from the diesel engine and electric motors in the stern. The center of the virtual aperture was located at $ ( x , y , z ) = ( 90 \u2009 m , \u2009 73 \u2009 m , \u2009 8 \u2009 m )$ and a single hydrophone was placed at $ ( x , y , z ) = ( 50 \u2009 m , \u2009 67 \u2009 m , \u2009 1 \u2009 m )$ similar to the arrangement shown in Fig. 9(a). The boundary conditions concerning the dry dock and the submarine were defined as *specular reflection* and *mixed diffuse and specular reflection*, respectively. Because the length of the dry dock in the *y*-direction is much larger than that of the submarine, the two surfaces at *y* = 0 m and *y* = 150 m were set to the *disappear* condition that provides anechoic termination.

The impulse response between the hydrophone and each virtual receiving element was calculated using the *Ray Tracing* study under the *Ray Acoustics* physics interface. For economy in finite element calculation, the acoustic reciprocity principle was invoked, where 1 000 000 rays were launched spherically from the hydrophone and a spherical 3-D receiver with a diameter of 0.3 m was placed at each virtual element recording the corresponding impulse response. A frequency range of 150 to 3000 Hz was covered using a parametric sweep to construct a band-limited impulse response for each virtual element, and a free triangular mesh with at least six degrees of freedom per wavelength at the highest frequency (3000 Hz) was used in all calculations.

All the 49 impulse responses were checked and passed the ergodicity test [Eq. (13)] given a peak cross correlation threshold of 18%, which completes a proof-of-concept for the inner aperture virtual array. This is rather remarkable, considering the dry dock is less than ideal as a chaotic cavity because of the partial loss of rays at the long ends and the relative symmetry and limited extent (leading to standing wave modes in the depth direction) of the space. The increase in peak cross correlation threshold (from 10% in Sec. III to 18% here) was due to the jaggedness of the cross correlation waveforms rather than the emergence of any noticeable cross correlation peaks. Figure 10 shows a case of ergodicity check for the impulse response associated with the 25th virtual element located at $ ( x , y , z ) = ( 90 \u2009 m , \u2009 73 \u2009 m , \u2009 8 \u2009 m )$ (shown with circle in the aperture). The first row shows the autocorrelation $ h 25 ( t + T ) \u2009 \u2605 \u2009 h \u0302 25 ( t )$, and the second row illustrates the cross correlation $ h 25 ( t + T ) \u2009 \u2605 \u2009 h \u0302 24 ( t )$ between the impulse responses for the 25th element and the adjacent 24th element (shown with square).

## V. CONCLUSIONS

The feasibility of a virtual receiving array based on a time-reversal chaotic cavity has been demonstrated. Employing a single receiving transducer in conjunction with a chaotic cavity, an array of 10 × 10 virtual receiving elements was built and tested in water. Having met the ergodicity conditions for the chaotic cavity [Eq. (13)], the virtual array performed on par with a physical array (here replaced with a 2-D raster scan) in terms of both waveform reproduction and source characterization. The key to the successful construction and operation of a virtual receiving array is the ergodicity of the chaotic cavity. The reverberation inside the cavity should be sufficiently ergodic such that all the significant rays emitted by the sound source are eventually captured by the single receiving transducer. With the ergodicity in place, the virtual array then becomes a more extensive time-reversal mirror that better “encloses” the sound source.

Finally, the concept of an inner aperture virtual array in fluids was examined via finite-element simulation. Requiring no separate reverberant cavity, it may be useful in some naturally reverberant spaces, where the physical constraints of the space strongly favor the use of a virtual array: for example, in a container filled with either contaminating, corrosive, or high-temperature fluid, for which only a few receiving transducers need to be sacrificed.

## ACKNOWLEDGMENTS

This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant number NRF2009-0077588).

### APPENDIX: MATHEMATICAL EQUIVALENCE OF TRA AND MFP

^{33,34}but their exact mathematical connection seems to have rarely been discussed in both TRA and MFP literature. Here, we show that TRA and MFP are mathematically equivalent to each other by a single mathematical identity. We begin with Eq. (12), repeated here for convenience,

*m*th transducer in a physical array (or a time-reversal mirror), $ h m j ( t )$ is the impulse response between the

*m*th transducer and the

*j*th location in the region of interest. In the case of a single-channel array ( $ m = 1$), the transducer/channel index $m$ in $ r m ( t )$ and $ h m j ( t )$ can be dropped as in Sec. II. If Eq. (A3) is summed over the channel index $ m \u2009 ( = 1 , \u2009 2 , \u2009 \u2026 , \u2009 M )$, we finally obtain

*j*th location via convolution with the corresponding impulse responses $ h m j ( t )$. The act of rebroadcast can be either physical (for a transmitting array) or computational (for a receiving array). Meanwhile, equivalently taking place on the right-hand side of Eq. (A4) is a rudimentary type of matched field beamforming, where the received signals $ r m ( t )$ are being matched via correlation with a series of model “replicas” $ h m j ( t )$ for source localization. This can be better understood in the frequency domain with the Fourier transform $F$ of the right-hand side of Eq. (A4),

*M*-dimensional vector containing the fast Fourier transforms of the received signals, $ w c H ( A \u0302 )$ is an

*M*-dimensional vector containing the replicas associated with the (unknown) location parameter $ A \u0302$, and $L$ is the number of snapshots to form an ensemble for averaging. Apart from squaring and averaging in Eq. (A6), Eqs. (A5) and (A6) are essentially the same, considering the following correspondence between the symbols in this paper (in component form) and Ref. 18 (in matrix form): received signals $ R m \u2194 R l$, replicas $ H m j \u2194 w c ( A \u0302 )$, location parameters $ \u2009 j \u2194 A \u0302$, conjugate transpose $ ( ) \xaf \u2194 ( ) H$, and inner products $ \u2211 m = 1 M H \xaf m j ( \omega ) R m ( \omega ) \u2194 w c H ( A \u0302 ) R l$. In ocean acoustics, MFP is most often carried out in the frequency domain using Eq. (A5) as the foundation for building more sophisticated matched field beamformers. To summarize, TRA and MFP can be regarded as mathematical twins that took two divergent paths with their own respective purposes, methods, traditions, triumphs, and limitations.

## REFERENCES

*The Fast Fourier Transform and Its Applications*

*Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography*

*Numerical Recipes in C: The Art of Scientific Computing*

*Statistical Physics: Part 1*