Patent Translate Powered by EPO and Google Notice This translation is machine-generated. It cannot be guaranteed that it is intelligible, accurate, complete, reliable or fit for specific purposes. Critical decisions, such as commercially relevant or financial decisions, should not be based on machine-translation output. DESCRIPTION JP2017530580 Abstract A method (500) for constructing a three-dimensional (3D) wavefield representation of a three-dimensional wavefield using a two-dimensional (2D) sensor array (110), comprising an omnidirectional sensor (340) arranged in a two-dimensional plane. Acquiring a three-dimensional wavefield signal using a two-dimensional array (110) of sensors (340, 350) including a) and a primary sensor (350), and digitizing the acquired three-dimensional wavefield signal Calculating the even coefficients of the spherical harmonics acquired by the omnidirectional sensor (340) and determined by the digitized three-dimensional wavefield signal, digitized three-dimensional wavefield acquired by the primary sensor (350) By calculating the odd coefficients of the spherical harmonics determined by the signal, and by the even and odd coefficients calculated for the acquired 3D wavefield signal The method comprising and that constructing a threedimensional wave field representation, which is determined. Flat sensor array [0001] The present invention relates generally to the field of signal processing, in particular threedimensional (3D) wavefield representation of an actual three-dimensional wavefield signal acquired and acquired using a two-dimensional (2D) sensor array. For building. [0002] In the field of signal processing, it is desirable to obtain a mathematical representation of a threedimensional wavefield consisting of actual three-dimensional wavefield signals, as such a 03-05-2019 1 representation enables accurate analysis of the three-dimensional wavefield. One such mathematical expression is the three-dimensional wavefield spherical harmonics expansion. [0003] A three-dimensional wave field signal of a spherical coordinate system (r, θ, φ) can be mathematically expressed by Equation 1 as an infinite sum of spherical harmonics. Here, C nm is a coefficient, j n (kr) is a spherical Bessel function, Y nm = P n | m | (cos θ) E m (φ) is a representation of spherical harmonics, and is normalized It is a Legendend function, which is a normalized exponential function. The normalized exponential function represents a spherical wave in the φ direction, and the normalized Legendre 陪 function represents a spherical wave in the θ direction. [0004] The spherical harmonics are orthonormal and therefore satisfy, where Y nm = P n | m | (cos θ) E m (φ) is a representation of spherical harmonics. [0005] FIG. 10 shows a plot of the 0 to 3 spherical harmonics, where the odd mode 1010 of the spherical harmonic is zero at θ = π / 2, and the even mode 1020 of the spherical harmonic is θ = π / 2 Indicates that it is not zero. FIG. 10 also shows a spherical coordinate system corresponding to the spherical harmonics. Even mode 1020 is only partially shown in FIG. 10 to avoid cluttering the figure. According to the spherical harmonics, only the even mode 1020 is observable in the xy plane (ie, the θ = π / 2 plane). That is, the odd mode 1010 can not be detected in the xy plane. Therefore, it is necessary to place the sensors at various vertical heights in order to obtain 3D wavefield signals in order to fully generate the mathematical representation of the 3D wavefield spherical harmonics expansion. 03-05-2019 2 [0006] One type of array arrangement that meets the above requirements is a spherical array. The shape of the spherical array is consistent with spherical harmonics that make the 3D wavefield signal acquired by the spherical array suitable for generating a 3D wavefield spherical harmonic expansion. There are two spherical array arrangement models: an open ball model (where the sensor is located in the open space) and a hard sphere model (where the sensor is located on the surface of the hard sphere). [0007] However, spherical arrays suffer from the problem that the array may be numerically defective due to nulls in the spherical Bessel function. This problem leads to the fact that the acquired three-dimensional wave signal is very sensitive to the diameter of the spherical array. Also, placing the sensors in a spherical array follows the strict rules of orthogonality of spherical harmonics and limits the flexibility of array placement (especially in terms of sensor volume). Furthermore, the spherical shape of the array is not only unrealistic, but also causes difficulties in implementation. [0008] Another limitation of spherical arrays is the narrow frequency band due to the properties of the spherical Bessel function. Thus, a spherical array can not process three-dimensional wavefield signals for a particular order of spherical harmonics. The design of the spherical array has to be done carefully so that for the target frequency band the active spherical Bessel function is not zero at the radius of the spherical array. [0009] Therefore, there is a need to provide a more practical array arrangement. [0010] A configuration is disclosed that attempts to solve the above problem using a two-dimensional sensor array to acquire a three-dimensional wavefield signal and construct a three-dimensional wavefield representation from the acquired three-dimensional wavefield signal. 03-05-2019 3 [0011] According to a first aspect of the present disclosure, a method of constructing a threedimensional (3D) wavefield representation of a three-dimensional wavefield using a twodimensional (2D) sensor array, the method comprising: Acquiring a three-dimensional wavefield signal using a two-dimensional array of sensors comprising an orientation sensor and a primary sensor; digitizing the acquired three-dimensional wavefield signal; and acquiring by the omnidirectional sensor Calculating even coefficients of spherical harmonics dependent on said digitized three-dimensional wave field signal, and said spherical harmonics dependent on said three-dimensional wave field signal acquired and digitized by said primary sensor Calculating odd-numbered coefficients of the three-dimensional wave-field representation depending on the even-numbered coefficient and the odd-numbered coefficient calculated for the obtained threedimensional wave-field signal The method comprising is provided. [0012] According to another aspect of the present disclosure, there is provided a computer program product comprising a computer readable medium having recorded thereon a computer program for performing the above method. [0013] Other aspects of the invention are also disclosed. [0014] At least one embodiment of the present invention will now be described with reference to the drawings. FIG. 7 shows a block diagram of a system that uses a two-dimensional (2D) array to obtain threedimensional (3D) wavefield signals and construct a three-dimensional wavefield representation in accordance with the present disclosure. A schematic block diagram of a general purpose computer system capable of implementing the above configuration is formed. 03-05-2019 4 A schematic block diagram of a general purpose computer system capable of implementing the above configuration is formed. FIG. 2 shows an arrangement of a two-dimensional sensor array of the system of FIG. FIG. 2 shows an arrangement of a two-dimensional sensor array of the system of FIG. FIG. 5 is a flow diagram of a method of designing a two-dimensional sensor array. FIG. 2 shows a block diagram of an implementation of the system of FIG. FIG. 2 shows a block diagram of an implementation of the system of FIG. FIG. 2 shows plots (a) and (c) of the sound field captured by the system of FIG. 1A and plots (b) and (d) of the reconstructed sound field. FIG. 2 is a flow diagram of a method of constructing a three-dimensional (3D) wavefield representation of a three-dimensional wavefield using the two-dimensional (2D) sensor array of FIG. 1; FIG. 2 is a block diagram of the system of FIG. 1 implemented in an active noise rejection application in accordance with the present disclosure. 7 is a plot of the error of the active noise removal system of FIG. 6A. FIG. 2 is a block diagram of the system of FIG. 1 implemented in a beam forming application in accordance with the present disclosure. FIG. 2 is a block diagram of the system of FIG. 1 implemented in an application for direction of arrival estimation in accordance with the present disclosure; FIG. 2 is a block diagram of the system of FIG. 1 implemented in a 3D sound field reproduction application in accordance with the present disclosure. The expression of the 3rd-order spherical harmonics is shown. [0015] When reference is made to steps and / or features having the same reference in any one or more of the accompanying drawings, those steps and / or features have the same function for the purpose of the present description unless the contrary intention is indicated. Or have an action. [0016] Two-dimensional with omnidirectional (non-directional) sensor and primary sensor (i.e. direction sensor) to acquire three-dimensional wavefield signal so that mathematical expression of acquired three-dimensional wavefield signal can be constructed A sensor array configuration is disclosed. The omnidirectional sensor functions to obtain even mode (ie, horizontal wave field) components, 03-05-2019 5 and the primary sensor measures odd mode (ie, vertical wave field) components. A threedimensional wavefield representation can then be constructed from the acquired even and odd mode components. [0017] FIG. 1 shows a system 100 for acquiring 3D wavefield signals and constructing a 3D wavefield representation of the acquired 3D wavefield signals. System 100 includes a two-dimensional sensor array 110 and a computer system 120. The sensor array 110 is arranged in a plurality of concentric circular arrays, each concentric circular array having a radius (different from the other concentric circular arrays) and a separate number of sensors. For the sake of consistency, the term "sensor" is used below, but in some cases the terms "antenna", "microphone" and "hydrophone" can be applied as well. Depending on the type of sensor used, the threedimensional wavefield signal obtained may be either an acoustic wavefield signal, a radio frequency wavefield signal, or a micro wavefield signal. The configuration of the two-dimensional sensor array 110 will be described in detail in connection with FIGS. [0018] The computer system 120 includes an array processing module 130 and an application module 140. The array processing module 130 processes the three-dimensional wave field signal acquired by the two-dimensional sensor array 110, and generates a three-dimensional wave field spherical harmonics expansion from the acquired three-dimensional wave field signal (ie, even mode coefficients and odd modes). Generate coefficients). The application module 140 uses the generated 3D wave field expansion in any of the active noise removal, beamforming, direction of arrival estimation, and 3D sound field recording / reproduction applications. [0019] Computer system 120 may be a general purpose computer system, or a dedicated computer for performing coding of array processing module 130 and application module 140. The method of acquiring 3D wavefield signals and generating a 3D wavefield spherical harmonic expansion by array processing module 130 is described below with reference to FIG. The implementation of application module 140 in using the three-dimensional wavefield expansion generated by array processing module 130 is described below in conjunction with FIGS. 6-9. 03-05-2019 6 [0020] Computer Description FIGS. 2A and 2B illustrate a general purpose computer system 120 that can implement the various configurations described. [0021] As seen in FIG. 2A, the computer system 120 includes a computer module 201, an input device such as a keyboard 202, a mouse pointer device 203, a scanner 226, a camera 227, a microphone 280, a printer 215, a display 214 and a speaker 217. And an output device. An external modulation demodulator (modem) transceiver 216 may be used by computer module 201 to communicate with communication network 220 via connection 221. Communication network 220 may be the Internet, a cellular communication network, or a wide area network (WAN), such as a private WAN. If the connection 221 is a telephone line, the modem 216 can be a conventional "dial-up" modem. Alternatively, if connection 221 is a high capacity (eg, cable) connection, modem 216 can be a broadband modem. A wireless modem may also be used for wireless connection to communication network 220. [0022] Computer module 201 generally includes at least one processor unit 205 and a memory unit 206. For example, memory device 206 may include semiconductor random access memory (RAM) and semiconductor read only memory (ROM). The computer module 201 may also include an audio video interface 207 coupled to a video display 214, a speaker 217 and a microphone 280, a keyboard 202, a mouse 203, a scanner 226, a camera 227, a 2D sensor array 110, and optionally a joystick or other. And a plurality of input / output (I / O) interfaces including an I / O interface 213 coupled to a human interface device (not shown) and an interface 208 for an external modem 216 and a printer 215. In some implementations, modem 216 can be incorporated into computer module 201, for example, within interface 208. Computer module 201 also has a local network interface 211 that enables coupling of computer system 120 to a local area communication network 222 known as a local area network (LAN) via connection 223. As shown in FIG. 2A, the local communication network 222 may also be coupled to the wide area network 220 via a connection 224 which generally includes so-called "firewall" devices or devices of similar functionality. The local network interface 211 can include an Ethernet circuit 03-05-2019 7 card, a Bluetooth wireless device or an IEEE 802.11 wireless device, but implementing many other types of interfaces for the interface 211 Can. [0023] The I / O interfaces 208 and 213 can provide either series connection or parallel connection or both, the former being generally implemented according to the Universal Serial Bus (USB) standard and corresponding USB connectors (not shown) ). A storage device 209 is provided and generally includes a hard disk drive (HDD) 210. Other storage devices such as floppy disk drives and magnetic tape drives (not shown) can also be used. In general, an optical disc drive 212 is provided to function as a non-volatile data source. Suitable sources of data to system 120 include, for example, optical disks (eg, CD-ROM, DVD, Blu-ray® disks), USB-RAM, portable, external hard drives, floppy disks, etc. Portable memory devices can be used. [0024] The components 205 through 213 of the computer module 201 generally communicate via the interconnected bus 204 to effect conventional operating modes of the computer system 120 known to those skilled in the art. For example, processor 205 is coupled to system bus 204 using connection 218. Similarly, memory 206 and optical disk drive 212 are coupled to system bus 204 by connection 219. Examples of computers that can implement the described configuration include an IBM-PC and compatible, Sun's sparstation, Apple's Mac® or similar computer system. [0025] The method 120 of processing the acquired three-dimensional wavefield signal to generate a three-dimensional wavefield expansion of spherical harmonics can be implemented using a computer system 120, and the process of FIG. 5 described is: One or more software application programs 233 executable in computer system 120 may be implemented. Specifically, the steps of the method for generating the configuration of the spherical harmonic three-dimensional wave field from the two-dimensional sensor array 110 are performed by the instructions 231 (see FIG. 2B) in the software 233 executed in the computer system 120. It will be. Software instructions 231 may be formed as one or more code modules, each for performing one or more particular tasks. The software can also be divided into two separate parts, in which case the first part and the corresponding code modules (ie modules 130 and 140) acquire three-dimensional wavefield 03-05-2019 8 signals from the two-dimensional sensor array 110 And performing a three-dimensional wave field expansion of the spherical harmonics, the second part and the corresponding code module managing the user interface between the first part and the user. [0026] The software may, for example, be stored on a computer readable medium including the storage device described below. Software is loaded into computer system 120 from computer readable media and is then executed by computer system 120. A computer readable medium having such software or computer program recorded on the computer readable medium is a computer program product. The use of computer program products in computer system 120 preferably results in an advantageous apparatus for generating a three-dimensional wavefield expansion of spherical harmonics from the three-dimensional wavefield signals acquired by two-dimensional sensor array 110. [0027] The software 233 is generally stored in the HDD 210 or the memory 206. The software is loaded from computer readable medium into computer system 120 and executed by computer system 120. Thus, for example, software 233 may be stored on an optically readable disk storage medium (e.g., CD-ROM) 225 that is read by optical disk drive 212. A computer readable medium having such software or computer program recorded on the computer readable medium is a computer program product. The use of a computer program product in computer system 120 preferably results in an apparatus for generating a three-dimensional wavefield expansion of spherical harmonics from the three-dimensional wavefield signals acquired by two-dimensional sensor array 110. [0028] In some cases, the application program 233 may be encoded on one or more CD-ROMs 225 and supplied to the user and read via the corresponding drive 212, or read by the user from the network 220 or 222 May be Additionally, software may be loaded into computer system 120 from other computer readable media. Computer readable storage medium refers to any nontransitory tangible storage medium that provides instructions and / or data recorded on computer system 120 for execution and / or processing. Examples of such storage media, 03-05-2019 9 whether or not such a device is internal or external to computer module 201, include floppy disks, magnetic tapes, CD-ROMs, DVDs, Blu-ray, etc. A computer readable card such as a disk, hard disk drive, ROM or integrated circuit, USB memory, magneto-optical disk, or PCMCIA card is included. Examples of temporary or non-tangible computer readable transmission media that can be involved in providing software, application programs, instructions and / or data to the computer module 201 include wireless or infrared transmission channels as well as other computers or networks. This includes the Internet or an intranet that includes information recorded on a network connection to the device, sending an e-mail or a website or the like. [0029] A code module corresponding to the second part of the application program 233 described above may be executed to implement one or more graphical user interfaces (GUIs) rendered or otherwise displayed on the display 214 it can. Generally, manipulation of the keyboard 202 and mouse 203 causes the user of the computer system 120 and the application to manipulate the interface in a functionally adaptable manner to provide control commands and / or input to the application associated with the GUI. can do. Other forms of functionally compatible user interface may also be implemented, such as an audio interface that utilizes speech prompt output via speaker 217 and user voice command input via microphone 280. [0030] FIG. 2B is a detailed schematic block diagram of processor 205 and “memory” 234. Memory 234 represents a logical collection of all memory modules (including HDD 209 and semiconductor memory 206) accessible by computer module 201 of FIG. 2A. [0031] When the computer module 201 is first powered on, a power on self test (POST) program 250 is executed. The POST program 250 is generally stored in the ROM 249 of the semiconductor memory 206 of FIG. 2A. Hardware devices such as the ROM 249 that store software may be referred to as firmware. The POST program 250 examines the hardware in the computer module 201 to ensure proper functionality, and in general, the processor 205, memory 234 (209, 206), and generally ROM 249 for correct operation. Check the basic input / output system software (BIOS) module 251 stored in. When the POST program 250 is successfully executed, the BIOS 03-05-2019 10 251 boots up the hard disk drive 210 of FIG. 2A. When the hard disk drive 210 is booted up, a bootstrap loader program 252 resident on the hard disk drive 210 is executed via the processor 205. As a result, the operating system 253 is loaded into the RAM memory 206, and the operating system 253 starts operation. Operating system 253 is a system level application that can be executed by processor 205 to perform various advanced functions, including processor management, memory management, device management, storage management, software application interfaces, and general purpose user interfaces. [0032] The operating system 253 manages the memory 234 (209, 206) so that each process or application running on the computer module 201 has enough memory to execute without conflicting with the memory allocated to other processes. Guarantee to have. In addition, the various types of memory available in system 120 of FIG. 2A must be properly used to efficiently execute each process. Thus, aggregated memory 234 does not indicate how a particular segment of memory is allocated (unless otherwise stated), but rather the general concept of memory accessible by computer system 120 and its use Intended to provide too. [0033] As shown in FIG. 2B, the processor 205 includes a plurality of functional modules, including a controller 239, an arithmetic and logic unit (ALU) 240, and a local or internal memory 248, sometimes referred to as a cache memory. Cache memory 248 generally includes a plurality of storage registers 244-246 in a register section. One or more internal buses 241 functionally interconnect these functional modules. Processor 205 also generally includes one or more interfaces 242 for communicating with external devices via system bus 204 using connection 218. Memory 234 is coupled to bus 204 using connection 219. [0034] The application program 233 includes a series of instructions 231 which can include conditional branch and loop instructions. Program 233 may also include data 232 used to execute program 233. The instructions 231 and data 232 are stored in memory locations 228, 229, 230 and 235, 236, 237, respectively. Depending on the relative size of instruction 231 and memory locations 228-230, particular instructions may be stored in a single memory location, as represented by 03-05-2019 11 the instructions shown in memory location 230. Alternatively, the instruction may be divided into multiple portions stored in separate memory locations, as represented by portions of the instruction shown at memory locations 228 and 229. [0035] Generally, processor 205 is provided with a set of instructions to be executed therein. Processor 1105 waits for the next input, and processor 205 responds to that input by executing another set of instructions. Each input is retrieved from data generated by one or more of the input devices 202, 203, 204, data received from an external source via one of the networks 220, 202, one of the storage devices 206, 209 The data may be provided from one or more of a plurality of sources, including data retrieved from storage medium 225 inserted into corresponding reader 212, all shown in FIG. 2A. Execution of a set of instructions can result in the output of data in some instances. Execution can also include storing data or variables in memory 234. [0036] The disclosed configuration uses input variables 254 stored in the memory 234 at corresponding memory locations 255, 256, 257. The disclosed arrangement produces output variables 261 which are stored in the memory 234 at corresponding memory locations 262, 263, 264. Intermediate variables 258 may be stored at memory locations 259, 260, 266 and 267. [0037] Referring to processor 205 of FIG. 2B, registers 244, 245, 246, arithmetic logic unit (ALU) 240, and controller 239 execute a “fetch, decode and execute” cycle for each instruction in the instruction set that makes up program 233. Work together to perform the sequence of microoperations needed to perform the Each fetch, decode and execute cycle is a fetch operation for fetching or reading the instruction 231 from the memory location 228, 229, 230, a decoding operation where the controller 239 determines whether or not the instruction is fetched, a controller 239 and / or Or an execution operation in which the ALU 240 executes an instruction. [0038] 03-05-2019 12 Thereafter, additional fetch, decode and execute cycles can be performed for the next instruction. Similarly, controller 239 can perform a storage cycle that stores or writes values to memory location 232. [0039] Each step or sub-process in the process of FIG. 5 is associated with one or more portions of program 233 and cooperates to perform a fetch, decode and execute cycle for each instruction in the instruction set of the target portion of program 233. It is executed by the register units 244, 245 and 247, the ALU 240 and the controller 239 in the processor 205. [0040] The method of acquiring the three-dimensional wave field signal from the two-dimensional sensor array 110 and generating the three-dimensional wave field expansion of the spherical harmonics based on the acquired three-dimensional wave field signal is alternatively the function of the method of FIG. Or may be implemented in dedicated hardware, such as one or more integrated circuits that perform the subfunctions. Such dedicated hardware may include field programmable gate arrays, graphics processing units, digital signal processors, or one or more microprocessors and associated memories. [0041] As mentioned in the paragraph above, the odd mode 1010 of the spherical harmonics of a threedimensional wave field can not be detected when the sensor array 110 is located in the xy plane. However, as described below, the spherical harmonics of the odd mode of the 3D wave field can be obtained by a sensor (i.e., a primary sensor) having a directional reception pattern perpendicular to the xy plane. Thus, the two-dimensional sensor array 110 has an omnidirectional sensor (for obtaining even harmonic coefficients of the mode) and a directional reception pattern perpendicular to the xy plane (for obtaining even coefficients of the spherical harmonic for odd modes). A combination of primary sensors having 球面 can be used to obtain spherical harmonics for both even and odd modes. 03-05-2019 13 [0042] 3A and 3B show two exemplary arrangements of a two-dimensional sensor array 110 for detecting three-dimensional wavefield signals. FIG. 3A is two-dimensional with primary sensors 350 arranged in a plurality of concentric arrays (310A, 310B, 310C, 310D, and 310E) having corresponding radii (320A, 320B, 320C, 320D, and 320E, respectively). 7 shows a sensor array 110A. Concentric arrays (310A, 310B, 310C, 310D, and 310E) are collectively referred to hereinafter as circular array 310. However, circular array 310 N or radius 320 N will be used in referring to the largest of circular array 310 below. The radius 320 N is also used to refer to the size of the two-dimensional sensor array 110 below. Each concentric array 310 includes a plurality of sensors 350. [0043] In the array 110, each of the primary sensors 350 is formed by two omnidirectional sensors 340 arranged in close proximity to one another. The distance between the two omnidirectional sensors 340 forming the primary sensor 350 is small compared to the array radius 320N. The primary sensor (i.e., the direction sensor) 350 has opposing bi-directional receive patterns and is oriented such that the bi-directional receive patterns are perpendicular to the plane of the array 110A. The output of primary sensor 350 is the difference between the outputs of omnidirectional sensor 340 that forms primary sensor 350. [0044] One of the omnidirectional sensors 340 that is part of the primary sensor 350 functions to receive even mode spherical harmonics. Primary sensor 350 functions to receive the spherical harmonics of the odd mode using both of the omnidirectional sensors 340 forming sensor 350. The functionality of array 110A is further described in connection with FIG. 4A. [0045] The number of primary sensors 350 on each circular array 310 is given by: Where N X is the number of sensors 350 on each circular array 310, N is the maximum order of observable spherical harmonics, k is the wave number of the design frequency, and R is the radius 320 of 03-05-2019 14 the circular array 310 Where c is the wave velocity. For audio applications, c = 340 m / s, and for RF applications, c = 300,000,000 m / s. For example, a 0.2 m radius circular array 310 designed to receive 900 MHz RF signals has 13 sensors (e.g., antennas). In another example, a 0.4 m radius circular array 310 designed to receive up to 1500 Hz audio signals has 33 sensors (eg, microphones). [0046] FIG. 3B shows omnidirectional sensors arranged in a plurality of concentric arrays (310A, 310B, 310C, 310D, 310E, and 310F) having corresponding radii (320A, 320B, 320C, 320D, 320E, and 320F, respectively). An array 110 B is shown having 340 and a primary sensor 350. Each concentric array 310 has either an omnidirectional sensor 340 or a primary sensor 350. Array 110 B is arranged such that concentric circular arrays with omnidirectional sensors 340 (ie, 310 A, 310 C and 310 E) alternate with concentric circular arrays with primary sensors 350 (eg, 310 B, 310 D and 310 F). The concentric circular arrays (310A, 310B, 310C, 310D, 310E, and 310F), like the circular array 310 of array 110A, are collectively referred to as circular array 310. [0047] In array 110 B, a cardioid sensor forms a primary sensor 350. In this manner, omnidirectional sensor 340 functions to receive even mode spherical harmonics, and primary sensor 350 functions to receive odd mode spherical harmonics. The functionality of array 110B is further described in connection with FIG. 4B. However, as implemented in the array 110A of FIG. 3A, the primary sensor 350 can also be formed by two omnidirectional sensors 340. [0048] FIG. 3C shows a flow chart of a method 300 of designing a two-dimensional sensor array 110. Method 300 begins at step 362 with selecting design parameters of two-dimensional sensor array 110. The design parameters include the maximum radius 320 N, the highest wave number k to be obtained and the type of primary sensor 350 used. For example, to design a twodimensional sensor array 110 for active noise removal in a room of dimensions 3 m × 3 m, the designer should select a radius of up to 320 N 1.5 m so that the array 110 fits on the ceiling of the room Can. The highest wave number k is selected according to the highest frequency to be acquired. The highest frequency of 850 Hz is, for example, k = k = 2πf / c = 15.7 for c = 340 m / 03-05-2019 15 s for audio applications. Method 300 then proceeds to step 364. [0049] At step 364, the maximum obtainable order of the wave field is calculated. The maximum obtainable degree N of the wave field can be obtained using the following equation. Method 300 continues at step 366. [0050] At step 366, the number of concentric circular arrays 310 is determined. Based on the maximum number of wavefield orders N calculated in step 364, the number of circular arrays 310 can be selected. For a two-dimensional sensor array 110A, the total number 350 of circular arrays for the primary sensor 310 is at least: For the two-dimensional sensor array 110B, the total number of circular arrays 310 for the omnidirectional sensor 340 is: The total number of circular arrays 310 for the primary sensor 350 is N first = N−N omni. The method 300 then proceeds to step 368. [0051] At step 368, the radius 320 of each concentric array 310 is determined. The radius 320 of each circular array 310 is chosen such that the circular array 310 is distributed and spherical Bessel zeros are avoided in the target frequency band. A spherical Bessel function (shown in Equation 10 below) is used to determine the radius 320 of each circular array 310. For a given frequency k, the spherical Bessel function j n (kr) is zero at some radius 320. If the circular array 310 is placed at a radius 320 where the spherical Bessel function is zero, the product of the spherical Bessel function (ie, “Bessel zero” will be zero, and the calculation of C nm will be very difficult. Therefore, the radius 320 at which the "Bessel zero" occurs should be avoided when designing the circular array 310. Method 300 then proceeds to step 370. [0052] At step 370, the number of sensors (340 and 350) is determined. The number of sensors (340 03-05-2019 16 and 350) on each circular array 310 is given by Equation 4 above. Method 300 ends after step 370. [0053] 4A shows an implementation of array 110A in system 100. FIG. As described in connection with FIG. 3A above, primary sensor 350 of array 110A is formed by two omnidirectional sensors (340A and 340B). The output of each of the omnidirectional sensors 340 is sent to the computer system 120 and then processed by the array processing module 130A when the array processing module 130A is executed by the processor 205. [0054] The array processing module 130A includes a fast Fourier transform (FFT) module (430A, 430B), a difference module 440, an even coefficient module 410A, and an odd coefficient module 420A. The FFT modules 430A and 430B are collectively referred to below as the FFT module 430. The FFT module 430 digitizes the three dimensional wave signal acquired by the omnidirectional sensor 340. The even coefficient module 410A processes the digitized three-dimensional wavefield signal acquired by the omnidirectional sensor 340B to calculate the coefficients of the even-mode spherical harmonics of the acquired three-dimensional wavefield signal. The odd coefficient module 420A processes the digitized three-dimensional wavefield signal acquired by the primary sensor 350 to calculate the coefficients of the even mode spherical harmonics of the acquired three-dimensional wavefield signal. [0055] To determine the coefficients of the even mode spherical harmonics, the output of omnidirectional sensor 340B is sent to FFT module 430B and the output of FFT module 430B is sent to even coefficient module 410A. The even coefficient module 410A then obtains the even mode coefficients using the following equation: Here, the integral of the m-th harmonic present in the X-th circular array 310. Is the output from the FFT module 430B. For example, the output from the FFT module 430B is the measured sound pressure at frequency k from each microphone unit 340B. Equation 5 is the inverse matrix of the coefficients of the even-mode spherical harmonics shown below in Equation 23. In order to calculate even mode coefficients, m is even. 03-05-2019 17 [0056] The output of omnidirectional sensor 340A is digitized by FFT module 430B to determine the coefficients of the spherical harmonics in odd mode. The difference module 440 then calculates the difference between the digitized outputs of the omnidirectional sensors 340A and 340B to obtain the output of the primary sensor 350, as described in connection with FIG. 3A above. The output of difference module 440 is sent to odd coefficient module 420A. [0057] The odd coefficient module 420A then obtains the odd mode coefficients by multiplying the input signal from the difference module 440 with the inverse matrix of the coefficients of the spherical harmonics of the odd mode shown below in Equation 23. [0058] Similar to obtaining the even mode coefficients, the odd mode coefficients are obtained using equation 5 above, but are the output from difference module 440. In order to calculate odd mode coefficients, m is odd. [0059] Alternatively, modules 410A and 420A may be implemented as a single module that receives inputs from FFT module 430B and difference module 440. The combined matrix solution is as follows. As the 2D array 110A has a circular array 310 with an omnidirectional sensor 340, Equation 6 is used to calculate U mm x. [0060] Equation 8 is similar to Equation 5, but the dimension of the matrix in Equation 8 is larger because the circular array 310 used here is the sum of both the omnidirectional sensor 340 B and the differential sensor 350. 03-05-2019 18 [0061] FIG. 4B shows an implementation of array 110 B in system 100. The output of each of the omnidirectional sensor 340 and the primary sensor 350 is sent to the computer system 120 and then processed by the array processing module 130B when the array processing module 130B is executed by the processor 205. [0062] Array processing module 130B includes FFT modules 432 and 434, even coefficient module 410B, and odd coefficient module 420B. The FFT modules 432 and 434 digitize the three dimensional wave signals acquired by the omnidirectional sensor 340 and the primary sensor 350, respectively. The even coefficient module 410B processes the digitized three-dimensional wavefield signal acquired by the omnidirectional sensor 340 to calculate the coefficients of the even-mode spherical harmonics of the acquired three-dimensional wavefield signal. The odd coefficient module 420 B processes the digitized three-dimensional wavefield signal acquired by the primary sensor 350 to calculate the coefficients of even mode spherical harmonics of the acquired three-dimensional wavefield signal. [0063] The output of the omnidirectional sensor 340 is sent to the FFT module 432 and the output of the FFT module 432 is sent to the even coefficient module 410B to determine the coefficients of the even mode spherical harmonics. The even coefficient module 410 B then obtains even mode coefficients using equation 5 above, but with the outputs from the FFT modules 432 and 434, m is even, and equation 6 to calculate U mm x Is used. [0064] The output of the primary sensor 350 is digitized by the FFT module 434 and sent to the odd coefficient module 420 B to determine the coefficients of the spherical harmonics in odd mode. 03-05-2019 19 [0065] The odd coefficient module 420 B then obtains odd mode coefficients using the following equation: Where is the alpha vector of equation 5 for difference sensor 350, is the alpha vector of equation 5 of cardioid sensor 350, is calculated using equation 6, and is obtained by even coefficient module 410B It is an even mode coefficient. [0066] The odd mode coefficients are then calculated using Equation 5, where is where U odd is obtained using Equation 7 and is as defined in the previous paragraph. [0067] Alternatively, the even coefficient module 410B and the odd coefficient module 420B may be implemented as a single module so that operations can be performed using one matrix operation. The combined matrix solution is the same as Equation 8, but in this case U mmX is β is a scaling factor defined by the reception pattern of the cardioid microphone 350. [0068] The inverse matrix used by modules 410A, 410B, 420A, and 420B depends on the position of sensors 340, 350. Thus, the inverse matrix is defined for the particular configuration of the twodimensional sensor array 110. A plot of the reconstructed and actual sound fields is shown in FIG. 4C. The plots (a) and (c) of FIG. 4C are the actual sound fields captured by the system of FIG. 1A (with the two-dimensional sensor array 110A) in the plane of z = 0 m and z = 0.2 m, respectively. And plots (b) and (d) are wave fields reconstructed in these two planes, respectively. 03-05-2019 20 [0069] FIG. 5 illustrates a method 500 for the system 100 to acquire a three-dimensional wave signal and construct a three-dimensional wave field representation of the acquired three-dimensional wave signal. Method 500 begins at step 510 with acquiring a three-dimensional wavefield signal using a two-dimensional sensor array 110. Two-dimensional sensor array 110 can be any of exemplary arrays 110A and 110B. Method 500 then proceeds to step 520. [0070] At step 520, the acquired three-dimensional wave field signal is digitized. The acquired threedimensional wavefield signals are digitized by FFT modules 430 or 432 and 434, as described above in connection with FIGS. 4A and 4B, depending on the configuration of array 110. Method 500 then proceeds to step 530. [0071] At step 530, calculate even coefficients of the spherical harmonics. The acquired and digitized three-dimensional wave field signal is transmitted to the even coefficient module 410A or 410B. The transmission of the digitized three-dimensional wavefield signal to the even coefficient module 410A or 410B is as described above in connection with FIGS. 4A and 4B. Method 500 proceeds to step 540. [0072] At step 540, calculate the odd coefficients of the spherical harmonics. The acquired and digitized three-dimensional wavefield signal is transmitted to the odd coefficient module 420A or 420B. The transmission of the digitized three-dimensional wavefield signal to the odd coefficient module 420A or 420B is as described above in connection with FIGS. 4A and 4B. Method 500 proceeds to step 550. [0073] 03-05-2019 21 At step 550, a three-dimensional wave field representation is constructed. The construction of the three-dimensional wavefield representation (defined in Equation 1) occurs by using the calculated even and odd coefficients. Method 500 ends after step 550. [0074] FIG. 6A shows an active noise removal system 600 having a two-dimensional sensor array 110, a computer system 120 having an array processing module 130 and an application module 140, and a speaker 640. Two-dimensional sensor array 110 and array processing module 130 are as described above in connection with FIGS. 3A, 3B, 4A, and 4B. Computer system 120 is as described above in connection with FIGS. 2A and 2B. [0075] The application module 140 includes an adaptation module 610, a filter module 620, and a signal generation module 630. The adaptation module 610 receives the coefficients (ie, both even mode coefficients and odd mode coefficients) of the 3D wavefield spherical harmonics from the array processing module 130. The adaptation module 610 then calculates a set of weights corresponding to the noise received by the array 110 and transmits the set of weights to the filter module 620. [0076] Filter module 620 receives the series of weights and adjusts the filter coefficients of filter module 620. The filter coefficients are sent to the signal generation module 630. [0077] The signal generation module 630 includes a reference signal matrix formed by multiplying the coefficients of the three-dimensional wave field spherical harmonics with the channel information (ie, position information) of the speaker 640. The signal generation module 630 then 03-05-2019 22 multiplies the received filter coefficients with the channel information to generate a series of discrete speaker drive signals for the speaker 640. The individual speaker drive signals are complex numbers with sinusoidal amplitude and phase. The time domain speaker drive signal is then generated by combining the individual speaker drive signals and sent to the speaker 640. [0078] Speaker channel information may be calculated based on theoretical models or measured during an off-line calibration process. The speaker 640 reproduces the drive signal generated by the signal generation module 630 and generates a sound field that is a phase-reversed version of the noise field, thus removing noise and generating a quiet zone. Such active noise removal applications can be used as noise control in cabins of vehicles (cars, aircrafts), industrial noise reduction (factory), etc. [0079] As conventional active noise removal techniques do not take into account spatial acoustic information, the performance of these systems is generally limited and limited by the situation. However, when performed in the wave domain (using sound field coefficients), the denoising algorithm can achieve much higher resolution in both the frequency domain and the spatial domain. [0080] For example, active noise removal can be used on a car. In this application, the microphone array 110 must be small enough to fit in the roof of the cabin but large enough to provide good resolution in all target frequency bands. As the majority of ambient noise power (e.g., traffic noise, wind noise, engine noise) is in the 50-850 Hz range, the microphone array 110 is designed to operate within this frequency band. [0081] Active noise removal requires high accuracy, but small errors can lead to significant degradation 03-05-2019 23 of the system 600 performance. Thus, the main design goal for this application is to maximize the accuracy of system 600 while keeping the number of microphones low. [0082] In this application, the omnidirectional microphone 340 has a smaller profile as compared to a cardioid microphone, so the array 110A is better. The omnidirectional microphones 340 can also be mounted close to one another without causing much distortion in the acquired sound field signal. In addition, separate calculations of the even and odd mode coefficients of array 110A minimize interference between microphones 340 on different circular arrays 310, resulting in better accuracy. [0083] The interior space of a car is often limited and irregular, and it is not possible to mount a spherical array in the car. However, with the two-dimensional sensor array 110, it is possible to integrate the planar array 110 into the ceiling of a car. If the maximum radius 320 N of the array is 0.46 meters, the array 110 can substantially cover the area where the passenger's head should be, while keeping the silhouette of the array 110 relatively small. For a maximum frequency of 850 Hz and a radius 320 N of 0.46 m, the array 110 can receive sound field harmonics to the order of: at least 2N + 1 = 27 microphone pairs (350) as the outermost It means that it should be arranged in a circular array 310E. The circular array 310 is non-uniformly placed closer to the outer circular array 310 E, with more circular arrays 310 in order to maximize the detection accuracy of higher order harmonics. In this example, the radius 320 is set to 0.46 m, 0.38 m, 0.30 m and 0.20 m. The quantities of microphones 350 of each circular array 310A, 310B, 310C, 310D, 310E are determined to be 21, 19, 15, and 11, respectively. [0084] The performance of the array 110 for active noise removal is evaluated using simulation. The simulation tests the response of the array 110 to a single point source at a frequency of 1501150 Hz operating from 1.6 meters away from the center of the array at θ = π / 4. The derived sound field coefficients are used to reconstruct a three dimensional wave field by comparing the reconstructed three dimensional wave field with the original sound field. 03-05-2019 24 [0085] FIG. 6B shows the performance of system 600 for different frequencies and angles of incidence. It can be seen from FIG. 6B that the error is relatively small below 850 Hz, which is the design maximum frequency of the array 110. Above 850 Hz, the error increases rapidly as the order of the spherical harmonics increases beyond the design frequency of the array 110 determined by the method 300. [0086] According to FIG. 6B, the reproduction accuracy does not largely change depending on the incident angle of the plane wave. At 850 Hz, the array 110 can capture waves incident from any direction with an error of less than 4%. [0087] FIG. 7 shows a beam forming system 700 having a two-dimensional sensor array 110, a computer system 120 having an array processing module 130 and an application module 140, and a speaker / antenna 740. Two-dimensional sensor array 110 and array processing module 130 are as described above in connection with FIGS. 3A, 3B, 4A, and 4B. Computer system 120 is as described above in connection with FIGS. 2A and 2B. [0088] The application module 140 includes a weighting module 710, a wave synthesis module 720, and an adaptation module 730. The weighting module 710 receives the three-dimensional wavefield coefficients from the array processing module 130 and multiplies each of the received coefficients with a weighting factor. The weighted coefficients are then transmitted to the wave synthesis module 720, which synthesizes a new time domain signal using the weighted coefficients. The combined time domain signal is the output of the beamformer. [0089] 03-05-2019 25 The adaptation module 730 optimizes the directivity of the system 700. An adaptation module 730 receives the beamforming output of the wave combining module 720 and compares the beamforming output to a target beamforming direction. The adaptation module 730 then updates the weighting factors in response to the comparison and sends the updated weighting factors to the weighting module 710. Typical applications of the beam forming system 700 are RF antenna arrays, directional recordings (such as for conference recordings), and the like. [0090] FIG. 8 shows a source localization / direction-of-arrival estimation system 800 comprising a twodimensional sensor array 110 and a computer system 120 comprising an array processing module 130 and an application module 140. Two-dimensional sensor array 110 and array processing module 130 are as described above in connection with FIGS. 3A, 3B, 4A, and 4B. Computer system 120 is as described above in connection with FIGS. 2A and 2B. [0091] The application module 140 includes a correlation matrix module 810 and an arrival direction module 820. The correlation matrix module 810 calculates a correlation matrix from the constructed 3D wave field representation and sends the correlation matrix to the direction of arrival module 820. Direction of arrival module 820 includes a direction of arrival algorithm (eg, MUSIC) that outputs a two dimensional plot corresponding to the possible directions of the incident wave using the correlation matrix. Applications in which the DOA system 800 may be used include tracking, sonar / radar scanning, and the like. [0092] FIG. 9 shows a three-dimensional sound field recording system 900 having a two-dimensional sensor array 110, a computer system 120 with an array processing module 130 and an application module 140, and speakers 940. Two-dimensional sensor array 110 and array processing module 130 are as described above in connection with FIGS. 3A, 3B, 4A, and 4B. Computer system 120 is as described above in connection with FIGS. 2A and 2B. 03-05-2019 26 [0093] The application module 140 includes a wave synthesis module 910 for receiving the built 3D wave field representation and generating a series of speaker drive signals. The wave synthesis module 910 requires prior knowledge of the position and channel information of the speaker 940 to generate a speaker drive signal. The wave synthesis module 910 then sends the generated speaker drive signal to the speaker 940. However, unlike system 600, the adaptation module is not required as the accuracy in reconstructing the acquired three-dimensional wavefield signal need not be as accurate as system 600. [0094] System 900 enables complete recording and synthesis of acoustic scenes. System 900 can be used for applications such as teleconferencing, performance / movie recording, and the like. [0095] Mathematical proof of obtaining coefficients of spherical harmonics in odd mode in Xy plane As mentioned in the background art above, the 3D wavefield spherical harmonics expansion is arranged in 3 dimensions (ie xyz plane) I need a sensor. That is, sensors placed only in the xy plane (i.e., in two dimensions) can not provide the spherical harmonic odd mode 1010 when the odd mode 1010 in the xy plane is zero. However, it was observed that FIG. 10 shows that the rate of change (ie,) of P nm with respect to θ is exactly the same for the even and odd modes. That is, even mode 1020 is equal to zero when θ = π / 2, and odd mode 1010 is not zero at θ = π / 2. [0096] This observation is proved by the Legendre 特性 function property. When x = cos (π / 2) = 0, equation 10 is P ′ nm (0) = (m + n) P n−1, m (0). This shows the observed relationship between P ′ nm (cos (π / 2)) and P n−1, m (cos (π / 2)). Therefore, P n−1, m (0) = 0 = P ′ n, m (0). Otherwise, both P n−1, m (0) and P ′ n, m (0) are non-zero. If the odd number P n | m | (cos (π / 2)) is zero and the even number P n | m | (cos (π / 2)) is not zero, then even P ′ n | m | 2)) is zero, and the odd number P ′ n | m | (cos (π / 2)) is not zero. 03-05-2019 27 [0097] Therefore, if the odd number P n | m | (cos (π / 2)) can not be observed on the xy plane, the odd number P ′ n | m | (cos (π / 2)) is observable. This means that the odd mode can be determined. [0098] Thus, the above mathematical relationships show that the odd modes of the three-dimensional wavefield spherical harmonic expansion can be obtained in the xy plane. [0099] In the primary sensor spherical coordinate system for obtaining the coefficients of spherical harmonics in odd mode in the xy plane, the particle velocity of the point (r, θ, φ) can be expressed in the frequency domain as the slope of Equation 1. [0100] For simplicity, it is assumed that the sensor measures particle velocity in the θ direction. Equation 11 can be modified as a result. [0101] Substituting equation 1 into equation 12 results in: [0102] Take the partial derivative of [0103] Since the coefficient C nm is treated as an unknown parameter in Equation 14, the derived representation should be suitable to represent any type of wave field. 03-05-2019 28 [0104] For a plane wave incident from the azimuth, the wave field can be expressed as: [0105] Combining Equations 14 and 15, the wave field captured by the primary sensor is: [0106] A portion (ie,) of the left side of Equation 16 is usually equal to the pattern of the primary sensor when arranged radially at θ = π / 2. That is, the primary sensor is directed to (θ + 2 / π, φ). The remaining terms on the left side (ie, P (r, θ, φ, k)) are the actual wave fields at the location of the sensor. By multiplying the terms of the two left sides, the wave field acquired by the primary sensor is obtained. [0107] In the right side of Equation 16, the Legendre 陪 function is replaced by the first derivative of the function, and the zero point of the term is changed. As mentioned above, the odd mode is not zero when x = 0. Thus, this new representation allows measurement of odd mode wavefield coefficients in the θ = π / 2 plane. 03-05-2019 29 [0108] Thus, according to paragraphs [00122] and [00123] above, the primary sensor obtains odd mode coefficients of spherical harmonics when the primary sensor is arranged vertically (ie perpendicular to the xy plane) be able to. [0109] One example of a primary sensor is two omnidirectional sensors placed in close proximity, as described in connection with FIG. 3A. Assuming that two omnidirectional sensors are located in spherical coordinates at the positions M 1 = (r, 0,0) and M 2 = (r, π, 0), the output signal is S = S 1 −S 2 (Eq. 17) where S1 and S2 are the signal outputs of two omnidirectional sensors located at M 1 and M 2 respectively. For a far field source located at a certain position (L, θ, φ) (L >> r), the two omnidirectional sensors have a reception pattern expressed as G = cos θ (equation 18). [0110] Although the magnitudes of the maxima are symmetrical, the maxima in the region of θ> π / 2 have negative gain, meaning that the phase of the signal is reversed when the signal source comes from this direction Do. The size of the local maxima is important because the orientation of the sensor pair should match Equation 16. Otherwise, the calculated wave field is inverted. [0111] In another embodiment, a cardioid sensor can be used as the primary sensor. Generally, the cardioid pattern of a cardioid sensor is realized by summing two acquired signals and adding a delay to one of the acquired signals. The added delay determines the gain pattern of the cardioid sensor. In the case of voice microphones, the delay is achieved by utilizing materials that reduce the speed of the sound waves. 03-05-2019 30 [0112] In one embodiment, the cardioid sensor can be implemented using two omnidirectional sensors and adding a delay to the output of one of the omnidirectional sensors. It should be noted that the cardioid sensor is not superior to the implementation of the two omnidirectional sensors. [0113] Since the cardioid pattern can also be obtained as a weighted sum of the difference pattern with the omnidirectional pattern, the gain can be written as G = (β + (1-β) cos (θ)) (Eq. 19) Where β is a weighting factor. For β = 0.5, a standard cardioid pattern is realized, which means that the two components have the same weight. Thus, the wave field captured by the cardioid sensor can be represented as a weighted sum of the omnidirectional sensor and the differential sensor. Combining Equations 1 and 14, the wavefield acquired by the cardioid sensor is: [0114] Similar to Equation 16, the left side of Equation 20 represents the received pattern of the Cardioid sensor multiplied by the actual wave field at the location of the sensor. The right side of Equation 20 is a weighted combination of wave field expansions of two reception patterns. For a given cardioid sensor (eg, a unidirectional microphone), the left hand side of equation 20 is defined. However, if the exact beam pattern is known, the weighting factor α can be adjusted to fit the beam pattern. [0115] The total wave field at the position (r, π / 2, φ) on the circular array 310 of the array arrangement array 110 can be obtained by the following equation. [0116] If both sides of Equation 21 are multiplied by Em (−φ) and integrated over [0, 2π] for φ, the total wave field on one concentric circular array (for example, 310A) is obtained. 03-05-2019 31 [0117] The total wavefield across multiple concentric arrays 310 is found by solving the determinant of Here, is the integral of the m th harmonic present in the circular array (eg, 310A). U NmX = jN (kr X) P N | m | (0) is a harmonic associated with each coefficient. [0118] Since the term P N | m | (0) = 0 for the odd mode, only even mode coefficients can be derived from the above equation. [0119] One example of deriving the odd mode coefficients is to use primary sensors arranged in a circular array (e.g., 310A) and oriented perpendicular to the [theta] = [pi] / 2 plane. Equation 16 is then integrated over [0, 2π] for φ: Where is the wavefield received by the primary sensor at (r, π / 2, φ). The coefficients can then be calculated by solving the matrix inversion shown in Equation 23. By the term P ′ n | m | (cos θ) = 0 when θ = π / 2, the matrix solution contains only odd mode coefficients when the primary sensors are arranged in a circular array (eg, 310A). [0120] Thus, the full three-dimensional wavefield spherical harmonic expansion can be retrieved by combining the retrieved even mode coefficients and odd mode coefficients. [0121] Equations 22 and 24 relate to the ideal situation where the sensors are continuous across a circular array (e.g., 310A). 03-05-2019 32 However, in a practical implementation, there are only discrete numbers of sensors arranged in a circular array (eg, 310A). Thus, the discrete forms of Equations 22 and 24 are [0122] With spatial sampling of the acquired three-dimensional wavefield signals, sensors (340 or 350) arranged in a circular array (e.g., 310A) observe a limited number of spherical harmonic orders. The relationship between the number N X of sensors and the maximum order of observable spherical harmonics is given by N X = 2N + 1. However, the properties of the spherical Bessel function mean that a limited number of orders can be observed in a circular array (e.g., 310A). The maximum number of observable harmonic orders is given by: Here, k is a wave number. The exact amount Nx of sensors used for each circular array (e.g., 310A) depends on the size of the circular array (e.g., 310A) and the wave number k of the target wavefield. [0123] Because the order of the spherical harmonics obtained by the circular array (eg, 310A) is limited, aliasing of higher harmonics (not detected by the circular array 310A) to lower harmonics occurs. Aliasing produces an error that is determined by the presence of undetected higher harmonics. However, Equation 27 provides each circular array 310 with a sufficient number of sensors to provide high accuracy for most applications. [0124] However, if the application does not require high accuracy, then lower accuracy can be used to design circular arrays (eg, 310A). The lower accuracy assumes observable harmonic orders at radius r, which means that fewer sensors are used in a circular array (eg, 310A). [0125] Since the quantity of sensors (340 or 350) on the circular array (eg 310A) is associated with the 03-05-2019 33 wave number k that can be converted to wavelength λ, the number of sensors (340 or 350) is the target frequency of the application And can be derived from Where c is the wave velocity. For audio applications, c = 340 m / s, and for RF applications, c = 300,000,000 m / s. For example, a 0.2 m radius circular array (eg, 310A) designed to receive 900 MHz RF signals has 13 sensors, but a 0.4 m radius designed for audio signals up to 1500 Hz. A circular array (eg, 310A) requires 33 sensors (ie, microphones). [0126] The total number of sensors (340 or 350) is doubled for array 110A by using two omnidirectional sensors 340 for each sensor 350. It should be noted that the distance between the two omnidirectional sensors 340 in the array 110A is small compared to the array radius 320N in order to approximate in Eq. [0127] For arrays 110A and 110B, even mode coefficients and odd mode coefficients can be calculated together using a single pseudo-inverse operation, instead of processing any of them separately. The calculations are performed as in Equation 23, where α mx is the sum of the m-order mode responses of the circular array (eg, 310A). U NmX is the wavefield representation of a circular array (eg, 310A), which must exactly match α mX on the left side of the equation. In other words, U NmX needs to take one of the following equations based on the sensor (340 or 350) used for the circular array (eg, 310A). The ability of a two-dimensional array to detect the vertical components (ie, odd mode coefficients) of a three-dimensional wave field spherical harmonic expansion is due to the combination of the zero order sensor (340) and the primary sensor (350). The planar array of omnidirectional sensors (340) can not distinguish the waves coming from both sides of the array 400. As a result, only even modes of wave field components that are symmetrical on the sensor plane can be captured by the planar array. The odd modes correspond to wave field components that are asymmetric on a plane, and these modes are essential to represent the wave incident from the out-of-plane direction. By using an omnidirectional sensor in combination with a vertically placed primary sensor, the array can distinguish between waves incident from both sides of the array plane, which is now the asymmetry of the wave field is now a hybrid sensor array It means that it can be recognized. [0128] 03-05-2019 34 As mentioned in paragraph [0008] above, spherical arrays operate in a narrow frequency band. However, the two-dimensional array 110 operates in a wide frequency band because the array 110 is uniformly distributed. A portion of the circular array 310 can always receive an active mode at a frequency. However, the highest order active mode is received by one or two largest circular arrays of radius 320N or 320N-1. Thus, the radius 320 of the circular array 310 needs to be designed carefully, otherwise the captured wavefield coefficients may be noisy. All in all, the two-dimensional array 110 has the ability to analyze the signals of the broadband threedimensional wave field. Such broadband capabilities make the 2D array 110 suitable for acoustic applications as sound waves such as human voice and engine sounds generally have a wide frequency band from 100 Hz to several thousand Hertz. [0129] Error Analysis Spatial Sampling In most sensor array configurations based on discrete sensors, spatial sampling is the main and root cause of error. In general, to avoid spatial sampling, the distance between each two sensors must be smaller than half the wavelength of the target signal (ie d ≦ λ / 2), where d is two sensors Is the distance between and λ is the wavelength of the highest frequency of the signal. [0130] However, in the case of the two-dimensional array 110 described above, the error due to spatial sampling is expressed by the following equation. [0131] Spatial sampling of the array 110 limits the maximum observable spatial frequency represented as the order of the spherical harmonics. The truncation error due to the restriction is expressed by the following equation. [0132] 03-05-2019 35 The power ΔE is determined by the spherical Bessel function, the value of which is a function of the wave number k and the radius 320 (ie, r). For a given degree of a spherical Bessel function, the value of j n (kr) approaches zero if kr is sufficiently small. On the other hand, for fixed kr, the active spherical harmonics are limited to N. [0133] The maximum number of active spherical harmonic orders is given by: Thus, the minimum amount of sensors used in the array is 2N + 1. However, the accuracy of the array can be improved by using additional sensors in the array. [0134] In Equation 25, the wavefield coefficients are calculated by solving the matrix equation α = UC. The aliasing error of α affects all sound field coefficients C nm. The error rate is given by [0135] If the amount of sensors (340 or 350) possessed by the circular array (e.g., 310A) is insufficient, all [alpha] mX obtained from the output of the array will be inaccurate. The algorithm uses matrix inversion to solve the coefficients, which follows the least mean square (LMS) fit, so the error of a single α mX is evenly distributed among all the calculated wavefield coefficients of mode m . The error of each obtained coefficient C nm is given by [0136] Equation 33 shows that the individual coefficients of the lower order modes tend to have less aliasing error because there are many coefficients that share the error. However, the total error power for each mode remains the same across all available modes. [0137] 03-05-2019 36 The error rate can be minimized as long as a sufficient amount of sensors are used for spatial sampling. It should be noted that the error rate only reflects the average expected error of the calculated coefficients due to space aliasing. Another source of error due to poor sampling is the loss of a certain amount of active higher harmonics (these harmonics caused aliasing errors in the lower order coefficients), but this type of error Can be recognized only during wave field reproduction, and the individual wave field coefficients are not directly affected by this phenomenon. [0138] As the array comprises a plurality of concentric circular arrays 310, each circular array (e.g., 310A) observes spherical harmonics of different maximum orders. For example, while all circular arrays 310 observe the zeroth harmonic, the highest harmonic is observable by the largest circular array 310N of radius 320N. Thus, the low order modes are more accurate because the low order modes are sampled by most, if not all, of the circular array 310. However, because the higher order modes are observed only by one or two circular arrays (eg, 310N and 310N-1) with the largest radii (eg, 320N and 320N-1 respectively), the higher order coefficients are exactly Not calculated Thereby, the central part of the wave field is calculated more accurately than the outer region. Thus, the reconstructed 3D wave field gradually loses accuracy at higher altitudes, as only higher harmonics are associated with these regions. Rather than using a uniformly distributed circular array 310 to compensate for such errors, more circular arrays 310 with larger radii 320 are placed in the system 100. [0139] The described arrangement is applicable to the field of signal processing. [0140] The foregoing describes only some embodiments of the present invention, modifications and / or changes of the present invention can be made without departing from the scope and spirit of the present invention, and the embodiments are exemplary. There is no limitation. [0141] In the context of the present specification, the term "comprising" means "including mainly but 03-05-2019 37 not necessarily necessarily" or "having" or "including" and not "consisting only of" . Variations of the word "comprising" such as "comprise" and "comprises" have correspondingly changed meanings. 03-05-2019 38

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