Method and system for determining the probability density function of the response of a reverberant system

Abstract

A method and system are provided for probabilistically determining an expected field response by a reverberation system that accounts for input conductance frequency uncertainty. One method includes determining a marginal probability density function of the input conductance frequency uncertainty of the reverberation system based at least in part on the physical dimensions, loss coefficient, and statistical average of the input excitation energy of the enclosure structure, determining an unconditional probability density function of the field response of the reverberation system to the excitation energy based at least in part on the input conductance frequency uncertainty, determining an expected field response by the reverberation system versus frequency in response to the input probability value of the excitation energy using the unconditional probability density function, and providing a graphical user interface (GUI) display including a graphical representation of the expected field response versus frequency.

Description

[Technical Field]

[0001] (CROSS-REFERENCE TO RELATED APPLICATIONS) This application claims priority to U.S. Provisional Patent Application No. 63 / 421,178, filed November 1, 2022, the entire contents of which are incorporated herein by reference.

[0002] The subject matter described herein generally relates to the interaction between a wave field and a reverberation system, and more particularly, embodiments of the subject matter relate to analytically determining the unconditional probability density function of the reverberation response versus frequency taking into account uncertainties. [Background technology]

[0003] Engineers often need to be able to estimate or predict the real-world dynamic environment in which a device or component will operate so that they can design and test the device or component for reliable operation in that environment. For example, electrical engineers may need to estimate the maximum electromagnetic field strength at which an electronic component, device, or system must operate so that they can design and test for immunity to electromagnetic interference. When electronic equipment is housed within an enclosure, the electromagnetic field within the enclosure becomes reverberant at higher frequencies (e.g., based on wavelength relative to the dimensions of the enclosure), at which point electromagnetic reflections accumulate to produce a multi-modal reverberation response that is typically quantified by a total wave field energy level. Due to random or uncertain excitation or uncertainty in the exact modal parameters of the enclosure (determined by the dimensions and electromagnetic properties of the enclosure), the reverberation energy level can typically only be quantified statistically.

[0004] As another example, in the field of vibro-acoustics, engineers may need to estimate the maximum vibration levels that sensitive equipment and / or payloads will experience during operation. For example, mechanical engineers designing a rocket or launch vehicle need to be able to estimate the maximum vibration levels likely to be experienced during transonic flight so that they can design and test the safe operation of the equipment during flight. The vibratory wave field response of a vehicle's structural panels can be driven by unsteady aerodynamic forces during transonic flight. Again, vibratory waves within the vehicle structural panel subsystems reflect and scatter at higher frequencies, at which point the vibrations accumulate to produce reverberant vibration energy levels. [Prior art documents] [Patent documents]

[0005] [Patent Document 1] U.S. Patent No. 10,565,326 Summary of the Invention [Problem to be solved by the invention]

[0006] While various statistical energy analysis methods exist and can be used to estimate mean or average reverberation energy levels, care must be taken not to underestimate the statistical variance about the mean and the resulting maximum expected reverberation energy level, since significantly underestimating the maximum expected response during the design phase can lead to equipment failure in the operating environment. At the same time, any significant overestimation of the reverberation energy level can make it cost- and / or weight-prohibitive to design a device or component for the estimated reverberation energy level. Therefore, it is desirable to calculate or otherwise estimate reverberation energy levels in an accurate and reliable manner without significantly overestimating or underestimating the expected reverberation energy levels. [Means for solving the problem]

[0007] A method and system are provided for probabilistically determining an expected field response by a reverberation system that accounts for input conductance frequency uncertainty. One method includes receiving user input defining characteristics of the reverberation system, the characteristics including physical dimensions, wave propagation velocity, and loss coefficient of an enclosure structure, identifying excitation characteristics of excitation energy input to the reverberation system, determining a conditional probability density function of the field response of the reverberation system to the excitation energy based at least in part on a statistical average of the magnitude-squared field response, determining a marginal probability density function of the input conductance frequency uncertainty of the reverberation system based at least in part on the physical dimensions, wave propagation velocity, and loss coefficient, determining an unconditional probability density function of the field response of the reverberation system to the excitation energy based at least in part on the marginal probability density of the input conductance frequency uncertainty, determining an expected field response by the reverberation system vs. frequency in response to the input probability value of the excitation energy using the unconditional probability density function, and providing a graphical user interface (GUI) display including a graphical representation of the expected field response vs. frequency. In one implementation, the method determines an unconditional probability density function for the expected field response based on a conditional probability density function having a marginal probability distribution for the uncertain input conductance and a marginal probability density function for at least one of an effective input current uncertainty related to the excitation energy and a Q-factor uncertainty related to a loss factor of the reverberation system. In another implementation, determining the marginal probability density function for the input conductance uncertainty of the reverberation system includes calculating a marginal probability density function of the input conductance frequency uncertainty based at least in part on the physical dimensions, the wave propagation speed, the loss factor, and a statistical mean of the field response.

[0008] An apparatus is also provided for a computer-readable medium having computer-executable instructions stored thereon. The computer-executable instructions, when executed by a processing system, cause the processing system to: receive user input defining characteristics of a reverberation system, including physical dimensions and loss coefficients of an enclosure structure; identify excitation characteristics of excitation energy input to the reverberation system; determine a marginal probability density function of input conductance-frequency uncertainty of the reverberation system based at least in part on statistical means of the physical dimensions, loss coefficients, and excitation energy; determine an unconditional probability density function of a field response of the reverberation system to the excitation energy based at least in part on the input conductance-frequency uncertainty; determine an expected field response by the reverberation system versus frequency in response to the input probability value of the excitation energy using the unconditional probability density function; and provide a graphical user interface (GUI) display including a graphical representation of the expected field response versus frequency. In one implementation, the GUI display includes a graph showing the maximum expected field response versus frequency for the input probability values.

[0009] Also provided is an apparatus for a computing device including a computer-readable medium having computer-executable instructions stored thereon and a processor coupled to the computer-readable medium for executing the computer-executable instructions, the computer-executable instructions receiving user input defining characteristics of a reverberation system, the characteristics including physical dimensions, wave propagation velocity, and loss coefficient of an enclosure structure; identifying excitation characteristics of excitation energy input to the reverberation system; determining a marginal probability density function of input conductance-frequency uncertainty of the reverberation system based at least in part on statistical means of the physical dimensions, loss coefficient, and excitation energy; determining an unconditional probability density function for the field response of the reverberation system to the excitation energy based at least in part on the input conductance-frequency uncertainty; using the unconditional probability density function to determine an expected field response by the reverberation system versus frequency in response to the excitation energy for the input probability values; and providing a graphical user interface (GUI) display including a graphical representation of the expected field response versus frequency. In one implementation, the GUI display includes a graph showing the maximum expected field response versus frequency for the input probability values.

[0010] This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. [Brief explanation of the drawings]

[0011] Exemplary embodiments of the presently disclosed subject matter are described below in conjunction with the following drawings, in which like numerals refer to like elements and in which: [Figure 1] FIG. 1 is a block diagram of a networked computing system in accordance with one or more exemplary implementations. [Figure 2]FIG. 1 is a block diagram of a local computing system in accordance with one or more exemplary implementations. [Figure 3] 3 is a flow diagram of a probabilistic response prediction process suitable for implementation by the computing system of FIG. 2 or a probabilistic response prediction application in the computing system of FIG. 1, according to one or more exemplary implementations. [Figure 4] 4 illustrates an exemplary graphical user interface (GUI) display suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3, according to one or more exemplary implementations. [Figure 5] 4 illustrates an exemplary graphical user interface (GUI) display suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3, according to one or more exemplary implementations. [Figure 6] 4 illustrates an exemplary graphical user interface (GUI) display suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3, according to one or more exemplary implementations. [Figure 7A] 4 illustrates an exemplary graphical user interface (GUI) display suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3, according to one or more exemplary implementations. [Figure 7B] 4 illustrates an exemplary graphical user interface (GUI) display suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3, according to one or more exemplary implementations. [Figure 8] 4 illustrates an exemplary graphical user interface (GUI) display suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3, according to one or more exemplary implementations. [Figure 9]4 illustrates an exemplary graphical user interface (GUI) display suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3, according to one or more exemplary implementations. [Figure 10] 4 illustrates an exemplary graphical user interface (GUI) display suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3, according to one or more exemplary implementations. [Figure 11] FIG. 4 is a block diagram illustrating statistical parameters of a reverberation wave field system suitable for analysis in connection with the probabilistic response prediction process of FIG. 3 in an exemplary implementation. [Figure 12] FIG. 4 is a block diagram illustrating uncertainty parameters of a reverberation wavefield system suitable for analysis in connection with the probabilistic response prediction process of FIG. 3 in an exemplary implementation. [Figure 13] FIG. 4 is a block diagram illustrating the relationships between connected reverberation systems suitable for analysis related to the probabilistic response prediction process of FIG. 3. [Figure 14] FIG. 4 is a block diagram illustrating the relationship between a reverberant system and a deterministic system suitable for analysis related to the probabilistic response prediction process of FIG. 3. [Figure 15] FIG. 1 is a schematic diagram illustrating the relationship between a mixed deterministic field and a reverberant field, according to one or more implementations. [Figure 16] FIG. 1 is a schematic diagram illustrating the relationship between a coupled deterministic system and a reverberant system, according to one or more implementations. [Figure 17] 4 illustrates an exemplary graphical user interface (GUI) display suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3, according to one or more exemplary implementations. DETAILED DESCRIPTION OF THE INVENTION

[0012] The following detailed description is merely exemplary in nature and is not intended to limit the subject embodiments or the application and uses of such embodiments. Furthermore, there is no intention to be bound by any expressed or implied theory presented in the preceding technical field, background, brief summary or the following detailed description.

[0013] Embodiments described herein generally relate to methods and systems for analytically determining the maximum expected spatial local wave-field response, i.e., the maximum expected level of reverberation field variable response that may exist at any location within an enclosure, cavity, or other reverberant structure or system, for a given confidence level (or probability percentile) for a frequency range of interest. In this regard, rather than providing a spatially averaged wave-field energy level, the maximum expected response energy and maximum expected spatial local field variable response are determined analytically or numerically in a manner that accounts for uncertainty while avoiding over- or underestimation of the maximum expected response, thereby enabling engineers to effectively tune or design a reverberant system or its components in a way that ensures that the reverberant system or its components are likely to withstand excitation energy while minimizing cost and weight.

[0014] The response amplitude of any wave field in a closed reverberant environment is typically complex and difficult to predict at high frequencies because the wave field amplitude is a highly variable function of both frequency and spatial location and can be excited by statistically uncorrelated excitation sources. For example, a typical application in which a designer or other engineer may be interested in predicting the maximum expected response is the prediction of the maximum expected electromagnetic interference to aircraft avionics due to the operation of multiple personal wireless devices in the cabin. Another application is the prediction of the maximum expected acoustic loads on a spacecraft and its sensitive electronic components within its launch vehicle fairing due to liftoff rocket plume acoustic loads and flight transonic aeroacoustic loads. Traditionally, there are two widely used low-order models for this problem. In electromagnetic field applications, the excitation is often treated as a deterministic single-frequency source, and the enclosure corresponding to the reverberant system or cavity is assumed to be electrically large (high modal overlap) so that frequency dispersion is negligible. In this case, the highly variable, spatially localized amplitude of the electric or magnetic field has a Rayleigh probability density distribution. However, for electrically small or low modal overlapping cavities, which are common in practice, this model underpredicts the amplitude response due to additional frequency dispersion.

[0015] In vibroacoustics, where broadband random excitation sources are more common, the complex spatial field response is reduced to a simpler total wave-field energy level through spatial averaging. This reduction allows the vibration and acoustic energy levels of highly complex interconnected wave-field systems to be predicted using a simple power balance principle. This reduction also simplifies the estimation of the frequency variance of the energy, which allows the maximum vibration and acoustic energy levels to be predicted with a log-normal probability density function. However, spatially localized field response amplitudes exhibit significant additional variance relative to the predicted spatially averaged energy estimate. From electromagnetism, the spatial field variability (conditional on the field energy level) is known to be Rayleigh distributed, but the sum of the combined spatial-temporal variability has a probability density function that is neither Rayleigh nor log-normal.

[0016] Therefore, reliable prediction of the maximum expected amplitude in the most general case remains undefined, often resulting in conservative upper-bound estimates that are problematic for efficient design. For example, U.S. Pat. No. 10,379,147, incorporated herein by reference in its entirety, teaches a method for predicting the variance of reverberation electric field energy levels due to uncertainties in the modal parameters of cavity resonances using a low-order model from vibroacoustics. However, this is a spatially averaged wavefield energy analysis that does not consider the Rayleigh spatial variance of local field variable responses. U.S. Pat. No. 10,565,325, incorporated herein by reference in its entirety, teaches that the maximum expected reverberation energy amplitude is subject to at least three sources of uncertainty: modal parameter uncertainty characterized by the frequency variance of the input conductance; statistically independent damping loss factor (Q-factor) variance; and statistically independent power input variance due to uncertainty in the excitation source amplitude. In this regard, the subject matter described herein defines a probability density function for predicting the maximum expected spatially localized wavefield response to account for uncertainties associated with each of the input conductance variance, the attenuation loss coefficient (or Q factor) variance, and the excitation source intensity variance, as opposed to the spatially averaged wavefield energy level.

[0017] 1 illustrates an exemplary embodiment of a networked computing system 100 including a probabilistic response prediction web application 102 that can be configured to analytically determine an unconditional probability density function of a device, component, or other reverberant system that receives an excitation and provides a corresponding indication of an expected spatially localized wavefield response, as described in more detail below. In the exemplary implementation, computing system 100 includes a server 104 that generates or otherwise provides an instance of probabilistic response prediction web application 102 that is accessed by corresponding instances of client devices 106 via a communications network 108, such as the Internet or any type or combination of wired and / or wireless computer networks, cellular networks, mobile broadband networks, wireless networks, etc. It should be understood that FIG. 1 is a simplified representation of computing system 100 and is not intended to be limiting.

[0018] Client device 106 generally represents an electronic device coupled to network 108 that may be utilized by a user to access an instance of web application 102 using an application 110 running on or thereon. In practice, client device 106 may be embodied as any type of personal computer, mobile phone, tablet, or other network-enabled electronic device coupled to network 108 that executes or otherwise supports a web browser or other client application 110 that enables a user to access one or more graphical user interface (GUI) displays provided by web application 102. In an exemplary implementation, client device 106 includes a display device, such as a monitor, screen, or another conventional electronic display, capable of graphically presenting data and / or information, along with a user input device, such as a touchscreen, touch panel, mouse, joystick, directional pad, motion sensor, or the like, that can receive input from a user of client device 106. The illustrated client device 106 executes or otherwise supports client application 110, which communicates with server 104 to access instances of web application 102. For example, in some implementations, the client application 110 is realized as a web browser or similar local client application executed by the client device 106 that contacts the server 104 using a network protocol such as the HyperText Transport Protocol (HTTP).In this manner, in one or more implementations, client application 108 may be utilized to access or otherwise initiate an instance of web application 102 hosted by server 104, and web application 102 provides one or more web page GUI displays within client application 110 that include GUI elements for interfacing with and / or interacting with web application 102 supported by server 104.

[0019] Server 104 generally represents one or more server computing devices, server computing systems, or other combinations of processing logic, circuitry, hardware, and / or other components configured to support instances of web application 102 provided to client devices 106 over network 108. In an exemplary implementation, server 104 generally includes at least one processing system 120, which may be implemented using any suitable processing system and / or device, such as, for example, one or more processors, central processing units (CPUs), controllers, microprocessors, microcontrollers, processing cores, application specific integrated circuits (ASICs), and / or other hardware computing resources configured to support the operations of the processing system described herein. Additionally, although not shown in FIG. 1 , in practice server 104 may also include one or more communication interfaces, including any number of transmitters, receivers, transceivers, wired network interface controllers (e.g., Ethernet adapters), wireless adapters, or another suitable network interface supporting communications with network 108 coupled thereto. The application server 104 also includes or otherwise has access to a data storage element 122 (or memory) that stores code or other computer-executable programming instructions that, when executed by the processing system 120, cause the processing system 120 to support or otherwise facilitate the web application 102 and associated software services that are configurable to support the subject matter described herein. Depending on the implementation, the memory 122 may be realized as random access memory (RAM), read-only memory (ROM), flash memory, magnetic or optical mass storage, or any other suitable non-transitory short-term or long-term data storage or other computer-readable medium, and / or any suitable combination thereof capable of storing code or other programming instructions executable by the processing system 120.

[0020] 2 illustrates an exemplary embodiment of a local computing system 200 suitable for supporting or otherwise implementing a probabilistic response prediction software application 202 that can be configured to analytically determine the unconditional probability density function of a device, component, or other reverberant system subjected to excitation and provide a corresponding indication of the expected spatially local wavefield response, as described in more detail below. The illustrated computing system 200 includes, without limitation, a user input device 204, a processing system 206, an output device 208, and a data storage element 210. It should be understood that FIG. 2 is a simplified representation of a local computing system for purposes of illustration and is not intended to limit the scope of the subject matter in any way. In this regard, in practice, the local computing system 200 may be implemented in a client device 220, such as an instance of the client device 106 of FIG. 1. That is, depending on the implementation, the probabilistic response prediction software application 202 may be implemented locally on a client device 106, 220, remotely as a web application 102, or in another distributed manner, and the subject matter described herein is not limited to any particular implementation of the probabilistic response prediction application 102, 202 and the corresponding processes, services, tasks, operations, and / or other functionality described herein.

[0021] User input device 204 generally represents hardware and / or other components configured to provide a user interface with computing system 200. Depending on the embodiment, user input device 204 may be embodied as a keypad, keyboard, mouse, one or more buttons, touch panel, touch screen, audio input device (e.g., microphone), etc. Output device 208 generally represents hardware and / or other components configured to provide output from computing system 200 to a user, as described in more detail below. In an exemplary embodiment, output device 208 is embodied as an electronic display device associated with client device 220, configured to graphically display information and / or content under the control of processing system 206, as described in more detail below. Thus, for purposes of description, output device 208 may alternatively be referred to herein as a display device. That said, in other implementations, output device 208 may be embodied as a communication interface or other input / output interface supporting communication with client device 220.

[0022] 2 , processing system 206 generally represents the hardware, circuitry, processing logic, and / or other components of computing system 200 coupled to user input device 204 and display device 208 to receive input from a user, perform various functions and / or processing tasks utilizing the input provided by the user, and provide output to the user, as described in more detail below. Depending on the embodiment, processing system 206 may be implemented or realized using a computer, a general-purpose processor, a microprocessor, a controller, a microcontroller, a state machine, an associative memory, an application-specific integrated circuit, a field-programmable gate array, any suitable programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof, designed to perform the functions described herein. Furthermore, the steps of a method or algorithm described in connection with the embodiments disclosed herein may be embodied directly in hardware, in firmware, in a software module executed by processing system 206, or in any practical combination thereof. Data storage element 210 (or memory) may be embodied as any type of non-transitory short-term or long-term storage medium capable of storing programming instructions, code, or other data for execution by processing system 206, including any type of random access memory (RAM), read-only memory (ROM), flash memory, registers, hard disk, removable disk, magnetic or optical mass storage, and / or any other suitable computer-readable medium. The computer-executable programming instructions, when read and executed by processing system 206, cause processing system 206 to run or otherwise provide probabilistic response prediction application 202, and to perform tasks, operations, and / or functions described in more detail below.

[0023] 1-2, in an exemplary implementation, the probabilistic response prediction application 102, 202 causes the client device 106, 220 to receive user input indicating the physical dimensions and / or physical layout of the reverberant subsystem to be analyzed along with the mechanical, electrical, chemical, material, and / or other physical properties (e.g., permeability and permittivity that define the wave propagation speed) of the reverberant subsystem and / or medium, and to display (e.g., on the display device 208) one or more GUI displays including one or more GUI elements (e.g., text boxes, etc.) adapted to affect the reverberant response to input excitation energy. Additionally, the probabilistic response prediction application 102, 202 may also display GUI elements adapted to receive user input indicating the amplitude, frequency, and / or other characteristics of excitation energy provided by an excitation source that are used to characterize and predict the reverberant response of the reverberant subsystem and / or medium, as described in more detail below. After providing the desired input information, the user may manipulate the user input device 204 to select a GUI element (e.g., a button, etc.), which causes the processing system 206 to continue executing programming instructions using the input received from the user to calculate or otherwise determine a maximum expected reverberation response for the reverberation subsystem and generate or otherwise provide one or more output indications on the display device 208 indicative of the maximum expected reverberation response, as described in more detail below.

[0024] FIG. 3 illustrates an exemplary probabilistic response prediction process 300 suitable for implementation in connection with the probabilistic response prediction application in the computing system of FIGS. 1-2. Various tasks performed in connection with the probabilistic response prediction process 300 may be performed by software, hardware, firmware, or any combination thereof. For illustrative purposes, the following description of the probabilistic response prediction process 300 may refer to elements described above in connection with FIGS. 1-2. In an exemplary embodiment, one or more aspects of the probabilistic response prediction process 300 are implemented in or by a probabilistic response prediction software application 102, 202 executing on a computing device 104, 220. It should be understood that the probabilistic response prediction process 300 may include any number of additional or alternative tasks, that the tasks illustrated in FIG. 3 need not be performed in the illustrated order, and that the probabilistic response prediction process 300 may be incorporated into a more comprehensive procedure or process having additional functionality not described in detail herein. Furthermore, one or more of the tasks illustrated in FIG. 3 may be omitted from an embodiment of the probabilistic response prediction process 300 without impairing the intended overall functionality.

[0025] The probabilistic response prediction process 300 initializes or otherwise begins by receiving or otherwise obtaining user input defining characteristics of the reverberation system of interest, characteristics of the excitation energy input to the reverberation system, and additional reverberation system conditions, including the damping coefficient (or Q factor) of the reverberation system and uncertainties associated with the excitation energy (task 302). For example, as described above, the probabilistic response prediction application 102, 202 may provide one or more GUI displays including text boxes, drop-down menus, and other GUI elements for receiving user input indicating the physical dimensions and / or physical layout of the reverberation system to be analyzed along with the mechanical, electrical, chemical, material, and / or other physical properties (e.g., permeability, permittivity, conductivity, etc.) of the structure and / or medium that define the reverberation system and affect the reverberation response to the input excitation energy, in addition to user input indicating the amplitude, frequency, and / or other characteristics of the excitation energy provided by the excitation source used to characterize and predict the reverberation response of the reverberation system and / or medium. For example, to predict the maximum expected electric field response in an aircraft's avionics bay when flying in close proximity to a high-power 5G cell tower, the received user inputs may include the volume and surface area of ​​the avionics bay, the absorbing section of the contents, and the transmitting section of the windows and openings to characterize the reverberant field enclosure, along with the external electric field strength (in V / m) and antenna impedance that characterize the excitation source. Another application is the prediction of the maximum expected acoustic loads on a spacecraft and its sensitive electronics components within a launch vehicle fairing due to liftoff rocket plume acoustic loads and flight transonic aeroacoustic loads. In this example, the received user inputs may include the volume and surface area of ​​the launch vehicle fairing and spacecraft, the absorbing section of the spacecraft and acoustic blanket on the fairing walls, and the transmitting section of the decompression vent and access hatch openings to characterize the reverberant field enclosure, along with the external liftoff acoustic sound pressure level (in dB) and transonic flight fluctuating surface pressure level (in dB) that characterize the excitation source.

[0026] In some implementations, the probabilistic response prediction application 102, 202 may receive user input in the form of a computer-aided design (CAD) file or another computer file in a suitable format that can be analyzed by the probabilistic response prediction application 102, 202 to extract or derive physical properties of the structure of the reverberation system along with mechanical, electrical, chemical, material and / or other physical properties of the medium or materials of the reverberation system defined in the CAD file.

[0027] 4-8 illustrate exemplary GUI displays 400, 500, 600, 700, and 800 that may be presented by the probabilistic response prediction application 102, 202 in connection with the probabilistic response prediction process 300 of FIG. 3. In this regard, FIG. 4 illustrates GUI display 400 including GUI elements for receiving user input defining the physical dimensions of the cavity, FIG. 5 illustrates GUI display 500 including GUI elements for receiving user input defining the permittivity, permeability, and potentially other material properties associated with the cavity, and FIG. 6 illustrates GUI display 600 including GUI elements for receiving user input defining the Q-factor and other losses associated with the cavity. FIGS. 7A and 7B illustrate different GUI displays illustrating Q-factor (or attenuation loss) uncertainty information for a reverberant system that may be received as user input as a probability density distribution, as shown in FIG. 7A, in the form of a table, histogram, or other suitable format (e.g., comma-separated values), or in the form of a mean, standard deviation, and potentially other statistical parameters defining a probability density distribution of the Q-factor. 8 shows a GUI display 800 including GUI elements for receiving user input defining the location or orientation of input excitation energy and the corresponding power or energy level associated with the input excitation energy. In an exemplary implementation, in addition to the GUI displays defining the reverberation system and excitation source, the probabilistic response prediction application 102, 202 also provides a GUI display, such as GUI display 900 of FIG. 9, for receiving user input identifying a desired probability or confidence with which the user wants the probabilistic response prediction application 102, 202 to probabilistically determine the minimum and / or maximum expected field reverberation response of the reverberation system to the input excitation energy.

[0028] 3, after receiving user input defining characteristics or conditions associated with the reverberation system of interest, the probabilistic response prediction process 300 proceeds to probabilistically determine the maximum expected reverberation field response of the reverberation system having the user-input conditions for input excitation energy with a desired probability level. To determine the expected field response, the probabilistic response prediction process 300 first calculates or otherwise determines the statistical mean of the wave field energy and the frequency variance of the wave field energy in the reverberation system resulting from the input excitation energy based on the user-input conditions (task 304). In an exemplary implementation, the probabilistic response prediction application 102, 202 calculates or determines a statistical mean of the wave field energy and a cumulative variance of the wave field energy with respect to frequency based at least in part on a first variance associated with the input excitation energy representing the uncertainty in the amplitude of the excitation energy with respect to frequency, a second variance associated with the uncertainty in the attenuation provided by the reverberation system with respect to frequency, and a third variance associated with the input modal power acceptance of the reverberation system, as described in more detail in U.S. Pat. Nos. 10,565,326 and 10,379,147 (both of which are incorporated by reference herein in their entireties).

[0029] After determining the statistical mean, the probabilistic response prediction process 300 determines a conditional probability density of the field response of the reverberant system based on the statistical mean of the wave field energy using a Rayleigh distribution model (task 306). In this regard, the conditional probability density may be expressed by one or more of the following equations (7)-(9), where E ξ (x,ω) represents the spatially local field response to any frequency (ω) at any point in space (x), and both the Rayleigh distribution for the amplitude of the field response and the exponential distribution for the magnitude-squared field response are subject to the statistical mean-square field

[0030]

number

[0031] After determining the conditional probability density of the field response, the probabilistic response prediction process 300 determines the unconditional probability density of the field response of the reverberant system based on the conditional probability density and the marginal distribution of the frequency variance in the mean field. Because the spatial mean-squared field response is proportional to the total wave field energy, a log-normal distribution of the frequency uncertainty of the energy can be used, and the relative variance can be calculated directly from the known reverberant field parameters using equations (16)-(18). It will be explained later that this energy-frequency variance model is for the specific case where the Q factor is known (e.g., measured) and the excitation strength is known (e.g., measured). Under these bounded uncertainty conditions, the aforementioned marginal distribution of the spatial mean-squared field response is determined by the point input conductance f{G ii (ω)} (the real part of the point input impedance) (task 308). The unconditional probability density may be determined by numerical integration of a user-input probability density function for the input conductance frequency uncertainty, such as a log-normal distribution represented by the following equations (16)-(18). In one example, the unconditional probability density is determined as a convenient closed-form analytical solution to the unconditional distribution integral equation using an inverse gamma distribution having a form represented by one of equations (21), (22), and (24), as described in more detail below. In this regard, in some implementations, the probability density function of the input conductance frequency uncertainty is calculated using the user-input physical dimensions of the enclosure structure for the reverberation system, the user-input loss coefficient of the reverberation system, and a statistical average of the excitation energy, and the unconditional probability density is calculated using the calculated probability density function of the input conductance frequency uncertainty and the inverse gamma distribution.

[0032] After determining the unconditional probability density due to uncertainty in the input conductance, probabilistic response prediction process 300 determines a probability function for the field response based on the unconditional probability density by incorporating additional user-input values ​​for the damping coefficient (or Q-factor) uncertainty and the excitation uncertainty (task 310). In this regard, as described in more detail below, the unconditional probability density function of the electric field (or field response) at a point in space is determined by the excitation energy

[0033]

number

[0034] Once the probability function of the field response has been determined based on the unconditional probability density, the stochastic response prediction process 300 calculates or otherwise determines an expected field response for any frequency at any point in space based on the input probability level (P) using an inverse function derived from the unconditional probability density function (task 312). In this regard, for corresponding user-input values ​​of minimum and maximum probability values ​​(or confidence values), the stochastic response prediction application 102, 202 calculates a corresponding stochastic maximum field response for the frequency having the input confidence level or probability maximum value, and a corresponding stochastic minimum field response for the frequency having the input confidence level or probability minimum value, for example, by evaluating the probability function at individual frequencies within the input frequency range of interest. Furthermore, the stochastic response prediction application 102, 202 calculates a corresponding stochastic mean field response with respect to frequency, for example, by using a 99% confidence level or probability percentile.

[0035] After calculating or determining the expected field response for the user-input probability level, the probabilistic response prediction process 300 outputs or provides an indication of the expected field response to the user (task 314). For example, in one or more exemplary implementations, the probabilistic response prediction application 102, 202 generates a graph or another graphical representation in a GUI display on the display device 208 showing the maximum, average, and minimum expected field response levels with respect to frequency across the frequency range of interest. In this manner, the probabilistic response prediction application 102, 202 allows the user to visually perceive and confirm what the expected maximum and minimum responses to an input excitation energy are likely to be for the reverberant system defined by the user within a desired confidence level or probability that accounts for uncertainties associated with the excitation energy, damping coefficient (or Q factor), modal conductance, etc. Furthermore, by analytically providing a probabilistic solution for the expected field response, the delay between the probabilistic response prediction application 102, 202 receiving user input defining the reverberant system of interest (e.g., in task 302) and providing an indication of the resulting field response (e.g., in task 314) is reduced relative to more computational approaches such as Monte Carlo simulation, while still accurately predicting the minimum and maximum expected field responses at any point in space within the reverberant system over the frequency range of interest.

[0036] 10 illustrates an exemplary probabilistic response GUI display 1000 including a graph 1002 illustrating the relationship between a maximum expected field response 1004, a minimum expected field response 1006, and an average expected field response 1008 versus frequency. In this regard, the graphical representation of the maximum expected field response 1004 represents the maximum expected field response at any location within the reverberation system at each frequency having a higher probability or confidence level input by the user (e.g., 99.5%), the graphical representation of the minimum expected field response 1006 represents the minimum expected field response at any location within the reverberation system at each frequency having a lower probability or confidence level input by the user (e.g., 0.5%), and the graphical representation of the average expected field response 1008 represents the average expected field response at any location within the reverberation system at each frequency having a probability or confidence level of 50%. The illustrated probabilistic response GUI display 1000 also includes a graphical representation of a target limit or constraint 1010 for the field response versus frequency, which may be input by the user or otherwise provided. In this regard, the target field response limit 1010 may correspond to a design specification or other regulatory requirement to which a user designing a reverberation system seeks to adhere. Thus, when the maximum expected field response 1004 exceeds the target field response limit 1010, the user may attempt to modify one or more characteristics of the reverberation system (e.g., the physical dimensions of the system, the material or medium through which the excitation energy reverberates, etc.) or otherwise modify or limit the input excitation energy until the maximum expected field response 1004 meets or otherwise remains below the target field response limit 1010 over the frequency range of interest.

[0037] Figure 11 shows an example relationship between the excitation field strength, related to input excitation, input conductance, and damping loss coefficient (or Q-factor), and the volume of the reverberation system for the expected field response versus frequency at any location within the volume of the reverberation system. Figure 12 depicts an example relationship between a probability density function representation of the uncertainty in the excitation field strength versus frequency, a probability density function representation of the input conductance, and a probability density function representation of the damping loss coefficient (or Q-factor), and the expected field response versus frequency at any location within the volume of the reverberation system. Figures 13-14 show example relationships between multiple connected or coupled reverberation systems suitable for analysis or use with an actual implementation of the probabilistic response prediction process 300 and corresponding uncertainties suitable for probabilistic modeling associated with an actual implementation of the probabilistic response prediction process 300. In this regard, it will be appreciated that while the present subject matter may be described herein in the context of a single reverberant system for simplicity and illustrative purposes, in practice the stochastic response prediction process 300 is extensible to accommodate any number or combination of systems, including multiple connected or coupled reverberant systems, or hybrid configurations including a reverberant system connected or coupled to a deterministic system, and any type of reverberant field, including, but not limited to, electric fields, magnetic fields, electromagnetic fields, acoustic fields, vibrational wave fields, etc.

[0038] 11-14, with continuing reference to FIGS. 1-10, the mathematical derivation of the unconditional probability density function and corresponding probability function for determining the expected field response for any particular frequency for any point within a volume associated with a reverberant system will now be described in more detail. Without loss of generality, a physical and mathematical explanation of the subject is provided for the exemplary case of a high frequency reverberant electromagnetic field within an open enclosure such as an avionics box. The total energy U(ω) of the reverberant field at frequency ω (rad / s) can be calculated by the magnitude squared total electric field U(ω), expressed by the following equation (1):

[0039]

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[0041] The volume integral of equation (1) is proportional to the spatial average of the square of the electric field amplitude and is expressed by the following equation (2):

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[0047] As described in U.S. Pat. No. 10,379,147, which is incorporated herein by reference, the standing wave modal density (modes / rad / s) in Equation (3) increases very rapidly (e.g., with the square of the frequency).

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[0055] Therefore, the statistical mean square electric field level can be obtained from equations (2) and (4), which can be expressed by equation (5) below:

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[0057] Compared with equation (3), this means that the power input P IN and requires only statistical mean estimates for the enclosure Q-factor uncertainties. Mean brackets without subscripts

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[0061] For electrically large enclosures (such as reverberation test chambers) that are overmoded,

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[0064] This is a single parameter distribution that scales to the mean field, expressed by equation (8) below.

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[0066] The corresponding mean square electric field has a chi-squared 2 degree of freedom (exponential) distribution, which can be expressed by equation (9) below:

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[0068] This single parameter distribution also scales to the mean electric field in equation (8) and by definition has a relative variance of 1, given by equation (10) below:

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[0070] On the other hand, electrically small enclosures are known to have probability density functions (PDFs) that diverge from Rayleigh because they typically do not satisfy the overmoded condition. Here, we introduce the parameter modal overlap to quantify the overmoded condition. In any given frequency range, the Δω modal overlap m(ω) is the modal attenuation bandwidth for the average frequency interval Δω = 1 / n(ω), as expressed by the following equation (11):

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[0075] The main effect of low modal overlap in electrically small enclosures is to make the reverberation field energy highly frequency dependent. U(ω) The corresponding mean square electric field has the same degree of frequency dispersion.

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[0078] Therefore, in the exemplary embodiment described herein, the unconditional PDF can thus be defined for an electrically small enclosure with low modal overlap and large frequency dispersion, as expressed by the following equation (13):

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[0085] In various implementations, the log-normal distribution

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[0090] These PDF parameters can be calculated directly from the known user-input physical parameters of the reverberant enclosure as follows: Frequency Variability Average

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[0096] Lognormal marginal distribution in the unconditional PDF integral of Eq. (14)

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[0099] It does not have a closed-form analytical solution and must therefore be solved using numerical integration. In applications involving large networks of interconnected reverberant field enclosures, such as compact packaging of electronics in consumer electronics or under deck environments on ships, limiting distributions that allow closed-form analytical solutions to the unconditional field integral equations are needed.

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[0101] In various implementations, the inverse gamma probability density function (PDF)

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[0105] The two parameters of the inverse gamma distribution with respect to the lognormal distribution are

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[0108] Inverse Gamma PDF

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[0113] Advantageously, the Lomax PDF of equation (22) also has an inverse solution that allows direct calculation of the quantile or percentile P bounds of the unconditional electric field PDF model, expressed by equation (24) below:

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[0115] The probability density function of the (non-squared) amplitude of the reverberation field is expressed by the following equation (25):

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[0117] mean field

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[0121] Using the modal expansion Green's function of equation (3), the deterministic input power is expressed as follows:

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[0123] The statistical average power input from equation (27)

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[0131] This statistical average of the frequency-dependent input conductance implied by equation (27) is given by the uncertainty modal parameter r and the frequency band Δω for any particular center frequency ω c It is evaluated as follows.

[0132] Considering the above, the mean field

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[0136] Therefore, the marginal PDF for the product of the three random variables in Eq. (14)

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[0140] Substituting equation (31) into equation (14) gives three different (statically independent) uncertainties, expressed by equation (32) below:

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[0145] This model has been found to have useful solutions for many particular cases of interest in practical applications of the subject stochastic model for reverberant wave fields.

[0146] No uncertainty in source current and Q factor

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[0150] This is the same PDF due to the frequency uncertainty of the mean field of the low modal overlap reverberation field in equations (16) and (20), and all of the frequency uncertainty of the mean field M(ω) is reflected in the reverberation field

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[0152] In one or more exemplary implementations, the input conductance

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[0160] The statistical distribution of the reverberant field Q-factor is the reflected scattering parameter of a simple monopole probe antenna in the field [Bremner, IEEE EMC Symp., 2018].

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[0162] All three mean field variables are PDF

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[0169] The limiting PDF for which all three mean field variables are at least approximately log-normally distributed

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[0174] Substituting into equation (33), this gives a more convenient closed-form analytical solution, which is the same as equation (22), but defined above in equation (39).

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[0177] The corresponding probability density function for the (non-squared) amplitude of the reverberation field is given by equation (41) below:

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[0179] 13-14, in one or more exemplary implementations, an unconditional PDF model is developed for the field response of a reverberant system that has an effective excitation source of additional uncertainty when it is part of a network of multiple connected reverberant systems. As described in more detail in U.S. Pat. Nos. 10,379,147 and 10,565,326, both of which are incorporated herein by reference, the higher energy

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[0185] Net Power Flow

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[0192] In a simple first-order perturbation analysis, previous researchers [Hoijer (Barton) and Kroon (Kroon), IEEE Trans, EMC 2013] found that the variance (uncertainty) of the coupling loss coefficient is zero and that it is proportional to the square of the amplitude of the electric field in the drive system.

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[0196] The corresponding probability density function for the (non-squared) amplitude of the reverberation field is given by equation (46) below:

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[0210] In such a network of connected reverberant wavefield systems, there may be multiple statistically uncorrelated net power inputs, typically

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[0214] In the most general case, the marginal probability density function (PDF) of the uncertainty for each excitation power input is different.

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[0219] 13-14, in some implementations, an unconditional PDF model is developed for the field response of a reverberation system that has additional uncertainty in the Q-factor loss when it is part of a network of multiple connected reverberation systems.

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[0231] Without loss of generality, the coupling loss factor

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[0239] In the most general case, the marginal probability density function (PDF) of the uncertainty in each of the power losses will be different.

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[0243] In various implementations, the amplitude PDF of the electric field, as developed in any one of the aforementioned models governed by a respective one of equations (7), (25), or (41), is

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[0250] It has also been found that when the subject's reverberant fields have low modal overlap, the PDF of the combined deterministic and statistical fields has a Pareto distribution (non-central Lomax distribution) as given by Equation (58) below.

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[0252] In various implementations, an unconditional PDF model is developed for the response of a low-dimensional deterministic wave-field system coupled to a higher-dimensional stochastic wave-field system, such as the aforementioned reverberant field in a three-dimensional bounding volume as shown graphically in Figure 16. A typical application is the transverse electromagnetic (TEM) mode propagation of currents on a multi-conductor cable (or transmission line) within an enclosure supporting a reverberant three-dimensional wave field, as more fully described in U.S. Patent Nos. 10,338,117 and 10,156,599. The coupled squared-amplitude magnetic field on the cable

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[0256] When the subject's reverberation fields have high modal overlap, the PDF of the combined deterministic and statistical fields has a Rician distribution (non-central Rayleigh distribution) expressed by Equation (60) below.

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[0258] It has also been found that when the subject's reverberation fields have low modal overlap, the PDF of the combined deterministic and statistical fields has a Pareto distribution (non-central Lomax distribution) as given by Equation (61) below.

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[0260] 17 illustrates a GUI display 1700 including GUI elements for receiving user input defining coupling loss coefficients between different systems, such as a combination of mixed or coupled reverberant and deterministic systems or other interconnected systems, in connection with an implementation of the probabilistic response prediction process 300 that is scalable to cover two or more structures or systems. In this regard, in a scalable implementation of the probabilistic response prediction process 300, the user input received at 302 may further include coupling loss coefficients between the systems, along with additional physical dimensions or other information characterizing openings or joints between different cavities, enclosures, or structures corresponding to each interconnected system. In a typical multiple connected system implementation, the probabilistic response prediction process 300 determines the statistical mean net power input and net power loss between the connected systems (e.g., as described at 304 in U.S. Pat. Nos. 10,338,117 and 10,156,599), in addition to the statistical mean and frequency dispersion of the connected systems. The probabilistic response prediction process 300 then considers the coupling energy levels between the systems, along with the uncertainty contributions of the net power losses to and / or net power inputs from the connected systems, to determine the expected reverberation field response of the multiple connected wave fields with input probability levels, as described above in the context of Equations (41)-(45). Meanwhile, for mixed deterministic and statistical wave field systems, the probabilistic response prediction process 300 can utilize the subject matter described in U.S. Pat. Nos. 10,338,117 and 10,156,599 to arrive at the field response levels of the deterministic wave fields based on the excitation strength and radiation loss coefficients entered by the user. Then, at 306, the conditional probability density of the total mixed field response level is determined, as described above in the context of Equations (46) and (49), before determining the unconditional probability density function of the total mixed field response level based on the user input for the Q-factor uncertainty and excitation uncertainty, as described above in the context of Equations (47) and (49).In this regard, it should be understood that the probabilistic response prediction process 300 is not limited to individual reverberation systems or any particular type, number, or configuration of systems having coupled or connected wave fields.

[0261] As used herein, the word "exemplary" means "serving as an example, instance, or illustration." Thus, any implementation described herein as exemplary is not necessarily to be construed as preferred or advantageous over other embodiments. All embodiments described herein are exemplary embodiments provided to enable any person skilled in the art to make or use the invention, and do not limit the scope of the invention, which is defined by the claims.

[0262] Those skilled in the art will understand that the various illustrative logic blocks, modules, circuits, and algorithm steps described in connection with the embodiments disclosed herein can be implemented as electronic hardware, computer software, or a combination of both. Some of the embodiments and implementations are described above in terms of functional and / or logic block components (or modules) and various processing steps. However, it should be understood that such block components (or modules) can be realized by any number of hardware, software, and / or firmware components configured to perform the specified functions. To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, modules, circuits, and steps have been described above generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends on the particular application and design constraints imposed on the overall system. Those skilled in the art may implement the described functionality in various ways for each particular application, and such implementation decisions should not be interpreted as causing a departure from the scope of the invention. For example, one embodiment of a system or component may use various integrated circuit components, such as memory elements, digital signal processing elements, logic elements, look-up tables, etc., that can perform various functions under the control of one or more microprocessors or other control devices. Furthermore, those skilled in the art will appreciate that the embodiments described herein are merely example implementations.

[0263] The various illustrative logic blocks, modules, and circuits described in connection with the embodiments disclosed herein may be implemented or performed using a general-purpose processor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A general-purpose processor may be a microprocessor, but alternatively, the processor may be any conventional processor, controller, microcontroller, or state machine. A processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, multiple microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration.

[0264] The steps of a method or algorithm described in connection with the embodiments disclosed herein may be embodied directly in hardware, in a software module executed by a processor, or in a combination of the two. The software module may reside in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, a hard disk, a removable disk, a CD-ROM, or any other form of non-transitory storage medium known in the art. An exemplary storage medium is coupled to the processor such that the processor can read information from, and write information to, the storage medium. Alternatively, the storage medium may be integral to the processor. The processor and the storage medium may reside in an ASIC.

[0265] The subject matter may be described herein in terms of functional and / or logical block components and with reference to symbolic representations of operations, processing tasks, and functions that can be performed by various computing components or devices. Such operations, tasks, and functions are sometimes referred to as being computer-executed, computerized, software-implemented, or computer-implemented. In practice, one or more processor devices may perform the described operations, tasks, and functions by manipulating electrical signals representing data bits in memory locations within a system memory, as well as performing other processing of the signals. The memory locations in which data bits are maintained are physical locations that have particular electrical, magnetic, optical, or organic properties corresponding to the data bits. It should be understood that the various block components illustrated in the figures may be realized by any number of hardware, software, and / or firmware components configured to perform the specified functions. For example, one embodiment of a system or component may employ various integrated circuit components, e.g., memory elements, digital signal processing elements, logic elements, look-up tables, etc., that can perform various functions under the control of one or more microprocessors or other control devices.

[0266] When implemented in software or firmware, the various elements of the systems described herein are essentially code segments or instructions that perform various tasks. Programs or code segments can be stored on a processor-readable medium or transmitted by a computer data signal embodied in a carrier wave over a transmission medium or communication path. A "computer-readable medium," "processor-readable medium," or "machine-readable medium" may include any medium capable of storing or transferring information. Examples of processor-readable media include electronic circuits, semiconductor memory devices, ROM, flash memory, erasable ROM (EROM), floppy diskettes, CD-ROMs, optical disks, hard disks, fiber optic media, radio frequency (RF) links, and the like. A computer data signal may include any signal capable of propagating over a transmission medium, such as an electronic network channel, optical fiber, air, an electromagnetic path, or an RF link. Code segments may also be downloaded over a computer network, such as the Internet, an intranet, or a LAN.

[0267] Some of the functional units described herein are referred to as “modules” to more specifically emphasize their implementation independence. For example, functionality referred to herein as a module may be implemented, in whole or in part, as hardware circuitry, including custom VLSI circuits or gate arrays, off-the-shelf semiconductors such as logic chips, transistors, or other discrete components. A module may also be implemented in programmable hardware devices, such as field programmable gate arrays, programmable array logic, programmable logic devices, etc. A module may also be implemented in software for execution by various types of processors. An identified module of executable code may comprise one or more physical or logical modules of computer instructions, which may be organized, for example, as an object, a procedure, or a function. Nevertheless, the executable files of an identified module need not be physically located together but may comprise heterogeneous instructions stored in different locations that, when logically joined together, constitute a module and achieve the stated purpose for the module. In fact, a module of executable code may be a single instruction or many instructions, distributed across several different code segments, among different programs, and across several memory devices. Similarly, operational data may be embodied in any suitable form and organized within any suitable type of data structure. The operational data may be collected as a single data set, or may be distributed across different locations, including different storage devices, and may exist, at least in part, solely as electronic signals on a system or network.

[0268] As used herein, relative terms such as "first" and "second" are used solely to distinguish one entity or action from another and do not necessarily require or imply any actual relationship or order between such entities or actions. Ordinal numbers such as "first," "second," and "third" simply indicate different ones of a plurality and do not imply any order or sequence unless specifically defined by the claim language. The order of text in any of the claims does not imply that process steps must be performed in chronological or logical order according to such order unless specifically defined by the claim language. Process steps may be interchanged in any order without departing from the scope of the invention, so long as such interchange is not inconsistent with the claim language and is logically consistent.

[0269] Moreover, the foregoing description may refer to elements or nodes or features that are "coupled" together. As used herein, unless otherwise specified, "coupled" means that one element / node / feature is directly or indirectly joined to (or in direct or indirect communication with) another element / node / feature, and need not necessarily be mechanically joined. For example, two elements may be physically, electronically, logically, or in any other manner coupled to each other via one or more additional elements. Thus, while a drawing may show one example arrangement of elements directly connected to each other, additional intervening elements, devices, features, or components may be present in one embodiment of the illustrated subject matter. Moreover, certain terminology may be used herein for reference purposes only and, therefore, is not intended to be limiting.

[0270] While at least one exemplary embodiment has been presented in the foregoing detailed description of the invention, it should be understood that a vast number of variations exist. It should also be understood that the exemplary embodiment(s) are merely examples and are in no way intended to limit the scope, applicability, or configuration of the invention. Rather, the foregoing detailed description will provide those skilled in the art with a convenient road map for implementing exemplary embodiments of the invention. It will be understood that various changes can be made in the function and arrangement of elements described in the exemplary embodiment without departing from the scope of the invention as set forth in the appended claims.

Claims

1. receiving user input defining characteristics of the reverberation system, said characteristics including physical dimensions, wave propagation velocity, and loss coefficient of the enclosure structure; identifying an excitation characteristic of excitation energy input to the reverberation system; determining a conditional probability density function of a field response of the reverberant system to the excitation energy based at least in part on a statistical average of magnitude-squared field responses; determining a marginal probability density function of an input conductance frequency uncertainty of the reverberation system based at least in part on the physical dimensions, the wave propagation velocity, and the loss factor; determining an unconditional probability density function of the field response of the reverberation system to the excitation energy based at least in part on the marginal probability density of input conductance frequency uncertainty; determining an expected field response by the reverberation system with respect to frequency in response to the excitation energy for input probability values ​​using the unconditional probability density function; providing a graphical user interface (GUI) display including a graphical representation of the expected field response versus frequency; A method comprising:

2. 2. The method of claim 1, further comprising determining the unconditional probability density function of the expected field response based on the conditional probability density function having a marginal probability distribution of an uncertain input conductance and a marginal probability density function of at least one of an effective input current uncertainty associated with the excitation energy and a Q-factor uncertainty associated with the loss factor of the reverberation system.

3. The unconditional probability density function is governed by the following equation: [Equation 1] During the ceremony, [Equation 2] is the unconditional probability density function of the field response at any position x and any frequency ω, [Equation 3] is the statistical average of the excitation energies, [Equation 4] is the square of the amplitude of the effective current of the excitation energy, [Equation 5] is the Q factor, [Equation 6] is the input conductance scaled with frequency ω, permittivity ε, and volume V, [Equation 7] is a first user-input probability density function of the effective current uncertainty associated with the excitation energy; [Equation 8] is a second user-input probability density function of Q-factor uncertainty associated with the Q-factor of the reverberant system; [Equation 9] The method of claim 2 , wherein: is the scaled marginal probability density function of the input conductance frequency uncertainty.

4. 2. The method of claim 1, wherein determining the marginal probability density function of the input conductance uncertainty of the reverberant system comprises calculating a marginal probability density function of the input conductance frequency uncertainty based at least in part on the physical dimensions, the wave propagation speed, the loss coefficient, and the statistical mean of the field response.

5. The unconditional probability density function is governed by the following equation: [Equation 10] During the ceremony, [0011] is the unconditional probability density function of the electric field response at any position x and any frequency ω, [0012] is the spatial average of the amplitude squared field response, [0013] 10. The method of claim 1, wherein σ is a user-input marginal probability density function of all uncertainties in the spatial mean square field.

6. The method of claim 5 , wherein determining the unconditional probability density function comprises calculating the unconditional probability density function by numerical integration of a user-input marginal probability density function for the input conductance frequency uncertainty.

7. An inverse gamma probability density function is used for the input conductance frequency uncertainty to obtain a closed-form analytical solution for the unconditional field response distribution governed by the following equation: [0014] During the ceremony, [Equation 15] is the unconditional probability density function of the field response, [0016] is a first parameter of the distribution, [Equation 17] is the mean square electric field, [Equation 18] is a first parameter of the distribution, [Equation 19] is the relative frequency dispersion of the mean square electric field, [Equation 20] is the modal overlap, [Equation 21] is the modal density, [Equation 22] is the spatial mode shape dispersion, N and L are estimated as statistics. [Equation 23] 6. The method of claim 5, wherein the number of receivers and source locations used to

8. The input probability value is a user input percentile probability P ｍａｘ , P ｍｉｎ and the expected field response is calculated from the inverse of the unconditional probability density function as percentile maximum and minimum values ​​of the user input percentile probabilities P ｍａｘ , P ｍｉｎ 6. The method of claim 5, comprising: a maximum and minimum expected field response to the excitation energy independent of position and frequency within the reverberant field at

9. The input probability value is a user input percentile probability P ｍａｘ , P ｍｉｎ and the expected field response is calculated as the respective percentile maximum and minimum values ​​from the inverse function governed by the following equation: ｍａｘ , P ｍｉｎ maximum and minimum expected field responses to the excitation energy independent of position and frequency within the reverberation field at [0000] During the ceremony, [Equation 25] is the squared rectangular component of the electric field amplitude at probability percentile P, α is the distribution [Equation 26] is the first parameter of β is the distribution [0000] The method of claim 7, wherein the second parameter is a second parameter of

10. 3. The method of claim 2, wherein determining the unconditional probability density function comprises calculating the unconditional probability density function by numerical integration of an exponential distribution of the conditional electric field, a user-input marginal probability density function of the input conductance frequency uncertainty, and at least one of a second user-input marginal probability density function of source effective current uncertainty and a probability density function of source effective current uncertainty.

11. User-input percentile probability P ｍａｘ , P ｍｉｎ 11. The method of claim 10, wherein the maximum expected field response and the minimum expected field response to the excitation energy independent of position and frequency within the reverberant field at comprises calculating the percentile maxima and minima from the inverse of the unconditional probability density function.

12. Determining the unconditional probability density function includes using a single lognormal distribution for a product of two or more component marginal uncertainties in the mean field random variables, and calculating the unconditional field response distribution governed by the following equation: [0000] During the ceremony, [0000] is the unconditional probability density function (PDF) of the squared rectangular component of the electric field amplitude at any position x and any frequency ω, [Equation 30] is the mean square electric field, [Equation 31] is the sum of the means of the uncertainties in the natural logarithms of each of the three random variables in M, [Equation 32] 3. The method of claim 2, wherein M is the sum of the variances of the uncertainties in the natural logarithms of each of the three random variables in M.

13. Determining the unconditional probability density function includes using a single inverse gamma distribution for the product of two or more component uncertainties in the mean field random variables, and calculating the closed-form analytical PDF formula [Equation 33] and calculating by [Equation 34] is the unconditional probability density function (PDF) of the squared rectangular component of the electric field amplitude at any position x and any frequency ω, [Equation 35] is the first PDF parameter that is a function of the logarithmic random variable uncertainty, [Equation 36] is a second parameter that is a function of the logarithmic random variable uncertainty, [Equation 37] is the sum of the means of the uncertainties in the natural logarithms of each of the three random variables in M, [Equation 38] 3. The method of claim 2, wherein M is the sum of the variances of the uncertainties in the natural logarithms of each of the three random variables in M.

14. User-input percentile probability P ｍａｘ , P ｍｉｎ wherein the maximum and minimum expected field responses to the excitation energy independent of position and frequency in the reverberation field at [Number 39] During the ceremony, [Equation 40] is the rectangular component of the squared amplitude of the electric field at probability percentile P, α is the distribution [Equation 41] is the first parameter of β is the distribution [Equation 42] The method of claim 13, wherein the second parameter is a second parameter of

15. For cases where uncertainty excitation arises from net mean power input from coupled reverberation wave fields, determining the unconditional probability density function includes numerical integration of an exponential distribution of the conditional electric field, a user-input marginal probability density function for frequency uncertainty in input conductance, and at least one user-input marginal probability density function for effective source current uncertainty associated with net mean power input from coupled reverberation wave fields, governed by the following equation: [Equation 43] During the ceremony, [0.0000] is the unconditional probability density function of the field response of the reverberation system 1 at any position x and any frequency ω, [Equation 45] is the statistical mean of the excitation energies, [Equation 46] is the square of the amplitude of the effective current associated with the excitation from at least one connected reverberation system 2, [Equation 47] is a first user input marginal probability density function for the uncertainty of the effective excitation current from at least one connected reverberation system 2; [Number 48] is a second user-input marginal probability density function for the uncertainty in the Q-factor of the reverberation system 1; [Number 49] is the input conductance G of the reverberation system 1 １１ 3. The method of claim 2, wherein the frequency distribution of (ω) is a scaled marginal probability density function.

16. For the case where the uncertainty excitation arises from the net mean power input from the connected reverberant wave fields, the limiting probability density function of the effective source current from at least one connected wave field is governed by the following equation: [Number 50] During the ceremony, [Equation 51] is the marginal probability density function (PDF) of the effective excitation current from at least one connected reverberation system 2; [Number 52] is a variable proportional to the average total energy in the subject reverberant wavefield system 1, [Number 53] is a variable proportional to the average total energy of the connected reverberation wave field system 2, [Number 54] is the uncertainty of the coupling loss factor [Number 55] is the first parameter of the inverse gamma marginal distribution of [Number 56] is the average of the coupling loss coefficients between reverberation system 1 and reverberation system 2, [Number 57] is the second parameter of the inverse gamma bound distribution of the coupling loss coefficient uncertainty [Number 58] [Number 59] is the statistical variance of the uncertainty coupling loss coefficient between reverberation system 1 and reverberation system 2, 16. The method of claim 15, wherein u(X) is a Heaviside unit step function.

17. For the case of multiple uncertain excitations arising from both direct and net mean power inputs from multiple connected reverberation wave fields, the limiting probability density function of the effective excitation current is governed by the following equation: [Number 60] During the ceremony, [Number 61] is the marginal probability density function (PDF) of the current squared of the total effective amplitude over multiple excitations; [Number 62] is the statistical mean input conductance of the subject's reverberant wave field, [Number 63] is the total net power input into the subject's reverberant wave field, [Number 64] is the limiting PDF of the total net power input into the subject's reverberant wave field, [Number 65] 16. The method of claim 15, wherein {overscore (R)} is the limiting PDF for each of the statistically uncorrelated net power input components.

18. If the Q-factor uncertainty arises from net mean power loss into the coupled reverberation wave field, determining the unconditional probability density function comprises numerical integration of an exponential distribution of the conditional electric field, a user-input marginal probability density function of frequency uncertainty in input conductance, and at least one user-input probability density function of effective Q-factor uncertainty associated with net power loss into the coupled reverberation wave field, and is governed by the following equation: [Number 66] During the ceremony, [Number 67] is the unconditional probability density function of the field response of the reverberation system 1 at any position x and any frequency ω, [Number 68] is the squared field of the amplitude of the statistical mean of the excitation energy, [Number 69] is the first user bound probability density function for the uncertainty in the effective source current from the external excitation of the reverberation system 1; [Number 70] is a second user-input marginal probability density function for the Q-factor uncertainty associated with losses to at least one connected reverberation system 2; [Number 71] is the input conductance G of the reverberation system 1 １１ 3. The method of claim 2, wherein the frequency distribution is a scaled marginal probability density function for the frequency dispersion in (ω).

19. If the Q-factor uncertainty arises from the net mean power loss from the connected reverberant wavefields, then the marginal probability density function of the Q-factor due to the loss to at least one connected reverberant wavefield system is: is governed by the following equation: [Number 72] During the ceremony, [Number 73] is the marginal probability density function of the Q-factor uncertainty associated with losses to at least one connected reverberation system 2, [Number 74] is the uncertainty of the coupling loss factor [Number 75] is the first parameter of the marginal distribution of [Number 76] is the average of the coupling loss coefficients between reverberation system 1 and reverberation system 2, [Number 77] is the uncertainty of the coupling loss factor [Number 78] is the first parameter of the marginal distribution of [Number 79] 19. The method of claim 18, wherein ∑ m is the variance of the coupling loss coefficient between reverberation system 1 and reverberation system 2.

20. For the case where the power losses of the uncertainties arise from both the power losses due to the internal dissipative Q-factor and the net power losses due to the connected reverberant wavefields, the limiting probability density function for the effective Q-factor is governed by the following equation: [Number 80] During the ceremony, [Number 81] is the marginal probability density function (PDF) of the total effective Q factor over multiple losses, [Number 82] is the total net power loss from the subject's reverberant wavefield, [Number 83] is the limiting PDF of the total net power loss from the subject reverberant wavefield, [Number 84] 20. The method of claim 18, wherein {overscore (R)} is the limiting PDF for each of the statistically uncorrelated net power loss components.

21. the user input includes defining a direct field characteristic; the user input characteristics include a physical location and radiation aperture size of a direct field source and a physical location of a sensor within the enclosure structure; The method comprises: Predicting the direct field response at the sensor location; formula [Number 85] and an estimate of the total mixed field at the sensor location, governed by [Number 86] is the probability density function of the total direct plus reverberant amplitude-squared electric field at response location x and frequency ω, [Number 87] is the DC electric field at response position x and frequency ω, [Number 88] The method of claim 1 , wherein ω is the reverberant electric field at all positions and frequencies ω.

22. For the case where the total electric field within an enclosure comprises the sum of a deterministic direct field and an uncertain reverberant field with high modal overlap, the unconditional probability density function is governed by the following equation: [Number 89] During the ceremony, [Number 90] is the response position x Ｏ and the unconditional probability density function (PDF) of the total magnitude squared of the rectangular component of the electric field at any frequency ω, [Number 91] is the mean amplitude-squared level of the reverberant field, [Number 92] is the response position x Ｏ is the amplitude of the deterministic electric field at [Number 93] 22. The method of claim 21 , wherein: is a modified Bessel function of the first kind of zeroth order.

23. For the case where the total electric field within an enclosure comprises the sum of a deterministic direct field and an uncertain reverberant field with low modal overlap, the unconditional probability density function is governed by the following equation: [Number 94] During the ceremony, [Number 95] is the response position x Ｏ and the unconditional probability density function of the total amplitude squared rectangular component of the electric field at any frequency ω, [Number 96] is the amplitude of the deterministic electric field at the response location x, [Number 97] is the first PDF parameter, a function of the logarithmic random variable uncertainty, [Number 98] is the second parameter, a function of the logarithmic random variable uncertainty, [Number 99] is the sum of the means of the uncertainties in the natural logarithms of each of the three random variables in M, [Number 100] 22. The method of claim 21 , wherein M is the sum of the variances of the uncertainties in the natural logarithms of each of the three random variables in M.

24. a user defining an additional low-dimensional wave field system, such as a multi-conductor transmission line, disposed within the reverberant wave field system, the method comprising: User input characteristics including conductor material and dimensions, cross-sectional material and dimensions, and terminal impedance load; user input of deterministic applied voltage or electric field excitation at the transmission line terminals; Predicting deterministic terminal magnetic field and current responses to applied terminal excitation; Predicting the statistical distribution of terminal magnetic field and current responses to applied reverberation field excitations; and a prediction of the total mixed deterministic-statistical response magnetic field and current in the terminal governed by the following equations: [Number 101] During the ceremony, [Number 102] is the amplitude-squared probability density function of the total mixed deterministic-statistical terminal magnetic field response, [Number 103] is the squared amplitude deterministic terminal magnetic field response to the applied terminal excitation, [Number 104] The method of claim 1 , wherein ρ is the probability density function of the magnitude-squared terminal magnetic field response to a reverberation field excitation along the transmission line.

25. If the total terminal magnetic field comprises the sum of a deterministic field from an applied terminal electric field and an uncertain terminal field excited by a reverberant field with high modal overlap, the unconditional probability density function is governed by the following equation: [Number 105] During the ceremony, [Number 106] is the unconditional probability density function (PDF) of the amplitude of the total magnetic field at the cable termination, [Number 107] is the component of the amplitude of the terminal magnetic field excited by the reverberation field, [Number 108] is the component of the amplitude of the terminal magnetic field excited by a deterministic electric field applied to terminal x, [Number 109] 25. The method of claim 24, wherein {right arrow over (x)} is a modified Bessel function of the first kind of zeroth order.

26. For the case where the total electric field within an enclosure comprises the sum of a deterministic direct field and an uncertain reverberant field with low modal overlap, the unconditional probability density function is governed by the following equation: [Number 110] During the ceremony, [Number 111] is the unconditional probability density function (PDF) of the amplitude of the total magnetic field, [Number 112] is the component of the amplitude of the terminal magnetic field excited by the deterministic electric field applied to the terminal, [Number 113] is the first PDF parameter that is a function of the log probability variance uncertainty, [Number 114] is the second parameter, which is a function of the logarithmic random variable uncertainty, [Number 115] is the sum of the means of the uncertainties in the natural logarithms of each of the three random variables in M, [Number 116] 25. The method of claim 24, wherein M is the sum of the variances of the uncertainties in the natural logarithms of each of the three random variables in M.

27. 1. A computer-readable medium having stored thereon computer-executable instructions that, when executed by a processing system, cause the processing system to: receiving user input defining characteristics of a reverberation system, said characteristics including physical dimensions and loss coefficients of an enclosure structure; identifying an excitation characteristic of excitation energy input to the reverberation system; determining a marginal probability density function of input conductance frequency uncertainty of the reverberation system based at least in part on the physical dimensions, the loss coefficient, and the statistical average of the excitation energy; determining an unconditional probability density function of the field response of the reverberation system to the excitation energy based at least in part on the input conductance frequency uncertainty; determining an expected field response by the reverberation system with respect to frequency in response to the excitation energy for input probability values ​​using the unconditional probability density function; and providing a graphical user interface (GUI) display including a graphical representation of the expected field response versus frequency.

28. 28. The computer-readable medium of claim 27, wherein the GUI display includes a graph illustrating maximum expected field response versus frequency of the input probability values.

29. 1. A computing device comprising: a computer-readable medium having computer-executable instructions stored thereon; a processor coupled to the computer-readable medium for executing the computer-executable instructions, the instructions comprising: receiving user input defining characteristics of a reverberation system, said characteristics including physical dimensions and loss coefficients of an enclosure structure; identifying an excitation characteristic of excitation energy input to the reverberation system; determining a marginal probability density function of input conductance frequency uncertainty of the reverberation system based at least in part on the physical dimensions, the loss coefficient, and the statistical average of the excitation energy; determining an unconditional probability density function of the field response of the reverberation system to the excitation energy based at least in part on the input conductance frequency uncertainty; determining an expected field response by the reverberation system with respect to frequency in response to the excitation energy for input probability values ​​using the unconditional probability density function; and providing a graphical user interface (GUI) display including a graphical representation of the expected field response versus frequency.

30. 30. The computer-readable medium of claim 29, wherein the GUI display includes a graph illustrating maximum expected field response versus frequency of the input probability values.

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