Methods and systems for determining a probability density function for a response of a reverberant system

Abstract

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

Description

ATTORNEY DOCKET NO.: 106.0007PC METHODS AND SYSTEMS FOR DETERMINING A PROBABILITY DENSITY FUNCTION FOR A RESPONSE OF A REVERBERANT SYSTEM CROSS REFERENCE TO RELATED APPLICATION

[0001] This application claims priority to United States Provisional Patent Application No. 63 / 421,178, filed November 1, 2022, the entire content of which is incorporated by reference herein. TECHNICAL FIELD

[0002] The subject matter described herein relates generally to the interaction between wave fields and reverberant systems, and more particularly, embodiments of the subject matter relate to analytically determining an unconditional probability density function of a reverberant response with respect to frequency that accounts for uncertainty. BACKGROUND

[0003] Engineers often need to be able to estimate or predict the real-world dynamic environment in which a device or component will operate in, so that an engineer 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 wavefield strength in which an electronic component, device, or system must operate, so that they can design and test for immunity to electromagnetic interference. When electronics are housed within an enclosure, the electromagnetic field within the enclosure will become reverberant at higher frequencies (e.g., based on the wavelength relative to the dimensions of the enclosure), at which point electromagnetic wave reflections accumulate to create a multi-modal, reverberant response usually quantified by the total wavefield energy level. The reverberant energy level can typically only be quantified statistically because either the excitation is random or uncertain or because the exact modal parameters of the enclosure (which are dictated by the enclosure’s dimensions and electromagnetic properties) are uncertain.

[0004] As another example, in the field of vibro-acoustics, engineers may need to estimate the maximum vibration level that sensitive equipment and / or payloads will experience during 1   ATTORNEY DOCKET NO.: 106.0007PC operation. For example, mechanical engineers designing a rocket or launch vehicle need to be able to estimate the maximum vibration level that is likely to be experienced in transonic flight, so that they can design and test for safe operation of the equipment in flight. The vibration wavefield response of the structural panels of the vehicle may be driven by unsteady aerodynamics forces during transonic flight. Again, the vibrational waves in the vehicle structural panel subsystems will reflect and scatter at higher frequencies, at which point the vibrations accumulate to create a reverberant vibrational energy level.

[0005] While various statistical energy analysis methods exist and can be employed to estimate the mean or average reverberant energy level, care must be taken so as not to underestimate the statistical variance about the mean, and the resulting maximum expected reverberant energy level, as any gross under estimation of maximum expected response at the design stage will lead to equipment failures in the operating environment. At the same time, any gross overestimate of the reverberant energy level can make it cost prohibitive and / or weight prohibitive to design devices or components for the estimated reverberant energy level. Accordingly, it is desirable to calculate or otherwise estimate the reverberant energy level in an accurate and reliable manner without grossly overestimating or underestimating the expected reverberant energy level. BRIEF SUMMARY

[0006] Methods and systems are provided for probabilistically determining an expected field response by a reverberant system that accounts for input conductance frequency uncertainty. One method involves receiving user input defining characteristics of a reverberant system, the characteristics including physical dimensions of an enclosure structure, a wave propagation speed and a loss factor, identifying excitation characteristics for excitation energy to be input to the reverberant system, determining a conditional probability density function for a field response of the reverberant system to the excitation energy based at least in part on a statistical mean of a magnitude squared field response, determining a marginal probability density function for input conductance frequency uncertainty of the reverberant system based at least in part on the physical dimensions, the wave propagation speed and the loss factor, determining an unconditional probability density function for the field response of the reverberant system to the excitation energy based at least in part on the marginal probability 2   ATTORNEY DOCKET NO.: 106.0007PC density of input conductance frequency uncertainty, determining an expected field response by the reverberant system with respect to frequency in response to the excitation energy for an input probability value using the unconditional probability density function, and providing a graphical user interface (GUI) display including a graphical representation of the expected field response with respect to frequency. In one implementation, the method determines the unconditional probability density function for the expected field response based on the conditional probability density function with marginal probability distribution for uncertain input conductance and a marginal probability density function for at least one of effective input current uncertainty associated with the excitation energy and Q factor uncertainty associated with the loss factor of the reverberant system. In another implementation, determining the marginal probability density function for the input conductance uncertainty of the reverberant system involves 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 the statistical mean of the field response.

[0007] An apparatus for a computer-readable medium having computer-executable instructions stored thereon is also provided. The computer-executable instructions, when executed by a processing system, cause the processing system to receive user input defining characteristics of a reverberant system, the characteristics including physical dimensions of an enclosure structure and a loss factor, identify excitation characteristics for excitation energy to be input to the reverberant system, determine a marginal probability density function for input conductance frequency uncertainty of the reverberant system based at least in part on the physical dimensions, the loss factor and a statistical mean of the excitation energy, determine an unconditional probability density function for the field response of the reverberant system to the excitation energy based at least in part on the input conductance frequency uncertainty, determine an expected field response by the reverberant system with respect to frequency in response to the excitation energy for an input probability value using the unconditional probability density function, and provide a graphical user interface (GUI) display including a graphical representation of the expected field response with respect to frequency. In one implementation, the GUI display includes a graph depicting a maximum expected field response with respect to frequency for the input probability value. 3   ATTORNEY DOCKET NO.: 106.0007PC

[0008] An apparatus is also provided for a computer device including a computer-readable medium having computer-executable instructions stored thereon and a processor coupled to the computer-readable medium to execute the computer-executable instructions to provide software configurable to receive user input defining characteristics of a reverberant system, the characteristics including physical dimensions of an enclosure structure, a wave propagation speed and a loss factor, identify excitation characteristics for excitation energy to be input to the reverberant system, determine a marginal probability density function for input conductance frequency uncertainty of the reverberant system based at least in part on the physical dimensions, the loss factor and a statistical mean of the excitation energy, determine an unconditional probability density function for the field response of the reverberant system to the excitation energy based at least in part on the input conductance frequency uncertainty, determine an expected field response by the reverberant system with respect to frequency in response to the excitation energy for an input probability value using the unconditional probability density function, and provide a graphical user interface (GUI) display including a graphical representation of the expected field response with respect to frequency. In one implementation, the GUI display includes a graph depicting a maximum expected field response with respect to frequency for the input probability value.

[0009] This summary is provided to describe select concepts in a simplified form that are further described in the Detailed Description. This summary is not intended to identify key 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 DESCRIPTION OF DRAWINGS

[0010] Exemplary embodiments of the subject matter of the present disclosure will hereinafter be described in conjunction with the following drawing figures, wherein like numerals denote like elements, and wherein:

[0011] FIG.1 is a block diagram of a networked computing system in accordance with one or more exemplary implementations;

[0012] FIG.2 is a block diagram of a local computing system in accordance with one or more exemplary implementations; 4   ATTORNEY DOCKET NO.: 106.0007PC

[0013] FIG.3 is a flow diagram of a probabilistic response prediction process suitable for implementation by a probabilistic response prediction application in the computing system of FIG. 2 or the computing system of FIG. 1 in accordance with one or more exemplary implementations;

[0014] FIGS. 4-10 and 17 depict exemplary graphical user interface (GUI) displays suitable for presentation by a probabilistic response prediction application in connection with the probabilistic response prediction process of FIG. 3 in accordance with one or more exemplary implementations;

[0015] FIG. 11 is a block diagram depicting statistical parameters of a reverberant wavefield system suitable for analysis in connection with the probabilistic response prediction process of FIG.3 in an exemplary implementation;

[0016] FIG. 12 is a block diagram depicting uncertain parameters of a reverberant wavefield system suitable for analysis in connection with the probabilistic response prediction process of FIG.3 in an exemplary implementation;

[0017] FIG. 13 is a block diagram depicting the relationship between connected reverberant systems suitable for analysis in connection with the probabilistic response prediction process of FIG.3;

[0018] FIG.14 is a block diagram depicting the relationship between a reverberant system and a deterministic system suitable for analysis in connection with the probabilistic response prediction process of FIG.3;

[0019] FIG.15 is a schematic depicting the relationships between mixed deterministic and reverberant fields in accordance with one or more implementations; and

[0020] FIG. 16 is a schematic depicting the relationships between coupled deterministic and reverberant systems in accordance with one or more implementations. DETAILED DESCRIPTION

[0021] The following detailed description is merely exemplary in nature and is not intended to limit the subject matter of the application and uses thereof. 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. 5   ATTORNEY DOCKET NO.: 106.0007PC

[0022] Embodiments described herein generally pertain to methods and systems for analytically determining the maximum expected spatially local wavefield response, that is, the maximum expected level of reverberant field variable response that may be present at any location within an enclosure, cavity or other reverberant structure or system with a given level of confidence (or probability percentile) with respect to a frequency range of interest. In this regard, rather than providing a space-averaged wavefield energy level, the maximum expected response energy and the maximum expected spatially local field variable response is analytically or numerically determined in a manner that accounts for uncertainty while avoiding overestimation or underestimation of maximum expected response, thereby enabling an engineer to effectively tune or design a reverberant system or a component thereof in a manner that ensures the reverberant system or a component will be likely to withstand excitation energy while minimizing costs and weight.

[0023] The response amplitude of any wavefield in an enclosed, reverberant environment is typically complex and difficult to predict at high frequencies, in part, because the wavefield amplitude is a highly variable function of both frequency and spatial location and may be excited by sources that are statistically uncorrelated. For example, a typical application where a designer or other engineer may be interested in predicting the maximum expected response is the prediction of the maximum expected electromagnetic interference on aircraft avionics due to the operation of multiple personal wireless devices in the passenger cabin. Another application is prediction of the maximum expected acoustic loads on a spacecraft and its sensitive electronics components inside a launch vehicle fairing, due to lift-off rocket plume acoustic loads and flight transonic aero-acoustic loads. Traditionally, there are two widely used reduced 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 electrically large (high modal overlap) such that the frequency variance can be ignored, in which case the highly variable spatially local amplitude of electric or magnetic field has a Rayleigh probability density distribution. However, for electrically small or low modal overlap cavities which are common in practice, this model underpredicts response amplitudes due to additional frequency variance. 6   ATTORNEY DOCKET NO.: 106.0007PC

[0024] In vibro-acoustics where broadband random excitation sources are more common, the complex spatial field response is reduced by space-averaging to a simpler total wavefield energy level. This reduction allows vibration and acoustic energy levels of quite complex interconnected wavefield systems to be predicted using a simple power balance principle. The reduction also simplifies the estimate of the frequency variance of the energy, which allows maximum vibration, acoustic energy levels to be predicted with the log normal probability density function. However, the spatially local field response amplitude exhibits large additional variance over the predicted spatially-averaged energy predictions. From electromagnetics, the spatial field variability (conditional on the field energy level) is known to be Rayleigh distributed, but the total combined spatio-temporal variability has a probability density function that is neither Rayleigh nor log normally distributed.

[0025] Reliable prediction of the maximum expected amplitude in the most general case has therefore remained undefined; often resulting in conservative upper bound estimates which are problematic to efficient design. For example, U.S. Patent No. 10,379,147, which is incorporated by reference herein in its entirety, teaches a method for predicting the variance of a reverberant electric field energy level due to uncertainty in the modal parameters of the cavity resonances using a reduced order model from vibro-acoustics. However, this is a spatially- averaged wavefield energy analysis that does not account for the Rayleigh spatial variance of the local field variable response. U.S. Patent No. 10,565,325, which is incorporated by reference herein in its entirety, teaches that the maximum expected reverberant energy amplitude is subject to at least three sources of uncertainty: modal parameter uncertainty characterized by frequency variance of input conductance, a statistically independent damping loss factor (Q factor) variance, plus a statistically independent power input variance attributable to uncertainty in excitation source amplitudes. In this regard, the subject matter described herein defines a probability density function for predicting the maximum expected spatially local wavefield response, as distinct from the space-averaged wavefield energy level, and in a manner that accounts for the uncertainties associated with each of the input conductance variance, the damping loss factor (or Q factor) variance and the excitation source strength variance. 7   ATTORNEY DOCKET NO.: 106.0007PC

[0026] FIG. 1 depicts an exemplary embodiment of a networked computing system 100 that includes a probabilistic response prediction web application 102 that is configurable to analytically determine an unconditional probability density function for a device, component or other reverberant system subject to excitation and providing corresponding indicia of the expected spatially local wavefield response(s), as described in greater detail below. In exemplary implementations, the computing system 100 includes a server 104 that generates or otherwise provides instances of a probabilistic response prediction web application 102 that are accessed by corresponding instances of client devices 106 over a communications network 108, such as the Internet or any sort or combination of wired and / or wireless computer network, cellular network, mobile broadband network, radio network, or the like. It should be appreciated that FIG. 1 is a simplified representation of a computing system 100 and is not intended to be limiting.

[0027] The client device 106 generally represents an electronic device coupled to the network 108 that may be utilized by a user to access an instance of the web application 102 using an application 110 executing on or at the client device 106. In practice, the client device 106 can be realized as any sort of personal computer, mobile telephone, tablet or other network- enabled electronic device coupled to the network 108 that executes or otherwise supports a web browser or other client application 110 that allows a user to access one or more graphical user interface (GUI) displays provided by the web application 102. In exemplary implementations, the 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, a touch panel, a mouse, a joystick, a directional pad, a motion sensor, or the like, capable of receiving input from the user of the client device 106. The illustrated client device 106 executes or otherwise supports a client application 110 that communicates with the server 104 to access an instance of the 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 networking protocol, such as the hypertext transport protocol (HTTP). In this manner, in one or more implementations, the client application 108 may be utilized to access or otherwise initiate an instance of a web application 102 hosted by the server 104, where the web application 102 provides one or more web page GUI displays within the 8   ATTORNEY DOCKET NO.: 106.0007PC client application 110 that include GUI elements for interfacing and / or interacting with the web application 102 supported by the server 104.

[0028] The server 104 generally represents the one or more server computing devices, server computing systems or other combination of processing logic, circuitry, hardware, and / or other components configured to support instances of a web application 102 provided to client devices 106 via the network 108. In exemplary implementations, the 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 operation of the processing system described herein. Additionally, although not illustrated in FIG. 1, in practice, the server 104 may also include one or more communications interfaces, which include any number of transmitters, receiver, transceivers, wired network interface controllers (e.g., an Ethernet adapter), wireless adapters or another suitable network interface that supports communications to / from the network 108 coupled thereto. The application server 104 also includes or otherwise accesses a data storage element 122 (or memory) that stores code or other computer-executable programming instructions that, when executed by the processing system 120, are configurable to cause the processing system 120 to support or otherwise facilitate the web application 102 and related software services that are configurable to support subject matter described herein. Depending on the implementation, the memory 122 may be realized as a random access memory (RAM), read only memory (ROM), flash memory, magnetic or optical mass storage, or any other suitable non-transitory short or long term data storage or other computer-readable media, and / or any suitable combination thereof capable of storing code or other programming instructions executable by the processing system 120.

[0029] FIG.2 depicts an exemplary embodiment of a local computing system 200 suitable for supporting or otherwise implementing a probabilistic response prediction software application 202 that is configurable to analytically determine an unconditional probability density function for a device, component or other reverberant system subject to excitation and providing corresponding indicia of the expected spatially local wavefield response(s), as 9   ATTORNEY DOCKET NO.: 106.0007PC described in greater 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 explanation 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 at a client device 220, such as an instance of the client device 106 in FIG.1. That is, depending on the implementation, the probabilistic response prediction software application 202 may be implemented locally at 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.

[0030] The user input device 204 generally represents the hardware and / or other components configured to provide a user interface with the computing system 200. Depending on the embodiment, the user input device 204 may be realized as a key pad, a keyboard, a mouse, one or more button(s), a touch panel, a touchscreen, an audio input device (e.g., a microphone), or the like. The output device 208 generally represents the hardware and / or other components configured to provide output to the user from the computing system 200, as described in greater detail below. In an exemplary embodiment, the output device 208 is realized as an electronic display device associated with the client device 220 that is configured to graphically display information and / or content under control of the processing system 206, as described in greater detail below. Accordingly, for purposes of explanation, the output device 208 may alternatively be referred to herein as a display device. That said, in other implementations, the output device 208 may be realized as a communications interface or other input / output interface that supports communications to / from the client device 220.

[0031] Still referring to FIG. 2, the processing system 206 generally represents the hardware, circuitry, processing logic, and / or other components of the computing system 200 coupled to the user input device 204 and the display device 208 to receive input from the user, utilize the input provided by the user to execute various functions and / or processing tasks, and provide an output to the user, as described in greater detail below. Depending on the 10   ATTORNEY DOCKET NO.: 106.0007PC embodiment, the processing system 206 may be implemented or realized with a computer, a general purpose processor, a microprocessor, a controller, a microcontroller, a state machine, a content addressable 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. The data storage element 210 (or memory) may be realized as any sort of non-transitory short or long term storage media capable of storing programming instructions, code or other data for execution by the processing system 206, including any sort of random access memory (RAM), read only memory (ROM), flash memory, registers, hard disks, removable disks, magnetic or optical mass storage, and / or any other suitable computer-readable medium. The computer- executable programming instructions, when read and executed by the processing system 206, cause the processing system 206 to execute or otherwise provide the probabilistic response prediction application 202 and perform the tasks, operations, and / or functions described in greater detail below.

[0032] Referring now to FIGS. 1-2, in exemplary implementations, the probabilistic response prediction application 102, 202 causes a client device 106, 220 to display (e.g., on display device 208) one or more GUI displays that include one or more GUI elements (e.g., text boxes or the like) that are adapted to receive user inputs indicative of the physical dimensions and / or physical layout of a reverberant subsystem to be analyzed along with the mechanical, electrical, chemical, material and / or other physical properties of the reverberant subsystem and / or the medium (e.g., the permeability and the permittivity that define the wave propagation speed) and influence the reverberant response to input excitation energy. Additionally, the probabilistic response prediction application 102, 202 may also display GUI elements that are adapted to receive user inputs indicative of the magnitude, frequency, and / or other characteristics of the excitation energy to be provided by an excitation source to be used for purposes of characterizing and predicting the reverberant response of the reverberant subsystem and / or the medium, as described in greater detail below. After providing the desired input information, the user may manipulate the user input device 204 to select a GUI 11   ATTORNEY DOCKET NO.: 106.0007PC element (e.g., a button or the like) that causes the processing system 206 to continue executing the programming instructions using the inputs received from the user to calculate or otherwise determine a maximum expected reverberant response for the reverberant subsystem and generate or otherwise provide one or more output indications indicative of the maximum expected reverberant response on the display device 208, as described in greater detail below.

[0033] FIG.3 depicts an exemplary probabilistic response prediction process 300 suitable for implementation in connection with a probabilistic response prediction application in a computing system of FIGS. 1-2. The 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 mentioned above in connection with FIGS. 1-2. In an exemplary embodiment, one or more aspects of the probabilistic response prediction process 300 are implemented at or by the probabilistic response prediction software application 102, 202 executed at a computing device 104, 220. It should be appreciated that the probabilistic response prediction process 300 may include any number of additional or alternative tasks, the tasks shown in FIG.3 need not be performed in the illustrated order, and the probabilistic response prediction process 300 may be incorporated into a more comprehensive procedure or process having additional functionality not described in detail herein. Moreover, one or more of the tasks shown in FIG.3 could be omitted from an embodiment of the probabilistic response prediction process 300 as long as the intended overall functionality remains intact.

[0034] The probabilistic response prediction process 300 initializes or otherwise begins by receiving or otherwise obtaining user input defining the characteristics of the reverberant system of interest and additional reverberant system conditions including characteristics of the excitation energy input to the reverberant system and uncertainties associated the damping factor (or Q factor) of the reverberant system and the excitation energy (task 302). For example, as described above, the probabilistic response prediction application 102, 202 may provide one or more GUI displays that include text boxes, drop-down menus and other GUI elements for receiving user inputs indicative of the physical dimensions and / or physical layout of a reverberant system to be analyzed along with the mechanical, electrical, chemical, material 12   ATTORNEY DOCKET NO.: 106.0007PC and / or other physical properties of the structure and / or the medium (e.g., the permeability, the permittivity, the conductivity, and / or the like) that define the reverberant system and influence the reverberant response to input excitation energy, in addition to user inputs indicative of the magnitude, frequency, and / or other characteristics of the excitation energy to be provided by an excitation source to be used for purposes of characterizing and predicting the reverberant response of the reverberant system and / or the medium. For example, to predict the maximum expected electric field response in the avionics bay of an aircraft when flying in close proximity to a high power 5G cell tower, the received user input may include the volume and surface area of the avionics bay, the absorption section of the contents and the transmission section of windows and apertures to characterize the reverberant field enclosure, along with the exterior electric field strength (in V / m) and antenna impedance characterizing the excitation sources. Another application is prediction of the maximum expected acoustic loads on a spacecraft and its sensitive electronics components inside a launch vehicle fairing, due to lift-off rocket plume acoustic loads and flight transonic aero-acoustic loads. In this example, the received user input may include the volume and surface area of the launch vehicle fairing and the spacecraft, the absorption section of the spacecraft and acoustic blankets on the fairing walls and the transmission section of de-pressurization vent holes and access hatch apertures to characterize the reverberant field enclosure, along with the exterior lift-off acoustic sound pressure levels (in dB) and transonic flight fluctuating surface pressure levels (in dB) characterizing the excitation sources.

[0035] In some implementations, the probabilistic response prediction application 102, 202 may receive the user input in the form of a computer-aided design (CAD) file or another computer file in a suitable format that can be parsed by the probabilistic response prediction application 102, 202 to extract or otherwise derive the physical characteristics of the structure of the reverberant system along with the mechanical, electrical, chemical, material and / or other physical properties of the medium or materials of the reverberant system defined within the CAD file.

[0036] FIGS. 4-8 depict exemplary GUI displays 400, 500, 600, 700, 800 that may be presented by a probabilistic response prediction application 102, 202 in connection with the probabilistic response prediction process 300 of FIG. 3. In this regard, FIG.4 depicts a GUI 13   ATTORNEY DOCKET NO.: 106.0007PC display 400 including GUI elements for receiving user input defining the physical dimensions of a cavity, FIG.5 depicts a 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 depicts a GUI display 600 including GUI elements for receiving user input defining the Q factor and other losses associated with the cavity. FIG.7A and FIG.7B depict different GUI displays depicting Q factor (or damping loss) uncertainty information for the reverberant system that may be received as user input as a probability density distribution in the form of a table, histogram or other suitable format (e.g., comma-separated values), as depicted in FIG. 7A, or in the form of a mean, standard deviation and potentially other statistical parameters defining the probability density distribution for the Q factor. FIG. 8 depicts a GUI display 800 including GUI elements for receiving user input defining the location or orientation of the input excitation energy and the corresponding power or energy level associated with the input excitation energy. In exemplary implementations, in addition to the GUI displays defining the reverberant 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 the desired probability or confidence for which the user would like the probabilistic response prediction application 102, 202 to probabilistically determine the minimum and / or maximum expected field reverberant response of the reverberant system to the input excitation energy.

[0037] Referring again to FIG.3, after receiving user input defining the characteristics or conditions associated with the reverberant system of interest, the probabilistic response prediction process 300 proceeds with probabilistically determining a maximum expected reverberant field response for the reverberant system having the user input conditions to the input excitation energy with the 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 wavefield energy and the frequency variance of the wavefield energy within the reverberant system resulting from the input excitation energy based on the user input conditions (task 304). In exemplary implementations, the probabilistic response prediction application 102, 202 calculates or otherwise determines a statistical mean for the wavefield energy and a cumulative variance of the wavefield energy with respect to frequency based at least in part on a first variance associated with the input excitation energy 14   ATTORNEY DOCKET NO.: 106.0007PC representing an uncertainty in an amplitude of the excitation energy with respect to frequency, a second variance associated with the uncertainty in the damping provided by the reverberant system with respect to frequency and a third variance associated with an input modal power acceptance of the reverberant system, as described in greater detail in U.S. Patent No. 10,565,326 and U.S. Patent No.10,379,147, both of which are incorporated by reference herein in their entirety.

[0038] After determining the statistical mean, the probabilistic response prediction process 300 determines a conditional probability density for the field response of the reverberant system based on the statistical mean for the wavefield energy using a Rayleigh distribution model (task 306). In this regard, the conditional probability density may be represented by one or more of equations (7)-(9) below where ^క^^,^^ represents the spatially local field response for any frequency (^) at any point in space (^) and both the Rayleigh distribution for field response magnitude and Exponential distribution for the magnitude squared field response are conditional on the statistical mean squared field 2 ^2^ E2r . In the wider application of these models to electrically small systems withoverlap, it is found that the normalizing mean field exhibits a high degree of statistical variation with frequency which needs to be incorporated as an additional independent random variable, as described in greater detail below.

[0039] After determining the conditional probability density for the field response, the probabilistic response prediction process 300 determines an unconditional probability density for 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. Since the spatial mean squared field response is proportional to total wavefield energy, the log normal distribution for frequency uncertainty of energy can be used, where the relative variance can be calculated directly from the known reverberant field parameters using equations (16) - (18) . It is explained later that this energy frequency variance model is for the specific case when Q factor is known (e.g., measured) and excitation strength is known (e.g., measured). Under these limited uncertainty conditions, the foregoing marginal distribution for spatial mean squared field response is attributable solely to the frequency uncertainty of the point input conductance 15   ATTORNEY DOCKET NO.: 106.0007PC f^Gii ^^^ ^of the reverberant field (the real part of point input impedance) (task 308). Theprobability density may be determined by numerical integration of a user inputdensity function for the input conductance frequency uncertainty such as the log normal distribution, represented by equations (16) - (18) below. In one example, the unconditional probability density is determined as a convenient closed form analytic solution to the unconditional distribution integral equation using an inverse-gamma distribution having the form represented by one of equations (21), (22) and (24), as described in greater detail below. In this regard, in some implementations, a probability density function of the input conductance frequency uncertainty is calculated using the user input physical dimensions of the enclosure structure for the reverberant system, the user input loss factor of the reverberant system, and the statistical mean of the excitation energy, with the unconditional probability density being calculated using the calculated probability density function of the input conductance frequency uncertainty and an inverse-gamma distribution.

[0040] After determining the unconditional probability density attributable to uncertainty in input conductance, the probabilistic response prediction process 300 determines a probability function for the field response based on the unconditional probability density by incorporating additional the user input values for the damping factor (or Q factor) uncertainty and the excitation uncertainty (task 310). In this regard, as described in greater detail below, the unconditional probability density function of the electric field (or field response) at a point in space may be represented as an integral of the conditional probability density function over the marginal distribution for product of three independent random variables corresponding to the excitation energy (^^ଶ), the modal conductance (^^^) and the damping factor (^).The resulting unconditional electric field probability density function is an integral of known probability density functions having both numerical solutions and a convenient closed form solution using the inverse-gamma distribution and the power balance equation. The probability function for the field response can be derived as the inverse of the unconditional cumulative distribution for any probability limit P .

[0041] Once the probability function for the field response has been determined based on the unconditional probability density, the probabilistic response prediction process 300 calculates or otherwise determines the expected field response for any frequency at any point 16   ATTORNEY DOCKET NO.: 106.0007PC in space based on the input probability level (P) using the inverse function derived from the unconditional probability density function (task 312). In this regard, for corresponding user input values for minimum and maximum probability values (or confidence values), the probabilistic response prediction application 102, 202 calculates the corresponding probabilistic maximum field response with respect to frequency with the input level of confidence or probability maxima and the corresponding probabilistic minimum field response with respect to frequency with the input level of confidence or probability minima, for example, by evaluating the probability function at individual frequencies within the input frequency range of interest. Additionally, the probabilistic response prediction application 102, 202 calculates the corresponding probabilistic mean field response with respect to frequency, for example, by using a level of confidence or probability percentile of 99%.

[0042] After calculating or otherwise determining the expected field response for the user input probability level, the probabilistic response prediction process 300 outputs or otherwise provides indicia 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 within a GUI display on the display device 208 that depicts the maximum, mean and minimum expected field response levels with respect to frequency over a frequency range of interest. In this manner, the probabilistic response prediction application 102, 202 allows the user to visually perceive and ascertain what the expected maximum and minimum response to the input excitation energy is likely to be for the reverberant system defined by the user within the desired level of confidence or probability that accounts for the uncertainty associated with the excitation energy, the damping factor (or Q factor), the modal conductance and / or the like. Additionally, by analytically providing a stochastic solution for the expected field response, the delay between the probabilistic response prediction application 102, 202 receiving the user input defining the reverberant system of interest (e.g., at task 302) and providing indicia of the resulting field response (e.g., at task 314) is reduced relative to more computational approaches such as Monte Carlo simulations or the like, while still accurately predicting the minimum and maximum expected field response at any point in space within the reverberant system across the frequency range of interest. 17   ATTORNEY DOCKET NO.: 106.0007PC

[0043] FIG.10 depicts an exemplary stochastic response GUI display 1000 that includes a graph 1002 depicting the relationship between the maximum expected field response 1004, the minimum expected field response 1006 and the mean expected field response 1008 with respect to 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 reverberant system at the respective frequency with an upper 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 reverberant system at the respective frequency with a lower probability or confidence level input by the user (e.g., 0.5%), and the graphical representation of the mean expected field response 1008 represents the mean expected field response at any location within the reverberant system at the respective frequency with a probability or confidence level of 50%. The illustrated stochastic response GUI display 1000 also includes a graphical representation of a target limit or constraint 1010 for the field response with respect to frequency that may be input or otherwise provided by the user. In this regard, the targeted field response limit 1010 may correspond to a design specification or other regulatory requirement that the user designing the reverberant system is attempting to comply with. Accordingly, when the maximum expected field response 1004 exceeds the targeted field response limit 1010, the user may modify one or more characteristics of the reverberant system (e.g., the physical dimensions of the system, the material or medium within which the excitation energy reverberates, and / or the like) or otherwise attempt to modify or limit the input excitation energy until the maximum expected field response 1004 satisfies or otherwise remains below the targeted field response limit 1010 over the frequency range of interest.

[0044] FIG. 11 depicts an exemplary relationship between the excitation field strength associated with the input excitation, the input conductance, and the damping loss factor (or Q factor) and the volume of the reverberant system with respect to the expected field response with respect to frequency at any location within the volume of the reverberant system. FIG.12 depicts exemplary relationship between the probability density function representation of the uncertainty of the excitation field strength with respect to frequency, the probability density function representation of the input conductance, and the probability density function representation of the damping loss factor (or Q factor) and the expected field response with 18   ATTORNEY DOCKET NO.: 106.0007PC respect to frequency at any location within the volume of the reverberant system. FIGS.13-14 depicts exemplary relationships between multiple connected or coupled reverberant systems suitable for analysis or use with a practical implementation of the probabilistic response prediction process 300 and the corresponding uncertainties suitable for stochastic modeling in connection with a practical implementation of the probabilistic response prediction process 300. In this regard, it will be appreciated that although the subject matter may be described herein in the context of a single reverberant system for simplicity and purposes of explanation, in practice, the probabilistic response prediction process 300 is extensible to accommodate any number or combination of systems, including multiple connected or coupled reverberant systems or hybrid configurations involving reverberant systems connected or coupled to deterministic systems, and any sort of reverberant field, including, but not limited to electric fields, magnetic fields, electromagnetic fields, acoustic fields, vibrational wavefields, and / or the like.

[0045] Referring now to FIGS. 11-14 with continued reference to FIGS. 1-10, mathematical derivations of the unconditional probability density function and corresponding probability function for determining the expected field response with respect to any particular frequency for any point within a volume associated with a reverberant system will now be described in greater detail. Without loss of generality, a physics and mathematics description of the subject matter is provided for the exemplary case of high frequency reverberant electromagnetic field in an aperture enclosure such as an avionics box. The total energy U^^^of the reverberant field at frequency ^ (rad / s) is proportional to the volume^V ^themagnitude squared total electric field E2T , represented by equation (1) below. U^ 1^2 2 2^ ^ E T ^ ^ H T ^ dV ^^^ E T dV (1)The volume integral inof the electric field, represented by equation (2) below: U^ ^ V E 2 2T^3 ^ V Er (2)19   ATTORNEY DOCKET NO.: 106.0007PC whereEis any rectang2 222rular component of the total electric field ET^ Ex^ Ey^ EzandxR denotes average of x over domainR. At high frequencies, when the free propagation wavelength ^ ^2 ^ c0^ is smaller than the characteristic dimensions of the enclosure, the reverberant electric field Er ^x, ^^at positionxcan be fully defined by a Green’s function expansion of thermodes. For an electrically small dipolecurrent source IS^xi, ^ ^ of length LSat xi, the electric field response at position xocan be equation (3) below.E^^x^ o ,^^^ i^LS^ r ^xo ^ I S ^xi,^ ^ ^ r ^xi ^^ ^^2 ^ ^ ^ ^ ^ ^ 2 (3)

[0046] Asherein, the standing wave modal density (modes per rad / s) in equation (3) increases so rapidly (e.g., with the square of frequency n ^ ^ ^ ^ V ^2^2c3) that the prediction of spatial mode shapes ^r ^ x ^, modal loss factors^^ 1r^x ^ ^ Q rand modal excitation factorsIS ^x i, ^ ^ ^ r ^ x i ^quicklybecomes an uncertain process, highly sensitive to details which arewithin design tolerances. The probabilistic response prediction process described herein provides a novel probabilistic model for the electric field under these conditions, characterized by a probability density function (PDF) denoted fE ^Er ^x, ^^ ^. It should be noted that the correspondingstatistical population is an ensemble of^^realizations (e.g., measurements) for a single enclosure design, but with variations in the uncertain parameters. The statistical mean of this uncertainty distribution can be predicted from known parameters, using the principle of power balance. The statistical mean power input PINto the reverberant field by excitation sources equals the statistical mean power losses PDISSfrom the reverberant field, characterized by the mean loss factor ^L^ Q^ 1, represented by equation (4) below. PIN^ PDISS^ ^ ^LU ^ ^ U Q(4)The statisticaland (4), which may be represented by equation (5) below. 20   ATTORNEY DOCKET NO.: 106.0007PC 2 P IN Q Er^^^(5)Compared with equation (3), anyuncertainties in the power input PINX with no subscript indicate averaging over a notional ensemble of many different physical instances of the same enclosure design. Since this is difficult to measure in practice, it is common to assume that ergodicity applies, so that more convenient alternative averaging ensembles can be substituted according to equation (6) below: E 2^ E2 2 2 rrr^ E rx^ E r^ (6)where Xris an in areverberation chamber with a mode stirrer.

[0047] For electrically large enclosures which are in an overmoded condition (such as reverberation test chambers) the un-averaged electric field magnitude Er ^x, ^^^ Re ^ E2r ^ ^ Im ^ E2r ^ at any location x and frequency ^ is Chi distributed with 2distribution), represented by equation (7) below. E^ 2 ^f^Er ^^ 2 r exp ^ ^Er2^(7) This is a single parameter(8) below. 2 ^2^ E2r (8) The corresponding mean squared(exponential) distribution, which may be represented by equation (9) below. 22^ 1 ^ E r ^^ ^^^^(9)This single parameterand – 21   ATTORNEY DOCKET NO.: 106.0007PC by definition - has a relative variance of unity represented by equation (10) below. r2 ^^E 2r^^^ 2 ^ 2 ^^E 2r^^E 2r^ 1(10)

[0048] Electricallydensity function (PDF) that diverges from Rayleigh, because they typically do not meet the overmoded condition. Here we introduce the parameter modal overlap to quantify the overmoded condition. In any given frequency range ^ ^ , modal overlapm ^ ^ ^is defined asthe ratio of the modal damping bandwidth ^L^ to the average frequency spacing ^^ ^1 n^^^represented by equation (11) below.m ^ ^ ^ ^ ^ ^Ln ^ ^ ^ ^ ^ n ^ ^ ^ Q(11)wheren ^ ^ ^ ^ V ^2^2c3 isis when a single frequency excitation of the cavity will excite many resonant modes. By definition, this condition is always true when modal overlap is significantly greater than unity m^^^^ 1

[0049] The principal effect of low modal overlap in electrically small enclosures is to make the reverberant field energy highly dependent on frequency U^^^. The corresponding mean squared electric field exhibits the same degree of frequency variance2Er^ ^ ^x , r . Since theRayleigh and exponential distributions scale on a mean field whichto be constant (deterministic), they are effectively only conditional distributions, which may be represented by equation (12) below. 2^ 2 ^^ 2 1 ^ E r ^ (12)

[0050] PDF can therefore be defined for electrically small enclosures with low modal overlap and large frequency variance, represented by equation (13) below: ^ ^ E^x ^^2^^^^E2E^^2^^ ^E^^2^^E^^2^ (13)22   ATTORNEY DOCKET NO.: 106.0007PC which will be simplified by using M ^ Er^ ^ ^x , r, resulting in equation (14) below. ^ fE2 ^Er^x,^^2^^^^ f^ 22^E2rM^fM^ M ^dM (14)0 Andform represented by equation (15) below^ ^^ 2fE21Er^^^ exp^ ^ r ^^f^ M ^dM (15)To evaluate this firstnecessary to define the marginal distribution for the frequency uncertainty of the mean field fM ^M^In various implementations, a Log-normal distributionM ^ LN ^ ^LM , ^ LM ^isutilized as a good fit to measured ensembles Er^ ^ ^2x , r ,(15) below^^^1^2 ^^ ^Ln^M^ ^^ L^M^ ^(16) where the mean^ 2 L M ^ Ln^M ^ ^^^ Ln^1 ^ r^M^^ ^^(17) These PDFparameters of the reverberant enclosure as follows. The frequency variability mean M^ ^can be calculated from power balance equation (5) and the relative variance r2^M^ ^ ^can be represented using Gaussian orthogonal ensemble (GOE) statistics, represented by equation (18)(17) below: 23   ATTORNEY DOCKET NO.: 106.0007PC r2^ M ^ ^^^ r2^ U ^22^^^ ^ ^ U ^ U 1 1^ ^K 1^ ^K 1^2^(18) wherethe frequency variance. The parameter K is a measure of the mode shape spatial variance, which for three dimensional volume wavefields is K ^ E^ ^^^ ^E^ ^^^ ^^ 3. L and N arerespectively the number of receiver and source positions used to estimate r2^M^ ^ ^as a statistical quantity.

[0052] The Log-normal marginal distributionfLN^ M ^in the unconditional PDF integralof equation (14) may take the form of^2 1 ^E2 ^ 1 2f^Er^^^Exp ^r^ Exp ^^^ ^ Ln ^ M ^ ^ ^L^2^ 2L^^d M (19)Thisnumerical integration. For applications involving a large network of inter-connected reverberant field enclosures – such as compact packaging of electronics in consumer electronics or the below decks environment on a ship – it is desirable to find an alternative PDF model for marginal distributionfM^M^ that does allow a closed form analytical solution to the unconditional field

[0053] In various implementations, the inverse-gamma probability density function (PDF) M ^ ^^1^^ , ^^has been found to fit theEr^^^2x, rensemble as well as the log-normal distribution, at least at the moreof M , represented by equation (20) (19) below. ^M ^ ^^^M ^^^ 1 ^^ ^^ ^^(20)As for the log-normalare defined in terms of the known mean field equation (5) and the known relative variance 24   ATTORNEY DOCKET NO.: 106.0007PC equation (18), represented by equation (21)(20) below. 1 ^ ^ 2 ^ 2 r^M^^ ^ (21)

[0054] The inverse-gammathe field integral equation (14) resulting in a Lomax distribution for the unconditional electric field PDF, represented by equation (22)below: ^ fLomax^2 ^^Er^x,^^ ^^^^^^^^ 2^+1(22) where the^ ^, ^ ^cumulative density function is represented by equation (23) below. 22 ^FLomax ^E^^x, ^ ^^^ 1 ^ ^ ^^^^^E^^ ^^ ^ ^ (23)Also conveniently, theallows direct calculation of the quantile or percentile P limits of the unconditional electric field PDF model, represented by equation (24) below. ^1^1I^P^^^1 ^ ^^^ ^ ^^1 ^ P^^^ ^^^^ ^ ^^1 ^ P^^(24)The probabilityis represented by equation (25) below. ^ ^^^ ^^E ^ ^^^ 2 E(25)

[0055] The PDF of the mean fieldMhas been found to encompass additional sources of uncertainty that are important to the wider application of the forgoing inventions for probabilistic modeling of realistic reverberant electric field environments. This starts by noting that the power input to a reverberant field is the Poynting vector and therefore will always take 25   ATTORNEY DOCKET NO.: 106.0007PC the form of the product of two conjugate variables in the frequency domain. For instance, power input can be defined as the product of the magnitude squared source current I2S^^^and the real part of the single point input impedance Gii ^^^, which is inputconductance, and may be represented by equation (26) below. Px, ^ ^ I2 x, ^ 2IN ^ i ^ S ^ i ^LSGii ^xi, ^^ (26)Using the modal isrepresented by equation L2P ,S ^ ^ 2 2r ^ r ^ r ^ x i ^I2S ^ ^ ^IN^xi^^ ^ ^ ^^222 4^(27)The statistical mean inmodal parameters ^r, ^r, ^r ^x to a very simple form dependent only on themean source current strength I2L2S , the field volume V and its modal density n^^^, which may be represented bybelow. I 2 L 2 ^ ^^s ^ ^^ ^(28)wheren ^ ^ ^ ^ V ^2^2c3 is the(26) and (28) has been shown to provide a particularly simple model for the statistical mean point input conductance Gii ^^^of any reverberant field, represented by equation (29) below.G ^ ii^ ^c^^ ^^^n ^ ^c^ (29)This statistical mean of the(27) is over both r uncertain modal parameters and over frequency band ^ ^ , evaluated at any specific center frequency^c .

[0056] In view of the foregoing, the mean field M ^ Er^ ^ ^x , ris the product of three variables – source current I2i , input conductance Giiand loss factor Q^ 1- each of which may 26   ATTORNEY DOCKET NO.: 106.0007PC have significant uncertainty in the practical application of the subject probabilistic model for reverberant fields, represented by equation (30) below. M^E 2 rr,^ ^^1 I 2 G Q 3^ ^ Vi ii (30)All three uncertainties can ,provided the marginal PDFfM^M^ can be defined for the product of three random variables in equation (30). Using the further simplification of terms x ^ I2; y ^ Q ; z ^ Gii3 ^ ^ V this has been shown to take the general form, represented by equation (31) below. f ^ M ^ ^ f ^ xyz ^ ^ ^ (31)field with three different (statically independent) sources of uncertainty f^x^, f^y^, f^M xy^^ f^z^, represented by equation (32) below.^ ^ ^^E2^^^^ E2M ^^ ^^x^ ^ ^ ^ M ^1 ^^dx dM(32)And usingform represented by equation (33) below.^ 2^ ^ E 2^ ^ 1 ^r^^ ^M^1 ^^^^ ^^^ ^ ^ ^ ^(33)It hasin practical application of the subject probabilistic model for reverberant wavefields.

[0057] For the case of no uncertainty in the source current and no uncertainty in the Q factor ^2^x^^ ^ 2^y^^ 0 , x and y are not random variables (RVs) and can be replaced by theirconstant values ^x^^ X,^y^^ Y, so that equation (31) reduces to PDF of a single RV,represented by equation (34) below. 27   ATTORNEY DOCKET NO.: 106.0007PC 1 f ^ M ^ ^ f ^ z ^ XY (34)This is the same PDFoverlap reverberant fields in , uncertainty in the mean fieldM ^ ^ ^is attributable to the frequency variability in the inputconductance of the reverberant field f^Gii ^^^ ^. As previously, this marginal PDF can bemodeled with an Inverse-gamma resulting in a Lomax distribution of equation(22) for the unconditional field response

[0058] In one or more exemplary implementations, an unconditional PDF model has been developed for the case where there is uncertainty in Q factor f^y^^ f^Q^in addition tofrequency variance in the input conductancef ^ z ^ ^ 3 ^ ^ V f ^ G ii ^ ^ ^ ^, but no uncertainty inthe source current ^2^x^^ 0 . For these conditions, the mean field PDF of equation (31) reduces to equation (35) below. ^^^M^ ^1M ^1 ^^ ^ ^ ^ (35)As an example of a(ormeasured) and the input conductance frequency uncertainty fZ ^z^takes the log-normal form, the unconditional reverberant field response is a numerical integration of equation (32), represented by equation (36) below. 2^ 1 ^^1 ^ E 2^r^ ^^^Xy ^^ ^ ^ ^ M ^^ ^ ^ 2 ^^ 2 ^ 1(36)by mathematically curve-fitting equation (3) to a measurement of the reflection scattering parameterS11 ^ ^ ^of a simple monopole probe antenna in the field [Bremner, IEEE EMCSymp., 2018]. 28   ATTORNEY DOCKET NO.: 106.0007PC

[0060] When all three mean field variables are uncertain with PDFsf ^ x ^, f ^ y ^ , f ^ z ^,the marginal PDFf ^ M ^ ^ f ^ xyz ^quickly converges to a log-normal distribution, as would bepredicted by the Central Limit Theorem (CLT). Since f^z^^ f^Gii ^^^ ^is known to be log-normal distributed andf ^ x ^, f ^ y ^can often be to log-normal distributed,the unconditional PDF equation (32) reduces to equation (37) below: ^^ E 2 ^ ^ 2 ^f^ E2 1 1r^^^Exp^r^ XyExp^ ^Ln^ M ^ ^22 ^dM^ ^^ ^ ^ ^ ^ ^LM ^ ^LM (37)whereLMLGLQLI2 2 2 2 2(38)

[0061] When all threePDF f^ M ^ ^ f ^ xyz ^which is at least approximately log-normal distributed, the marginal PDFf^ M ^can also be modeled with an inverse-gamma distribution of equation (20) with thefollowing^^M, ^M ^parameters of equation (39) below.^22E^ ^Lm^ ^^ ^^^ ^ ^(39) When substituted inaLomax distribution the same as equation (22) but with the^ ^M , ^ M ^parameters defined inequation (39) above, resulting in equation (40) below. ^ ^ 2 ^^^ ^ ^^^M M (40) The correspondingthe 29   ATTORNEY DOCKET NO.: 106.0007PC reverberant field is represented by equation (41) below. E^^ M f^^Er ^x,^^ ^^^^^ 2Er^M M2^M+1(41) ^

[0062] With reference to FIGS. 13-14, in one or more exemplary implementations, an unconditional PDF model is developed for the field response of a reverberant system which has additional uncertain effective excitation sources when it is part of a network of multiple connected reverberant systems. There is a statistical mean net power flow Pˆ ^ ^ ^ U ^ U from a connected field with higher energyU ^221 21 2 1 2 3 ^ ^ V 2 E 2 xinto the21^3 ^ ^ V1E1 x, as morePatent 10,379,147 and U.S. both of which are incorporated by referenceherein. The coupling loss factor ^21for an aperture between the connected fields is related to the aperture transmission section ^12 ^^^ 2 ^ , paveraged over 2 ^ steradians and polarity p , as shown in equation (42) below ^ c 1 12 ^ ^2^V122 ^ , p (42)A single instance of the net poweri , source current I21 ^xi, ^^2exciting the subject field E21 at input location xi. The marginaldistributioncan therefore be written as represented by equation (43)2 ^ I 21^x i,^^^^ G 11 f ^ Pˆ21 ^ (43) and varioustheunconditional field response of the subject system f 2^E1 ^x , ^^ ^.30   ATTORNEY DOCKET NO.: 106.0007PC In a simple first order perturbation analysis, previous researchers [Hoijer and Kroon, IEEE Trans. EMC 2013] have assumed zero variance (uncertainty) in the coupling loss factor and an effective current source with exponential distribution scaled on the mean field level of the driving systemelectric field magnitude squared f^I 221 ^^ Exponential ^^E ^^. If the marginal distribution ofinput conductance remains as the marginal PDF for the spatial meanfield can be represented by ^ ^2^ 1^E2 ^^ E2^^^ ^^ ^(44) and thesolution represented by equation (45) below 2^ ^ ^ ^ ^1 ^ ^^ E 1 ^x, ^ 2^ ^^^ ^^^^ ^(45) Thefield is represented by equation (46) below. 2^ ^ ^ Ex, ^ ^^^ ^ ^^^ ^ ^^ ^ ^1 ^ ^^^ 1 ^ ^(46)A more rigorous model incorporates uncertainty in the coupling loss factorf ^ ^21 ^in equation (43)and uses the difference between the two reverberant fieldat the coupling 2 2 ^21^ ^2V2E2^ ^1V1E1to define the marginal distribution of the effective source current. assumptions, the marginal distribution for the effective current isproduct of two independent distributions and is represented by equation (47) below 2 G ^ 111^ ^ 21 ^ 21 ^(47)31   ATTORNEY DOCKET NO.: 106.0007PC Assuming f^E 21^, f^E 22^have approximately exponential distributions, the marginalin magnitude squared electric field across the coupling is representedf^ ^ ^ ^ ^^ ^21^ ^ 1Exp^ ^21u Exp^ 21 ^u^ ^ ^^^^^^ ^ ^21^ ^ ^^^ ^ ^ ^21^ ^^ (48)where ^coupling loss factor f^^21 ^can be represented by anInverse Gamma marginal distribution (20) with user- ^221and variance ^^^^,21which when used with (48) in equation (47) yields a closed form analytic solutiondistribution for the coupling excitation current source, represented by equation (49) below ^ ^ 1 ^^^ 1 ^^f 2 ^ ^I ^^^^21^ ^2^2^ ^ 1^^^2 2^ ^ ^ 1^^^2^^(49 ^^ ^ ^I21^ ^^^2^^u^ ^I21^ ^ ^I21^ ^^^2^^u^I21^) ^In such a network of connected reverberant wavefield systems, there may be multiple, statistically uncorrelated net power inputs, typically consisting the direct source power input to the subject wavefield PIN1 plus the sum of net power inputs from multiple connected reverberant fields with higher energy levels^PˆIN k1so that the total power input is represented by equation (50) belowkP IN 1,Tot^ PIN 1^^PˆIN k1(50). In the most general case, theeach of the excitation power inputs will be different f^PIN1^, f^Pˆ IN21 ^, f^P ˆ IN31 ^, ... f^P ˆ INk1 ^. The marginalPDF of the total net excitation power input to the subject reverberant wavefield in that case is defined by Bayes’ law of total probability, represented by equation (51) below 32   ATTORNEY DOCKET NO.: 106.0007PC P IN IN f^PIN11,Tot ^ ^If^PINPk 11^ ^^INf^P ˆINk1 ^(51)and the(52) below f^ITot^IN 21 ^ ^f^P ^1,Tot ^ ^^^ ^^^G11 ^ ^ ^f^PIN1,Tot^(52). where G11 ^^^is

[0063] Additionally with reference to FIGS. 13-14, in some implementations, an unconditional PDF model is developed for the field response of a reverberant system which has additional uncertain Q factor losses when it is part of a network of multiple connected reverberant systems. When there is a statistical net mean power flow P1ˆ2^ ^ ^12U1^ U2from the subject reverberant fieldU21 ^3 ^ ^ V1E1 xinto a2 U2^3 ^ ^ V2E2 x- where ^12is the coupling loss factor between the connected fields - the net power flow Pˆ12defines an uncertain effective Q factor f^Q^which effects the field response PDF f 2E^E1 ^of the subject system. From a first order perturbation analysisconsidering that a single instance perturbation of uncertain Q12^ U1^12 ^U1^ U2 ^will not be significant compared with other losses unless U2^ U1of f^Q^will be the same as the distribution of f^^^112 ^, represented by equation (53) below.^U^ ^ ^ ^1^ ^ ^ ^^U^ ^ ^1^ ^^(53) Without loss of33   ATTORNEY DOCKET NO.: 106.0007PC by an Inverse Gamma marginal distribution (20) with user-defined mean ^12and variance ^2 ^ ^12^ . The unconditional field response f^E1 ^x , ^^2^can then be defined using equations(38) -(41)

[0064] In such a network of connected reverberant wavefield systems, there may be multiple, statistically uncorrelated net power losses, typically consisting the internal energy dissipation losses of the subject wavefield PLoss1 plus the sum of net power losses to multiple connected reverberant fields with lower energy levels ^PLoss 1ˆkso that the total power loss isk represented by equation (54) belowP Loss 1,Tot^ PLoss 1^^PLoss 1ˆk(54).k In the most general case, the functions (PDFs) for uncertainty ineach of these power losses will be different f^PLoss1^, f^Pˆ Loss12 ^, f^P ˆ Loss13 ^, ... f^P ˆ Loss1k ^. Themarginal PDF of the total net power lossin that case defined by Bayes’ law of total probability, represented by equation (55) below Loss Loss LossP1P1 k^P^ ^ ^PLoss^^^P ˆLoss^(55)and theequation (56) below ^2^3 ^ ^VLos21E 1 ^^ ^Ps ^ (56)

[0065] In various implementations, an unconditional PDF model is developed for the combination of a deterministic direct field EDirr^x, ^^mixed with an uncertain reverberant field with electric field magnitude PDF f RevE^Er ^^^ ^, graphically depicted in FIG. 15, as34   ATTORNEY DOCKET NO.: 106.0007PC developed in any one of the preceding models governed by a respective one of equations (7), (25) or (41). The total field isETot Dir Revr^x o, ^ ^ ^ E r ^ x o , ^ ^ ^ E r ^ ^ ^. It will be probabilistic dueto the uncertainty of the f RevE^Er ^^^ ^and it will depend onlocation due to the direct field componentEDirr^x o, ^ ^. If the subject reverberant field is highmodal overlap, the PDF of the combined deterministic and statistical field will have a Rice distribution (non-central Rayleigh distribution), represented by equation (57) below. 2 ETot ^ Dir ^ ^ 2 2 ^r ^x^E^x^ERevEDir^x^ ^ERev^Tot^ ^ ^^ o ^ r o r ^ ^ ^ r o r ^(57)It has PDFof the combined deterministic and statistical field will have a Pareto distribution (non-central Lomax distribution), represented by equation (58) below. f^ Er ^xo ^ ^^2 E Totr ^x ^ ^ ^ Toto ^^ (58)

[0066] In various implementations, an unconditional PDF model is developed for the response of a low dimensionality deterministic wavefield system coupled to a high dimensionality probabilistic wavefield system, such as the foregoing reverberant field in a three dimensional bounding volume as graphically depicted in FIG.16. A typical application is the transverse electromagnetic (TEM) mode propagation of currents on a multi-conductor cable (or transmission line) inside an enclosure that supports a reverberant three dimensional wavefield, as more fully described in U.S. Patent No. 10,338,117 and U.S. Patent No. 10,156,599. The coupled squared magnitude magnetic field h2tot(proportional to current) on the cable will be the sum of a deterministic response component due terminal electric field 2 excitations eapp(proportional to voltage) plus the mean of a probabilistic response due to additional excitation by a coupled reverberant field with energy U. For a total impedance ZTat the interface and associated radiation resistance RRad, represented by equation (59) below. 35   ATTORNEY DOCKET NO.: 106.0007PC ^^4 U^ ^h2Z^1^e2 ^ ^T*tot^ T ^^app ^ ^^ ^2^ ^^R Rad^Z ^^T(59) If the subjectand statistical field a , represented by equation (60) below. 2htot^^ happhrev^ ^^ ^ h 2app^ h2rev^^ h ^^I^ ^(60) It has also been PDFof the combined deterministic and statistical field will have a Pareto distribution (non-central Lomax distribution), represented by equation (61) below. f^ tot ^ 2^ ^ ^ h ^ htot^ (61)

[0067] FIG.17input defining coupling loss factors between different systems, such as the combinations 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 extensible to cover more than one structure or system. In this regard, in extensible implementations of the probabilistic response prediction process 300, the user input received at 302 may further include coupling loss factors between systems along with additional physical dimensions or other information characterizing the apertures or junctions between different cavities, enclosures or structures corresponding to the respective interconnected systems. For general multiple connected system implementations, the probabilistic response prediction process 300 determines the statistical mean net power inputs and net power losses between connected systems in addition to the statistical mean and frequency variance for the connected systems (e.g., at 304 as described in U.S. Patent No. 10,338,117 and U.S. Patent No.10,156,599). Thereafter, the probabilistic response prediction process 300 accounts for the coupled energy levels between systems, along with the uncertainty contributions of net power 36   ATTORNEY DOCKET NO.: 106.0007PC losses to and / or net power inputs from connected systems to determine the expected reverberant field response of multiple connected wavefields with the input probability level as described above in the context of equations (41)-(45). On the other hand, for mixed deterministic and statistical wavefield systems, the probabilistic response prediction process 300 may utilize the subject matter described in U.S. Patent No. 10,338,117 and U.S. Patent No. 10,156,599 to arrive field response levels of the deterministic wavefields based on excitation strength and radiation loss factors input by the user. The conditional probability density of the total mixed field response level is then determined at 306 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 Q factor uncertainty and excitation uncertainty as described above in the context of equations (47) and (49). In this regard, it should be appreciated that the probabilistic response prediction process 300 is not limited to an individual reverberant system or any particular type, number or configuration of systems having coupled or connected wavefields.

[0068] As used herein, the word “exemplary” means “serving as an example, instance, or illustration.” Thus, any embodiment described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments. All of the embodiments described herein are exemplary embodiments provided to enable persons skilled in the art to make or use the invention and not to limit the scope of the invention which is defined by the claims.

[0069] Those of skill in the art will appreciate that the various illustrative logical blocks, modules, circuits, and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both. Some of the embodiments and implementations are described above in terms of functional and / or logical block components (or modules) and various processing steps. However, it should be appreciated that such block components (or modules) may 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 37   ATTORNEY DOCKET NO.: 106.0007PC hardware or software depends upon the particular application and design constraints imposed on the overall system. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention. For example, an embodiment of a system or a component may employ various integrated circuit components, e.g., memory elements, digital signal processing elements, logic elements, look-up tables, or the like, which may carry out a variety of functions under the control of one or more microprocessors or other control devices. In addition, those skilled in the art will appreciate that embodiments described herein are merely exemplary implementations.

[0070] The various illustrative logical blocks, modules, and circuits described in connection with the embodiments disclosed herein may be implemented or performed with 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 in the alternative, 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, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration.

[0071] 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. A software module may reside in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, 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. In the alternative, the storage medium may be integral to the processor. The processor and the storage medium may reside in an ASIC. 38   ATTORNEY DOCKET NO.: 106.0007PC

[0072] 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 may 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 can carry out the described operations, tasks, and functions by manipulating electrical signals representing data bits at memory locations in the system memory, as well as other processing of signals. The memory locations where data bits are maintained are physical locations that have particular electrical, magnetic, optical, or organic properties corresponding to the data bits. It should be appreciated that the various block components shown in the figures may be realized by any number of hardware, software, and / or firmware components configured to perform the specified functions. For example, an embodiment of a system or a component may employ various integrated circuit components, e.g., memory elements, digital signal processing elements, logic elements, look-up tables, or the like, which may carry out a variety of functions under the control of one or more microprocessors or other control devices.

[0073] When implemented in software or firmware, various elements of the systems described herein are essentially the code segments or instructions that perform the various tasks. The program or code segments can be stored in a processor-readable medium or transmitted by a computer data signal embodied in a carrier wave over a transmission medium or communication path. The “computer-readable medium”, “processor-readable medium”, or “machine-readable medium” may include any medium that can store or transfer information. Examples of the processor-readable medium include an electronic circuit, a semiconductor memory device, a ROM, a flash memory, an erasable ROM (EROM), a floppy diskette, a CD- ROM, an optical disk, a hard disk, a fiber optic medium, a radio frequency (RF) link, or the like. The computer data signal may include any signal that can propagate over a transmission medium such as electronic network channels, optical fibers, air, electromagnetic paths, or RF links. The code segments may be downloaded via computer networks such as the Internet, an intranet, a LAN, or the like.

[0074] Some of the functional units described in this specification have been referred to as “modules” in order to more particularly emphasize their implementation independence. For 39   ATTORNEY DOCKET NO.: 106.0007PC example, functionality referred to herein as a module may be implemented wholly, or partially, as a hardware circuit comprising 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, or the like. Modules may also be implemented in software for execution by various types of processors. An identified module of executable code may, for instance, comprise one or more physical or logical modules of computer instructions that may, for instance, be organized as an object, procedure, or function. Nevertheless, the executables of an identified module need not be physically located together, but may comprise disparate instructions stored in different locations that, when joined logically together, comprise the module and achieve the stated purpose for the module. Indeed, a module of executable code may be a single instruction, or many instructions, and may even be distributed over 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 over different locations including over different storage devices, and may exist, at least partially, merely as electronic signals on a system or network.

[0075] In this document, relational terms such as first and second, and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Numerical ordinals such as “first,” “second,” “third,” etc. simply denote different singles of a plurality and do not imply any order or sequence unless specifically defined by the claim language. The sequence of the text in any of the claims does not imply that process steps must be performed in a temporal or logical order according to such sequence unless it is specifically defined by the language of the claim. The process steps may be interchanged in any order without departing from the scope of the invention as long as such an interchange does not contradict the claim language and is logically coherent.

[0076] Furthermore, the foregoing description may refer to elements or nodes or features being "coupled" together. As used herein, unless expressly stated otherwise, "coupled" means that one element / node / feature is directly or indirectly joined to (or directly or indirectly 40   ATTORNEY DOCKET NO.: 106.0007PC communicates with) another element / node / feature, and not necessarily mechanically. For example, two elements may be coupled to each other physically, electronically, logically, or in any other manner, through one or more additional elements. Thus, although the drawings may depict one exemplary arrangement of elements directly connected to one another, additional intervening elements, devices, features, or components may be present in an embodiment of the depicted subject matter. In addition, certain terminology may also be used herein for the purpose of reference only, and thus are not intended to be limiting.

[0077] While at least one exemplary embodiment has been presented in the foregoing detailed description of the invention, it should be appreciated that a vast number of variations exist. It should also be appreciated that the exemplary embodiment or exemplary embodiments are only examples, and are not intended to limit the scope, applicability, or configuration of the invention in any way. Rather, the foregoing detailed description will provide those skilled in the art with a convenient road map for implementing an exemplary embodiment of the invention. It being understood that various changes may be made in the function and arrangement of elements described in an exemplary embodiment without departing from the scope of the invention as set forth in the appended claims. 41

Claims

ATTORNEY DOCKET NO.: 106.0007PC CLAIMS What is claimed is:

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

2. The method of claim 1, further comprising determining the unconditional probability density function for the expected field response based on the conditional probability density function with marginal probability distribution for uncertain input conductance and a marginal probability density function for at least one of effective input current uncertainty associated with the excitation energy and Q factor uncertainty associated with the loss factor of the reverberant system. 42   ATTORNEY DOCKET NO.: 106.0007PC 3. The method of claim 2, wherein the unconditional probability density function is governed by the equation, ^2^ ^ f21^ ^^E^M^r^x,^E^^^^ exp^ ^ r^^ ^f^x^f^y^f z1^ ^dy dx dMwherein:field response at anylocationxand any frequency^;M^ E2 r^ xyzis the statistical mean of the excitation energy;magnitude of an effective current of excitation energy;y^ Qis the Q factor;conductance scaled by frequency^, permittivity^and volumeV;f^ x ^ ^ f 2^ IS ^is a first user input probability density function of effective currentwith the excitation energy; f^ y ^ ^ f ^ Q ^is a second user input probability density function of Q factor uncertaintyfactor of the reverberant system; and f^ M xy ^ ^ f ^ z ^ ^3 ^ ^ V f ^ G ii ^ ^ ^ ^is the scaled marginal probability density functionfor the input conductance frequency uncertainty.

4. The method of claim 1, wherein determining the marginal probability density function for 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 factor and the statistical mean of the field response. 43   ATTORNEY DOCKET NO.: 106.0007PC 5. The method of claim 1, wherein the unconditional probability density function is ^2overned by the equation f ^ E21 r ,^ E^ ^^ r ^g^x^^ ^exp^f^ M ^dM^^^, wherein:f 2^ Er ^x, ^ ^ ^field response atany locationxand any frequency^;M^ E2 ris the spatial mean of the magnitude squared field response; andf^ M ^ ^ f 2^ Er ^is a user input marginal probability density function of allmean squared 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. The method of claim 5, wherein 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 equation ^ fL^2 ^^Er^x,^^ ^^^^^^^, wherein:rdensity function for the field response;^ ^1 ^is the first parameter of the distribution;M ^ E 2 2r^ ISGii ^^^Q 3 ^ ^ V is mean squared electric field;of the distribution;44   ATTORNEY DOCKET NO.: 106.0007PC r2 ^ Gii^^ ^^1 1 K 1 K 1 2 ^^ ^ ^1^ ^ ^ ^1^ ^ ^ ^1^ is relative frequeLN^m ^ ω ^ ^^ ^^N N^^ ^^L L^^LN^^ncymω nis modal overlap;density;spatial mode shape variance; andthe number of receiver and source positions used to estimater2^M^ ^ ^as a statistical quantity.

8. The method of claim 5, the input probability value comprising user input percentileprobabilitiesP max, P min, wherein the expected field response comprises maximum and minimumexpected field responses to the excitation energy independent of location and frequency within thereverberant field at the user input percentile probabilitiesP max, P mincalculated as respectivepercentile maximum and minimum from the inverse function of the unconditional probability density function.

9. The method of claim 7, the input probability value comprising user input percentileprobabilitiesP max, P min, wherein the expected field response comprises maximum and minimumexpected field responses to the excitation energy independent of location and frequency within thereverberant field at the user input percentile probabilitiesP max, P mincalculated as respectivepercentile maximum and minimum from the inversegoverned by the equation, ^1^1^ ^ ^ ^^ ^^ ^^ ^ ^ ^ ^^^ ^^ ^ ^component of electric field at he first parameter of the distribution ^^E 2^ is t ^ ^ ^^ ^ ^;45   ATTORNEY DOCKET NO.: 106.0007PC esecond parameter of the distributionLomax^ 2^ is th f^ ^Er^x, ^ ^ ^^^ ^ ^.

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

11. The method of claim 10, wherein the maximum and minimum expected field response to the excitation energy independent of location and frequency within the reverberantfield at user input percentile probabilitiesP max, P mincomprises calculating the percentile maximumand minimum from the inverse function of the unconditional probability density function.

12. The method of claim 2, wherein determining the unconditional probability density function comprises using a single lognormal distribution for the product of two or more component marginal uncertainties in the mean field random variable and calculating the unconditional field response distribution governed by the equation ^E 2 ^^x ^ ^^^^1 ^E2E ^r^^ 1 ^ 2^^^ ^ ^ ^^ 2^ 2 ^ M ,fE x, 2^ r ^ ^ ^ ^is the unconditional probability density function (PDF) of magnitudeof electric field, at any locationxand any frequency^;M^ E 2r ^ I 2S G ii ^ ^ ^ Q 3 ^ ^ Vof uncertainty in the46   ATTORNEY DOCKET NO.: 106.0007PC ^2LM^ ^ 2 ^^Ln^Gii^^^^ ^ 2 ^^Ln^Q^^^^ ^ 2 ^^Ln^I 2S^^^ is sum of the variances of variables inM.

13. The method of claim 2, wherein determining the unconditional probability density function comprises using a single inverse gamma distribution for the product of two or more component uncertainties in the mean field random variable and calculation by the closed form 2^^ Manalytic PDF equation fM E2 ^^^Er ^x,^^ ^^^^^M ^^+1^2 M , wherein: f^ Er x 2^ , ^ ^ ^is(PDF) of magnitude squared,^of electric field, at any locationxand any frequency ;^M^Exp^^^Lm^ ^2Lm^^^^ ^ 1^ is the first PDF parameter, a function of log random^^^22^^Lm^ ^^ ^ ^ ^ ^log^LM^ Ln ^ Gii^ ^ Ln ^ Q ^ ^ Ln ^^I 2S^^is sum of the means of uncertainty in thenatural logarithm of each of the three random variables inM; and ^2 2L^ ^ 2 ^^Ln ^ Gii^ ^^^ ^ 2 ^^Ln ^ Q ^ ^ ^ ^ 2 ^^Ln^IS ^^^is sum of the variances ofvariables inM.

14. The method of claim 13, wherein the maximum and minimum expected field response to the excitation energy independent of location and frequency within the reverberantfield at user input percentile probabilitiesP max, P mincomprises calculating the percentile maximum47   ATTORNEY DOCKET NO.: 106.0007PC and minimum from the inverse function governed by the equation, 1 ^1I P^ ^1^ ^^ ^P^ ^^^ ^P^E2r^ ^^^^^ 1 ^ ^^ ^ ^^ 1 ^ ^^ wherein:^ ^component of electric field at probability2^ is the first parameter of the distribution fLomax^^^Er^x, ^ ^ ^^^ ^ ^; and2^ is the second parameter of the distribution^ ^^^ ^ ^.

15. The method of claim 2, for the case where uncertain excitation arises from net mean power input from a connected reverberant wavefield wherein determining the unconditional probability density function comprises numerical integration of an exponential distribution for the conditional electric field and user input marginal probability density functions for frequency uncertainty of input conductance and at least one user input marginal probability density function for effective source current uncertainty associated with net power input from a connected reverberant wavefield, governed by the equation ^2^ ^ 2 1^ E ^ ^ M ^^ ^ r2 ^ 1 ^ 1 2^ ^^ ^ ^^ ^ ^ ^1fE x , ^ 2^ 1 ^ ^ ^is the unconditional probability density function for the field response ofat any locationxand any frequency^;M 2 1^ E 1 ^ I2 21 G 11 Q 1 3 ^ ^ 1 V 1fromatf2^ I21 ^is a first user input marginal probability density function for uncertainty incurrent from at least one connected reverberant system 2; 48   ATTORNEY DOCKET NO.: 106.0007PC f^ Q1 ^is a second user input marginal probability density function for uncertainty in Qfactor of the reverberant system 1; and f^M1 I 221 Q 1 ^ ^3 ^ ^ 1 V 1 f ^ G 11 ^ ^ ^ ^is the scaled marginal probability density functionforG11 ^ ^ ^of reverberant system 1.

16. The method of claim 15, for the case where uncertain excitations arise from net mean power input from connected reverberant wavefields wherein the marginal probability density function for effective source current from at least one connected wavefield is governed by the equation ^ 2 ^ ^^2^ ^^ 1 ^^^ 1 ^^1^^2 2^ ^ ^ 1^^2^f^I21^ ^^ ^2^^^ ^ ^I21^ ^^^^2^^u^ ^I21^ ^ ^I21^ ^^^^2^^u^I21^ ^^^f2^ I21 ^is marginal probability density function (PDF) of effective excitation currentfrom at least one connected reverberant system 2 ^1^ ^1V1E21 is a variable proportional to mean total energy in the subject reverberant^2^ ^2V2E2 2is a variable proportional to mean total energy in the connected2; ^^^ ^21 ^^^^ 1^is the first parameter of the Inverse Gamma marginal distribution forcoupling loss factor uncertaintyf ^ ^21 ^;^21is mean of theloss factor between reverberant system 1 and reverberant ^^^2 ^^^21^^^ 221 ^ ^is the second parameter of the Inverse Gamma marginalfactor uncertaintyf ^ ^21 ^;49   ATTORNEY DOCKET NO.: 106.0007PC ^2^^21 ^is statistical variance of the uncertain coupling loss factor between reverberant system 1 and reverberant system 2; and u^X^is the Heaviside unit step function.

17. The method of claim 15, for the case where multiple uncertain excitations arise from both direct power inputs and net mean power inputs from multiple connected reverberant wavefields wherein the marginal probability density function for effective excitation current is ^ P IN^governed by the equationf^ITot21^^f^^1,Tot^^ ^^G11^ ^ ^f^P IN 1,Tot^, wherein: ^ITot 21^ismagnitudethe total over multiple excitations; G11^^^^ ^n1^^^is the statistical mean input conductance of the subject reverberant2^V1PIN IN1, Tot^ P1^^PˆINk 1is total net power input to the subject reverberant wavefield ;k to thef^PIN1^, f^Pˆ IN21 ^, f^Pˆ IN31 ^, ... f^Pˆ INk1 ^are the marginal PDFs of each of the statistically uncorrelated net power input components.

18. The method of claim 2, for the case where Q factor uncertainty arises from net mean power loss to a connected reverberant wavefield wherein determining the unconditional probability density function comprises numerical integration of an exponential distribution for the conditional electric field and user input marginal probability density functions for frequency uncertainty of input conductance and at least one user input probability density function for effective Q factor uncertainty associated with net power loss to a connected reverberant wavefield, governed by the equation 50   ATTORNEY DOCKET NO.: 106.0007PC ^2^ ^ f^ E1 ,21 ^^^ exp^ E^r^ 2 ^ ^ M^^ ^^1 ^1^ 12^ 1 2^x ^ ^^f I f^Q^fG ii^2^2 dQ 12 d I 1 d M 1fE 2^ 1 ^x , ^ ^ ^is the unconditional probability density function for the field response ofat any locationxand any frequency^;M^2 2 1E 1 ^ I 1 G 11 Q 21 3 ^ ^ 1 V 1is the statistical mean magnitude squared field of thefI 2^ 1 ^is a first user input marginal probability density function for uncertainty ineffective source current from external excitation of reverberant system 1; f^ Q12 ^is a second user input marginal probability density function for uncertainty in Qfactor with the losses to at least one connected reverberant system 2; and f^M1 I 21 Q 12 ^ ^3 ^ ^ 1 V 1 f ^ G 11 ^ ^ ^ ^is the scaled marginal probability density functionforG11 ^ ^ ^of the reverberant system 1.

19. The method of claim 18, for the case where Q factor uncertainty arises from net mean power loss from connected reverberant wavefields wherein the marginal probability density function of Q factor attributable to losses to at least one connected reverberant wavefield system ^^^ ^^ ^ ^1 ^ ^ ^ 2^^ ^^^^ 1 ^ ^^^^is governed by the equation ^ ^^ ^, wherein:12 ^f^ Q12 ^is the marginal probability density function of Q factor uncertainty associated with thelosses to at least one connected reverberant system 2; ^^^ ^12 ^^^^ 1^is the first parameter of the marginal distribution for coupling loss factor f^ Q12 ^;51   ATTORNEY DOCKET NO.: 106.0007PC ^12is mean of the coupling loss factor between reverberant system 1 and reverberant ^^^2 ^^^12^ ^ ^212^^is the first parameter of the marginal distribution for coupling loss factor^ ^21 ^; and^2^^12 ^of the coupling loss factor between reverberant system 1 andreverberant system 2.

20. The method of claim 18, for the case where multiple uncertain power losses arise from both power loss to internal dissipation Q factor and net power losses to multiple connected reverberant wavefields wherein the marginal probability density function for effective Q factor is ^ 3 ^ ^V2 Loss 2 ^^^1 E 1 ^^P^^ ^1,Tot ^Lossthe total over multiple losses; P Loss 1, Tot^ PLoss 1^^PLoss 1ˆ kis total net power loss from the subject reverberant wavefield ;Loss^^ ^Loss^^ 1^ ˆLoss^is the marginal PDF of total net powerlossf^PLoss1^, f^Pˆ Loss12 ^, f^P ˆ Loss13 ^, ... f^P ˆ Loss1k ^are the marginal PDFs of each of the statistically21. The method of claim 1, wherein the user input includes defining characteristics of a direct field, the user input characteristics including physical location and radiation aperture dimensions of a direct field source and physical locations of sensors in the enclosure structure; 52   ATTORNEY DOCKET NO.: 106.0007PC the method further comprising, prediction of the direct field response at sensor locations; prediction of the total mixed field at sensor locations, governed by the equation 2^ Tot Dir Re 2f Er ^x o, ^ ^ ^ ^ f ^ E vr ^ x o , ^ ^ ^ E r ^ ^ ^ ^,whereinof total direct plus reverberantand frequency^;EDirr ^x o, ^ ^is the direct electric field at response locationxand frequency^; andERe vr ^ ^ ^is the reverberant electric field at all locations and frequency^.

22. The method of claim 21, for the case where the total electric field within an enclosure comprises the sum of a deterministic direct field plus an uncertain reverberant field with high modal overlap, wherein the unconditional probability density function is governed by the 2 ETot ^ Dir Rev ^ ^ Dir 2 Re 2 ^r ^x^E^x^E E^x^ ^Evo ^ r o r ^ ^ r o ris unconditional probability density function (PDF) of total magnitudeof electric field, at response locationx oand any frequency^;2ERevr is mean magnitude squared level of the reverberant field;o ^is magnitude of deterministic electric field at response locationo; andI0 ^ ^is the modified Bessel function of the first kind with order zero.

23. The method of claim 21, for the case where the total electric field within an enclosure comprises the sum of a deterministic direct field plus an uncertain reverberant field with 53   ATTORNEY DOCKET NO.: 106.0007PC low modal overlap, wherein the unconditional probability density function is governed by the ^ ^ ^ M equationf^ETotr^x o^ ^^2ETotr^xM Mo^22^ M +1 , wherein: Tot^ ^^ Dir^ ^^total magnitude squared,of electric field, at response locationx oand any frequency^;EDirr ^ x o ^is magnitude of deterministic electric field at response locationx;^ ^^^M^ 1^ is the first PDF parameter, a function of log random^^^22^^Lm^ ^^ ^ ^ ^ ^log^LM^ Ln^Gii ^^ Ln^Q^^ Ln ^^I 2S^^is sum of the means of uncertainty in theinM; and ^2LM^ ^ 2 ^^Ln ^ Gii^ ^^^ ^ 2 ^^Ln ^ Q ^ ^^^ ^ 2 ^^Ln^I 2S^^^is sum of the variances ofvariables inM.

24. The method of claim 1, wherein user defines an additional low dimensionality wavefield system - such as a multi-conductor transmission line - located within the reverberant wavefield system, the method further comprising: user inputs characteristics including materials and dimensions of conductors and cross section materials and dimensions and terminal impedance loads; user inputs the deterministic applied voltage or electric field excitation at the transmission line terminals; prediction of the deterministic terminal magnetic field and current response to the applied 54   ATTORNEY DOCKET NO.: 106.0007PC terminal excitations prediction of the statistical distribution of terminal magnetic field and current response to the applied reverberant field excitation prediction of the total mixed deterministic-statistical response magnetic field and current at the terminals, governed by the equationf 2 2 2^htot ^^ happ^ f^hrev ^ , whereinf^h 2tot ^ is the probability density function of magnitude squared, total mixedstatistical terminal magnetic field response;h 2 appis the magnitude squared, deterministic terminal magnetic field response to applied excitations; andf^h 2rev ^ is the probability density function of magnitude squared terminal magnetic fieldto reverberant field excitation along the transmission line.

25. The method of claim 24, for the case where the total terminal magnetic field comprises the sum of a deterministic field from applied terminal electric field plus an uncertain terminal field excited by reverberant field with high modal overlap, wherein the unconditional probability density function is governed by the equation 2h^^ happhrev^ ^^ ^ h 2app^ h2rev^^(PDF) of total magnetic fieldtermination; hrevis component of terminal magnetic field magnitude of excited by reverberant field; happis component of terminal magnetic field magnitude excited by deterministic electricfield applied at terminalsx; and55   ATTORNEY DOCKET NO.: 106.0007PC I0 ^ ^is the modified Bessel function of the first kind with order zero.The method of claim 24, for the case where the total electric field within an enclosure comprises the sum of a deterministic direct field plus an uncertain reverberant field with low modal overlap, wherein the unconditional probability density function is governed by the ^ ^ Mequationf^h tot^^2h M ^ M tot 22^ M +1 , wherein:function (PDF) of total magnetic fieldhappis component of terminal magnetic field magnitude excited by deterministic electric field at terminals; ^M^Exp^^^Lm^ ^2Lm^^^^M^ 1^ is the first PDF parameter, a function of log randomvariable^^ ^22^ ^^ ^^^Lm ^^log^L^ Ln^Gii ^^ Ln^Q^^ Ln ^^I 2 ^^is sum of the means of uncertainty in thevariables inM; and ^2 2L^ ^ 2 ^^Ln^Gii^^^^ ^ 2 ^^Ln^Q^^ ^ ^ 2 ^^Ln^IS ^^^ is sum of the variances ofvariables inM.

27. A computer-readable medium having computer-executable instructions stored thereon that, when executed by a processing system, cause the processing system to: 56   ATTORNEY DOCKET NO.: 106.0007PC receive user input defining characteristics of a reverberant system, the characteristics including physical dimensions of an enclosure structure and a loss factor; identify excitation characteristics for excitation energy to be input to the reverberant system; determine a marginal probability density function for input conductance frequency uncertainty of the reverberant system based at least in part on the physical dimensions, the loss factor and a statistical mean of the excitation energy; determine an unconditional probability density function for the field response of the reverberant system to the excitation energy based at least in part on the input conductance frequency uncertainty; determine an expected field response by the reverberant system with respect to frequency in response to the excitation energy for an input probability value using the unconditional probability density function; and provide a graphical user interface (GUI) display including a graphical representation of the expected field response with respect to frequency.

28. The computer-readable medium of claim 27, wherein the GUI display comprises a graph depicting a maximum expected field response with respect to frequency for the input probability value.

29. A computer device comprising: a computer-readable medium having computer-executable instructions stored thereon; and a processor coupled to the computer-readable medium to execute the computer-executable instructions to provide software configurable to: receive user input defining characteristics of a reverberant system, the characteristics including physical dimensions of an enclosure structure and a loss factor; identify excitation characteristics for excitation energy to be input to the reverberant system; determine a marginal probability density function for input conductance frequency uncertainty of the reverberant system based at least in part on the physical dimensions, the loss factor and a statistical mean of the excitation energy; 57   ATTORNEY DOCKET NO.: 106.0007PC determine an unconditional probability density function for the field response of the reverberant system to the excitation energy based at least in part on the input conductance frequency uncertainty; determine an expected field response by the reverberant system with respect to frequency in response to the excitation energy for an input probability value using the unconditional probability density function; and provide a graphical user interface (GUI) display including a graphical representation of the expected field response with respect to frequency.

30. The computer-readable medium of claim 29, wherein the GUI display comprises a graph depicting a maximum expected field response with respect to frequency for the input probability value. 58

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