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A method utilizes time-domain measurements of a nonlinear device to produce or extract a behavioral model from embeddings of these measurements. The resulting behavioral model of the nonlinear device is a black-box model that accommodates nonlinear devices with one or more input ports and one or mor

A method utilizes time-domain measurements of a nonlinear device to produce or extract a behavioral model from embeddings of these measurements. The resulting behavioral model of the nonlinear device is a black-box model that accommodates nonlinear devices with one or more input ports and one or more output ports. The black-box model is a functional form that is a closed form function of input variables that produces an output as opposed to a structural form. The method of producing a behavioral model comprises the steps of applying an input signal to the nonlinear device, sampling the input signal to produce input data, measuring a response of the device to produce output data, creating an embedded data set, fitting a function to the embedded data set, and verifying the fitted function. The method may apply a CDMA type input signal in the step of applying, and/or may fit a novel radial basis function in the step of fitting a function. The input signal is constructed from a single CDMA signal representation. The method of constructing is not dependent on knowledge of the behavioral model of the nonlinear device. The novel radial basis function may be determined using a modified gaussian function.

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1. A method of producing a behavioral model of a nonlinear device from embeddings of time-domain measurements, the device being real or virtual and having one or more input ports and one or more output ports, the method comprising:applying an input signal to the input port of the nonlinear device;sa

1. A method of producing a behavioral model of a nonlinear device from embeddings of time-domain measurements, the device being real or virtual and having one or more input ports and one or more output ports, the method comprising:applying an input signal to the input port of the nonlinear device;sampling the input signal to produce input data;measuring a response to the input signal at the output port of the device to produce output data corresponding to the input data;creating an embedded data set using a first subset of the input data and a first subset of the output data;fitting a function to the embedded data set; andverifying the fitted function using a second subset of the input data and a second subset of the output data, such that the verified fitted function is the behavioral model of the nonlinear device,wherein applying an input signal comprises constructing a CDMA type signal u(t) as the input signal, the CDMA type signal being given by where f c is a center frequency, b i (t) are random vectors, p i are coefficients of a bandpass finite impulse response (FIR) filter, N is a vector dimension expressed as an integer, and A is a constant amplitude value, the constructed CDMA type input signal being applied to the input port of the nonlinear device. 2. The method of claim 1, wherein constructing a CDMA type signal u(t) comprises:selecting the center frequency f c , the center frequency f c being a center frequency of an operational range of the nonlinear device;choosing the vector dimension N;generating the random vectors b i , the random vectors b i (t) being uniformly distributed binary random vectors of 1's and 0's;computing the filter coefficients p i ;picking the constant amplitude value A; andcalculating the input signal u(t). 3. The method of claim 1 wherein the step of fitting a function comprises using a radial basis function given bywherein β, α, and ω i are coefficients that are determined during the step of fitting, z(t) is output data, and c i are centers of the radial basis function and wherein φ(•) is a function of the centers c i and the output data z(t). 4. The method of claim 3 wherein the function φ(•) is a modified gaussian function given bywherein x represents a part of z reconstructed using prior output values, u represent the parts of z constructed from the input signal, the coefficient c is an output center associated with and having a same dimension as x, the coefficient d is an input center associated with and having a same dimension as u, and the terms v and w are fixed-widths of output and input gaussian functions, respectively. 5. A method of producing a behavioral model of a nonlinear device from embeddings of time-domain measurements, the device being real or virtual having one or more input ports and one or more output ports, the method comprising:applying an input signal to the input port of the nonlinear device;sampling the input signal to produce input data;measuring a response to the input signal at the output port of the device to produce output data corresponding to the input data;creating an embedded data set using a first subset of the input data and a first subset of the output data;fitting a function to the embedded data set; andverifying the fitted function using a second subset of the input data and a second subset of the output data, such that the verified fitted function is the behavioral model of the nonlinear device,wherein fitting a function comprises using a radial basis function to fit to the embedded data set, the radial basis function being given by where β, α, and ω i are coefficients that are determined during fitting a function, term z(t) comprises input and output data, c i are centers of the radial basis function, and φ(•) is a function of the centers c i and the term z(t). 6. The method of claim 5 wherein the function φ(•) is a modified gaussia n function given bywherein x represents a part of z reconstructed using prior output values, u represent the parts of z constructed from the input signal, the coefficient c is an output center associated with and having a same dimension as x, the coefficient d is an input center associated with and having a same dimension as u, and the terms v and w are fixed-widths of output and input gaussian functions, respectively. 7. A method of producing a behavioral model of a nonlinear device from embeddings of time-domain measurements, the device being real or virtual having one or more input ports and one or more output ports, the method comprising:applying an input signal to the input port of the nonlinear device;sampling the input signal to produce input data;measuring a response to the input signal at the output port of the device to produce output data corresponding to the input data;creating an embedded data set using a first subset of the input data and a first subset of the output data;fitting a function to the embedded data set; andverifying the fitted function using a second subset of the input data and a second subset of the output data, such that the verified fitted function is the behavioral model of the nonlinear device,wherein fitting a function comprises using a radial basis function to fit to the embedded data set, the radial basis function being given by where β, α, and ω i are coefficients that are determined during fitting a function, term z(t) comprises input and output data, c i are centers of the radial basis function, and φ(•) is a function of the centers c i and the term z(t), andwherein applying an input signal comprises constructing a CDMA type signal u(t) as the input signal, the CDMA type signal being given by where f c is a center frequency, b i (t) are random vectors, p i are coefficients of a bandpass finite impulse response (FIR) filter, N is a vector dimension expressed as an integer, and A is a constant amplitude value, the constructed CDMA type input signal being applied to the input port of the nonlinear device. 8. A method of determining a radial basis function for fitting to data that corresponds to a nonlinear device in response to an input signal applied to an input port of the nonlinear device during modeling of the nonlinear device, the method comprising:determining centers c and d;calculating coefficients v and w; andfinding the radial basis function where β, α, and ω i are coefficients that are determined as the radial basis function is fitted to data obtained from sampling the input signal applied to the nonlinear device and from measuring a response to the input signal from the nonlinear device, term z(t) comprises input and output data, the output data corresponding to the measured response from the nonlinear device, and c i are centers of the radial basis function,wherein φ(•) is a modified gaussian function given by where x represents a part of z reconstructed using prior output values, u represent the parts of z constructed from the input signal, the coefficient c is an output center associated with and having a same dimension as x, the coefficient d is an input center associated with and having a same dimension as u, and the terms v and w are fixed-widths of output and input gaussian functions, respectively. 9. The method of claim 8, wherein the term v is a standard deviation of a first subset of the output data and obtained from the nonlinear device wherein the term w is a standard deviation of a first subset of the input data obtained from the sampled input signal applied to the nonlinear device. 10. An apparatus for producing a behavioral model of a nonlinear device from embeddings of time domain measurements, the device having one or more inputs and one or more outputs comprising:a signal generator for generating an input signal that is applied to the input of the device, the generat ed input signal being a CDMA type signal being given by where f c is a center frequency, b i (t) are random vectors, p i are coefficients of a bandpass finite impulse response (FIR) filter, N is a vector dimension expressed as an integer, and A is a constant amplitude value;a data acquisition system for sampling the input signal, measuring an output signal at the output of the device in response to the input signal and producing input data and output data from the sampled input signal and from the measured output signal; anda signal processing computer for creating an embedded data set from a first subset of the input data and a first subset of the output data, for fitting a function to the embedded data set and for verifying that the fitted function models the device using a second subset of the input data and a second subset of the output data. 11. The apparatus of claim 10, wherein the random vectors b i (t) are uniformly distributed binary random vectors of 1's and 0's. 12. The apparatus of claim 10, wherein the center frequency f c is a center frequency of an operational range of the nonlinear device. 13. The apparatus of claim 10, wherein the function used by the signal processing computer to fit to the embedded data comprises a radial basis function given bywhere β, α, and ω i are coefficients that are determined during fitting a function, term z(t) comprises input and output data, c i are centers of the radial basis function, and φ(•) is a function of the centers c i and the term z(t). 14. The apparatus of claim 13, wherein the function φ(•) is a modified gaussian function given bywherein x represents a part of z reconstructed using prior output values, u represent the parts of z constructed from the input signal, the coefficient c is an output center associated with and having a same dimension as x, the coefficient d is an input center associated with and having a same dimension as u, and the terms v and w are fixed-widths of output and input gaussian functions, respectively. 15. The apparatus of claim 14, wherein the term v is a standard deviation of the first subset of the output data and wherein the term w is a standard deviation of the first subset of the input data. 16. An apparatus for producing a behavioral model of a nonlinear device from embeddings of time domain measurements, the device having one or more inputs and one or more outputs comprising:a signal generator for generating an input signal that is applied to the input of the device;a data acquisition system for sampling the input signal, measuring an output signal at the output of the device in response to the input signal and producing input data and output data from the sampled input signal and from the measured output signal; anda signal processing computer for creating an embedded data set from a first subset of the input data and a first subset of the output data, for fitting a function to the embedded data set and for verifying that the fitted function models the device using a second subset of the input data and a second subset of the output data,wherein the function used by the signal processing computer to fit to the embedded data comprises a radial basis function given by where β, α, and ω i are coefficients that are determined during fitting a function, term z(t) comprises input and output data, c i are centers of the radial basis function, and φ(•) is a function of the centers c i and the term z(t). 17. The apparatus of claim 16, wherein the function φ(•) is a modified gaussian function given bywhere x represents a part of z reconstructed using prior output values, u represent the parts of z constructed from the input signal, the coefficient c is an output center associated with and having a same dimension as x, the coefficient d is an input center associated with and having a same dimension as u, and the terms v and w are fixed- widths of output and input gaussian functions, respectively. 18. The apparatus of claim 17, wherein the term v is a standard deviation of the first subset of the output data and wherein the term w is a standard deviation of the first subset of the input data. 19. An apparatus for producing a behavioral model of a nonlinear device from embeddings of time domain measurements, the device having one or more inputs and one or more outputs comprising:a signal generator for generating an input signal that is applied to the input of the device;a data acquisition system for sampling the input signal, measuring an output signal at the output of the device in response to the input signal and producing input data and output data from the sampled input signal and from the measured output signal; anda signal processing computer for creating an embedded data set from a first subset of the input data and a first subset of the output data, for fitting a function to the embedded data set and for verifying that the fitted function models the device using a second subset of the input data and a second subset of the output data,wherein the function used by the signal processing computer to fit to the embedded data comprises a radial basis function given by where β, α, and ω i are coefficients that are determined during fitting a function, term z(t) comprises input and output data, c i are centers of the radial basis function, and φ(•) is a function of the centers c i and the term z(t), andwherein the signal generator generates a CDMA type signal as the input signal, the CDMA type input signal being given by where f c is a center frequency, b i (t) are random vectors, p i are coefficients of a bandpass finite impulse response (FIR) filter, N is a vector dimension expressed as an integer, and A is a constant amplitude value. 20. The apparatus of claim 19, wherein the random vectors b i (t) are uniformly distributed binary random vectors of 1's and 0's, and wherein the center frequency f c is a center frequency of an operational range of the nonlinear device.