Nonlinear modeling of a physical system using lookup table with polynomial interpolation

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First Claim
1. A digital predistortion (DPD) system, comprising:
 a digital signal processor configured to evaluate a nonlinear function for a complex input value, comprising;
a memory configured to store at least one lookup table with polynomial interpolation that represents said nonlinear function, wherein entries in said lookup table comprise polynomial coefficients of at least degree two for different segments of said nonlinear function, said polynomial coefficients configured to provide at least one of a rectangular or triangular basis function for each segment of said nonlinear function;
at least one processor, coupled to the memory, operative to;
obtain a point from said looktip table that is near a magnitude of said complex input value; and
generate a complex output value by evaluating said polynomial coefficients at said point to perform a Taylor Series expansion from said point.
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Abstract
Methods and apparatus are provided for nonlinear modeling of a physical system using lookup tables with polynomial interpolation. A nonlinear function is evaluated for a complex input value by obtaining at least one lookup table with polynomial interpolation that represents the nonlinear function, wherein entries in the lookup table comprise polynomial coefficients of at least degree two for different segments of the nonlinear function; obtaining a point from the lookup table that is near a magnitude of the complex input value; and generating a complex output value by evaluating the polynomial coefficients at the point to perform a Taylor Series expansion from said point. The nonlinear function characterizes, for example, a power amplifier or an inverse of a power amplifier and the lookup tables can be used, for example, to implement digital predistortion.
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PROCESSOR HAVING INSTRUCTION SET WITH USERDEFINED NONLINEAR FUNCTIONS FOR DIGITAL PREDISTORTION (DPD) AND OTHER NONLINEAR APPLICATIONS  
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20 Claims
 1. A digital predistortion (DPD) system, comprising:
a digital signal processor configured to evaluate a nonlinear function for a complex input value, comprising; a memory configured to store at least one lookup table with polynomial interpolation that represents said nonlinear function, wherein entries in said lookup table comprise polynomial coefficients of at least degree two for different segments of said nonlinear function, said polynomial coefficients configured to provide at least one of a rectangular or triangular basis function for each segment of said nonlinear function; at least one processor, coupled to the memory, operative to; obtain a point from said looktip table that is near a magnitude of said complex input value; and generate a complex output value by evaluating said polynomial coefficients at said point to perform a Taylor Series expansion from said point.  View Dependent Claims (3, 4, 5, 6, 7, 8, 9, 10)
 2. A digital predistortion (DPD) system, comprising:
a digital signal processor configured to evaluate a nonlinear function for a complex input value, comprising; a memory configured to store at least one lookup table with polynomial interpolation that represents said nonlinear function, wherein entries in said lookup table comprise polynomial coefficients of at least degree two for different segments of said nonlinear function; at least one processor, coupled to the memory, operative to; obtain a point from said looktip table that is near a magnitude of said complex input value; and generate a complex output value by evaluating said polynomial coefficients at said point to perform a Taylor Series expansion from said point, wherein values in said lookup table are configured to be obtained using a direct optimization technique that obtains physical data for said nonlinear function by applying a set of complex input samples x(n) to said nonlinear function and measuring a complex output y(n) of said nonlinear function and directly computes parameters φ
of a memory model for said nonlinear function from said physical data, said memory model comprises one or more lookup tables having polynomial interpolation, and said parameters φ
produce a substantially minimum mean square error, said parameters φ
are provided for storage as entries in said one or more lookup tables.
 11. A power amplifier, comprising:
a digital signal processor that evaluates a nonlinear function for a complex input value, comprising; a memory configured to store storing at least one lookup table with polynomial interpolation that represents said nonlinear function, wherein entries in said lookup table comprise polynomial coefficients of at least degree two for different segments of said nonlinear function, said polynomial coefficients configured to provide at least one of a rectangular or triangular basis function for each segment of said nonlinear function; and at least one processor, coupled to the memory, operative to; obtain a point from said looktip table that is near a magnitude of said complex input value; and generate a complex output value by evaluating said polynomial coefficients at said point to perform a Taylor Series expansion from said point.  View Dependent Claims (13, 14, 15, 16, 17, 18, 19, 20)
 12. A power amplifier, comprising:
a digital signal processor that evaluates a nonlinear function for a complex input value, comprising; a memory configured to store at least one lookup tablewith polynomial interpolation that represents said nonlinear function, wherein entries in said lookup table comprise polynomial coefficients of at least degree two for different segments of said nonlinear function; and at least one processor, coupled to the memory, operative to; obtain a point from said looktip table that is near a magnitude of said complex input value; and generate a complex outputvalue by evaluating said polynomial coefficients at said point to perform a Taylor Series expansion from said point, wherein values in said lookup table are configured to be obtained using a direct optimization technique that obtains physical data for said nonlinear function by applying a set of complex input samples x(n) to said nonlinear function and measuring a complex output y(n) of said nonlinear function and directly computes parameters φ
of a memory model for said nonlinear function from said physical data, said memory model comprises one or more lookup tables having polynomial interpolation, and said parameters φ
produce a substantially minimum mean square error, said parameters φ
are provided for storage as entries in said one or more lookup tables.
1 Specification
The present application claims priority to U.S. Patent Provisional Application Ser. No. 61/812,858, filed Apr. 17, 2013, entitled “Digital Front End (DFE) Signal Processing,” incorporated by reference herein.
The present application is related to U.S. patent application Ser. No. 14/168,621, filed Jan. 30, 2014, entitled “NonLinear Modeling of a Physical System Using Direct Optimization of LookUp Table Values;” PCT Patent Application No. PCT/US12/62179, filed Oct. 26, 2012, entitled “Software Digital Front End (SoftDFE) Signal Processing;” and PCT Patent Application No. PCT/US12/62186, filed Oct. 26, 2012, entitled “Processor Having Instruction Set with UserDefined NonLinear Functions for Digital PreDistortion (DPD) and Other NonLinear Applications,” each incorporated by reference herein.
The present invention is related to digital signal processing techniques and, more particularly, to techniques for modeling of nonlinear systems.
Digital predistortion (DPD) is a technique used to linearize a power amplifier in a transmitter to improve the efficiency of the power amplifier. A digital predistortion circuit inversely models the gain and phase characteristics of the power amplifier and, when combined with the amplifier, produces an overall system that is more linear and reduces distortion than would otherwise be caused by the power amplifier. An inverse distortion is introduced into the input of the amplifier, thereby reducing any nonlinearity that the amplifier might otherwise exhibit.
PCT Patent Application No. PCT/US12/62186, filed Oct. 26, 2012, entitled “Processor Having Instruction Set with UserDefined NonLinear Functions for Digital PreDistortion (DPD) and Other NonLinear Applications,” discloses nonlinear functions that include one or more parameters specified by a user, such as filter coefficient values or values from a lookup table. The disclosed DPD techniques are based on a generalized memory polynomial (GMP) model and implemented using lookup tables. Polynomial models, however, do not provide adequate performance for a number of common situations. For example, polynomial models do not capture functions that have discontinuities in them (such as a discontinuity of amplitude or derivative, or a higher order derivative).
U.S. patent application Ser. No. 14/168,621, filed Jan. 30, 2014, entitled “NonLinear Modeling of a Physical System Using Direct Optimization of LookUp Table Values,” provides improved techniques for nonlinear modeling of a physical system, such as a DPD system or a power amplifier, using direct optimization of lookup table values.
Nonetheless, a need remains for improved techniques for modeling nonlinear systems using lookup tables with polynomial interpolation.
Generally, improved techniques are provided for nonlinear modeling of a physical system using lookup tables with polynomial interpolation. According to one aspect of the invention, a nonlinear function is evaluated for a complex input value by obtaining at least one lookup table with polynomial interpolation that represents the nonlinear function, wherein entries in the lookup table comprise polynomial coefficients of at least degree two for different segments of the nonlinear function; obtaining a point from the lookup table that is near a magnitude of the complex input value; and generating a complex output value by evaluating the polynomial coefficients at the point to perform a Taylor Series expansion from said point. The nonlinear function characterizes, for example, a power amplifier or an inverse of a power amplifier. The twodimensional lookup tables can be used, for example, to implement digital predistortion.
According to another aspect of the invention, values in the lookup table are obtained using a direct optimization technique that obtains physical data for the nonlinear function by applying a set of complex input samples x(n) to the nonlinear function and measuring a complex output y(n) of the nonlinear function; and directly computes parameters Φ of a memory model for the nonlinear function from the physical data, wherein the memory model comprises one or more lookup tables having polynomial interpolation and wherein the parameters Φ produce a substantially minimum mean square error, wherein the parameters Φ are provided for storage as entries in the one or more lookup tables.
According to yet another aspect of the invention, the lookup tables with polynomial interpolation are used in a processor instruction as part of an instruction set of one or more of a scalar processor and a vector processor. The lookup table can be stored in a memory of a digital processor, and the polynomial interpolation is performed as part of a userdefined nonlinear instruction that takes a complex number as an input, x, and computes ƒ(x).
A more complete understanding of the present invention, as well as further features and advantages of the present invention, will be obtained by reference to the following detailed description and drawings.
Aspects of the present invention provide improved techniques for modeling nonlinear systems, such as a DPD system or a power amplifier. According to one aspect of the invention, a physical system, such as a DPD system, is modeled using a nonlinear model with lookup table values having polynomial interpolation. In one exemplary embodiment, the complex input and complex output data of a physical system are used to directly determine the polynomial values that are stored in the exemplary lookup tables that implement the nonlinear model. The model comprises one or more lookup tables having polynomial interpolation.
Existing DPD techniques are typically based on a generalized memory polynomial (GMP) model and are implemented using lookup tables. GMP itself models an ideal Volterra series. GMP models an actual physical system such as a reverse power amplifier response (i.e., output to input response). The lookup tables are obtained by first computing the coefficients of each of the GMP polynomials using conventional techniques (e.g., a least squares approach) and then sampling each of the polynomials to create the lookup tables for each polynomial. This can be considered modeling a model, or approximating an approximation. The lookup tables approximate the GMP polynomial model, which is itself an approximation of an ideal infinite tap Volterrabased predistorter.
Aspects of the present invention recognize that polynomial models do not work well over an extended interval of input signal and do not work well in the presence of a discontinuity (e.g., a derivative discontinuity). In addition, aspects of the present invention recognize that lookup tables will perform better in these cases.
With conventional one dimensional lookup tables, two values neighboring a desired value, x, are typically obtained from the lookup table, and then the result is obtained by performing a linear interpolation between the two obtained values.
Thus, aspects of the present invention provide a nonlinear model with lookup tables having polynomial interpolation. In one exemplary embodiment, the polynomial values for the lookup tables are directly optimized from physical data. The lookup table is defined as segments of polynomials defined by the coefficients of the polynomials stored in the lookup table for each segment. The output is computed by evaluating the polynomial for the lookup table point that is closest to the desired value, x.
According to a further aspect of the invention, in order to improve continuity of the output results between segments, linear interpolation is optionally performed between segments. For a given input, x, the two points in the lookup table that are closest to the given input, x, are obtained and the polynomial value corresponding to the two points are computed to form two intermediate values. Linear interpolation is then performed between these two intermediate values to obtain the final output.
The disclosed digital processors 100 may have a scalar architecture that processes a single complex number at a time, or a vector architecture that processes one or more complex numbers simultaneously. In the case of a vectorbased digital processor implementation, the input complex number is a vector comprised of a plurality of scalar complex numbers that are processed in parallel.
A vectorbased digital signal processor 100 may contain plural functional units for performing the functions described herein. For example, a dual digital signal processor 100 contains two functional units that are capable of performing two independent nonlinear function operations concurrently.
Generally, aspects of the present invention extend conventional digital signal processors to provide an enhanced instruction set that supports nonlinear functions using one or more lookup tables. The digital signal processor 100 in accordance with the present invention receives at least one number as an input, applies a function to the input and generates an output value.
ƒ(x)=y.
The function ƒ(x) is thus an unknown function defining the response of the actual physical system to be modeled, such as a reverse power amplifier response. Aspects of the present invention model the response ƒ(x) of the actual physical system using a model {circumflex over (ƒ)}(x) and estimate the parameters (polynomial LUT entries) of this model {circumflex over (ƒ)}(x) by performing direct optimization using the physical data.
As discussed hereinafter, the parameters (polynomial LUT entries) of the model {circumflex over (ƒ)}(x) are obtained using measured data x(n) and y(n). For example, to model the response of a power amplifier, x(n) is used as the complex input of the power amplifier and y(n) is used as the complex output of the power amplifier. Likewise, to model the inverse response of the power amplifier, a conceptual nonlinear system or black box is used that has as its complex input the power amplifier output and as its complex output the power amplifier input.
Thus, the nonlinear modeling process 110 applies the complex input samples x(n) to the physical system to measure a set of N complex output samples y(n) (n=0 to N−1) during step 220 to obtain the physical data for the system:
ƒ(x_{n})=y_{n }
The nonlinear modeling process 110 then directly optimizes the parameters Φ for the model during step 230 to find the best fit for the measured physical data. Generally, as discussed further below, the best fit identifies the parameters that provide a substantially minimum value for an exemplary mean square error using a least squares algorithm. In further variations, alternative algorithms can use, for example, recursive least squares (RLS) or least mean square (LMS) that achieve various tradeoffs between complexity and conversion speed.
As discussed further below, the model parameters comprise the polynomial entries for the LUT(s), referred to as {circumflex over (ƒ)}(x) of ƒ(x). The LUTs are then stored during step 240.
Segmented Polynomial Static NonLinear Model
The goal is to identify a static nonlinear function (real) ƒ(x)=y with a segmented polynomial lookup table.
As indicated above, a set of N samples n=0 to N−1 are measured:
ƒ(x_{n})=y_{n }
This optimization is used to find the segmented polynomial approximation {circumflex over (ƒ)}(x) of ƒ(x).
The segmented (or piecewise) polynomial approximation of the function ƒ can be written as:
where the rectangular basis function Π 300 (shown in
Alternatively, the segmented polynomial could use a triangular basis function Λ(x) 350, as shown in
where the triangular basis function Λ(x) 350 (shown in
for values of x that are less than δ, and equal to 0 for values of x that are greater than δ.
Consider a DPD (using a static model) with direct optimization of LUT with polynomial interpolation and linear interpolation between segments, as discussed above. Let the model have P element lookup tables, and order K polynomial interpolation. Let x(n) be the complex input to the nonlinear system, y(n) complex output, and ŷ(n) be the output of the model (to be estimated).
The approximation error can be written as:
e_{n}=ŷ_{n}−y_{n }
The cost function to be minimized can be expressed as follows:
Let U be the N×J (where J=P×K) matrix, where column elements are the
x(n)(x(n)−pδ)^{k}Λ(x(n)−pδ)
i.e., u(index,n)=x(n)(x(n)−pδ)^{k}Λ(x(n)−pδ),
where index=pK+k
The solution is given by a standard Least Square solution:
Φ=(U^{H}U)^{−1}·U^{H}Y
Generalized Memory Polynomial (GMP) Model Using Segmented Polynomials
The DPD model used for this analysis is based on the generalized memory polynomials (GMP). The GMP model is a simplified form of the more general but complex Volterra Series.
The GMP NL filter operation is described by the following equation (where x(n) is the input to DPD and y(n) is the output of DPD nonlinear filter):
This equation can be rewritten as:
In this mathematically equivalent form, polynomial terms appear:
Finally, the inputoutput relationship of the DPD is expressed as:
The ƒ_{m,l}(.) term can be approximated with segmented polynomials defined by a LUT, as follows
The GMP model can be written as:
The approximation error can be written as:
e_{n}=ŷ_{n}−y_{n }
The cost function to be minimized can be expressed as follows:
Let U be the N×J (where J=M×M×P×K) matrix, where column elements are expressed as follows:
x(n−m)(x(n−l)−pδ)^{k}Π(x(n−l)−pδ).
The solution is given by a standard Least Square solution:
Φ=(U^{H}U)^{−1}·U^{H}Y
It is again noted that an aspect of the invention improves continuity between points using linear interpolation between segments. For derivations, replace the rectangular basis function Π with the triangular basis function Λ in the above equations, as would be apparent to a person of ordinary skill in the art.
While exemplary embodiments of the present invention have been described with respect to digital logic blocks and memory tables within a digital processor, as would be apparent to one skilled in the art, various functions may be implemented in the digital domain as processing steps in a software program, in hardware by circuit elements or state machines, or in combination of both software and hardware. Such software may be employed in, for example, a digital signal processor, application specific integrated circuit or microcontroller. Such hardware and software may be embodied within circuits implemented within an integrated circuit.
Thus, the functions of the present invention can be embodied in the form of methods and apparatuses for practicing those methods. One or more aspects of the present invention can be embodied in the form of program code, for example, whether stored in a storage medium, loaded into and/or executed by a machine, wherein, when the program code is loaded into and executed by a machine, such as a processor, the machine becomes an apparatus for practicing the invention. When implemented on a generalpurpose processor, the program code segments combine with the processor to provide a device that operates analogously to specific logic circuits. The invention can also be implemented in one or more of an integrated circuit, a digital processor, a microprocessor, and a microcontroller.
It is to be understood that the embodiments and variations shown and described herein are merely illustrative of the principles of this invention and that various modifications may be implemented by those skilled in the art without departing from the scope and spirit of the invention.