rf.PAmemory

Create nonlinear power amplifier with memory

Since R2024a

Description

Use the rf.PAmemory System object™ to model a two-port power amplifier (PA) using a memory polynomial expression derived from the Volterra series. The Volterra series models the nonlinear relationship between input and output signals. This System object:

• Includes memory effects, where the output response depends on the current input signal and the input signal at previous times.

• Does not generate new frequency components and includes memory effects since the output response at any instance of time depends on both current and previous input signal values.

Use this System object when transmitting narrowband signals into your RF system.

To create a nonlinear power amplifier with memory and use it:

1. Create the rf.PAmemory object and set its properties.

2. Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

Creation

Syntax

rfpa = rf.PAmemory

rfpa = rf.PAmemory (Name=Value)

Description

rfpa = rf.PAmemory creates a narrowband power amplifier with memory using a two-dimensional polynomial expression derived from a Volterra series to model the nonlinear relationship between input and output signals.

rfpa = rf.PAmemory (Name=Value) sets properties of the rf.PAmemory object using one or more name-value arguments. For example, rfpa = rf.PAmemory(Model='Cross-term memory') creates a power amplifier element designed using the cross-term memory type. Properties you do not specify retain their default values.

Properties

Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.

If a property is tunable, you can change its value at any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

Model — Model type to design power amplifier

Memory polynomial (default) | Cross-term memory

Model to design power amplifier, specified as Memory polynomial or Cross-term memory. This table summarizes the characteristics of the two models.

Model	Characterization Data	Type of Coefficients	In-Band Spectral Regrowth	Out-of-Band Harmonic Generation
Memory polynomial (default)	Bandpass (I,Q)	2-D Complex matrix	Yes	No
Cross-term memory	Bandpass (I,Q)	2-D Complex matrix	Yes	No

For more information, see Model Types in rf.PAmemory System object.

Coefficientmatrix — Coefficient matrix

1+0i (default) | 2-D complex matrix

Coefficient matrix, specified as a 2-D complex matrix for the Memory polynomial and Cross-term memory models.

For the Memory polynomial and Cross-Term Memory models, you can identify the complex coefficient matrix based on the measured complex (I,Q) output-vs.-input amplifier characteristic. As an example, see the helper function in Coefficient Matrix Computation.

The size of the matrix depends on the number of delays and the degree of the system nonlinearity.

• For the Memory polynomial model, the matrix has dimensions Memory Depth (mem)-by-Voltage Order (deg).

• For the Cross-term memory model, the matrix has dimensions Memory Depth (mem)-by-{Memory Depth (mem)·(Voltage Order (deg)-1)+1}.

Measured interval of PA data (s) — Sample time of measured input-output data

1e-6 (default) | positive scalar

Sample time of the input-output data that the rf.PAmemory System object uses to construct the coefficient matrix, specified in seconds.

Dependencies

To enable this parameter, input rf.PAmemory System object to a MATLAB System (Simulink) block.

Usage

Syntax

out = rfpa(in)

Description

out = rfpa(in) returns a time-dependent amplified signal out based on the time-dependent input signal in.

Input Arguments

in — Time-dependent input signal

column

Time-dependent input signal, specified as a column. A column represents consecutive points in time.

Data Types: double | single

Output Arguments

out — Time-dependent output signal

complex column

Time-dependent output signal, returned as a complex column. The output time-dependent signal is equal in size to the input time dependent signal.

Data Types: double | single

Object Functions

To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named obj, use this syntax:

release(obj)

Common to All System Objects

step	Run System object algorithm
release	Release resources and allow changes to System object property values and input characteristics
reset	Reset internal states of System object

Examples

Create Nonlinear Power Amplifier with Memory

Create a nonlinear power amplifier using the cross-term memory model. Extract the coefficient matrix from the PAcoefficients.mat file.

load('PAcoefficients2.mat')
rfpa = rf.PAmemory (Model='Cross-term Memory',...
    CoefficientMatrix=fitCoefMat);

Generate the modulated symbols.

modData = qammod(randi([0 15],1000,1),16,'UnitAveragePower',true);

Return a time-dependent amplified signal out based on the time-dependent input signal modData.

out = rfpa(modData);

Plot the data in a scatter plot.

scatterplot(out)

Compute PA Coefficient Matrix

Load the measured power amplifier (PA) data.

load('PAdata.mat')

Extract and map the inputs and output signals of the PA data into local variables.

paInput = results.InputWaveform;
paOutput = results.OutputWaveform;

Perform the fit and RMS error calculation for these values. In this example, you use only half the data to compute the fitting coefficients, as the whole data set will be used to compute the relative error.

numDataPts = length(paInput);
halfDataPts = round(numDataPts/2);

Specify memory length, nonlinearity degree, and model type as inputs to compute PA coefficient matrix.

modType = "memPoly";
memLen = 5;
degLen = 5;

Compute the PA coefficient matrix using the fit_memory_poly_model helper function. For more information, see the helper function attached in this example or refer Coefficient Matrix Computation.

warning('off','MATLAB:rankDeficientMatrix')
fitCoefMatMem = fit_memory_poly_model(             ...
  paInput(1:halfDataPts),paOutput(1:halfDataPts),memLen,degLen,modType)

fitCoefMatMem = 5×5 single matrix

  22.9311 + 3.2123i   8.4207 - 2.7365i -16.4113 - 6.9913i  12.5001 + 4.5502i  -3.0464 - 1.0330i
   0.0000 + 0.0000i   9.4202 + 7.0544i -18.2650 -19.1580i   9.5445 +21.1397i  -1.6043 - 5.3108i
 -18.1518 +10.5096i  -4.9763 -16.1023i  10.7680 +23.3746i  -3.8337 -21.8603i   0.3766 + 5.0550i
  30.5309 -11.4868i  -0.1403 + 8.4035i  -1.9676 - 9.2857i  -0.7808 +10.6709i   0.4287 - 2.5254i
 -14.7750 + 4.2989i   1.4906 - 1.8336i  -1.4513 + 1.4996i   1.4281 - 2.5631i  -0.3660 + 0.6397i

Input this coefficient matrix to the rf.PAmemory System object to create a nonlinear power amplifier.

rfpa = rf.PAmemory (CoefficientMatrix=fitCoefMatMem)

rfpa = 
  rf.PAmemory with properties:

                Model: 'Memory polynomial'
    CoefficientMatrix: [5×5 double]

Algorithms

Model Types in rf.PAmemory System object

The rf.PAmemory System object supports two model types.

• Memory polynomial — The narrowband memory polynomial implementation (equation (19) of [1]) used in this model operates on the envelope of the input signal, does not generate new frequency components, and captures in-band spectral regrowth. Use this model to create a narrowband amplifier operating at high frequency.

  The output signal, at any instant of time, is the sum of all the elements of the complex matrix of dimensions Memory Depth (mem)-by-Voltage Order (deg):

  $$\left[\begin{array}{cccc}{C}_{11}{V}_0& C_{12}{V}_0\left|V_0\right|& \cdots & C_{1,\text{deg}}{V}_0{\left|V_0\right|}^{\text{deg}-1}\\ C_{21}{V}_1& C_{22}{V}_1\left|V_1\right|& \cdots & C_{2,\text{deg}}{V}_1{\left|V_1\right|}^{\mathrm{deg}-1}\\ ⋮& ⋮& \ddots & ⋮\\ C_{\text{mem},1}{V}_{\text{mem}-1}& C_{\text{mem},2}{V}_{\text{mem}-1}\left|V_{\text{mem}-1}\right|& \cdots & C_{\text{mem},\text{deg}}{V}_{\text{mem}-1}{\left|V_{\text{mem}-1}\right|}^{\mathrm{deg}-1}\end{array}\right].$$

  In the matrix, the number of rows equals the number of memory terms, and the number of columns equals the degree of the nonlinearity. The signal subscript represents the amount of delay.

• Cross-Term Memory — The narrowband memory polynomial implementation (equation (23) of [1]) used in this model also operates on the envelope of the input signal, does not generate new frequency components, and captures in-band spectral regrowth. Use this model to create a narrowband amplifier operating at high frequency. This model includes leading and lagging memory terms and provides a generalized implementation of the memory polynomial model.

  The output signal, at any instant of time, is the sum of all the elements of the matrix specified by the element-by-element product

  C .* M_CTM,

  where C is a complex coefficient matrix of dimensions Memory Depth (mem)-by-{Memory Depth (mem)·(Voltage Order (deg)-1)+1} and

  $$\begin{array}{c}{M}_{\text{CTM}}=\left[\begin{array}{c}{V}_0\\ V_1\\ ⋮\\ V_{\text{mem}-1}\end{array}\right]\left[\begin{array}{cccccccccccc}1& \left|V_0\right|& \left|V_1\right|& \cdots & \left|V_{\text{mem}-1}\right|& {\left|V_0\right|}^2& \cdots & {\left|V_{\text{mem}-1}\right|}^2& \cdots & {\left|V_0\right|}^{\mathrm{deg}-1}& \cdots & {\left|V_{\text{mem}-1}\right|}^{\mathrm{deg}-1}\end{array}\right]\\ =\left[\begin{array}{cccccccccccc}{V}_0& V_0\left|V_0\right|& V_0\left|V_1\right|& \cdots & V_0\left|V_{\text{mem}-1}\right|& V_0{\left|V_0\right|}^2& \cdots & V_0{\left|V_{\text{mem}-1}\right|}^2& \cdots & V_0{\left|V_0\right|}^{\mathrm{deg}-1}& \cdots & V_0{\left|V_{\text{mem}-1}\right|}^{\mathrm{deg}-1}\\ V_1& V_1\left|V_0\right|& V_1\left|V_1\right|& \cdots & V_1\left|V_{\text{mem}-1}\right|& V_1{\left|V_0\right|}^2& \cdots & V_1{\left|V_{\text{mem}-1}\right|}^2& \cdots & V_1{\left|V_0\right|}^{\mathrm{deg}-1}& \cdots & V_1{\left|V_{\text{mem}-1}\right|}^{\mathrm{deg}-1}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & ⋮& \ddots & ⋮& \ddots & ⋮\\ V_{\text{mem}-1}& V_{\text{mem}-1}\left|V_0\right|& V_{\text{mem}-1}\left|V_1\right|& \cdots & V_{\text{mem}-1}\left|V_{\text{mem}-1}\right|& V_{\text{mem}-1}{\left|V_0\right|}^2& \cdots & V_{\text{mem}-1}{\left|V_{\text{mem}-1}\right|}^2& \cdots & V_{\text{mem}-1}{\left|V_0\right|}^{\mathrm{deg}-1}& \cdots & V_{\text{mem}-1}{\left|V_{\text{mem}-1}\right|}^{\mathrm{deg}-1}\end{array}\right].\end{array}$$

  In the matrix, the number of rows equals the number of memory terms, and the number of columns is proportional to the degree of the nonlinearity and the number of memory terms. The signal subscript represents the amount of delay. The additional columns that do not appear in the Memory polynomial model represent the cross terms.

Coefficient Matrix Computation

To compute coefficient matrices, the block solves an overdetermined linear system of equations. Consider the Memory polynomial model for the case where the memory length is two and the system nonlinearity is of third degree. The matrix that describes the system is $$\left[\begin{array}{ccc}{C}_{11}{V}_0& C_{12}{V}_0\left|V_0\right|& C_{13}{V}_0{\left|V_0\right|}^2\\ C_{21}{V}_1& C_{22}{V}_1\left|V_1\right|& C_{23}{V}_1{\left|V_1\right|}^2\end{array}\right],$$and the sum of its elements is equivalent to the inner product $$\left[\begin{array}{cccccc}{V}_0& V_1& V_0\left|V_0\right|& V_1\left|V_1\right|& V_0{\left|V_0\right|}^2& V_1{\left|V_1\right|}^2\end{array}\right]\left[\begin{array}{c}{C}_{11}\\ C_{21}\\ C_{12}\\ C_{22}\\ C_{13}\\ C_{23}\end{array}\right].$$

If the input to the amplifier is the five-sample signal [x(1) x(2) x(3) x(4) x(5)] and the corresponding output is [y(1) y(2) y(3) y(4) y(5)], then the solution to $$\left[\begin{array}{cccccc}x(2)& x(1)& x(2)\left|x(2)\right|& x(1)\left|x(1)\right|& x(2){\left|x(2)\right|}^2& x(1){\left|x(1)\right|}^2\\ x(3)& x(2)& x(3)\left|x(3)\right|& x(2)\left|x(2)\right|& x(3){\left|x(3)\right|}^2& x(2){\left|x(2)\right|}^2\\ x(4)& x(3)& x(4)\left|x(4)\right|& x(3)\left|x(3)\right|& x(4){\left|x(4)\right|}^2& x(3){\left|x(3)\right|}^2\\ x(5)& x(4)& x(5)\left|x(5)\right|& x(4)\left|x(4)\right|& x(5){\left|x(5)\right|}^2& x(4){\left|x(4)\right|}^2\end{array}\right]\left[\begin{array}{c}{C}_{11}\\ C_{21}\\ C_{12}\\ C_{22}\\ C_{13}\\ C_{23}\end{array}\right]=\left[\begin{array}{c}y(2)\\ y(3)\\ y(4)\\ y(5)\end{array}\right],$$which can be found using the MATLAB^® backslash operator, provides an estimate of the coefficient matrix.

The treatment of the Cross-Term Memory model is similar. The matrix that describes the system is $$\left[\begin{array}{ccccc}{C}_{11}{V}_0& C_{12}{V}_0\left|V_0\right|& C_{13}{V}_0\left|V_1\right|& C_{14}{V}_0{\left|V_0\right|}^2& C_{15}{V}_0{\left|V_1\right|}^2\\ C_{21}{V}_1& C_{22}{V}_1\left|V_0\right|& C_{23}{V}_1\left|V_1\right|& C_{24}{V}_1{\left|V_0\right|}^2& C_{25}{V}_1{\left|V_1\right|}^2\end{array}\right],$$and the sum of its elements is equivalent to the inner product$$\left[\begin{array}{cccccccccc}{V}_0& V_1& V_0\left|V_0\right|& V_1\left|V_0\right|& V_0\left|V_1\right|& V_1\left|V_1\right|& V_0{\left|V_0\right|}^2& V_1{\left|V_0\right|}^2& V_0{\left|V_1\right|}^2& V_1{\left|V_1\right|}^2\end{array}\right]\left[\begin{array}{c}{C}_{11}\\ C_{21}\\ C_{12}\\ C_{22}\\ C_{13}\\ C_{23}\\ C_{14}\\ C_{24}\\ C_{15}\\ C_{25}\end{array}\right].$$

If the input to the amplifier is the five-sample signal [x(1) x(2) x(3) x(4) x(5)] and the corresponding output is [y(1) y(2) y(3) y(4) y(5)], then the solution $$\left[\begin{array}{cccccccccc}x(2)& x(1)& x(2)\left|x(2)\right|& x(1)\left|x(2)\right|& x(2)\left|x(1)\right|& x(1)\left|x(1)\right|& x(2){\left|x(2)\right|}^2& x(1){\left|x(2)\right|}^2& x(2){\left|x(1)\right|}^2& x(1){\left|x(1)\right|}^2\\ x(3)& x(2)& x(3)\left|x(3)\right|& x(2)\left|x(3)\right|& x(3)\left|x(2)\right|& x(2)\left|x(2)\right|& x(3){\left|x(3)\right|}^2& x(2){\left|x(3)\right|}^2& x(3){\left|x(2)\right|}^2& x(2){\left|x(2)\right|}^2\\ x(4)& x(3)& x(4)\left|x(4)\right|& x(3)\left|x(4)\right|& x(4)\left|x(3)\right|& x(3)\left|x(3)\right|& x(4){\left|x(4)\right|}^2& x(3){\left|x(4)\right|}^2& x(4){\left|x(3)\right|}^2& x(3){\left|x(3)\right|}^2\\ x(5)& x(4)& x(5)\left|x(5)\right|& x(4)\left|x(5)\right|& x(5)\left|x(4)\right|& x(4)\left|x(4)\right|& x(5){\left|x(5)\right|}^2& x(4){\left|x(5)\right|}^2& x(5){\left|x(4)\right|}^2& x(4){\left|x(4)\right|}^2\end{array}\right]\left[\begin{array}{c}{C}_{11}\\ C_{21}\\ C_{12}\\ C_{22}\\ C_{13}\\ C_{23}\\ C_{14}\\ C_{24}\\ C_{15}\\ C_{25}\end{array}\right]=\left[\begin{array}{c}y(2)\\ y(3)\\ y(4)\\ y(5)\end{array}\right]$$ to provides an estimate of the coefficient matrix.

Use this helper function to compute coefficient matrices for the Memory polynomial and Cross-Term Memory models. The inputs to the function are the given input and output signals, the memory length, the degree of nonlinearity, and the absence or presence of cross terms.

function a_coef = fit_memory_poly_model(paInput,paOutput,memLen,degLen,modType)
% FIT_MEMORY_POLY_MODEL
%   Procedure to compute a coefficient matrix given input and output
%   signals, memory length, nonlinearity degree, and model type.
%
%   Copyright 2017 MathWorks, Inc.

x = paInput(:);
y = paOutput(:);
xLen = length(x);

switch modType
  case 'memPoly'  % Memory polynomial
    xrow = reshape((memLen:-1:1)' + (0:xLen:xLen*(degLen-1)),1,[]);
    xVec = (0:xLen-memLen)' + xrow;
    xPow = x.*(abs(x).^(0:degLen-1));
    xVec = xPow(xVec);
  case 'ctMemPoly'  % Cross-term memory polynomial
    absPow = (abs(x).^(1:degLen-1));
    partTop1 = reshape((memLen:-1:1)'+(0:xLen:xLen*(degLen-2)),1,[]);
    topPlane = reshape(                                                 ...
        [ones(xLen-memLen+1,1),absPow((0:xLen-memLen)' + partTop1)].',  ...
        1,memLen*(degLen-1)+1,xLen-memLen+1);
    sidePlane = reshape(x((0:xLen-memLen)' + (memLen:-1:1)).',          ...
        memLen,1,xLen-memLen+1);
    cube = sidePlane.*topPlane;
    xVec = reshape(cube,memLen*(memLen*(degLen-1)+1),xLen-memLen+1).';
end

coef = xVec\y(memLen:xLen);
a_coef = reshape(coef,memLen,numel(coef)/memLen);

References

[1] Morgan, Dennis R., Zhengxiang Ma, Jaehyeong Kim, Michael G. Zierdt, and John Pastalan. "A Generalized Memory Polynomial Model for Digital Predistortion of Power Amplifiers." IEEE Transactions on Signal Processing. Vol. 54, No. 10, October 2006, pp. 3852–3860.

[2] Gan, Li, and Emad Abd-Elrady. "Digital Predistortion of Memory Polynomial Systems using Direct and Indirect Learning Architectures". Proceedings of the Eleventh IASTED International Conference on Signal and Image Processing (SIP) (F. Cruz-Roldán and N. B. Smith, eds.), No. 654-802. Calgary, AB: ACTA Press, 2009.

Version History

Introduced in R2024a