Power Amplifier

Model narrowband power amplifier with memory

Since R2022a

• 

Libraries:
RF Blockset / Idealized Baseband

Description

The Power Amplifier block models a two-port power amplifier using a memory polynomial expression derived from the Volterra series. The Volterra series models the nonlinear relationship between input and output signals. This block includes memory effects: the output response depends on the current input signal and the input signal at previous times. Use this block when transmitting narrowband signals into your RF system.

The Power Amplifier block mask icons are dynamic and show the model specified in the Model parameter.

Model: Memory polynomial	Model: Cross-term memory

Examples

Speed Up PA and DPD simulation

Speed up your PA and DPD simulation using Idealized Baseband Power Amplifier block.

• Open Script

Ports

Input

Port_1 — 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

Port_1 — 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

Parameters

Model — Model to design amplifier

Memory polynomial (default) | Cross-term memory

Use either the Memory polynomial model or the Cross-term memory model to design the power amplifier. The following 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 Type in Power Amplifier Block.

The Model Type parameter has been renamed to Model. (since R2024a)

Coefficient matrix — 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 $$\text{Memory Depth (mem)}\times \text{Voltage Order (deg)}$$.

• For the Cross-term memory model, the matrix has dimensions $$\text{Memory Depth (mem)}\times \{\text{Memory Depth (mem)}\cdot (\text{Voltage Order (deg)}-1)+1\}$$.

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 block uses to construct the coefficient matrix, specified in seconds.

The Time interval of measured PA data (s) parameter has been renamed to Measured interval of PA data (s). (since R2023b)

Simulate using — Type of simulation to run

Code generation (default) | Interpreted execution

Type of simulation to run, specified as one of the following:

• Code generation — Simulate model using generated C code. The first time you run a simulation, Simulink^® generates C code for the block. The C code is reused for subsequent simulations, as long as the model does not change. This option requires additional startup time, but the speed of the subsequent simulations is faster than Interpreted execution.

• Interpreted execution — Simulate model using the MATLAB^® interpreter. This option shortens startup time speed, but the speed of the subsequent simulations is slower than Code generation. In this mode, you can debug the source code of the block.

Algorithms

Model Type in Power Amplifier Block

The Power Amplifier block in the Idealized Baseband library 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 $$\text{Memory Depth (mem)}\times \text{Voltage Order (deg)}$$:

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

  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 $$\text{Memory Depth (mem)}\times \{\text{Memory Depth (mem)}\cdot (\text{Voltage Order (deg)}-1)+1\}$$ 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 2 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 to $$\left[\begin{array}{cccccccccc}x(2)& x(1)& x(2)\left|x(2)\right|& x(1)\left|x(2)\right|& x(2)\left|x(1)\right|& x(1)\left|x(1)\right|& x(2){\left|x(2)\right|}^2& x(1){\left|x(2)\right|}^2& x(2){\left|x(1)\right|}^2& x(1){\left|x(1)\right|}^2\\ x(3)& x(2)& x(3)\left|x(3)\right|& x(2)\left|x(3)\right|& x(3)\left|x(2)\right|& x(2)\left|x(2)\right|& x(3){\left|x(3)\right|}^2& x(2){\left|x(3)\right|}^2& x(3){\left|x(2)\right|}^2& x(2){\left|x(2)\right|}^2\\ x(4)& x(3)& x(4)\left|x(4)\right|& x(3)\left|x(4)\right|& x(4)\left|x(3)\right|& x(3)\left|x(3)\right|& x(4){\left|x(4)\right|}^2& x(3){\left|x(4)\right|}^2& x(4){\left|x(3)\right|}^2& x(3){\left|x(3)\right|}^2\\ x(5)& x(4)& x(5)\left|x(5)\right|& x(4)\left|x(5)\right|& x(5)\left|x(4)\right|& x(4)\left|x(4)\right|& x(5){\left|x(5)\right|}^2& x(4){\left|x(5)\right|}^2& x(5){\left|x(4)\right|}^2& x(4){\left|x(4)\right|}^2\end{array}\right]\left[\begin{array}{c}{C}_{11}\\ C_{21}\\ C_{12}\\ C_{22}\\ C_{13}\\ C_{23}\\ C_{14}\\ C_{24}\\ C_{15}\\ C_{25}\end{array}\right]=\left[\begin{array}{c}y(2)\\ y(3)\\ y(4)\\ y(5)\end{array}\right]$$ 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(x,y,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 = x(:);
y = y(:);
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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2022a

R2024a: Parameter of Power Amplifier block from Idealized Baseband library renamed

The following in the Power Amplifier block in the Idealized baseband library has been renamed or changed:

• The Model Type parameter has been renamed to Model.

• The Cross-Term Memory value in the Model parameter has been renamed to Cross-term memory.

• The default value of the Coefficient Matrix parameter has been updated to 1+0i from 1.

When you open a model created before R2024a containing the Power Amplifier block, the software replaces the:

• Model Type parameter with the Model parameter.

• Cross-Term Memory value with the Cross-term memory value.

• Default value of the Coefficient Matrix parameter from 1 to 1+0i.

R2023b: Parameter of Power Amplifier block from Idealized Baseband library renamed

The Time interval of measured PA data (s) parameter in the Power Amplifier block in the Idealized baseband library has been renamed to Measured interval of PA data (s).

When you open a model created before R2023b containing the Power Amplifer block, the software replaces the Time interval of measured PA data (s) parameter name with Measured interval of PA data (s).