Power Amplifier

Model power amplifier with memory

• Library:

• RF Blockset / Circuit Envelope / Elements

• 

Description

The Power Amplifier block models two-port power amplifiers. A memory polynomial expression derived from the Volterra series models the nonlinear relationship between input and output signals. This power amplifier includes memory effects because the output response depends on the current input signal and the input signal at previous times. These power amplifiers are useful when transmitting wideband or narrowband signals.

Parameters

Model — Model type

Memory polynomial (default) | Generalized Hammerstein | Cross-Term Memory | Cross-Term Hammerstein

Model type, specified as Memory polynomial, Generalized Hammerstein, Cross-Term Memory, or Cross-Term Hammerstein. The following table summarizes the characteristics of the different models:

Model	Characterization Data	Type of Coefficients	In-Band Spectral Regrowth	Out-of-Band Harmonic Generation
Memory polynomial (default)	Bandpass (I,Q)	Complex	Yes	No
Generalized Hammerstein	True passband	Real	Yes	Yes
Cross-Term Memory	Bandpass (I,Q)	Complex	Yes	No
Cross-Term Hammerstein	True passband	Real	Yes	Yes

• Memory polynomial – This narrowband memory polynomial implementation (equation (19) of [1]) 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 following 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 amount of delay.

• Generalized Hammerstein – This wideband memory polynomial implementation (equation (18) of [1]) operates on the envelope of the input signal, generates frequency components that are integral multiples of carrier frequencies, and captures in-band spectral regrowth. Increasing the degree of the nonlinearity increases the number of out-of-band frequencies generated. Use this model to create wideband amplifiers operating at low frequency.

  The output signal, at any instant of time, is the sum of all the elements of the following real matrix of dimensions $$\text{Memory Depth (mem)}\times \text{Voltage Order (deg)}$$:

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

  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 amount of delay.

• Cross-Term Memory – This narrowband memory polynomial implementation (equation (23) of [1]) 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 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 a 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 amount of delay. The additional columns that do not appear in the Memory polynomial model represent the cross terms.

• Cross-Term Hammerstein – This wideband memory polynomial implementation operates on the envelope of the input signal, generates frequency components that are integral multiples of carrier frequencies, and captures in-band spectral regrowth. Increasing the order of the nonlinearity increases the number of out-of-band frequencies generated. Use this model to create wideband amplifiers operating at low frequency.

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

  C .* M_CTH,

  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

  $$\begin{array}{c}{M}_{\text{CTH}}=\left[\begin{array}{c}{V}_0\\ V_1\\ ⋮\\ V_{\text{mem}-1}\end{array}\right]\left[\begin{array}{cccccccccccc}1& V_0& V_1& \cdots & V_{\text{mem}-1}& V_0^2& \cdots & V_{\text{mem}-1}^2& \cdots & V_0^{\mathrm{deg}-1}& \cdots & V_{\text{mem}-1}^{\mathrm{deg}-1}\end{array}\right]\\ =\left[\begin{array}{cccccccccccc}{V}_0& V_0^2& V_0{V}_1& \cdots & V_0{V}_{\text{mem}-1}& V_0^3& \cdots & V_0{V}_{\text{mem}-1}^2& \cdots & V_0^{\mathrm{deg}}& \cdots & V_0{V}_{\text{mem}-1}^{\mathrm{deg}-1}\\ V_1& V_1{V}_0& V_1^2& \cdots & V_1{V}_{\text{mem}-1}& V_1{V}_0^2& \cdots & V_1{V}_{\text{mem}-1}^2& \cdots & V_1{V}_0^{\mathrm{deg}-1}& \cdots & V_1{V}_{\text{mem}-1}^{\mathrm{deg}-1}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & ⋮& \ddots & ⋮& \ddots & ⋮\\ V_{\text{mem}-1}& V_{\text{mem}-1}{V}_0& V_{\text{mem}-1}{V}_1& \cdots & V_{\text{mem}-1}^2& V_{\text{mem}-1}{V}_0^2& \cdots & V_{\text{mem}-1}^3& \cdots & V_{\text{mem}-1}{V}_0^{\mathrm{deg}-1}& \cdots & V_{\text{mem}-1}^{\mathrm{deg}}\end{array}\right].\end{array}$$

  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 amount of delay. The additional columns that do not appear in the Generalized Hammerstein model represent the cross terms.

Coefficient Matrix — Coefficient matrix

1 (default) | complex matrix | real matrix

Coefficient matrix, specified as a complex matrix for the Memory polynomial and Cross-Term Memory models and as a real matrix for the Generalized Hammerstein and Cross-Term Hammerstein 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.

• For the Generalized Hammerstein and Cross-Term Hammerstein models, you can identify the real coefficient matrix based on the measured real passband output-vs.-input amplifier characteristic.

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

• For the Memory polynomial and Generalized Hammerstein models, the matrix has dimensions $$\text{Memory Depth (mem)}\times \text{Voltage Order (deg)}$$.

• For the Cross-Term Memory and Cross-Term Hammerstein models, the matrix has dimensions $$\text{Memory Depth (mem)}\times \{\text{Memory Depth (mem)}\cdot (\text{Voltage Order (deg)}-1)+1\}$$.

Coefficient Sample Time (s) — Sample interval of input-output data

1e-6 (default) | real positive scalar

Sample interval of input-output data used to identify the coefficient matrix, specified as a real positive scalar.

The accuracy of the model can be affected if the coefficient sample time differs from the simulation step size specified in the Configuration block. For best results, use a coefficient sample time at least as large as the simulation step size.

Rin (ohm) — Input resistance

50 (default) | real positive scalar

Input resistance, specified as a real positive scalar.

Rout (ohm) — Output resistance

50 (default) | real positive scalar

Output resistance, specified as a real positive scalar.

Ground and hide negative terminals — Ground RF circuit terminals

on (default) | off

Select this parameter to ground and hide the negative terminals. Clear the parameter to expose the negative terminals. By exposing these terminals, you can connect them to other parts of your model.

Algorithms

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." In 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.