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

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`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 $Memory Depth (mem) \times Voltage Order (deg)$ :
$[\begin{matrix} C_{11} V_{0} & C_{12} V_{0} | V_{0} | & \dots & C_{1, deg} V_{0} {| V_{0} |}^{deg - 1} \\ C_{21} V_{1} & C_{22} V_{1} | V_{1} | & \dots & C_{2, deg} V_{1} {| V_{1} |}^{\deg - 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ C_{mem, 1} V_{mem - 1} & C_{mem, 2} V_{mem - 1} | V_{mem - 1} | & \dots & C_{mem, deg} V_{mem - 1} {| V_{mem - 1} |}^{\deg - 1} \end{matrix}] .$
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 $Memory Depth (mem) \times Voltage Order (deg)$ :
$[\begin{matrix} C_{11} V_{0} & C_{12} V_{0}^{2} & \dots & C_{1, deg} V_{0}^{deg} \\ C_{21} V_{1} & C_{22} V_{1}^{2} & \dots & C_{2, deg} V_{1}^{deg} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ C_{mem, 1} V_{mem - 1} & C_{mem, 2} V_{mem - 1}^{2} & \dots & C_{mem, deg} V_{mem - 1}^{\deg} \end{matrix}] .$
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 $Memory Depth (mem) \times {Memory Depth (mem) \cdot (Voltage Order (deg) - 1) + 1}$ and
$\begin{matrix} M_{CTM} = [\begin{matrix} V_{0} \\ V_{1} \\ ⋮ \\ V_{mem - 1} \end{matrix}] [\begin{matrix} 1 & | V_{0} | & | V_{1} | & \dots & | V_{mem - 1} | & {| V_{0} |}^{2} & \dots & {| V_{mem - 1} |}^{2} & \dots & {| V_{0} |}^{\deg - 1} & \dots & {| V_{mem - 1} |}^{\deg - 1} \end{matrix}] \\ = [\begin{matrix} V_{0} & V_{0} | V_{0} | & V_{0} | V_{1} | & \dots & V_{0} | V_{mem - 1} | & V_{0} {| V_{0} |}^{2} & \dots & V_{0} {| V_{mem - 1} |}^{2} & \dots & V_{0} {| V_{0} |}^{\deg - 1} & \dots & V_{0} {| V_{mem - 1} |}^{\deg - 1} \\ V_{1} & V_{1} | V_{0} | & V_{1} | V_{1} | & \dots & V_{1} | V_{mem - 1} | & V_{1} {| V_{0} |}^{2} & \dots & V_{1} {| V_{mem - 1} |}^{2} & \dots & V_{1} {| V_{0} |}^{\deg - 1} & \dots & V_{1} {| V_{mem - 1} |}^{\deg - 1} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ V_{mem - 1} & V_{mem - 1} | V_{0} | & V_{mem - 1} | V_{1} | & \dots & V_{mem - 1} | V_{mem - 1} | & V_{mem - 1} {| V_{0} |}^{2} & \dots & V_{mem - 1} {| V_{mem - 1} |}^{2} & \dots & V_{mem - 1} {| V_{0} |}^{\deg - 1} & \dots & V_{mem - 1} {| V_{mem - 1} |}^{\deg - 1} \end{matrix}] . \end{matrix}$
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 $Memory Depth (mem) \times {Memory Depth (mem) \cdot (Voltage Order (deg) - 1) + 1}$ and
$\begin{matrix} M_{CTH} = [\begin{matrix} V_{0} \\ V_{1} \\ ⋮ \\ V_{mem - 1} \end{matrix}] [\begin{matrix} 1 & V_{0} & V_{1} & \dots & V_{mem - 1} & V_{0}^{2} & \dots & V_{mem - 1}^{2} & \dots & V_{0}^{\deg - 1} & \dots & V_{mem - 1}^{\deg - 1} \end{matrix}] \\ = [\begin{matrix} V_{0} & V_{0}^{2} & V_{0} V_{1} & \dots & V_{0} V_{mem - 1} & V_{0}^{3} & \dots & V_{0} V_{mem - 1}^{2} & \dots & V_{0}^{\deg} & \dots & V_{0} V_{mem - 1}^{\deg - 1} \\ V_{1} & V_{1} V_{0} & V_{1}^{2} & \dots & V_{1} V_{mem - 1} & V_{1} V_{0}^{2} & \dots & V_{1} V_{mem - 1}^{2} & \dots & V_{1} V_{0}^{\deg - 1} & \dots & V_{1} V_{mem - 1}^{\deg - 1} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ V_{mem - 1} & V_{mem - 1} V_{0} & V_{mem - 1} V_{1} & \dots & V_{mem - 1}^{2} & V_{mem - 1} V_{0}^{2} & \dots & V_{mem - 1}^{3} & \dots & V_{mem - 1} V_{0}^{\deg - 1} & \dots & V_{mem - 1}^{\deg} \end{matrix}] . \end{matrix}$
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 $Memory Depth (mem) \times Voltage Order (deg)$ .
For the Cross-Term Memory and Cross-Term Hammerstein models, the matrix has dimensions $Memory Depth (mem) \times {Memory Depth (mem) \cdot (Voltage Order (deg) - 1) + 1}$ .

`Coefficient Sample Time (s)` — Sample interval of input-output data
`1e-6` (default) | real positive scalar

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

Note

To improve model accuracy, specify the Coefficient Sample Time (s) parameter to be equal or larger than the Step size parameter in the Configuration block.

`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.

Model Examples

Digital Predistortion to Compensate for Power Amplifier Nonlinearities

Use digital predistortion (DPD) in a transmitter to offset the effects of nonlinearities in a power amplifier. This example use power amplifier models that were obtained from Power Amplifier Characterization (Communications Toolbox) example to simulate two cases. In the first simulation, the RF transmitter sends two tones. In the second simulation, the RF transmitter sends a 5G-like OFDM waveform with 100 MHz bandwidth.

Open Script

Power Amplifier Characterization

Characterize a power amplifier (PA) using measured input and output signals of an NXP Airfast PA. Optionally, you can use a hardware test setup including an NI PXI chassis with a vector signal transceiver (VST) to measure the signals at run time.

Open Live Script

Tips

To avoid the nonlinear power amplifier to operate in an undesirable region, the Simulink^® input signal must be scaled. This happens when a nonlinear power amplifier in the RF domain is used to amplify a Simulink signal.

Algorithms

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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

$[\begin{matrix} C_{11} V_{0} & C_{12} V_{0} | V_{0} | & C_{13} V_{0} {| V_{0} |}^{2} \\ C_{21} V_{1} & C_{22} V_{1} | V_{1} | & C_{23} V_{1} {| V_{1} |}^{2} \end{matrix}],$

and the sum of its elements is equivalent to the inner product

$[\begin{matrix} V_{0} & V_{1} & V_{0} | V_{0} | & V_{1} | V_{1} | & V_{0} {| V_{0} |}^{2} & V_{1} {| V_{1} |}^{2} \end{matrix}] [\begin{matrix} C_{11} \\ C_{21} \\ C_{12} \\ C_{22} \\ C_{13} \\ C_{23} \end{matrix}] .$

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

$[\begin{matrix} x (2) & x (1) & x (2) | x (2) | & x (1) | x (1) | & x (2) {| x (2) |}^{2} & x (1) {| x (1) |}^{2} \\ x (3) & x (2) & x (3) | x (3) | & x (2) | x (2) | & x (3) {| x (3) |}^{2} & x (2) {| x (2) |}^{2} \\ x (4) & x (3) & x (4) | x (4) | & x (3) | x (3) | & x (4) {| x (4) |}^{2} & x (3) {| x (3) |}^{2} \\ x (5) & x (4) & x (5) | x (5) | & x (4) | x (4) | & x (5) {| x (5) |}^{2} & x (4) {| x (4) |}^{2} \end{matrix}] [\begin{matrix} C_{11} \\ C_{21} \\ C_{12} \\ C_{22} \\ C_{13} \\ C_{23} \end{matrix}] = [\begin{matrix} y (2) \\ y (3) \\ y (4) \\ y (5) \end{matrix}],$

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

$[\begin{matrix} C_{11} V_{0} & C_{12} V_{0} | V_{0} | & C_{13} V_{0} | V_{1} | & C_{14} V_{0} {| V_{0} |}^{2} & C_{15} V_{0} {| V_{1} |}^{2} \\ C_{21} V_{1} & C_{22} V_{1} | V_{0} | & C_{23} V_{1} | V_{1} | & C_{24} V_{1} {| V_{0} |}^{2} & C_{25} V_{1} {| V_{1} |}^{2} \end{matrix}],$

and the sum of its elements is equivalent to the inner product

$[\begin{matrix} V_{0} & V_{1} & V_{0} | V_{0} | & V_{1} | V_{0} | & V_{0} | V_{1} | & V_{1} | V_{1} | & V_{0} {| V_{0} |}^{2} & V_{1} {| V_{0} |}^{2} & V_{0} {| V_{1} |}^{2} & V_{1} {| V_{1} |}^{2} \end{matrix}] [\begin{matrix} C_{11} \\ C_{21} \\ C_{12} \\ C_{22} \\ C_{13} \\ C_{23} \\ C_{14} \\ C_{24} \\ C_{15} \\ C_{25} \end{matrix}] .$

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

$[\begin{matrix} x (2) & x (1) & x (2) | x (2) | & x (1) | x (2) | & x (2) | x (1) | & x (1) | x (1) | & x (2) {| x (2) |}^{2} & x (1) {| x (2) |}^{2} & x (2) {| x (1) |}^{2} & x (1) {| x (1) |}^{2} \\ x (3) & x (2) & x (3) | x (3) | & x (2) | x (3) | & x (3) | x (2) | & x (2) | x (2) | & x (3) {| x (3) |}^{2} & x (2) {| x (3) |}^{2} & x (3) {| x (2) |}^{2} & x (2) {| x (2) |}^{2} \\ x (4) & x (3) & x (4) | x (4) | & x (3) | x (4) | & x (4) | x (3) | & x (3) | x (3) | & x (4) {| x (4) |}^{2} & x (3) {| x (4) |}^{2} & x (4) {| x (3) |}^{2} & x (3) {| x (3) |}^{2} \\ x (5) & x (4) & x (5) | x (5) | & x (4) | x (5) | & x (5) | x (4) | & x (4) | x (4) | & x (5) {| x (5) |}^{2} & x (4) {| x (5) |}^{2} & x (5) {| x (4) |}^{2} & x (4) {| x (4) |}^{2} \end{matrix}] [\begin{matrix} C_{11} \\ C_{21} \\ C_{12} \\ C_{22} \\ C_{13} \\ C_{23} \\ C_{14} \\ C_{24} \\ C_{15} \\ C_{25} \end{matrix}] = [\begin{matrix} y (2) \\ y (3) \\ y (4) \\ y (5) \end{matrix}]$

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);

Power Amplifier Block Icons

Power Amplifier block mask icons are dynamic and show the current type of model. This table shows you how the icons on this block vary based on the model type you set on the Model parameter on the block.

Model: `Generalized Hammerstein`	Output: `Memory polynomial`	Output: `Cross-Term Memory`	Output: `Cross-Term Hammerstein`

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.

Version History

Introduced in R2017b

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R2021b: Power Amplifier block icon updated

Starting in R2021b, the Power Amplifier block icon is now updated. When you open a model created before R2021b containing a Power Amplifier block, the software replaces the block icon with the R2021b version.

Power Amplifier

Description

Parameters

`Model` — Model type
`Memory polynomial` (default) | `Generalized Hammerstein` | `Cross-Term Memory` | `Cross-Term Hammerstein`

`Coefficient Matrix` — Coefficient matrix
1 (default) | complex matrix | real matrix

`Coefficient Sample Time (s)` — Sample interval of input-output data
`1e-6` (default) | real positive scalar

`Rin (ohm)` — Input resistance
`50` (default) | real positive scalar

`Rout (ohm)` — Output resistance
`50` (default) | real positive scalar

`Ground and hide negative terminals` — Ground RF circuit terminals
on (default) | off

Model Examples