CN110765720A - Power amplifier predistortion method of complex value assembly line recurrent neural network model - Google Patents

Power amplifier predistortion method of complex value assembly line recurrent neural network model Download PDF

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CN110765720A
CN110765720A CN201910865377.6A CN201910865377A CN110765720A CN 110765720 A CN110765720 A CN 110765720A CN 201910865377 A CN201910865377 A CN 201910865377A CN 110765720 A CN110765720 A CN 110765720A
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李明玉
蔡振东
靳一
代志江
徐常志
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Abstract

A power amplifier predistortion method of a complex value assembly line recurrent neural network model is characterized in that a power amplifier behavior model is modeled through the complex value assembly line recurrent neural network model, a predistorter module is solved, and therefore predistortion operation is carried out on a power amplifier input signal. Firstly, part of input and output signals of the power amplifier are used as test signals, forward modeling is carried out on the power amplifier, model weight is optimized through an enhanced complex value real-time recursive learning algorithm to obtain optimal model weight, and the nonlinear and memorability of the power amplifier represented by the model is checked; secondly, the model is inverted, so that the reverse modeling is carried out on the power amplifier, and a predistorter structure is obtained; finally, the input signal of the power amplifier passes through the predistorter structure to obtain a signal subjected to predistortion compensation, and the signal is sent into the power amplifier, so that the adjacent channel power ratio of the obtained output signal of the power amplifier can be obviously improved.

Description

Power amplifier predistortion method of complex value assembly line recurrent neural network model
Technical Field
The invention relates to the field of digital predistortion in digital signal processing, in particular to a power amplifier predistortion method of a complex value assembly line recurrent neural network model.
Background
With the rapid development of wireless communication technology, especially when 4G and 5G mobile communication are widely used, the demand of modern communication system for spectrum resources is increasing. To meet this requirement, high-order modulation techniques are widely used in communication systems, but these modulation techniques make the design of radio frequency power amplifiers, which play a crucial role in the radio frequency front-end section, more difficult and also cause spectral regrowth. In order to meet the requirements of modern wireless communication technology, a series of power amplifier linearization and efficiency enhancement technologies are proposed, such as feed-forward technology feedback, negative feedback technology feedback, linear linearity with Nonlinear Components technology, and predistortion technology. The predistortion technique is divided into analog predistortion and digital predistortion. Compared with other schemes, digital predistortion has become the mainstream technology of power amplifier linearization because of its advantages of low cost, good linearization performance, high flexibility and the like. The basic principle of digital predistortion is to insert a predistorter before the power amplifier, and the predistorter is inverse to the nonlinear characteristic of the power amplifier, so as to obtain linearly amplified radio frequency output at the output end of the power amplifier.
An Artificial Neural Network (ANN) is a nonlinear system with a powerful intelligent information processing function, has simple learning rule, strong robustness, memory capability and self-learning capability, and can map any complex nonlinear relation. The combination of the Neural Network technology and the power amplifier predistortion technology benefits from the strong approximation effect of the Neural Network in the nonlinear system modeling, and the Neural Network model for power amplifier behavior model modeling and digital predistortion mainly comprises a feed-forward (feed forward) Neural Network, a Radial Basis Function (RBF) Neural Network, a Time Delay Neural Network (TDNN) and the like. The method adopts a neural network learning method based on real-value information processing in the application of the communication system. However, in modern science, the nature of efficient model data has increased complex values, which has led to many learning algorithms being extended to the complex domain to process signals
When a wideband signal powers a power amplifier, the signal bandwidth may be comparable to the inherent bandwidth of the power amplifier, where the output of the power amplifier is related not only to the current input, but also to the previous input and output. Therefore, an excellent power amplifier model not only can solve the problem that an input signal is a complex value, but also can well represent the memory effect of the power amplifier. However, the traditional neural network model cannot solve the problem of complex-valued signals or completely express the memory effect of the power amplifier, and is difficult to realize both the problem and the memory effect.
Disclosure of Invention
The invention provides a power amplifier predistortion method of a complex value assembly line recurrent neural network model, which can not only express the memory effect of a power amplifier, but also process complex value signals and quickly update parameters of the model, aiming at the problems that the existing neural network model can not well express the memory effect of the power amplifier and when an input signal is a complex value, a heavy training algorithm is introduced to cause long training time, and the specific technical scheme is as follows:
a power amplifier predistortion method of a complex value assembly line recurrent neural network model is characterized in that:
the method comprises the following steps:
s1: generating an original signal, wherein the original signal is used as an input signal of a target power amplifier;
s2: establishing a neural network model, training the neural network model to obtain a power amplifier positive model, and inverting the power amplifier positive model to obtain a predistortion structure, wherein the predistortion structure is an inverse model obtained after the positive model is inverted;
s3: modulating an original signal to obtain a radio frequency signal;
s4: passing the radio frequency signal through a predistortion structure to obtain a predistortion signal;
s5: and obtaining a compensated power amplifier output signal by the pre-distortion signal through a target power amplifier.
Further: the S2 includes the following steps:
s2-1: selecting a segment from the original signal as a test signal;
s2-2: modulating the test signal to obtain a radio frequency signal;
s2-3: the radio frequency signal is amplified to obtain an output signal of the power amplifier;
s2-4: reducing the power of the output signal of the power amplifier through an attenuator to obtain an attenuated signal;
s2-5: carrying out down-conversion and filtering on the attenuated signal to obtain a target output signal;
s2-6: establishing a complex value assembly line recurrent neural network model, and training the complex value assembly line recurrent neural network model by using a test signal as an input parameter and a target output signal as an output parameter to obtain a power amplifier positive model;
s2-7: and optimizing the positive model of the power amplifier by using an enhanced complex value real-time recursive learning algorithm to obtain an optimized positive model of the power amplifier, and then inverting the optimized positive model of the power amplifier to obtain a predistortion structure of the power amplifier.
Further: the S2-6 comprises the following steps:
s2-6-1: establishing a complex value assembly line recurrent neural network model structure, wherein the model comprises M repeated modules;
each module is provided with N neurons, wherein the first M-1 modules are non-fully-connected recurrent neural networks, N-1 outputs in output neurons of the first module are used for feeding back to inputs, the outputs of the rest neurons, namely the output of the first neuron, are directly applied to the next module, the last module is a fully-connected recurrent neural network, and the outputs of all the neurons are fed back to the inputs;
s2-6-2: introducing augmented complex statistics, taking into account cross statistics of two variables, by inputting the conjugate x of variable x*To enhance the integrity of the information, a linear vector of Λ ═ x is obtainedT,xH]T；
S2-6-3: the weight vector of the l-th neuron of the network output layer of each module is
Figure BDA0002201108820000041
M modules all use the same weight matrix
Figure BDA0002201108820000042
S2-6-4: the mathematical expression of the model structure of the recurrent neural network of the complex value assembly line is
Figure BDA0002201108820000044
Where ψ (-) is an activation function,and
Figure BDA0002201108820000046
respectively representing the output of the ith neuron of the tth module at the moment k and the network node input of an activation function;
augmented input for the tth module at time kAnd input vector of Mth module
Figure BDA0002201108820000048
Is composed of
Figure BDA0002201108820000049
It(k)＝[s(k-t),...,s(k-t-p+1),1+j
yt+1,1(k-1),yt,2(k-1),...,yt,N(k-1)]
Figure BDA00022011088200000410
IM(k)＝[s(k-M),...,s(k-M-p+1),1+j
yM,1(k-1),yM,2(k-1),...,yM,N(k-1)]
Wherein p is the number of external inputs, i.e. delaying the input sample by p unit durations, and the upper right label a indicates that the expression is an augmented expression;
s2-5-5: as seen in S2-5-4, the inputs of the first M-1 modules include the output of the next module
Figure BDA00022011088200000411
And replacing the first feedback delay of the module, and the last module M has no input from the latter module, so the input retains the feedback delay of all the outputs of the module;
the mathematical expression of the model in S2-6-4 is simplified as:
Figure BDA00022011088200000412
s2-6-6: and (3) taking the first neuron output of the first module to represent the final output signal of the positive power amplifier model by using the formula obtained in S2-6-5:
further: the S2-7 comprises the following steps:
s2-7-1: the test signal in S2-1 is used to subtract the target output signal in S2-5 to obtain the t-th module error expression at the time k asWherein
Figure BDA0002201108820000053
Is the error of the real value of the signal,
Figure BDA0002201108820000054
is an imaginary error, d (k-t +1) is a target output signal, and since the input and output signals of the power amplifier are complex, the cost function of the complex value pipeline recurrent neural network model is as follows:
Figure BDA0002201108820000055
the lambda in the formula is an exponential weighting factor and is used for controlling the influence of the memory effect of each module on the whole structure;
s2-7-2: the weight coefficient is changed along the whole complex value assembly line recurrent neural network model by adopting a gradient descent learning method to minimize the cost function, and the method is as follows:
Figure BDA0002201108820000056
Figure BDA0002201108820000057
in the above formula
Figure BDA0002201108820000059
Is a complex-valued augmented weight function,
Figure BDA00022011088200000510
updating the formula for the weight of the nth weight of the ith neuron at time k,
Figure BDA00022011088200000511
for gradient expressions, η is the learning rate, here a constant;
s2-7-3: calculating the gradient of the real part and the imaginary part of the cost function to the complex value amplification type weight function as follows:
Figure BDA0002201108820000061
Figure BDA0002201108820000062
Figure BDA0002201108820000063
representing the sensitivity of the ith module at the time k of the ith neuron;
s2-7-4: according to the cauchy-riemann equation, the real and imaginary part sensitivities are obtained as a result of the sensitivities:
Figure BDA0002201108820000064
Figure BDA0002201108820000065
Figure BDA0002201108820000066
s2-7-5: substituting the formulas of S2-7-4 and S2-7-3 into S2-7-2
Figure BDA0002201108820000067
A gradient can be obtained as:
Figure BDA0002201108820000068
the output of the sensitivity function at the time k of the jth neuron of the tth module is:
Figure BDA0002201108820000069
δlnis a function of Crohn
S2-7-6: the final weight updating equation of the optimized power amplifier positive model is as follows:
Figure BDA0002201108820000071
s2-7-7: and (5) inverting the optimized power amplifier positive model to obtain a predistortion structure of the power amplifier.
The invention has the beneficial effects that: firstly, the invention combines the advantages of the traditional neural network model, popularizes the pipeline recurrent neural network model which can effectively express the memory effect to a complex field to obtain a complex value pipeline recurrent neural network model (CPRNN) for power amplifier modeling, popularizes the real-time recurrent learning algorithm used for model parameter optimization updating and further deduces to obtain an enhanced complex value real-time recurrent learning Algorithm (ACRTRL) for the CPRNN model. The effect in practical application shows that the complex value assembly line recurrent neural network model based on the enhanced complex value real-time recurrent learning algorithm can accurately model the power amplifier and effectively compensate the signal.
Secondly, the invention can utilize the CPRNN model to carry out more accurate modeling on the power amplifier.
Thirdly, the RTRL algorithm is popularized to a complex field and then an enhanced ACRTRL algorithm is deduced, so that the parameters of the model can be effectively updated, the complexity can be reduced, and the time required by convergence can be shortened.
Fourthly, the method can be applied to the power amplifier modeling of the F type, the Doherty type and the like which are popular at present, and has wide applicability.
Drawings
FIG. 1 is a block diagram of the present invention;
FIG. 2 is a structural diagram of the CPRNN of the present invention;
FIG. 3 is a graph of the modeling effect of the model of the present invention;
fig. 4 is a diagram of the effect of the present invention after predistortion.
Detailed Description
The following detailed description of the preferred embodiments of the present invention, taken in conjunction with the accompanying drawings, will make the advantages and features of the invention easier to understand by those skilled in the art, and thus will clearly and clearly define the scope of the invention.
As shown in fig. 1 and 2: a power amplifier predistortion method of recurrent neural network model of assembly line of complex value, wherein the predistortion system used includes RF PAs, the computer, vector signal source VSG, signal analyzer VSA, implement the power amplifier used for F class power amplifier, in this embodiment, recurrent neural network model of assembly line of complex value is CPRNN for short, the real-time recurrent learning algorithm of enhanced complex value is ACRTRL for short;
the method comprises the following steps:
s1: using MATLAB to generate an LTE double-carrier signal, wherein the LTE double-carrier signal is an input signal of a target power amplifier;
s2: establishing a neural network model, training the neural network model to obtain a power amplifier positive model, and inverting the power amplifier positive model to obtain a predistortion structure, wherein the predistortion structure is used for compensating nonlinearity of a power amplifier;
wherein, S2 specifically adopts the following steps:
s2-1: selecting one section from the LTE double-carrier signal as a test signal;
s2-2: downloading the test signal into a signal analyzer, and modulating the test signal to obtain a radio frequency signal;
s2-3: the radio frequency signal is amplified to obtain an output signal of the power amplifier;
s2-4: reducing the power of the output signal of the power amplifier through an attenuator to obtain an attenuated signal;
s2-5: carrying out down-conversion and filtering on the attenuated signal to obtain a target output signal;
s2-6: establishing a complex value assembly line recurrent neural network model by using MATLAB, training the complex value assembly line recurrent neural network model by using a test signal as an input parameter and a target output signal as an output parameter to obtain a power amplifier positive model;
s2-7: and optimizing the positive model of the power amplifier by using an enhanced complex value real-time recursive learning algorithm to obtain an optimized positive model of the power amplifier, and then inverting the optimized positive model of the power amplifier to obtain a predistortion structure of the power amplifier.
The optimized modeling effect is as shown in fig. 3, and fig. 3 shows that a target output signal is obtained through down-conversion and filtering, the prediction output of a complex value pipeline recurrent neural network model is obtained, and the error of the complex value pipeline recurrent neural network model is the error between the prediction and an actual value.
As can be seen from FIG. 2, the model can accurately model the power amplifier, and the modeling error is kept below-50 dB;
s3: modulating the LTE dual-carrier signal to obtain a radio frequency signal;
s4: passing the radio frequency signal through a predistortion structure to obtain a predistortion signal;
s5: and obtaining a compensated power amplifier output signal by the pre-distortion signal through a target power amplifier.
As shown in fig. 4, fig. 4 shows a predistortion signal processed by a predistortion structure, i.e. a compensated target power amplifier output signal, and an output signal without predistortion, i.e. a power amplifier output signal after an input signal directly passes through a target power amplifier, and an original signal.
The implementation result shows that the compensated target power amplifier output signal has the ACPR corrected by 19.95dB, and the implementation result and fig. 4 show that the CPRNN model based on the ACRTRL algorithm can effectively compensate the memory effect and the nonlinear characteristic of the power amplifier.
Wherein, in the above steps, the S2-6 includes the following steps:
s2-6-1: establishing a complex value assembly line recurrent neural network model structure, wherein the model comprises M repeated modules;
each module is provided with N neurons, wherein the first M-1 modules are non-fully-connected recurrent neural networks, N-1 outputs in output neurons of the first module are used for feeding back to inputs, the outputs of the rest neurons, namely the output of the first neuron, are directly applied to the next module, the last module is a fully-connected recurrent neural network, and the outputs of all the neurons are fed back to the inputs;
s2-6-2: introducing augmented complex statistics, taking into account cross statistics of two variables, by inputting the conjugate x of variable x*To enhance the integrity of the information, a linear vector of Λ ═ x is obtainedT,xH]T；
S2-6-3: the weight vector of the l-th neuron of the network output layer of each module is
Figure BDA0002201108820000101
M modules all use the same weight matrix
Figure BDA0002201108820000102
S2-6-4: the mathematical expression of the model structure of the recurrent neural network of the complex value assembly line is
Figure BDA0002201108820000103
Where ψ (-) is an activation function,
Figure BDA0002201108820000105
andrespectively representing the output of the ith neuron of the tth module at the moment k and the network node input of an activation function;
augmented input for the tth module at time k
Figure BDA0002201108820000107
And input vector of Mth module
Figure BDA0002201108820000108
Is composed of
Figure BDA0002201108820000109
It(k)＝[s(k-t),...,s(k-t-p+1),1+j
yt+1,1(k-1),yt,2(k-1),...,yt,N(k-1)]
Figure BDA00022011088200001010
IM(k)＝[s(k-M),...,s(k-M-p+1),1+j
yM,1(k-1),yM,2(k-1),...,yM,N(k-1)]
Wherein p is the number of external inputs, i.e. delaying the input sample by p unit durations, and the upper right label a indicates that the expression is an augmented expression;
s2-6-5: the mathematical expression of the model in S2-6-4 is simplified as:
Figure BDA00022011088200001011
s2-6-6: and (3) taking the first neuron output of the first module to represent the final output signal of the positive power amplifier model by using the formula obtained in S2-6-5:
Figure BDA0002201108820000111
wherein the S2-7 comprises the following steps:
s2-7-1: the test signal in S2-1 is used to subtract the target output signal in S2-5 to obtain the t-th module error expression at the time k as
Figure BDA0002201108820000112
Wherein
Figure BDA0002201108820000113
Is the error of the real value of the signal,
Figure BDA0002201108820000114
is an imaginary error, d (k-t +1) is a target output signal, and since the input and output signals of the power amplifier are complex, the cost function of the complex value pipeline recurrent neural network model is as follows:
Figure BDA0002201108820000115
the lambda in the formula is an exponential weighting factor and is used for controlling the influence of the memory effect of each module on the whole structure;
s2-7-2: the weight coefficient is changed along the whole complex value assembly line recurrent neural network model by adopting a gradient descent learning method to minimize the cost function, and the method is as follows:
Figure BDA0002201108820000116
Figure BDA0002201108820000117
Figure BDA0002201108820000118
in the above formula
Figure BDA0002201108820000119
Is a complex-valued augmented weight function,
Figure BDA00022011088200001110
updating the formula for the weight of the nth weight of the ith neuron at time k,
Figure BDA00022011088200001111
for gradient expression, η is the learning rate, taking a constant;
s2-7-3: calculating the gradient of the real part and the imaginary part of the cost function to the complex value amplification type weight function as follows:
Figure BDA0002201108820000121
Figure BDA0002201108820000122
Figure BDA0002201108820000123
representing the sensitivity of the ith module at the time k of the ith neuron;
s2-7-4: according to the cauchy-riemann equation, the real and imaginary part sensitivities are obtained as a result of the sensitivities:
Figure BDA0002201108820000124
Figure BDA0002201108820000125
Figure BDA0002201108820000126
s2-7-5: substituting the formulas of S2-7-4 and S2-7-3 into S2-7-2A gradient can be obtained as:
Figure BDA0002201108820000128
the output of the sensitivity function at the time k of the jth neuron of the tth module is:
Figure BDA0002201108820000129
δlnis a function of Crohn
S2-7-6: the final weight updating equation of the optimized power amplifier positive model is as follows:
s2-7-7: and (5) inverting the optimized power amplifier positive model to obtain a predistortion structure of the power amplifier.

Claims (4)

1. A power amplifier predistortion method of a complex value assembly line recurrent neural network model is characterized in that:
the method comprises the following steps:
s1: generating an original signal, wherein the original signal is used as an input signal of a target power amplifier;
s2: establishing a neural network model, training the neural network model to obtain a power amplifier positive model, and inverting the power amplifier positive model to obtain a predistortion structure;
s3: modulating an original signal to obtain a radio frequency signal;
s4: passing the radio frequency signal through a predistortion structure to obtain a predistortion signal;
s5: and obtaining a compensated power amplifier output signal by the pre-distortion signal through a target power amplifier.
2. The power amplifier predistortion method of the complex value assembly line recurrent neural network model as claimed in claim 1, characterized in that:
the S2 includes the following steps:
s2-1: selecting a segment from the original signal as a test signal;
s2-2: modulating the test signal to obtain a radio frequency signal;
s2-3: the radio frequency signal is amplified to obtain an output signal of the power amplifier;
s2-4: reducing the power of the output signal of the power amplifier through an attenuator to obtain an attenuated signal;
s2-5: carrying out down-conversion and filtering on the attenuated signal to obtain a target output signal;
s2-6: establishing a complex value assembly line recurrent neural network model, and training the complex value assembly line recurrent neural network model by using a test signal as an input parameter and a target output signal as an output parameter to obtain a power amplifier positive model;
s2-7: and optimizing the positive model of the power amplifier by using an enhanced complex value real-time recursive learning algorithm to obtain an optimized positive model of the power amplifier, and then inverting the optimized positive model of the power amplifier to obtain a predistortion structure of the power amplifier.
3. The power amplifier predistortion method of the complex value assembly line recurrent neural network model as claimed in claim 1, characterized in that:
the S2-6 comprises the following steps:
s2-6-1: establishing a complex value assembly line recurrent neural network model structure, wherein the model comprises M repeated modules;
each module is provided with N neurons, wherein the first M-1 modules are non-fully-connected recurrent neural networks, N-1 outputs in output neurons of the first module are used for feeding back to inputs, the outputs of the rest neurons, namely the output of the first neuron, are directly applied to the next module, the last module is a fully-connected recurrent neural network, and the outputs of all the neurons are fed back to the inputs;
s2-6-2: introducing augmented complex statistics, taking into account cross statistics of two variables, by inputting the conjugate x of variable x*To enhance the integrity of the information, a linear vector of Λ ═ x is obtainedT,xH]T；
S2-6-3: the weight vector of the l-th neuron of the network output layer of each module is
Figure FDA0002201108810000021
M modules all use the same weight matrix
S2-6-4: the mathematical expression of the model structure of the recurrent neural network of the complex value assembly line is
Figure FDA0002201108810000023
Where ψ (-) is an activation function,
Figure FDA0002201108810000024
and
Figure FDA0002201108810000025
respectively representing the output of the ith neuron of the tth module at the moment k and the network node input of an activation function;
augmented input for the tth module at time k
Figure FDA0002201108810000026
And input vector of Mth module
Figure FDA0002201108810000027
Is composed of
Figure FDA0002201108810000028
It(k)＝[s(k-t),...,s(k-t-p+1),1+j
yt+1,1(k-1),yt,2(k-1),...,yt,N(k-1)]
Figure FDA0002201108810000029
IM(k)＝[s(k-M),...,s(k-M-p+1),1+j
yM,1(k-1),yM,2(k-1),...,yM,N(k-1)]
Wherein p is the number of external inputs, i.e. the input sample is delayed by p unit durations, the upper right label a indicates that the formula is an augmented expression, and the upper right label x indicates that the formula is a conjugate;
s2-6-5: the mathematical expression of the model in S2-6-4 is simplified as:
Figure FDA0002201108810000031
s2-6-6: and (3) taking the first neuron output of the first module to represent the final output signal of the positive power amplifier model by using the formula obtained in S2-6-5:
Figure FDA0002201108810000032
4. the power amplifier predistortion method of the complex value assembly line recurrent neural network model as claimed in claim 1, characterized in that:
the S2-7 comprises the following steps:
s2-7-1: by using the measurement in S2-1Subtracting the target output signal in S2-5 from the test signal to obtain the t-th module error expression at the moment kWherein
Figure FDA0002201108810000034
Is the error of the real value of the signal,
Figure FDA0002201108810000035
is an imaginary error, d (k-t +1) is a target output signal, and since the input and output signals of the power amplifier are complex, the cost function of the complex value pipeline recurrent neural network model is as follows:
Figure FDA0002201108810000036
the lambda in the formula is an exponential weighting factor and is used for controlling the influence of the memory effect of each module on the whole structure;
s2-7-2: the weight coefficient is changed along the whole complex value assembly line recurrent neural network model by adopting a gradient descent learning method to minimize the cost function, and the method is as follows:
Figure FDA0002201108810000041
Figure FDA0002201108810000042
Figure FDA0002201108810000043
in the above formula
Figure FDA0002201108810000044
Is a complex-valued augmented weight function,
Figure FDA0002201108810000045
updating the formula for the weight of the nth weight of the ith neuron at time k,
Figure FDA0002201108810000046
for the gradient expression, η is the learning rate, which is usually taken to be a small constant;
s2-7-3: calculating the gradient of the real part and the imaginary part of the cost function to the complex value amplification type weight function as follows:
Figure FDA0002201108810000047
Figure FDA0002201108810000048
Figure FDA0002201108810000049
representing the sensitivity of the ith module at the time k of the ith neuron;
s2-7-4: according to the Cauchy-Riemann equation, a real part sensitivity function and an imaginary part sensitivity function which are formed by sensitivity are obtained:
Figure FDA00022011088100000410
Figure FDA00022011088100000411
Figure FDA00022011088100000412
s2-7-5: substituting the formulas of S2-7-4 and S2-7-3 into S2-7-2
Figure FDA00022011088100000413
A gradient can be obtained as:
Figure FDA0002201108810000051
the output of the sensitivity function at the time k of the jth neuron of the tth module is:
Figure FDA0002201108810000052
δlnis a function of Crohn
S2-7-6: the final weight updating equation of the optimized power amplifier positive model is as follows:
Figure FDA0002201108810000053
s2-7-7: and (5) inverting the optimized power amplifier positive model to obtain a predistortion structure of the power amplifier.
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