CN201878098U - Volterra-level based indirect learning type pre-distortion linearization system - Google Patents

Abstract

Translated fromChinese

本实用新型属数字通信预失真处理领域，尤其涉及一种基于Volterra级数间接学习型预失真线性化系统，包括：预失真信号生成模块、预失真信号处理模块、反馈模块及参数辨识模块；预失真信号处理模块由预失真信号经过处理后，得到射频功率放大器的输入信号；射频功率放大器输出信号中的一小部分功率经衰减后，进入反馈模块；反馈模块将反馈信号进行处理后，得到参数辨识模块的输入信号；参数辨识模块将其输出信号与预失真信号进行比较，从而得到误差信号；通过调整辨识模块与预失真器中的参数，逐渐缩小误差信号并使其归零。本实用新型稳定性高，自适应能力强，不用考虑其稳定性问题，同时能够处理多载波信号，互调失真改善效果好，可调范围大。

The utility model belongs to the field of digital communication pre-distortion processing, in particular to an indirect learning pre-distortion linearization system based on Volterra series, comprising: a pre-distortion signal generation module, a pre-distortion signal processing module, a feedback module and a parameter identification module; The distortion signal processing module obtains the input signal of the RF power amplifier after the predistortion signal is processed; a small part of the power of the output signal of the RF power amplifier is attenuated and enters the feedback module; the feedback module processes the feedback signal to obtain the parameter Identify the input signal of the module; the parameter identification module compares its output signal with the predistortion signal to obtain the error signal; by adjusting the parameters in the identification module and the predistorter, the error signal is gradually reduced and returned to zero. The utility model has high stability and strong self-adaptive ability, does not need to consider the stability problem, can process multi-carrier signals at the same time, has good intermodulation distortion improvement effect, and has a large adjustable range.

Description

Translated fromChinese

基于Volterra级数间接学习型预失真线性化系统Indirect Learning Predistortion Linearization System Based on Volterra Series

技术领域technical field

本实用新型属数字通信预失真处理领域，尤其涉及一种射频功率放大器线性化技术的基于Volterra级数间接学习型预失真线性化系统。The utility model belongs to the field of digital communication predistortion processing, in particular to a radio frequency power amplifier linearization technology based on Volterra series indirect learning type predistortion linearization system.

背景技术Background technique

随着数字通信技术的发展和3G技术的成熟，频带资源显得越来越珍贵。因此就要求对频带的利用率有所提高，这就迫切要求功率放大器有很好的线性度。在移动通信系统中，为了保证移动通信系统在一定范围内有信号覆盖，在信号通过射频前端和天线系统发射出去之前，通常使用功率放大器来进行信号放大。功率放大器的线性度直接影响着发射和接受信号的好坏程度，因此采用数字预失真技术是为了很好的解决线性度问题，同时可以提高功放效率，从而满足3G发展的需求。With the development of digital communication technology and the maturity of 3G technology, frequency band resources are becoming more and more precious. Therefore, it is required to increase the utilization rate of the frequency band, which urgently requires the power amplifier to have good linearity. In a mobile communication system, in order to ensure that the mobile communication system has signal coverage within a certain range, a power amplifier is usually used to amplify the signal before the signal is transmitted through the radio frequency front end and the antenna system. The linearity of the power amplifier directly affects the quality of the transmitted and received signals, so the use of digital pre-distortion technology is to solve the linearity problem well, and at the same time improve the efficiency of the power amplifier to meet the needs of 3G development.

一般的射频功率放大器都会产生频谱再生效应，这些现象都是由于功率放大器的非线性产生的，因此我们必须对功率放大器进行线性化处理也就是提高功率放大器的线性度。这就要求我们采用一些线性化技术来实现。对于线性化技术本身来讲可以很好的解决需求信道内的信号对其它临近信道的干扰。在3G的基站建设中，功率放大器的成本占到总成本的1/3以上，因此功率放大器如果解决了线性度和效率问题，这无疑给基站的成本带来大量的消减。General RF power amplifiers will produce spectrum regeneration effects, these phenomena are due to the nonlinearity of the power amplifier, so we must linearize the power amplifier, that is, improve the linearity of the power amplifier. This requires us to employ some linearization techniques to achieve. For the linearization technology itself, it can well solve the interference of signals in the required channel to other adjacent channels. In the construction of 3G base stations, the cost of the power amplifier accounts for more than 1/3 of the total cost. Therefore, if the power amplifier solves the linearity and efficiency problems, it will undoubtedly bring a lot of cost reduction to the base station.

目前国内外主要有：前馈法、功率回退、反馈法、预失真等射频功率放大器的线性化技术。其中前馈技术的优点在于，性能稳定、能够很好的改善功率放大器的线性化指标，但它同时也存在着成本高、器件特性随时间的变化不能够得到补偿、环路的设计比较复杂等缺点；功率回退法把工作电压从1dB回退到了线性工作区，因此它有较好的线性度，但同时也牺牲了功率放大器的效率，使得直流功耗非常大，这样就造成功放散热的问题，而散热是功率放大器的研究难点，故这种技术已经被其他线性化技术逐渐取代。而负反馈技术要求输入信号和反馈信号是同一时刻的信号，而系统本身是有延迟的，从这点来说是很难实现。At present, there are mainly linearization technologies of RF power amplifiers such as feedforward method, power back-off method, feedback method and pre-distortion at home and abroad. Among them, the advantage of feedforward technology is that it has stable performance and can improve the linearization index of the power amplifier, but it also has high cost, the change of device characteristics over time cannot be compensated, and the design of the loop is relatively complicated. Disadvantages: The power back-off method backs the working voltage from 1dB to the linear working area, so it has better linearity, but at the same time it also sacrifices the efficiency of the power amplifier, making the DC power consumption very large, which causes the heat dissipation of the successful radiator problem, and heat dissipation is a difficult point in the research of power amplifiers, so this technology has been gradually replaced by other linearization technologies. The negative feedback technology requires the input signal and the feedback signal to be at the same moment, and the system itself has a delay, which is difficult to achieve from this point of view.

实用新型内容Utility model content

本实用新型旨在克服现有技术的不足之处而提供一种成本较低，能够处理多载波信号，稳定性高，自适应能力强，互调失真改善效果好，可调范围大，整体结构简单的基于Volterra级数间接学习型预失真线性化系统。The utility model aims to overcome the deficiencies of the prior art and provide a low cost, capable of processing multi-carrier signals, high stability, strong self-adaptive ability, good intermodulation distortion improvement effect, large adjustable range, overall structure Simple predistortion linearization system based on Volterra series indirect learning.

为达到上述目的，本实用新型是这样实现的：In order to achieve the above object, the utility model is achieved in that:

基于Volterra级数间接学习型预失真线性化系统，其特征在于，包括：预失真信号生成模块、预失真信号处理模块、反馈模块及参数辨识模块；所述预失真信号生成模块、预失真信号处理模块、反馈模块及参数辨识模块的传输端口依次分别串接；Based on the Volterra series indirect learning type predistortion linearization system, it is characterized in that it includes: a predistortion signal generation module, a predistortion signal processing module, a feedback module and a parameter identification module; the predistortion signal generation module, predistortion signal processing The transmission ports of the module, the feedback module and the parameter identification module are respectively serially connected in sequence;

所述预失真信号生成模块包括预失真器，其由输入信号经过预失真器后，形成预失真信号；The predistortion signal generation module includes a predistorter, which forms a predistortion signal after the input signal passes through the predistorter;

所述预失真信号处理模块由预失真信号经过D/A转换、调制及上变频后，得到射频功率放大器的输入信号；所述射频功率放大器输出信号中的一小部分功率经衰减后，作为反馈信号进入反馈模块；The pre-distortion signal processing module obtains the input signal of the radio frequency power amplifier after the pre-distortion signal undergoes D/A conversion, modulation and up-conversion; after a small part of power in the output signal of the radio frequency power amplifier is attenuated, it is used as a feedback The signal enters the feedback module;

所述反馈模块将反馈信号经下变频、解调及A/D转换后，得到参数辨识模块的输入信号；The feedback module obtains the input signal of the parameter identification module after the feedback signal is down-converted, demodulated and A/D converted;

所述参数辨识模块将其输出信号与预失真信号进行比较，从而得到误差信号；通过调整辨识模块与预失真器中的参数，逐渐缩小误差信号。The parameter identification module compares its output signal with the predistortion signal to obtain an error signal; by adjusting the parameters in the identification module and the predistorter, the error signal is gradually reduced.

本实用新型可通过调整辨识模块与预失真器中的参数，逐渐缩小误差信号并使其归零。The utility model can gradually reduce the error signal and make it return to zero by adjusting the parameters in the identification module and the predistorter.

本实用新型可通过RLS算法调整辨识模块与预失真器中的参数，逐渐缩小误差信号并使其归零。The utility model can adjust the parameters in the identification module and the predistorter through the RLS algorithm, gradually reduce the error signal and make it return to zero.

本实用新型所述射频功率放大器输出信号中的一小部分功率经增益为1/G的衰减器衰减后，作为反馈信号进入反馈模块，其中G为放大器的期望增益。A small part of the power in the output signal of the radio frequency power amplifier described in the utility model is attenuated by an attenuator with a gain of 1/G, and then enters the feedback module as a feedback signal, where G is the expected gain of the amplifier.

本实用新型当参数辨识算法收敛后，将反馈模块及参数辨识模块断开；当预失真器与射频功率放大器之间失去原有的匹配关系时，将反馈回路和参数辨识模块重新接入。In the utility model, when the parameter identification algorithm converges, the feedback module and the parameter identification module are disconnected; when the original matching relationship between the predistorter and the radio frequency power amplifier is lost, the feedback loop and the parameter identification module are reconnected.

本实用新型所述预失真信号处理模块包括D/A转换部分、调制部分、上变频部分及射频功率放大器部分；所述D/A转换部分、调制部分、上变频部分及射频功率放大器的端口依次串接。The pre-distortion signal processing module of the utility model includes a D/A conversion part, a modulation part, an up-conversion part and a radio frequency power amplifier part; the ports of the D/A conversion part, modulation part, up-conversion part and radio frequency power amplifier are sequentially in series.

本实用新型所述反馈模块包括下变频部分、解调部分及A/D转换部分；所述下变频部分、解调部分及A/D转换部分的端口依次串接。The feedback module of the utility model includes a down-conversion part, a demodulation part and an A/D conversion part; the ports of the down-conversion part, demodulation part and A/D conversion part are sequentially connected in series.

本实用新型所述预失真信号生成模块或参数辨识模块可采用FPGA模块。The pre-distortion signal generation module or parameter identification module of the utility model can adopt FPGA module.

本实用新型结构简单，稳定性高，自适应能力强，不用考虑其稳定性问题，同时能够处理多载波信号，互调失真改善效果好，可调范围大，是目前性价比较高的一种功率放大器线性化技术。The utility model has the advantages of simple structure, high stability and strong self-adaptive ability. It does not need to consider its stability problem. At the same time, it can process multi-carrier signals. It has a good improvement effect on intermodulation distortion and a large adjustable range. amplifier linearization techniques.

本实用新型在深入研究功放基带预失真技术的理论和算法的基础上，提出了一种基于Volterra级数的线性预失真方法，并利用了RLS即递归最小二乘法进行自适应预失真调整，提高了功放线性拟合的收敛速度以及功放线性输出的稳定性。The utility model proposes a linear pre-distortion method based on Volterra series on the basis of in-depth research on the theory and algorithm of power amplifier baseband pre-distortion technology, and utilizes the RLS (recursive least squares method) for adaptive pre-distortion adjustment, improving The convergence speed of the linear fitting of the power amplifier and the stability of the linear output of the power amplifier are obtained.

附图说明Description of drawings

下面结合附图和具体实施方式对本实用新型作进一步说明。本实用新型的保护范围不仅局限于下列内容的表述。Below in conjunction with accompanying drawing and specific embodiment, the utility model is further described. The protection scope of the present utility model is not limited to the expression of the following content.

图1为本实用新型的整体结构示意图；Fig. 1 is the overall structural representation of the utility model;

图2为本实用新型基于FPGA的硬件结构示意图；Fig. 2 is the hardware structural representation based on FPGA of the utility model;

图3为有记忆非线性系统的分解。Figure 3 shows the decomposition of a nonlinear system with memory.

具体实施方式Detailed ways

Volterra级数理论是分析非线性系统的一种有效的数学工具。对于线性时不变系统，其零状态响应等于单位冲击响应h(t)与输入信号x(t)的卷积：Volterra series theory is an effective mathematical tool for analyzing nonlinear systems. For a linear time-invariant system, its zero-state response is equal to the convolution of the unit impulse response h(t) with the input signal x(t):

$\mathrm{<span><span>y</span>the\; y</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>)</span>)</span><span><span>=</span>=</span>{<span><span>\&Integral;</span>\&Integral;</span>}_{<span><span>-</span>-</span><span><span>\&infin;</span>\&infin;</span>}^{<span><span>\&infin;</span>\&infin;</span>}\mathrm{<span><span>h</span>h</span>}<span><span>(</span>(</span>\mathrm{<span><span>\&tau;</span>\&tau;</span>}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>t</span>t</span>}<span><span>-</span>-</span>\mathrm{<span><span>\&tau;</span>\&tau;</span>}<span><span>)</span>)</span>\mathrm{<span><span>d\&tau;</span>d\&tau;</span>}$

Volterra级数模型是一种泛函级数模型，它将上述形式的关系加以推广，用于描述有记忆非线性系统。The Volterra series model is a functional series model, which generalizes the relationship of the above form, and is used to describe nonlinear systems with memory.

由非线性动态系统的分解定理可知，连续泛函F(·)所表征的非线性动态系统，当其输入信号的能量有限时，总可以分解为有记忆线性系统和一个无记忆非线性系统的级联，如图3所示。将线性子系统分别记为F_L1(·)，F_L2(·)，…，F_LN(·)，每个子系统的输出依次为w₁(t)，w₂(t)，…，w_N(t)，无记忆非线性系统记为F_NL(·)，则整个系统的输出可表示为：According to the decomposition theorem of nonlinear dynamic systems, the nonlinear dynamic system represented by the continuous functional F( ) can always be decomposed into a linear system with memory and a nonlinear system without memory when the energy of its input signal is limited Cascade, as shown in Figure 3. The linear subsystems are denoted as F_L1 ( ), F_L2 ( ), ..., F_LN ( ), and the output of each subsystem is w₁ (t), w₂ (t), ..., w_N (t), the memoryless nonlinear system is denoted as F_NL (·), then the output of the whole system can be expressed as:

y(t)＝F_NL[w₁(t)，w₂(t)，…，w_N(t)]y(t)=F_NL [w₁ (t), w₂ (t), ..., w_N (t)]

对于预失真线性化系统，预失真器的设计十分重要。Volterra级数可以将满足一定条件的有记忆非线性系统逼近到任意准确的程度，它不仅可用于射频功率放大器的建模，而且也可以用于构造预失真器。一般形式的Volterra预失真器的输入、输出信号之间的关系如式(1-1)所示。For predistortion linearization systems, the design of the predistorter is very important. Volterra series can approximate the nonlinear system with memory satisfying certain conditions to any degree of accuracy. It can be used not only for modeling RF power amplifiers, but also for constructing predistorters. The relationship between the input and output signals of a general form of Volterra predistorter is shown in formula (1-1).

$\mathrm{<span><span>y</span>the\; y</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span><span><span>=</span>=</span>{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>0</span>0</span>}}<span><span>+</span>+</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span><span><span>+</span>+</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span><span><span>+</span>+</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>$

$<span><span>+</span>+</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}<span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>k</span>k</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>k</span>k</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>k</span>k</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>k</span>k</span>}}<span><span>)</span>)</span><span><span>+</span>+</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>4</span>4</span>}<span><span>)</span>)</span>$

在(1-1)式中，Volterra核参数的数量为：In formula (1-1), the number of Volterra kernel parameters is:

${\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>para</span>para</span>}}<span><span>=</span>=</span>\frac{{\mathrm{<span><span>M</span>m</span>}}^{\mathrm{<span><span>k</span>k</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}<span><span>)</span>)</span>$

可见在Volterra级数模型中，参数的数量和记忆长度成幂函数关系，和模型阶数成指数函数关系。受计算复杂度的限制，不经任何简化的Volterra模型只适用于低阶弱非线性的情形。而对于宽带系统中的有记忆射频功放，低阶弱非线性模型很难精确描述它的特性或逆特性。因此，直接将一般形式的volterra级数运用于功放建模或预失真器设计是十分困难的，必须进行简化改进。It can be seen that in the Volterra series model, the number of parameters has a power function relationship with the memory length, and an exponential function relationship with the model order. Due to the limitation of computational complexity, the Volterra model without any simplification is only suitable for low-order weak nonlinear situations. As for the RF power amplifier with memory in the broadband system, it is difficult to accurately describe its characteristics or inverse characteristics by the low-order weak nonlinear model. Therefore, it is very difficult to directly apply the general form of volterra series to power amplifier modeling or predistorter design, and simplification and improvement must be carried out.

对放大器非线性特性的分析可知，奇次项产生输出信号的奇阶谐波频率分量和奇阶互调频率分量，偶次项产生直流分量、偶阶谐波频率分量和偶阶互调频率分量。一般情况下，只有奇阶互调频率分量落在通带内，而其它失真分量都落在通带以外，可以容易地用滤波器滤除。尽管在预失真器中包含偶次项，对提高线性化效果有一定的作用，但出于降低模型复杂度的考虑，仍然剔除了预失真器中的偶次项。去除直流项和偶次项后，(1-1)式可写为The analysis of the nonlinear characteristics of the amplifier shows that the odd-order term produces the odd-order harmonic frequency component and the odd-order intermodulation frequency component of the output signal, and the even-order term produces the DC component, the even-order harmonic frequency component and the even-order intermodulation frequency component . Generally, only odd-order intermodulation frequency components fall within the passband, while other distortion components fall outside the passband, which can be easily filtered out with a filter. Although the inclusion of even-order terms in the predistorter has a certain effect on improving the linearization effect, the even-order terms in the pre-distorter are still removed for the sake of reducing the complexity of the model. After removing the DC term and the even-order term, (1-1) can be written as

$\mathrm{<span><span>y</span>the\; y</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span><span><span>+</span>+</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>i</span><figure-callout\; label="i"\; filenames="HSA00000160412100012.png,HSA00000160412100021.png"\; state="\{\{state\}\}">i</figure-callout></span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>)</span>)</span>$

$<span><span>+</span>+</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>+</span>+</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}<span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>kd</span>kd</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>d</span>d</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>i</span><figure-callout\; label="i"\; filenames="HSA00000160412100012.png,HSA00000160412100021.png"\; state="\{\{state\}\}">i</figure-callout></span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>d</span>d</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>d</span>d</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>3</span>3</span>}<span><span>)</span>)</span>$

如果运用具有对称核的Volterra级数来构造预失真器，则参数数量将进一步减少。Volterra核的对称性的含意如下。如果k阶Volterra核h_k(i₁，i₂，…，i_k)满足If a Volterra series with a symmetric kernel is used to construct the predistorter, the number of parameters will be further reduced. The implications of the symmetry of the Volterra nucleus are as follows. If the k-order Volterra kernel h_k (i₁ , i₂ ,..., i_k ) satisfies

h_k(i₁，i₂，…，i_k)＝h_k(i_π(1)，i_π(2)，…，i_π(k))         (1-4)h_k (i₁ , i₂ ,...,i_k )=h_k (i_π(1) , i_π(2) ,...,i_π(k) ) (1-4)

称h_k(i₁，i₂，…，i_k)为对称核。式中，π(·)表示1，2，…，k的任意一种排列。例如，设h₃(i₁，i₂，i₃)为三阶对称核，则有：Call h_k (i₁ , i₂ ,…, i_k ) a symmetric kernel. In the formula, π(·) represents any permutation of 1, 2, ..., k. For example, if h₃ (i₁ , i₂ , i₃ ) is a third-order symmetric kernel, then:

h₃(i₁，i₂，i₃)＝h₃(i₁，i₃，i₂)＝h₃(i₂，i₁，i₃)＝h₃(i₂，i₃，i₁)＝h₃(i₃，i₁，i₂)＝h₃(i₃，i₂，i₁)h₃ (i₁ , i₂ , i₃ )=h₃ (i₁ , i₃ , i₂ )=h₃ (i₂ , i₁ , i₃ )=h₃ (i₂ , i₃ , i₁ )=h₃ (i₃ , i₁ , i₂ )=h₃ (i₃ , i₂ , i₁ )

具有对称核的时域Volterra级数满足如下唯一性定理：如果一个非线性系统的输入输出关系可以用Volterra级数来描述，且它的各阶核为对称的，则描述这一非线性系统输入输出关系的Volterra级数是唯一的。The time-domain Volterra series with a symmetric kernel satisfies the following uniqueness theorem: If the input-output relationship of a nonlinear system can be described by a Volterra series, and its kernels of each order are symmetric, then describe the nonlinear system input The Volterra series of the output relation is unique.

利用核的对称性，可以合并Voiterra级数预失真器中的冗余项，使参数数量大幅度减少。对(1-3)式利用对称性进行简化后可以写为：Utilizing the symmetry of the kernel, the redundant items in the Voiterra series predistorter can be combined, so that the number of parameters can be greatly reduced. After simplifying the formula (1-3) by using symmetry, it can be written as:

${\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>p</span>p</span>}}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span><span><span>+</span>+</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>i</span><figure-callout\; label="i"\; filenames="HSA00000160412100012.png,HSA00000160412100021.png"\; state="\{\{state\}\}">i</figure-callout></span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>)</span>)</span>$

$<span><span>+</span>+</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>+</span>+</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}<span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>\underset{{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>kd</span>kd</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>d</span>d</span>}}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>d</span>d</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>i</span><figure-callout\; label="i"\; filenames="HSA00000160412100012.png,HSA00000160412100021.png"\; state="\{\{state\}\}">i</figure-callout></span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>,</span>,</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>,</span>,</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>d</span>d</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>{\mathrm{<span><span>i</span>i</span>}}_{\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>d</span>d</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>5</span>5</span>}<span><span>)</span>)</span>$

但当系统的阶数较高或者记忆效应较强时，Voiterra核的数量仍较庞大。这使得Voiterra级数用于构造高阶强记忆预失真器时，仍会产生较大的计算量，所以要考虑进一步简化。But when the order of the system is high or the memory effect is strong, the number of Voiterra kernels is still large. This makes the Voiterra series still produce a large amount of calculation when used to construct a high-order strong memory predistorter, so further simplification should be considered.

在相关文献中，常可以见到一种被称作记忆多项式的功放或预失真器模型，它的表达式如下：In the relevant literature, it is often possible to see a power amplifier or predistorter model called a memory polynomial, and its expression is as follows:

$\mathrm{<span><span>y</span>the\; y</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\underset{\mathrm{<span><span>k</span>k</span>}<span><span>=</span>=</span>\mathrm{<span><span>1</span>1</span>}}{\overset{\mathrm{<span><span>K</span>K</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{\mathrm{<span><span>i</span>i</span>}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}}{\overset{\mathrm{<span><span>M</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>a</span>a</span>}}_{\mathrm{<span><span>ki</span>the\; ki</span>}}\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>\mathrm{<span><span>i</span>i</span>}<span><span>)</span>)</span>{<span><span>|</span>|</span>\mathrm{<span><span>x</span>x</span>}<span><span>(</span>(</span>\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>\mathrm{<span><span>i</span>i</span>}<span><span>)</span>)</span><span><span>|</span>|</span>}^{\mathrm{<span><span>k</span>k</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>6</span>6</span>}<span><span>)</span>)</span>$

式中，k为模型阶数，M为记忆长度，a_ki为多项式系数。它实际上是Volterra级数模型的一种特殊情形。在Volterra级数模型中，如果只保留对角核(diagonal kemel)，而将所有的非对角核置零，就得到了记忆多项式模型。记忆多项式模型过于简化，用它来设计预失真器，难以精确地描述有记忆功放的逆特性。In the formula, k is the order of the model, M is the memory length, and a_ki is the polynomial coefficient. It is actually a special case of the Volterra series model. In the Volterra series model, if only the diagonal kernel (diagonal kemel) is kept, and all non-diagonal kernels are set to zero, the memory polynomial model is obtained. The memory polynomial model is too simplified, and it is difficult to accurately describe the inverse characteristics of the power amplifier with memory if it is used to design the predistorter.

Volterra级数中，非对角核实际上代表了不同时刻输入信号间的“耦合”效应。比如，h₃(1，1，3)代表了n-1时刻与n-3时刻输入信号间的“耦合”。如果放大器的几个输入信号的采样时刻相距越远，其间的“耦合”效应也应该越弱，那么它们对应的Volterra核的值会越小，对输出的贡献也越小。In the Volterra series, the off-diagonal kernel actually represents the "coupling" effect between input signals at different times. For example, h₃ (1, 1, 3) represents the "coupling" between the input signals at time n-1 and time n-3. If the sampling moments of several input signals of the amplifier are farther apart, the "coupling" effect between them should be weaker, then the value of their corresponding Volterra cores will be smaller, and the contribution to the output will be smaller.

出于以下两点考虑，我们没有必要在Volterra模型中保留那些模很小的核。For the following two considerations, it is not necessary for us to keep those kernels with small modules in the Volterra model.

(1)这些核对模型的输出贡献很小，而对它们进行辨识要增加较大的计算量；(1) The output contribution of these check models is very small, and the identification of them requires a large amount of calculation;

(2)由于计算机的字长有限，对这些模很小的核进行辨识不可避免地会引入误差。(2) Due to the limited word length of the computer, it is inevitable to introduce errors to identify these small kernels.

因此将它们保留在模型中实际上未必能明显提高模型的精确度。考虑到功放模型和预失真器之间存在的对应关系，我们采用如下算法对(1-5)式所描述的Volterra预失真器进行进一步简化。将(1-5)式中的第l阶核记为h_l(i₁，i₂，…，i_l)，其中1＝1，3，…，2d+1。设定阈值λ∈{1，2，…，M}。当1＝1时，h₁(i₁)＝h₁(i₁)。当l≥3时，

如果max{|i_s-i_t|}≥λ，则令h_l(i₁，i₂，…，i_l)＝0；否则h₁(i₁，i₂，…，i_l)＝h₁(i₁，i₂，…，i_l)。So keeping them in the model may not actually improve the accuracy of the model significantly. Considering the corresponding relationship between the power amplifier model and the predistorter, we use the following algorithm to further simplify the Volterra predistorter described by equation (1-5). The first-order kernel in formula (1-5) is recorded as h_l (i₁ , i₂ , ..., i_l ), where 1=1, 3, ...,2d+1. Set the threshold λ ∈ {1, 2, ..., M}. When 1=1, h₁ (i₁ )=h₁ (i₁ ). When l≥3,

If max{|i_s -i_t |}≥λ, then let h_l (i₁ , i₂ ,…,i_l )=0; otherwise h₁ (i₁ , i₂ ,…,i_l )=h₁ (i₁ , i₂ ,..., i_l ).

这种算法实质上是在一般形式的Volterra预失真器和记忆多项式预失真器之间进行“折衷”。选取的阈值λ越小，则预失真器结构越简单，精确度越差。如果将该算法运用于(1-1)式，当λ＝1时，预失真器就退化为记忆多项式预失真器。选取的阈值λ越大，则保留的核参数越多，预失真器的精确度也越高。当λ＝M时，所有的核都被保留，预失真器等同于一般Volterra预失真器。This algorithm is essentially a "compromise" between a general form Volterra predistorter and a memory polynomial predistorter. The smaller the selected threshold λ, the simpler the structure of the predistorter and the worse the accuracy. If this algorithm is applied to (1-1) formula, when λ=1, the predistorter degenerates into a memory polynomial predistorter. The larger the selected threshold λ, the more kernel parameters are preserved, and the higher the accuracy of the predistorter is. When λ=M, all cores are preserved, and the predistorter is equivalent to a general Volterra predistorter.

归纳起来，依次按照如下三个步骤对一般形式Volterra预失真器进行简化。(1)去除直流项和偶次项，只保留奇次项。(2)利用Volterra核的对称性，合并预失真器模型中的冗余项。(3)运用简化算法，使预失真器中的参数数量进一步减少。最后，通过Volterra核对输入信号不断地调整，使其通过功放后，线性输出。To sum up, the general form Volterra predistorter is simplified according to the following three steps. (1) Remove the DC term and the even-order term, and only keep the odd-order term. (2) Taking advantage of the symmetry of the Volterra kernel, the redundant terms in the predistorter model are merged. (3) A simplified algorithm is used to further reduce the number of parameters in the predistorter. Finally, the input signal is continuously adjusted through the Volterra check, so that it is linearly output after passing through the power amplifier.

对基于工作函数的预失真线性化系统，实现方式大致可分为两类。由于预失真器特性是放大器特性的逆，因此可以先建立功率放大器的模型，然后再求解预失真器模型。当功率放大器的模型较简单时，这种方法是可行的，但对于有记忆高阶非线性系统，辨识它的逆模型十分困难。此外，这种方法很难实现预失真器参数的自适应调整，当放大器特性发生变化时，系统性能会迅速下降。另一种方法则可以不建立放大器的模型，直接获得预失真器参数。该方法在线性化系统中增加一条信号反馈回路，将反馈信号与预失真器输出信号对比，得到一个误差信号，在辨识过程中，通过不断地调整预失真器的参数来减小误差信号。当误差信号足够小时，就得到了预失真器的参数。这就是以下要采用的间接学习结构。For the predistortion linearization system based on the work function, the implementation methods can be roughly divided into two categories. Since the predistorter characteristic is the inverse of the amplifier characteristic, the power amplifier model can be established first, and then the predistorter model can be solved. This method is feasible when the model of the power amplifier is relatively simple, but for high-order nonlinear systems with memory, it is very difficult to identify its inverse model. In addition, this method is difficult to achieve adaptive adjustment of predistorter parameters, and when the characteristics of the amplifier change, the system performance will drop rapidly. Another method is to directly obtain the parameters of the predistorter without establishing a model of the amplifier. This method adds a signal feedback loop to the linearization system, and compares the feedback signal with the output signal of the predistorter to obtain an error signal. During the identification process, the error signal is reduced by continuously adjusting the parameters of the predistorter. When the error signal is small enough, the parameters of the predistorter are obtained. This is the indirect learning structure to be adopted below.

采用间接学习结构的预失真线性化系统实现方案如图1所示。信号的预失真过程在基带内完成，输入信号x(n)经过预失真器后，形成预失真信号x_p(n)。预失真信号经过D/A转换、调制和上变频后，得到射频功率放大器的输入信号x_RF(t)。放大器输出信号y_RF(t)中的一小部分功率经过增益为1/G的衰减器后形成反馈，其中G为放大器的期望增益。反馈信号经过下变频、解调和A/D转换后，得到参数辨识模块的输入信号u(n)。参数辨识模块具有和预失真器完全相同的结构和参数，它的输出信号记为u_p(n)。u_p(n)和预失真信号x_p(n)进行比较，得到误差信号e(n)。在工作过程中，通过RLS算法调整辨识模块和预失真器中的参数，不断地减小误差信号。在理想的情况下，当误差信号e(n)等于零时，可得y(n)＝Gx(n)，其中y(n)为放大器输出的基带等效信号。The implementation scheme of the predistortion linearization system using the indirect learning structure is shown in Figure 1. The pre-distortion process of the signal is completed in the baseband, and the input signal x(n) forms a pre-distortion signal x_p (n) after passing through the pre-distorter. After the predistortion signal is converted by D/A, modulated and up-converted, the input signal x_RF (t) of the radio frequency power amplifier is obtained. A fraction of the power in the amplifier output signal y_RF (t) is fed back through an attenuator with a gain of 1/G, where G is the desired gain of the amplifier. After the feedback signal is down-converted, demodulated and A/D converted, the input signal u(n) of the parameter identification module is obtained. The parameter identification module has exactly the same structure and parameters as the predistorter, and its output signal is marked as_up (n). u_p (n) is compared with the predistortion signal x_p (n) to obtain an error signal e(n). During the working process, the parameters in the identification module and the predistorter are adjusted through the RLS algorithm to continuously reduce the error signal. In an ideal situation, when the error signal e(n) is equal to zero, y(n)=Gx(n) can be obtained, where y(n) is the baseband equivalent signal output by the amplifier.

在上述预失真系统中，不需要预先辨识出功率放大器的模型，就可以直接获得预失真器的参数。参数辨识算法收敛后，就可将反馈回路和参数辨识模块暂时断开。在发射机工作过程中，功放特性会发生变化。当这种变化达到一定程度时，预失真器和放大器之间会失去原有的匹配关系。此时，可将反馈回路和参数辨识模块重新接入，以对预失真器参数进行自适应更新。In the above predistortion system, the parameters of the predistorter can be obtained directly without identifying the model of the power amplifier in advance. After the parameter identification algorithm converges, the feedback loop and the parameter identification module can be temporarily disconnected. During the operation of the transmitter, the characteristics of the power amplifier will change. When this change reaches a certain level, the original matching relationship between the predistorter and the amplifier will be lost. At this point, the feedback loop and the parameter identification module can be reconnected to perform adaptive update on the parameters of the predistorter.

本实用新型所述预失真信号处理模块包括D/A转换部分、调制部分、上变频部分及射频功率放大器部分；所述D/A转换部分、调制部分、上变频部分及射频功率放大器的端口依次串接。The pre-distortion signal processing module of the utility model includes a D/A conversion part, a modulation part, an up-conversion part and a radio frequency power amplifier part; the ports of the D/A conversion part, modulation part, up-conversion part and radio frequency power amplifier are sequentially in series.

本实用新型所述反馈模块包括下变频部分、解调部分及A/D转换部分；所述下变频部分、解调部分及A/D转换部分的端口依次串接。The feedback module of the utility model includes a down-conversion part, a demodulation part and an A/D conversion part; the ports of the down-conversion part, demodulation part and A/D conversion part are sequentially connected in series.

本实用新型所述预失真信号生成模块或参数辨识模块采用FPGA模块。The pre-distortion signal generation module or parameter identification module described in the utility model adopts an FPGA module.

对于预失真器以及参数辨识模块，本实用新型采用了基于FPGA的硬件开发逻辑，其硬件原理图如图2所示。For the predistorter and the parameter identification module, the utility model adopts FPGA-based hardware development logic, and its hardware schematic diagram is shown in FIG. 2 .

可以理解地是，以上关于本实用新型的具体描述，仅用于说明本实用新型而并非受限于本实用新型实施例所描述的技术方案，本领域的普通技术人员应当理解，仍然可以对本实用新型进行修改或等同替换，以达到相同的技术效果；只要满足使用需要，都在本实用新型的保护范围之内。It can be understood that the above specific description of the utility model is only used to illustrate the utility model and is not limited to the technical solutions described in the embodiments of the utility model. Those of ordinary skill in the art should understand that they can still understand the utility model Modifications or equivalent replacements are carried out to achieve the same technical effect; as long as the needs of use are met, they are all within the protection scope of the present utility model.

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Translated fromChinese

1.一种基于Volterra级数间接学习型预失真线性化系统，其特征在于，包括：预失真信号生成模块、预失真信号处理模块、反馈模块及参数辨识模块；所述预失真信号生成模块、预失真信号处理模块、反馈模块及参数辨识模块的传输端口依次分别串接；1. a kind of indirect learning type predistortion linearization system based on Volterra series, is characterized in that, comprises: predistortion signal generation module, predistortion signal processing module, feedback module and parameter identification module; Described predistortion signal generation module, The transmission ports of the pre-distortion signal processing module, the feedback module and the parameter identification module are respectively connected in series in sequence;所述预失真信号生成模块包括预失真器，其由输入信号经过预失真器后，形成预失真信号；The predistortion signal generation module includes a predistorter, which forms a predistortion signal after the input signal passes through the predistorter;所述预失真信号处理模块由预失真信号经过D/A转换、调制及上变频后，得到射频功率放大器的输入信号；所述射频功率放大器输出信号中的一小部分功率经衰减后，作为反馈信号进入反馈模块；The pre-distortion signal processing module obtains the input signal of the radio frequency power amplifier after the pre-distortion signal undergoes D/A conversion, modulation and up-conversion; after a small part of power in the output signal of the radio frequency power amplifier is attenuated, it is used as a feedback The signal enters the feedback module;所述反馈模块将反馈信号经下变频、解调及A/D转换后，得到参数辨识模块的输入信号。The feedback module obtains the input signal of the parameter identification module after the feedback signal is down-converted, demodulated and A/D converted.2.根据权利要求1所述的基于Volterra级数间接学习型预失真线性化系统，其特征在于：所述预失真信号处理模块包括D/A转换部分、调制部分、上变频部分及射频功率放大器部分；所述D/A转换部分、调制部分、上变频部分及射频功率放大器的端口依次串接。2. The indirect learning type predistortion linearization system based on Volterra series according to claim 1, wherein the predistortion signal processing module includes a D/A conversion part, a modulation part, an upconversion part and a radio frequency power amplifier part; the ports of the D/A conversion part, the modulation part, the up-conversion part and the radio frequency power amplifier are sequentially connected in series.3.根据权利要求2所述的基于Volterra级数间接学习型预失真线性化系统，其特征在于：所述反馈模块包括下变频部分、解调部分及A/D转换部分；所述下变频部分、解调部分及A/D转换部分的端口依次串接。3. the indirect learning type predistortion linearization system based on Volterra series according to claim 2, is characterized in that: the feedback module includes a down-conversion part, a demodulation part and an A/D conversion part; the down-conversion part The ports of the demodulation part and the A/D conversion part are sequentially connected in series.4.根据权利要求3所述的基于Volterra级数间接学习型预失真线性化系统，其特征在于：所述预失真信号生成模块或参数辨识模块采用FPGA模块。 4. The indirect learning predistortion linearization system based on Volterra series according to claim 3, characterized in that: said predistortion signal generation module or parameter identification module adopts FPGA module. the

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