Новые приложения модели нелинейного усилителя мощности

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Smirnov Andrei Vladimirovich,
PhD student, Moscow Technical University of Communications and Informatics, MTUCI, Moscow, Russia, sandrew2k@yandex. ru
Keywords: power amplifier, memoryless nonlinearity, AM-AM decomposition, efficiency, intermodulation noise, spectrum regrowth, output current DC component, Gaussian random process, single tone model, AM-AM linearization.
The use of the AM-AM and AM-PM functions for the modeling of nonlinear power amplifier distortion is widespread in the wireless system development process. The use of these models aims upon estimation of the out of band emission and the impact of intermodulation noise on the system energy budget. Based on the derived parameterized AM-AM function that was obtained after the detailed comparative analysis of the nonlinearities applied both to RF signal instantaneous amplitude and to the signal'-s low-pas equivalent, the present paper proposes the new applications of the power amplifier nonlinearity model. The parameters of the derived AM-AM function are the percent of time that the amplifier spends in the saturation region of its transfer characteristic and the percent of time that it works with the output current cut-off. The AM-AM function is derived in the form of the function basis expansion with the coefficients that are determined by the corresponding band-pass power of the intermodulation distortion components. This permits the model to be used for the system BER performance prediction in presence of partial nonlinear distortion compensation for a specified amplifier parameters set. The same approach is applied to obtain the nonlinear interconnection between the output current DC component and the input signal lowpass amplitude envelope, which is then used to carry out the theoretical PA energy efficiency estimation for the Gaussian model of the input signal. The obtained results of the mean efficiency estimation for various PA parameters are compared with the corresponding estimates of the intermodulation noise power.
Для цитирования:
Смирнов А. В. Новые приложения модели нелинейного усилителя мощности // T-Comm: Телекоммуникации и транспорт. — 2015. -Том 9. — № 9. — С. 76−84.
For citation:
Smirnov A.V. The novel applications of nonlinear power amplifier model. T-Comm. 2015. Vol 9. No. 9, pp. 76−84.
I. Introduction
The power amplifier models are commonly considered as a part of channel medium in modern communication system simulation environment [12]. The purpose of these models is to approximate the signal waveform distortion during the amplification process with a good precision. Following this, many baseband nonlinear power amplifier (NLPA) models were introduced in the last couple of decades, which can be classified as [2, 3]:
1) strictly memoryiess models — defined by the means of AM-AM function,
2) efficiently memoryiess models — defined by the means of AM-AM and AM-PM functions,
3) models with memory — defined by the Volterra operator kernels that do not admit the separate treatment of the amplitude envelope and phase distortions.
The NLPA models are most often used to predict the out of band emission and to evaluate the bit-error rate (BER) performance in presence of the intermodulation noise resulted from nonlinear amplification [I 1]. However, there exist some other possible applications of NLPA models, which might be useful in the early stage of the wireless system design process and which are not covered by the existing NLPA modelling approaches. Confining to the strictly memoryiess NLPA model assumption, the two following applications are considered in the present article.
The first application relates to the system performance evaluation under the use of PA linearization. As the linearization process is often takes place in a digital processing domain [I], only finite number of intermodulation distortion (IMD) components may be removed due to the fundamental bandwidth constraints of a digital signal processing path. So generally, there is always a residual intermodulation noise contaminating the output signal of the PA, This residua! noise comprises the higher order IMD products, which cannot be practically compensated by the means of linearization techniques. The effect of the partial IMD components suppression on the BER and the out-of-band system performances might be studied if the appropriate NLPA model would be at disposal during the design process.
The second possible application of NLPA model is the energy efficiency estimation over the range of PA parameters. If the AM-AM function is parameterized to account the cut-off and saturation effects in the PA, then it is possible to analyze statistically the distorted signal amplitude envelope behaviour. If, moreover, it is possible to derive the relation between the PA output current DC component and the input signal amplitude envelope (referred to as AM-DC function), then PA efficiency can also be the subject of statistical estimation and its mean value might be predicted [9]. Moreover, a comparative study of the efficiency and corresponding IMD power might be carried out to aid the compromise choice of the PA parameters.
The main purpose of the following article is to derive a parameterized NLPA model, that could be useful in both aforementioned applications and to provide the examples that expose these applications. The article is structured as follows. The 2™* section contains an overview of the OFDM
signal distortion problem in a strictly memoryiess nonlinear device. In the 3rd section an approach to AM-AM and AM-DC functions derivation process is presented. The main feature of the described model is that the AM-AM function is derived in the form of linear superposition, where the IMD components determine the expansion coefficients. The described method is strongly related to the assumption that the input signal can be referred to as stationary Gaussian random process, which is, for instance, the case for the conventional OFDM waveform. The interconnection between the spectrum regrowth and the AM-AM function shape is analyzed. The obtained AM-AM functions are analyzed and compared to the corresponding nonlinear characteristics derived by a simplistic single-tone approach.
The 4* section presents the examples of the derived NLPA model applications. First, the OFDM waveform BER evaluation for the punctured IMD product set AM-AM characteristic is carried out using the Monte-Carlo simulation method. Then, based on the derived AM-AM and AM-DC functions, the theoretical PA mean efficiency and output IMD power are evaluated through the set of PA parameters.
2. Nonlinear OFDM signal amplification
A typical equivalent representation scheme of NLPA is represented in Fig. I (a). Besides the nonlinear transfer function T (. v) of the amplifying device (AD) there are also the input and output circuits with the finite-duration impulse responses /j,-«® and hout ® that result from the parasitic
reactances of the AD and cause the memory effects to manifest during the amplification [5]. When the impulse response duration 7*0 is far lesser than the reciprocal of the input signal bandwidth (7~n „-), it can be assumed that its shape B
do not vary for a fixed level of the input signal amplitude envelope and induces a determined phase shift to the signal waveform [3]. Adopting such a model for NLPA, the amplitude and phase distortion of the signal can be modeled separately by the AM-AM and AM-PM functions respectively [8]. However, when the AD affords small perturbations of parasitic reactances over the signal dynamic range and the carrier frequency is of order of 1−2 GHz or less, then the NLPA memory effects can be neglected [8] and the equivalent NLPA scheme permits the following substitution that is depicted on Fig. I (b).
hm® -k& gt- - htmt{T~) ->
Fig. I. Power amplifier structure scheme: (a) with device input/output memory effect account, (b) strictly memoryiess case
The output bandpass filter bandwidth is assumed to be much greater than signal bandwidth, so the PA output signal is obtained from the input by consecutive application the memoryless nonlinearityti ('-)] '-?riear filtration
?'-3 (/¦):= /*[$?('-)]'- w^ere the filter removes all the spurious emissions from the distorted signal. Considering the PA with the parallel resonant output circuit, the signal can be
associated with the AD output current and the signal. ^(f) -with the AD output voltage RF component.
The transfer function may be used to set the de-
sired PA parameters concerning the saturation and the cutoff condition that determines the class of PA operation. It is then possible to define liJ (. v) in the following way:
*sat ~Tq, S & gt- Tsat,
T{,)= s-T9. Tsat<-s<-Td, (I)
0., v & lt- Tg.
So the AD is in cut-off mode when input RF signal instantaneous amplitude is less than Tf), in saturation when it is
greater than Tsal, and produces linear amplification for all
the other input signal amplitudes. The threshold values J1
and T0 are determined using the given quantiles qsat and c/q
specifying relative amount of time spent by the PA in each nonlinear region:
is the stationary autocorrelation function of 5j (/), a2 is the power of Jf/(/) and Without loss of generality, we
further suppose a2 = / to facilitate the notation. The correlation coefficient /-® determines the shape of the normalized power spectral density (PSD) function 9{f) of the low-pass equivalent of A'-|(/) — Qt) is the slow-varying amplitude envelope of the signal, and (p (t) is its phase. Under the Gaussian zero-mean process assumption the univariate pdf of
a{t) follows the Rayleigh law = a• e 2, and the univariate pdf of (p{t) follows the uniform law: = - ¦ Ir, [10].
The bivariate Gaussian pdf w (jC|, X2) admits the decomposition in the Hermite polynomial basis [7]:
, i***!? 00 ,
k=0 k! where H^f. v) are the Hermite polynomials of probabilistic
a 2
form: nk{x) = {-fe 2 2.
dx& quot-
where F (q) is the inverse of the univariate probability 9t® = ^
HicfriKntinn funrtirtn rf tha cirrrtol nrnwirlciH rh^f fhfi cfofinn- n
This allows the integral in (2) to be computed straightforwardly, which yields the following expression for ACF of
s2(& amp-.
00 h, 2
distribution function of the signal, provided that the station-arity condition holds.
Given parameterized AD transfer function lIJ (s), it is possible to determine the autocorrelation function (ACF) of
/f®=E{, 2(0-S2(/ + r)}=E{4'-[Sl (/)]-?[, l{f+r)]} (2)
The expectation in (2) is calculated using the bivariate law of the input signal instantaneous amplitude defined by the probability density function (pdf) w (s2 r))& gt- which
will be referred to as w{x[, x2} for simplicity. When the input signal is formed by the superposition of many (tens and more) independent data carriers, it is due to the central limit theorem of probability theory that a Gaussian random process model may be invoked to properly describe such a signal. For instance, this is the case for the OFDM data transmission and the CDMA systems group traffic channels [12]. The main properties of the input signal model ?](/) can be defined resorting to quadrature representation of the RF signal:
Here *,(/) and x^(^) are equal-power slowly varying quadrature components of the signal that contain all the information from the message source. /?®= cr2r® — cos ((ot)
and the PSD function of A'-2(/) can then found immediately:
00 h2 & quot-n
where p"(f) is the n-fold convolution of the normalized
PSD of the input RF signal. The expansion coefficients are computed as:
h =-j= /^(w^/^-^jHtfw)^. (4)
The part of f) that is centered around the carrier frequency represents the PSD shape of the signal j-jlf J bypassing the output filter circuits. It is determined by excluding from (3) all the components, which lay outside the carrier frequency spectrum region. Using the notation for the is the I& quot- harmonic Fourier coefficient for a periodic function {cosaif, it may be verified that
00 .2 00
Si (/)= StH^iW^I — ifl & lt-5>-
n=Q '-'- n-Q
where (2“ + t) is the number of IMD product and is the corresponding power.

The part of PSD of ?^(O centered around zero frequency represents the slowly varying DC current component Id that affects PA efficiency. It can be derived similarly:
S) l2 00
3o (/)= It-n (f), (6)
A typical picture of the components of 3|(/'-) and 30(/) in decibels is depicted in Fig. 2,
When AM-AM and AM-PM functions are predefined, it is straightforward to obtain the PSD expression for 53(f) [6].
In order to do so, it is sufficient to calculate ACF for the lowpass equivalent of 53(f):
= Efa®] - C{a[t + r)]-j (7) The calculation of expectation in (7) implies the use of joint pdf for amplitude and phase which is
available in the explicit form for the Gaussian process [10]. The calculations yield the following expression for the PSD function:
% (/)=1Ы & lt-W/).
f 2 ] (2
a .1 a
¦ 4
2 2 V & gt-
da (9)
o v f“
Fig. 2, The first 5 IMD components of 3] (/) and 3q (/)
It can be seen that the signals resulted on odd harmonics of carrier frequency comprise the odd order IMD components (where the 0th order component of 3] (/) stands for the non-distorted signal replica). Using the output signal PSD representation it is useful to analyze which IMD components provide the major part of the total intermodulation noise, which is of importance for the PA linearization subsystem design. The even order IMD components constitute the signals that are centered around the even carrier frequency harmonics (here the 0, h order component of 3o (/) is the
output current DC component). Hence, the signals at even harmonics have no phase modulation.
3. AM-AM and AM-DC functions derivation
The aforementioned approach may be used to obtain the PSD of the RF signal passed through a strictly memoryless nonlinear device. However, the use of a specific transfer function T in the digital waveform simulations is often undesirable due to limited bandwidth of digital simulation environment [4]. It is more convenient to resort to a baseband NLPA model that operates on the low-pass equivalent of the RF signal and have lesser requirements to the bandwidth.
As was noted in [3], when the NLPA may be regarded as effectively memoryless device, the complex envelope of can be determined using the AM-AM and AM-PM functions:
?3 {() = ??["('-)]'- cos{coQt + (p{t) + & lt-t[a (i)]), where is referred to as AM-AM function and
— as AM-PM function.
where the expansion coefficients cn determine the passband IMD components power and are computed using Laguerre functions of the I& quot- order:
As for the strictly memoryless NLPA model one can neglect AM-PM distortion, it is further assumed that & amp-{a) = 0. As such model was shown to be characterized by the transfer function in the RF domain it is then possible to use the correspondence between the PSD expressions defined in (5) and (8) and to derive the expression for AM-AM function Q (a) for a given
V{q0,qJ). The 5) of (8) stands for the PSD of the lowpass equivalent of 53(f), and one can note that = 2 ¦ 3|. Putting an equation between the IMD power coefficients, taking
у = - and using ?_i
V n +1
+1exp~jkw as the
orthonormal functions set in L^fO. +oc), the following expansion does make sense:
After the reverse change of variables and the calculus, the expression for AM-AM function can be finally derived in the form

fito-eEii vi*

'-2n+J ¦
(n + l2n + l)!'-
which can be interpreted as a linear combination
Q (a)= XKKW'-
where each component of the sum corresponds to a particular nonlinearity IMD product. It may be observed that there is no dependency between the PSD shape and the signs of coefficients an. This means that to a single KVfi0, C[sat) correspond an infinite number of the AM-AM functions differing in the signs of an. In other terms, there is infinite
number of possible AM-AM functions that provide the same amount of IMD noise.
The expression for AM-DC function is derived similarly. The sole difference is in the absence of phase modulation, as was concluded in the 2nd section. Taking D{a) to be an AM-DC function, the corresponding ACF is:
R (T)=E{D[a (t)}. D[4t+T)l
Then the PSD of DC component can be derived as:
%{f)=tdn2S2n (f) (11)
and the expansion coefficients d» are computed using Laguerre functions of the 0th order:
* (2 a '- (12)
dk= I D (a)-ae 2 -L°k
Comparing the corresponding IMD power coefficients in (6) and (I I) it can be noted that the AM-DC function can be expressed in the following way:
?(«)= IM-m & lt-i3>-
n = 0
where фп (а) = Ь°п

The first five basis functions of Q (a) and D{a) are depicted on Fig. 3, where the scaled version of the signal amplitude envelope pdf is also shown to aid the comparison.
5 ¦ w (a)/ / -Д-^& quot-
У? v --- 1 II il& quot-
TX& quot-"-"-*"-^/ /
сЧ,"ч -- / /
\ / г ~ - -1__1
\ / /
V /
\ л /
\ /V VX, А /
. 1
О 0.5 ! I. S 2 23 3 3. 5

/ / г Ж / 5 • w (a) / • / N /V / у

h. s
2. 5
3. 5
Fig. 3. The first 5 basis functions- a) for AM-AM expansion- b) for AM-DC expansion
The alternative method to obtain AM-AM and AM-DC functions consists in using a simplistic single-tone approach [9]. In this method the AM-AM function Q (ci) is substituted by the nonlinear gain function Qsine{a) computed for the
harmonic input to the nonlinear device over the desired range of amplitudes. Since it is convenient in this case to obtain i using the Fourier series method, the gain function
for the fundamental harmonic Qsint,(a), as well as for the DC component Dsl/]e (a), may both be found analytically for a specified cut-off and saturation parameters of PA transfer function ^(. v).
As was shown In [9], for the computation of Qsjni. {a) and Dslne (a) it is sufficient to introduce the angular parameters 6 and 19, which stand the for cut-off angle and saturation angle [5] and correspond to relative part of RF period spent by PA in the cut-off or saturation mode respectively. This values are determined by the amplitude of the harmonic input and the threshold values Tsat and Tg. The Fig. 4 illustrates the single-tone model parameters.
-IT Ttfl 0 JT/2 jr
Fig. 4. Single tone model parameters illustration
Using the adopted notation, the gain function expressions are determined as follows:
QsiM^ia-To+Auya^-ia-TsJ-a,^). (14)
DsJa) = (a ~Tg+ Au) ¦ a"{t& gt-) — [a — Tsa,)'-Ct0{& amp-)-Ai, (15) where the angular parameters need to be calculated for each value of a:

0, Тд& lt-0,а<-Тв
л, Тд& gt-0,а<-Т@,
arccos-, а:
а & gt- Ъ
0. a& lt-TsM.
and JuJa + Tf n'- ¦ The Burg coefficients used in
[0, 6{a)& lt-n.
(14−15) to facilitate the notation are determined as
6-sindcosO sinO-OcosO ri-i aiU) = -Ti-sr» ?w) = -77-?r- PJ'-
— COS 6) 7CI — cosO)
To compare the two described approaches to AM-AM and AM-DC functions derivation the intermodulation power

from the class C operation (qg & gt-0. 5) produces signal decompression, which besides the losses in power gain, inherit for class C mode, may also cause an efficiency degradation for the fixed Vdd PA implementation.
qf b)
Fig. 8. Theoretical estimation of the keen PA parameters for a range of cut-off percentage of time: a) mean efficiency- b) normalized output IMD noise power
5. Conclusions
The strictly memoryless model of nonlinear power amplifier with the Gaussian input was studied. Using the equivalent representation of the output signal PSD calculated for the nonlinearity applied to instantaneous signal amplitude and for the nonlinearity defined in terms of AM-AM function, the approach to derivation of AM-AM function for a given transfer characteristic of an amplifier was proposed. Foliowing this, the obtained parameterized AM-AM function was used in the OFDM waveform simulation process. The representation of the AM-AM function in form of linear expansion in a defined basis function set proved to be useful to evaluate
system performance with partial suppression of intermodulation noise. Using the proposed methods it was demonstrated that the out of band emission is by far a more severe problem than the BER performance degradation when using saturated class C amplifier in the OFDM signal amplifications. The comparative study of mean efficiency and IMD noise power was performed for a range of PA classes of operation that include class A, class AB, class B and class C. It was shown that without PA linearization the theoretical mean efficiency of class AB operation might exceed the one attained in ideal class B amplifier, but at a price of IMD power augmentation.
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2. jeruchim, M, Balaban, P, et al. 2002, Simulation of Communication Systems — Modeling, Methodology, and Techniques, 2nd ed. Kiuwer, New-York.
3. Minkoff, J 1982, '-Intermodulation noise in solid state power amplifiers for wideband signal transmission'-, 9'-& quot- AIAA International Communications Satellite System Conference, 1982, pp. 304−3 13.
4. Pedro, J, Maas, S 2005, '-A comparative overview of microwave and wireless power-amplifier behavioral modelling approaches'-, IEEE Trans. Microw. Theory Tech., vol, 53, no. 4, pp. I 150−1 163.
5. Shahgildyan, V, Kozyrev, V, Lyahovkin, A 2003, Radiopereda-yuschie Ustrojstva (in Russian), 3rd ed. Radio and Svyaz, Moscow.
6. S/i/nokov, Yu 2013, '-Power spectral density of interference caused by nonlinear distortions in devices with amplitude-phase conversion'-. Journal of Communications Technology and Electronics, vol. 58. no. 10. pp. 1024−1034.
7. Sm/rnov, A 2015, '-OFDM signal AM-AM distortion for a given power amplifier transfer characteristic'- (in Russian), The Radio-Electronic Devices and Systems for the Infocommunication Technologies (REDS-2015), Moscow, Russia, pp. 84−87,
8. Sm/rnov, A, Gorgadze, S 2015, '-AM-PM distortion in high-efficiency power amp!? fier'-, Int. conference «Synchroinfo-2015» (& quot-CnHxpoMH ())o-2015"-).
9. Smirnov, A, Gorgadze, S 2015, & quot-Power amplifier efficiency estimation in applications to amplitude modulated group signals& quot-, Electrosvyaz (to be published).
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I I. Varlomov, 0 2014. '-Research of influence of DRM broadcast transmitter nonlinearities onto the output signal parameters'-, T-Comm, vol. 8, no. 2, pp. 59−60,
12. Volkov, L, Nemirovskiy, M, Shinakov, Yu 2005, Digital radio systems basic methods and characteristics / Sis te m y tsifrovoy radios-vyazi bazovye metody i kharakteristiki (in Russian), Eko-Trendz, Moscow.
Смирнов Андрей Владимирович, Московский Технический Университет Связи и Информатики, аспирант кафедры РОС, Москва, Россия, sandrew2k@yandex. ru
Использование характеристик AM-AM и AM-PM для моделирования нелинейных искажений сигнала в усилителе мощности получило широкое распространение при разработке систем радиосвязи. Основной целью использования таких моделей является оценка уровня внеполосных излучений и влияния шума интермодуляционных искажений на энергетический бюджет радиолинии. Предлагаются новые приложения модели нелинейности усилителя на основании выведенной параметризованной АМ-АМ характеристики, которая получена после детального сопоставления между собой применений нелинейности к радиочастотному сигналу и к его низкочастотной комплексной огибающей. Параметрами полученной АМ-АМ характеристики являются процент времени работы усилителя в области насыщения своей проходной характеристики и процент времени его работы в состоянии отсечки выходного тока. Показано, что АМ-АМ характеристика представима в виде разложения по базисным функциям с коэффициентами разложения, определяемыми мощностью внутриполосной помехи интермодуляционных искажений соответствующего порядка. Этим обосновано использование модели для оценки влияния частичного подавления нелинейных искажений на помехоустойчивость системы при работе в заданном классе усиления и в заданном режиме по насыщению. Также получена параметризованная функция зависимости постоянной составляющей выходного тока усилителя от амплитудной огибающей входного сигнала, с помощью которой проведена теоретическая оценка средней эффективности усиления применительно к гауссовской модели входного сигнала. Полученная зависимость средней эффективности усиления от параметров усилителя сопоставляется с соответствующей зависимостью мощности сопутствующих интермодуляционных искажений.
Ключевые слова: усилитель мощности, безынерционная нелинейность, разложение АМ-АМ характеристики, эффективность, интермодуляционные искажения, внеполосное излучение, постоянная составляющая выходного тока, гауссовский случайный процесс, однотоновая модель, линеаризация AM-AM характеристики.
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2. Jeruchim, M, Balaban, P, et al. 2002, Simulation of Communication Systems — Modeling, Methodology, and Techniques, 2nd ed. Kluwer, New-York.
3. Minkoff, J 1982, '-Intermodulation noise in solid state power amplifiers for wideband signal transmission'-, 9th AIAA International Communications Satellite System Conference, 1982, pp. 304−313.
4. Pedro, J, Maas, S 2005, '-A comparative overview of microwave and wireless power-amplifier behavioral modelling approaches'-, IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1150−1163.
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8. Смирнов A, Горгадзе С. AM-PM distortion in high-efficiency power amplifier // Труды конференции & quot-Синхроинфо-2015"-. (in Russian)
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