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10-31-2020 A Simplified Accuracy Enhancement to the Saleh AM/AM Modeling and Linearization of Solid-State RF Power

Haider Al-kanan Portland State University, [email protected]

Fu Li Portland State University, [email protected]

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Citation Details Al-Kanan, H.; Li, F. A Simplified Accuracy Enhancement to the Saleh AM/AM Modeling and Linearization of Solid-State RF Power Amplifiers. Electronics 2020, 9, 1806

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Article A Simplifed Accuracy Enhancement to the Saleh AM/AM Modeling and Linearization of Solid-State RF Power Amplifers

Haider Al-Kanan * and Fu Li *

Department of Electrical and Computer Engineering, Portland State University, Portland, OR 97207-0752, USA * Correspondence: [email protected] (H.A.); [email protected] (F.L.)

 Received: 8 October 2020; Accepted: 29 October 2020; Published: 31 October 2020 

Abstract: The Saleh behavioral model exhibits high prediction accuracy for nonlinearity of traveling-wave tube power amplifers (TWT-PAs). However, the accuracy of the Saleh model degrades when modeling solid-state power amplifers (SSPAs) technology. In addition, the polynomial expansion of the Saleh model consists of only odd-order terms as analyzed in this work. This paper proposes a novel model accuracy enhancement for the Saleh amplitude-to-amplitude (AM/AM) model when applied to radio (RF) SSPAs. The proposed model enhancement accounts for the second-order intermodulation , which is an important nonlinearity challenge in wideband wireless communications. The proposed static AM/AM model is a three-parameter rational function, which exhibits low complexity compared to the state-of-the-art behavioral models. A transpose architecture of fnite-impulse digital flter is used to quantify the memory effect in SSPAs. A least-squares method is used for extracting all the model parameters. A linearization technique using a three-parameter digital predistortion model is also calculated to compensate for the AM/AM nonlinear distortion in SSPAs. Finally, the identifcation and evaluation of the enhanced Saleh model is presented based on measurements of RF SSPAs.

Keywords: solid-state power amplifers; behavioral models; digital predistortion; nonlinear distortion; memory estimation

1. Introduction The rapid evolution of wireless communications requires high dynamic range RF power amplifers (PAs) to efficiently operate on high-amplitude fuctuations in modern digital modulation [1]. Solid-state power amplifers (SSPAs) are intensively deployed devices in the front ends of mobile systems [2–4]. Therefore, empirical models for SSPAs have been utilized for decades to study the distortion effects on wireless communications systems. Accurate and low-complexity behavioral models for SSPAs have become important techniques in simulating and linearizing communication systems [5–7]. In particular, an accurate estimation of the model smoothness in the amplitude-to-amplitude (AM/AM) transition from the linear to the saturation regions has been a topic of interest in the state-of-the-art PA modeling [8–12]. This is because operating SSPAs near the compression region is often required for achieving high power efficiency and transmitting signals of high peak-to-average power ratio (PAPR) [13]. Rapp, Ghorbani, and Cann behavioral models are popular mathematical functions that match well the AM/AM characteristics of SSPAs. However, these models often consist of complicated rational functions; therefore, starting points and iterative estimation methods are typically required for such model estimation. In addition, these models exhibit a lack of relevance in modeling digital predistortion (DPD) for linearizing SSPAs.

Electronics 2020, 9, 1806; doi:10.3390/electronics9111806 www.mdpi.com/journal/electronics Electronics 2020, 9, 1806 2 of 13

The polynomial-based models are commonly used for SSPAs, because polynomials provide sufficient modeling fexibility in estimating the parameters and adjusting the model accuracy. However, polynomial models are subject to estimation errors such as curve under-ftting and over-ftting model prediction when truncated nonlinear coefficients are employed. The AM/AM characteristics of traveling-wave tube power amplifers (TWT-PAs) exhibit high nonlinearity and roll-over effects near the saturation region. Thus, the accuracy of the Saleh model signifcantly degrades when modeling the compression smoothness of the SSPAs. The proposed modifed Saleh model by O’Droma achieves substantial accuracy improvement for modeling the AM/AM nonlinearity for laterally-diffused metal-oxide semiconductor (LDMOS) SSPAs [9]. However, O’Droma proposed a six-parameter rational formula, which is computationally expensive. In addition, the direct inversion of the O’Droma model is mathematically difficult for calculating the predistortion model. Hence, this paper presents a simplifed behavioral model with an adequate AM/AM accuracy improvement for SSPA technologies. The amplitude-to-phase distortion is often minor in SSPAs and is assumed insignifcant in this simplifed modeling approach [10]. The presented modeling approach consists of three parameters, which allows better nonlinearity modeling for a large signal amplitude beyond the 1-dB compression point. In addition, the presented expression is convenient to characterize the even-order intermodulation distortion (IMD) for wideband and multiband communications. A simplifed and low complexity DPD model is also calculated to compensate for the AM/AM nonlinearity in SSPAs. This paper is organized as follows: Section 2 describes the proposed enhancement approach to the Saleh AM/AM model, followed by an approach to characterize the dynamic memory effect in Section 3. Section 4 presents a linearization model for SSPA using DPD. Section 5 includes the experiment and modeling results for both the SSPA and DPD model. Finally, conclusions are presented in Section 6.

2. Enhanced Approach for SSPAs The Saleh AM/AM model is a two-parameter function. The frst parameter α represents the small signal gain, and the second parameter β is used to adjust the transition sharpness (i.e., gain compression) from the linear region to saturation region. The AM/AM Saleh model F[u(t)] can be expressed as [8]: αu(t) F[u(t)] = (1)  p 2 1 + βu(t) where u(t) is the envelope of the PA input baseband signal. The Saleh model is an odd function {F[ u(t)] − = F[u(t)]}, and the polynomial expansion of the rational function in Equation (1) is given by: −

X∞ F[u(t)] = ( 1) kαβku(2k+1)(t) (2) − k=0 where k is an integer number {k = 0, 1, 2, 3, ...... }

F[u(t)] = αu(t) αβu3(t) + αβ2u5(t) + ...... (3) − The polynomial series in Equation (3) consists of the nonlinear coefficients {α, αβ, ... , αβk , ... }. Equation (3) shows a mathematical relationship of a magnitude-dependency among all the polynomial coefficients. In comparison with the typical odd-order Taylor model in Equation (4), the Taylor coefficients are often considered statistically and linearly independent parameters. Therefore, the Taylor model exhibits a higher degree of freedom for ftting the polynomial coefficients to the observed Electronics 2020, 9, 1806 3 of 13 data. However, the polynomial curve ftting is subject to uncertainty and often exhibits high prediction errors beyond the range of the observed data.

3 5 T[u(t)] = c1u(t) + c3u (t) + c5u (t) + ...... (4)

The magnitude of the Taylor coefficients {c1, c3, ...... } are considered device-dependent, but the independent coefficients are insufficient for describing the effect of the higher order IMDs on the lower order IMDs. For example, the ffth-order IMD affects the third-order IMD and so on [14]. The higher order Taylor polynomial can achieve better model accuracy to ft the nonlinear curvature of the AM/AM conversion, because using additional independent coefficients increases modeling the dimensions of the nonlinear system, but this often leads to a noisy estimate and poor coverage of confdence intervals. The typical output of the AM/AM Saleh model increases almost linearly in the small signal magnitude at the rate α. However, the gain of the Saleh model decreases monotonically after the 1/2 maximum curvature (this occurs at |u| = β− ) [15] and approaches zero for very large theoretical values of |u|. This is one of the main weaknesses in the Saleh model when applied to SSPAs, because the saturation level in SSPAs is almost constant and specifed by the supply voltage on the drain/collector of the output . Power amplifers are typically designed to operate over a dynamic range of the load-line with the DC quiescent point “Q-point” which is set in the active region of the transistor on a small amplitude signal (e.g., Q-point is set to the middle of the load-line in a class-A design). However, on a large input amplitude signal, the output signal becomes extensively large and forces the output signal to surpass the power supply voltage and becomes fattened. Therefore, the output signal is limited by the supply voltage in the saturation region [16]. A simple model enhancement is introduced in this work to address the model limitations by including an additional quadratic term within the numerator of the Saleh model [17] as illustrated

αu(t) + λu2(t) Fes[u(t)] = (5) 1 + βu2(t) where Fes[u(t)] is the proposed enhanced Saleh AM/AM model. The parameter λ is a real positive number introduced here for adjusting both gain and roll-over near the saturation region. The small signal gain α of the model in Equation (5) is approximately equal to the linear gain of the Saleh model as depicted: ∂(Fes[u]) lim = α (6) u 0 ∂(u) → The parameter λ causes the function in Equation (5) to be mathematically non-monotonic and the output increases asymptotically toward the saturation for a large signal amplitude (saturation level ≈ λ/β). Finally, the polynomial expansion of Equation (5) becomes a mixed nonlinear combination of odd and even terms as follows:

X ∞ k k (2k)  2  Fes[u(t)] = ( 1) β u (t) αu(t) + λu (t) (7) − k=0

2 3 4 2 5 2 6 3 7 Fes[u(t)] = αu(t) + λu (t) αβu (t) λβu (t) + αβ u (t) + λβ u (t) αβ u (t) + ...... (8) − − − Equation (8) illustrates that the magnitude of the second-order IMD is completely characterized by the parameter λ. The other higher order even-terms in this model are characterized by the nonlinear coefficients {λ, λβ, ... , λβk, ... }. In addition, the proposed model enhancement can improve the fatness in the saturation region, because the mathematical expression in Equation (8) converges asymptotically to the constant value (λ/β) [17]. The second-order IMD of SSPAs is an ongoing concern in wideband and multiband communications [14,18,19]. Figure 1 shows the typical two-tone IMDs, which illustrates the locations Electronics 2020, 9, 1806 4 of 13 of the second-order distortion (f f , f + f ) with respect to the fundamental tones for wideband Electronics 2020, 9, x FOR PEER REVIEW 2 − 1 2 1 4 of 13 communications [14].

wideband nonlinear distortion Magnitude Magnitude

Frequency

f2–f1 2f1–f2 f1 f2 2f2–f1 f2 + f1 Figure 1. Wideband output spectrum of RF power illustrating the two-tone harmonic Figure 1. Wideband output spectrum of RF power amplifer illustrating the two-tone harmonic distortion effect. distortion effect.

Model Estimation A least-squares method is used for calculating the parameters of the enhanced Saleh model as described in this section. Substituting z(t) = Fes[[uu((tt)])] inin EquationEquation (5)(5) andand re-arranging the equation variables results in 2 2 z(t) = αu(t) + λu22(t) z(t)βu (t) (9) zt() =αut+ ()λut ()−− zt ()βut () (9) where z(t) is the envelope of the PA baseband output signal. A matrix equation can be formulated where z(t) is the envelope of the PA baseband output signal. A matrix equation can be formulated from Equation (9) by substituting discrete-time samples of both u(t) and z(t) as illustrated from Equation (9) by substituting discrete-time samples of both u(t) and z(t) as illustrated ⎡ ⎤ ⎡ ⎤ ⎡ zu(0) (0) − 22 α ⎤ α ⎢ z(0) ⎥ ⎢ u(0) zu(0) (0)2 u (0) 2 ⎥⎢ ⎥ ⎢ ⎥ ⎢ z(0)u (0) u (0) ⎥⎢ ⎥ ⎢ ⎥ ⎢ − 22 ⎥⎢ ⎥ ⎢ ⎥ ⎢ −zu(1)z( )u (1)2( ) u (1) u2( ) ⎥⎢ ⎥ ⎢ ⎥ ⎢ zu(1)= (1) 1 1 β 1 ⎥⎢ β ⎥ ⎢ z(1) ⎥ = ⎢ u(1) −  ⎥⎢ ⎥ (10)(10) ⎢ ⎥ ⎢  . . ⎥⎢ ⎥ ⎢ . ⎥ ⎢ . . . ⎥⎢ ⎥ ⎢ . ⎥ ⎢ . 22 ⎥⎢ ⎥ ⎢ . ⎥ ⎢ zn(). un () −znu() ()2 n u () n λ 2 ⎥⎢ ⎥ ⎣⎢ ⎦⎥ ⎣⎢ z(n)u (n)  u (n) ⎦⎥⎢ ⎥ z(n) u(n) − ⎣ λ ⎦ For simplicity, a matrix notation is used in representing (10) For simplicity, a matrix notation is used in representing (10) z=Uc (11) where z is a column vector of ((n + 1) × 1) elements,z = Uc c is a column vector of the model parameters (11)(3 × 1), and U is a matrix consisting of ((n + 1) × 3) elements of the input and output samples. Finally, the where z is a column vector of ((n + 1) 1) elements, c is a column vector of the model parameters (3 vector c is calculated as: × × 1), and U is a matrix consisting of ((n + 1) 3) elements of the input and output samples. Finally, the × T-1T vector c is calculated as: c=(UU) U z (12) c = (UTU) 1UTz (12) where ()T denotes the operator of a matrix transposition− [17]. where ()T denotes the operator of a matrix transposition [17]. 3. Memory Modeling 3. Memory Modeling Memory effect is a popular hysteresis phenomenon in RF PAs which causes dynamic nonlinear dispersionMemory in theeffect AM/AM is a popular conversion hysteresis [20,21]. phenomenon Memory effect in RF is PAs considered which causes device-dependent dynamic nonlinear due to thedispersion impedances’ in the variationAM/AM conversionof both the [ 20matching,21]. Memory and biasing effect isnetworks considered with device-dependent respect to operating due frequency,to the impedances’ and to the variation temperature of both variation the matching in ’ and biasing junctions networks as depicted with respectin Figure to 2.operating frequency, and to the temperature variation in transistors’ junctions as depicted in Figure 2.

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Figure 2. Simplified architecture of the memory component elements of the RF power amplifier.

A digital filter is a straightforward approach for modeling the memory effect in RF PAs. In this work a transpose finite-impulse response (FIR) digital filter is used in cascade with the enhanced Saleh model based on Hammerstein modeling theory as depicted in Figure 3. The architecture of the transpose FIR filter is computationally less complex than the direct FIR filter in real-time implementation because it supports multiple constant multiplication technique. TheFigureFigure FIR filter 2.2. SimplifedSimplified mimics architecturearchitecturethe output ofsignalof thethe memorymemory dependency componentcomp onenton the elementselements past samples ofof thethe RF RFof powerthepower input amplifer. amplifier. signal using a finite number of coefficients for controlling the memory depth in the model. A digital filter is a straightforward approach for modeling the memory effect in RF PAs. In this A digital flter is a straightforward approachM for modeling the memory effect in RF PAs. In this − workwork a a transpose transpose fnite-impulse finite-impulse response z(ˆresponsen= ) (FIR) hm(FIR) ( digital )F [digital un ( flter− m filter )] is e junmused is() used in cascade in cascade with thewith enhanced the enhanced Saleh  es (13) modelSaleh modelbased on based Hammerstein on Hammerstein modeling modeling theorym=0 as theory depicted as depictedin Figure in3. TheFigure architecture 3. The architecture of the transpose of the FIRtranspose flter is computationallyFIR filter is computationally less complex than less the codirectmplex FIR thanflter inthe real-time direct implementationFIR filter in real-time because where h(m) represents the filter impulse response, M is the memory depth in the model, and 𝑧̂𝑛is itimplementation supports multiple because constant it supports multiplication multiple technique. constant multiplication technique. the outputThe FIR signal filter of mimics the dynamic the output nonlinear signal model. dependency on the past samples of the input signal using a finite number of coefficients for controlling the memory depth in the model.

M − z(ˆ n= ) hm ( )F [ un (− m )] ejunm ()  es (13) m=0 where h(m) represents the filter impulse response, M is the memory depth in the model, and 𝑧̂𝑛is the output signal of the dynamic nonlinear model.

Figure 3. 3. Hammerstein Hammerstein model model structure structure for RF forpower RF amplifier power amplifer using the usingenhanced the Saleh enhanced amplitude- Saleh to-amplitudeamplitude-to-amplitude (AM/AM) model. (AM/AM) model.

4. ModelingThe FIR offlter Digital mimics Predistortion the output signal dependency on the past samples of the input signal using a fniteDPDs number have ofbecome coefficients popular for approachescontrolling thefor improvingmemory depth the linearity in the model. of RF PAs for a reliable and an efficient signal transmission [22,23]. The DPD implementation in the digital domain as in Figure 4 XM j]u(n m) is practically more efficient thanzˆ( nanalog) = predistorh(m)Festion[u(n for m compensating)]e − the nonlinear distortion(13) in Figure 3. Hammerstein model structure for RF power amplifier− using the enhanced Saleh amplitude- RF PAs. The DPD models based on Taylorm=0 polynomials are popular linearization techniques to to-amplitude (AM/AM) model. compensate for a weak-nonlinear distortion in RF PAs [24]. ( ) whereA hcomplex(m) represents structure the flterof a impulseDPD model response, is not M desired is the memory in wireless depth communication, in the model, and because zˆ n is this the 4. Modeling of Digital Predistortion requiresoutput signal a complex of the anddynamic high-computational-cost nonlinear model. hardware system. Hence, a predistortion model in DPDs have become popular approaches for improving the linearity of RF PAs for a reliable and this4. Modeling work is calculated of Digital directly Predistortion from a simplified rational function of the enhanced Saleh model using feweran efficient coefficients. signal transmission [22,23]. The DPD implementation in the digital domain as in Figure 4 is practicallyDPDs have more become efficient popular than approaches analog predistor for improvingtion for thecompensating linearity of RFthe PAs nonlinear for a reliable distortion and an in eRFfficient PAs. signal The DPDtransmission models [based22,23]. on The Taylor DPD implementationpolynomials are in popular the digital linearization domain as techniquesin Figure 4 isto practicallycompensate more for eaffi weak-nonlinearcient than analog distortion predistortion in RF for PAs compensating [24]. the nonlinear distortion in RF PAs. The DPDA complex models structurebased on Taylorof a DPD polynomials model is arenot popular desired linearization in wireless communication,techniques to compensate because thisfor arequires weak-nonlinear a complex distortion and high-computational-cost in RF PAs [24]. hardware system. Hence, a predistortion model in this work is calculated directly from a simplified rational function of the enhanced Saleh model using fewer coefficients.

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Figure 4.4. SimplifedSimplified diagram of a digital predistortion transmittertransmitter system.system.

AThe complex proposed structure DPD modelof a DPD can model be deployed is not desired for the in PA wireless of strong communication, nonlinearity becauseover a wide this requiresamplitude a complex range. The and mathematical high-computational-cost inversion of thhardwaree enhanced system. Saleh Hence, model ais predistortion calculated as modelfollows: in this work is calculated directly from a simplifed rational function of the enhanced Saleh model using −1 fewer coefficients. ut()=Fes [ dt ()] (14) The proposed DPD model can be deployed for the PA of strong nonlinearity over a wide amplitude where d(t) and u(t) are the DPD model input and output magnitudes, respectively, as depicted on the range. The mathematical inversion of the enhanced1 Saleh model is calculated as follows: simplified transmitter architecture in Figure. 4. Fes is the proposed enhanced DPD model, which is calculated by exchanging the input and output variables1 of the enhanced Saleh model [11] Fes[u(t)] u(t) = Fes− [d(t)] (14) in Equation (5) and simplifying the expression as: where d(t) and u(t) are the DPD model input−− and 2output magnitudes, respectively, as depicted on the ()dβλu 1 αu+d =0 (15) simplifed transmitter architecture in Figure 4. Fes− [] is the proposed enhanced DPD model, which is calculatedFinally, by the exchanging DPD model the is input calculated and output by solving variables a quadratic of the enhanced formula in Saleh Equation model (15) [11 with] Fes[ respectu(t)] in Equationto the variable (5) and u. simplifyingAn equation the solution expression of a negativeas: sign is used in this model, since the DPD input and output magnitudes are real and normalized in this work. (dβ λ)u2 αu + d = 0 (15) − − 2 2 − αα−−44 βd+dλ F[]=1 d (16) Finally, the DPD model is calculatedes by solving2 ()ad quadraticβλ− formula in Equation (15) with respect to the variable u. An equation solution of a negative sign is used in this model, since the DPD input and outputEquation magnitudes (16) represents are real thande DPDnormalized model, in which this work. outputs a magnitude of the predistorted baseband signal. The maximum input amplitude of the DPD model is calculated to satisfy a positive p real value inside the root term of Equation (16) as follows:2 2 1 α α 4βd + 4dλ Fes− [d] = − − (16) 2 2(d≥β λ2) α +4dmaxλ 4−βd max (17) Equation (16) represents the DPD model, which outputs a magnitude of the predistorted baseband The maximum amplitude of the DPD input |dmax| to satisfy Equation (17) is given by: signal. The maximum input amplitude of the DPD model is calculated to satisfy a positive real value inside the root term of Equation (16) as follows: λλβα++22 = dmax (18) 2 2β 2 α + 4dmaxλ 4βd (17) ≥ max where |dmax| represents a threshold of the DPD baseband input signal. Therefore, a clipping modelThe in maximum Equation amplitude(19) is required of the DPDon the input input |dmax of | theto satisfy DPD Equationmodel to (17)maintain is given the by: condition in Equation (17). p λ + λ2 + βα2 dmax = < (18) dt(), if2 dt () dmax | | β dtˆ()= (19) where |dmax| represents a clipping threshold of the DPD baseband≥ input signal. Therefore, a clipping  ddtdmax, if ( ) max model in Equation (19) is required on the input of the DPD model to maintain the condition in Equationwhere |𝑑 (17).𝑡|is a clipped amplitude version of the time domain signal d(t). Figure 5 describes the characteristics of the clipping model in Equation (19) for the amplitude relationships between the input and output signals. This model forces the large amplitudes of the input signal to a constant threshold according to the value of |dmax|, which is specified by the model parameters (α,λ,β). Equation (18) illustrates that the level magnitude of the clipping model is directly proportional to the power amplifier gain (α,λ) and inversely proportional to the parameter β. Although the model in

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⎧ d(t) , if d(t) < d ⎪ max ˆ ⎨⎪ | | d(t) = ⎪ (19) ⎪ ⎩ dmax , if d(t) dmax | | ≥ | | where dˆ(t) is a clipped amplitude version of the time domain signal d(t). Figure 5 describes the characteristics of the clipping model in Equation (19) for the amplitude relationships between the input and output signals. This model forces the large amplitudes of the input signal to a constant Electronics 2020, 9, x FOR PEER REVIEW 7 of 13 threshold according to the value of |dmax|, which is specifed by the model parameters (α,λ,β). Equation (18)Electronics illustrates 2020, 9 that, x FOR the PEER level REVIEW magnitude of the clipping model is directly proportional to the power7 of 13 Figure 5 is linear with unit gain in the amplitude range (0 to dmax), the clipping model is another amplifernonlinear gain function (α,λ) andrequired inversely on the proportional input of the to DPD the parametermodel. The β .model Although clipping the model characteristics in Figure are 5 Figure 5 is linear with unit gain in the amplitude range (0 to dmax), the clipping model is another isdepicted linear with in Figureunit gain 6, whichin the amplitudedescribe the range variation (0 to dinmax the), the input clipping signal model PAPR is withanother respect nonlinear to the functionnonlinear required function on required the input on of thethe inputDPD model.of the DPD The modelmodel. clipping The model characteristics clipping characteristics are depicted arein clipping level |dmax| using an orthogonal frequency division multiplexing (OFDM) signal. The output depicted in Figure 6, which describe the variation in the input signal PAPR with respect to the Figureamplitude 6, which becomes describe equal the to variation a non-clipped in the inputamplitude signal for PAPR a large with clipping respect levelto the as clipping illustrated. level |dmax| usingclipping an orthogonal level |dmax| frequency using an orthogonal division multiplexing frequency division (OFDM) multiplexing signal. The output (OFDM) amplitude signal. The becomes output equalamplitude to a non-clipped becomes equal amplitude to a non-clipped for a large amplitudeclipping level for aas large illustrated. clipping level as illustrated.

Figure 5. Signal amplitude model required on the input of the digital predistortion (DPD) model. FigureFigure 5. 5. Signal Signal amplitude amplitude limiter limiter model model required required on the on input the inputof the digitalof the predistortiondigital predistortion (DPD) model. (DPD) model.

FigureFigure 6. 6. The The model model clipping clipping level level and and the the peak-to-average peak-to-average power power ratio ratio (PAPR) (PAPR) of of the the clipped clipped signal. signal. TheFigure impact 6. The of model the clipping clipping nonlinearitylevel and the peak-to-avon the distortionerage power effects ratio is (PAPR) not severe, of the because clipped the large signal. peaks Thetypically impact occur of the with clipping low probability. nonlinearity Therefore, on the di distortionfferent clipping effects is approaches not severe, have because been the widely large usedpeaks in typicallywireless communicationoccur with low to probability. reduce the Therefore,signal PAPR, different and they clipping are considered approaches an importanthave been The impact of the clipping nonlinearity on the distortion effects is not severe, because the large designwidely aspect used whenin wireless operating communication RF PAs on OFDM to reduce signals. the Insignal addition, PAPR, reducing and they the are signal considered PAPR can an peaks typically occur with low probability. Therefore, different clipping approaches have been improveimportant the design power aspect efficiency when in operating RF PAs, because RF PAs theon OFDMmaximum signals. power In addition,efficiency reducing is obtained the when signal widely used in wireless communication to reduce the signal PAPR, and they are considered an operatingPAPR can RF improve PAs near thethe compressionpower efficiency region in [14 RF]. Therefore,PAs, because applying the maximumsignal clipping power in combinationefficiency is important design aspect when operating RF PAs on OFDM signals. In addition, reducing the signal withobtained DPD oftenwhen leads operating to performance RF PAs nearimprovement the compressi in poweron region efficiency [14]. and Therefore, linearity [applying13,25–27]. signal PAPR can improve the power efficiency in RF PAs, because the maximum power efficiency is clipping in combination with DPD often leads to performance improvement in power efficiency and obtained when operating RF PAs near the compression region [14]. Therefore, applying signal linearity [13,25–27]. clipping in combination with DPD often leads to performance improvement in power efficiency and 5.linearity Evaluation [13,25–27]. Results

5. EvaluationTwo different Results SSPA technologies, gallium arsenide (GaAs) (ZFL-1000LN from Mini-Circuits, Brooklyn, NY, USA) and gallium nitride (GaN) (ZX60-8008E from Mini-Circuits) devices under test Two different SSPA technologies, gallium arsenide (GaAs) (ZFL-1000LN from Mini-Circuits, (DUT) are used in this experiment for model evaluation and linearity enhancement of the DUT PA. Brooklyn, NY, USA) and gallium nitride (GaN) (ZX60-8008E from Mini-Circuits) devices under test The experiment set-up consists of a signal generator (E4438C from Keysight Technologies, Santa (DUT) are used in this experiment for model evaluation and linearity enhancement of the DUT PA. Rosa, CA, USA), a (RSA6120A from Tektronix, Beaverton, OR, USA), a high- The experiment set-up consists of a signal generator (E4438C from Keysight Technologies, Santa power attenuator, and a computer (Dell computer with Intel Xeon CPU E3-1241 v3 @ 3.50 GHz and Rosa, CA, USA), a spectrum analyzer (RSA6120A from Tektronix, Beaverton, OR, USA), a high- 16 GB of ram) with MATLAB (R2018a from MathWorks, Natick, MA, USA) as shown in Figure 7. A power attenuator, and a computer (Dell computer with Intel Xeon CPU E3-1241 v3 @ 3.50 GHz and 16 GB of ram) with MATLAB (R2018a from MathWorks, Natick, MA, USA) as shown in Figure 7. A

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5. Evaluation Results Two different SSPA technologies, gallium arsenide (GaAs) (ZFL-1000LN from Mini-Circuits, Brooklyn, NY, USA) and gallium nitride (GaN) (ZX60-8008E from Mini-Circuits) devices under test (DUT) are used in this experiment for model evaluation and linearity enhancement of the DUT PA. The experiment set-up consists of a signal generator (E4438C from Keysight Technologies, Santa Rosa, CA, USA), a spectrum analyzer (RSA6120A from Tektronix, Beaverton, OR, USA), a high-power attenuator, and a computer (Dell computer with Intel Xeon CPU E3-1241 v3 @ 3.50 GHz and 16 GB of ram) with MATLAB (R2018a from MathWorks, Natick, MA, USA) as shown in Figure 7. A 64-QAM dual-bandElectronics 2020 modulated, 9, x FOR PEER OFDM REVIEW signal was applied on the input of the SSPA, and the output signal8 of is 13 RF down-converted, demodulated, and acquired by the spectrum analyzer.

(a) (b) 64-QAM dual-band modulated OFDM signal was applied on the input of the SSPA, and the output Figure 7. Experimental validation for modeling and digital predistortion of a solid-state power amplifer signal is RF down-converted, demodulated, and acquired by the spectrum analyzer. (SSPA). (a) Schematic diagram. (b) Photograph of the test set-up. Figure 7. Experimental validation for modeling and digital predistortion of a solid-state power amplifier 5.1.(SSPA). SSPA (a Modeling) Schematic Performance diagram. (b) Photograph of the test set-up. The AM/AM conversion of a GaAs RF PA is measured by sweeping the input amplitude of a 5.1. SSPA Modeling Performance two-tone signal from the signal generator at a 1 GHz center frequency and a 10 KHz tone spacing. The AM/AMThe nonlinear AM/AM conversionconversion of thea GaAs SSPA RF measurements PA is measured as wellby sweeping as the model the input estimation amplitude using of both a thetwo-tone Saleh and signal enhanced from the Saleh signal models generator is illustrated at a 1 GHz in center Figure frequency 8. The enhanced and a 10 KHz Saleh tone model spacing. shows improvedThe AM/AM curve-ftting nonlinear resultsconversion in the of linearthe SSPA and measurements compression regionsas well ascompared the model to estimation the original using Saleh model.both the In Salehaddition, and enhancedthe residual Saleh square models errors is illustrated between thein Figure measured 8. The and enhanced model Salehconversions model shows(|zmeas| improved2 curve-fitting results in the linear and compression regions compared to the original Saleh − |zmode|) clearly show lower fuctuation errors and accuracy improvement of the enhanced Saleh model withmodel. respect In addition,to the input the amplitude. residual square errors between the measured and model conversions 2 (|zmeasThe|− |modelzmode|) evaluation clearly show for lowerpredicting fluctuation the nonlinear errors an distortiond accuracy in improvementa frequency domain of the enhancedis shown in FigureSaleh 9model, using with the respectpower spectrumto the input density amplitude. of a dual-band long-term evolution (LTE) signal. Table 1 presents the obtained accuracy improvement in the enhanced Saleh model using the normalized mean square-errors (NMSE) based on GaAs and GaN DUT PA technologies. These results show a model improvement of around 4dB NMSE compared to the original Saleh model. In addition, the enhanced model achieved better model accuracy compared to the ffth-order Taylor model.

Figure 8. Measured and modeled static AM/AM results of the SSPA using a two-tone test. The square residual errors of the Saleh model and enhanced Saleh model are overlaid with respect to the input amplitude.

Electronics 2020, 9, x FOR PEER REVIEW 8 of 13

(a) (b) 64-QAM dual-band modulated OFDM signal was applied on the input of the SSPA, and the output signal is RF down-converted, demodulated, and acquired by the spectrum analyzer.

Figure 7. Experimental validation for modeling and digital predistortion of a solid-state power amplifier (SSPA). (a) Schematic diagram. (b) Photograph of the test set-up.

5.1. SSPA Modeling Performance The AM/AM conversion of a GaAs RF PA is measured by sweeping the input amplitude of a two-tone signal from the signal generator at a 1 GHz center frequency and a 10 KHz tone spacing. The AM/AM nonlinear conversion of the SSPA measurements as well as the model estimation using both the Saleh and enhanced Saleh models is illustrated in Figure 8. The enhanced Saleh model shows improved curve-fitting results in the linear and compression regions compared to the original Saleh model. In addition, the residual square errors between the measured and model conversions 2 Electronics(|zmeas|−| 2020zmode, |)9, 1806 clearly show lower fluctuation errors and accuracy improvement of the enhanced9 of 13 Saleh model with respect to the input amplitude.

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The model evaluation for predicting the nonlinear distortion in a frequency domain is shown in Figure 9, using the power spectrum density of a dual-band long-term evolution (LTE) signal. Table 1 presents the obtained accuracy improvement in the enhanced Saleh model using the normalized meanFigure square-errors 8.8. MeasuredMeasured (NMSE) and and modeled modeledbased static on static GaAs AM/AM AM and/AM results GaN results of DUT the of SSPA thePA SSPAtechnologies. using using a two-tone a two-tone These test. Theresults test. square The show a modelresidualsquare improvement residual errors of errors theof aroundSaleh of the mode Saleh 4dBl modeland NMSE enhanced and compared enhanced Saleh model toSaleh the model areoriginal overlaid are overlaidSaleh with model. respect with respect Into theaddition, toinput the the enhancedamplitude.input amplitude.model achieved better model accuracy compared to the fifth-order Taylor model.

Figure 9.9. MeasuredMeasured and and modeled modeled output output spectrum spectrum of ofthe the solid-state solid-state power power amplifer amplifier using using a dual-band a dual- LTEband signal. LTE signal.

Table 1. Accuracy comparison in normalized mean square-errors (NMSE) of the model evaluation Table 1. Accuracy comparison in normalized mean square-errors (NMSE) of the model evaluation using gallium arsenide (GaAs) and gallium nitride (GaN) technologies. using gallium arsenide (GaAs) and gallium nitride (GaN) technologies. Intermodulation Distortion NMSE (dB) Model Number of Coeff. NMSE (dB) Number of Intermodulation(IMD) Distortion Model GaAsGaAs PAGaN GaN PA Coeff. (IMD) Saleh 2 odd terms PA27.12 PA 28.54 − − Enhanced Saleh 23 odd odd and terms even terms −27.1231.37 − 28.54 32.71 − − 5th orderEnhanced Polynomial Saleh 35 oddodd and and even even terms terms −31.3722.16 − 32.71 21.47 5th order − − 5 odd and even terms −22.16 −21.47 Polynomial The overall accuracy in NMSE of the enhanced Saleh model in cascade with FIR flter is depicted in Figure 10, with respect to the FIR flter number of coefficients (i.e., memory depth). This shows The overall accuracy in NMSE of the enhanced Saleh model in cascade with FIR filter is depicted the obtained optimal model accuracy and the required flter number of coefficients to quantify the in Figure 10, with respect to the FIR filter number of coefficients (i.e., memory depth). This shows the obtained optimal model accuracy and the required filter number of coefficients to quantify the memory effect of the SSPA. The FIR filter in this modeling approach reaches the optimal accuracy at a memory depth around 15 for practical implementation, and the accuracy in NMSE changes slightly and becomes almost flat after in the higher order memory depth as illustrated in Figure 10. The dynamic enhanced model achieves a NMSE below −47dB.

Electronics 2020, 9, 1806 10 of 13 memory effect of the SSPA. The FIR flter in this modeling approach reaches the optimal accuracy at a memory depth around 15 for practical implementation, and the accuracy in NMSE changes slightly and becomes almost fat after in the higher order memory depth as illustrated in Figure 10. The dynamic Electronicsenhanced 2020 model, 9, x FOR achieves PEER REVIEW a NMSE below 47dB. 10 of 13 −

Figure 10. The model accuracy in normalized mean square-errorsquare-error (NMSE) of the dynamic enhanced Saleh model in terms of the memory depth of the fnite-impulsefinite-impulse responseresponse (FIR)(FIR) flter.filter.

5.2. ResultsResults of DigitalDigital Predistortion The static AMAM/AM/AM conversion causes most the nonlinearnonlinear distortion in RF PAs. Therefore, the efficiencyefficiency of the DPD to compensate for the AMAM/AM/AM distortion is a signifcantsignificant factor in linearizing the nonlinearity in communication systems. TheThe DPDDPD modelmodel basedbased onon the enhanced Saleh model is evaluated in the time and frequency domains usinusingg a dual-band LTE signal. The AM/AMAM/AM conversion of thethe DPDDPD model model exhibits exhibits signifcant significant gain gain expansion expansion characteristics characteristics over over a wide a wide range range of the of PA the input PA inputamplitude amplitude as shown as inshown Figure in 11 Figure. In addition, 11. In theaddition, DPD model the DPD compensated model compensated for the gain compressionfor the gain compressionbehavior of the behavior SSPA and of improvedthe SSPA theand balance improved between the balancein-phase between and quadrature-phase in-phase and inquadrature- the signal phaseconstellation in the signal diagram, constellation as shown diagram,in Figure as12 .shown in Figure 12.

Figure 11. The dynamic AM/AMAM/AM conversion of the RF power amplifieramplifer and DPD model conversions based inverting the enhanced Saleh model.

Electronics 2020, 9, 1806 11 of 13 ElectronicsElectronics 20202020,, 99,, xx FORFOR PEERPEER REVIEWREVIEW 1111 ofof 1313

((aa)) ((bb))

FigureFigure 12.12. ConstellationConstellation diagramdiagram ofof aa 64-QAM64-QAM basebandbaseband ouououtputtputtput signalsignal ofof thethe RFRF powerpowerpower amplifier,amplifier,amplifer, ((aa)) withoutwithout DPD,DPD, andand ((bb)) withwith DPD.DPD.

TheThe linearizationlinearizationlinearization capability capabilitycapability of theofof theDPDthe DPDDPD model modelmodel in a frequency inin aa frequencyfrequency domain showsdomaindomain signifcant showsshows significantsignificantmitigation mitigationofmitigation the spectrum ofof thethe regrowth spectrumspectrum in regrowthregrowth the adjacent inin thethe channels adjacentadjacent of channelsthechannels dual-band ofof thethe spectrum dual-banddual-band as spectrumspectrumdepicted in asas Figure depicteddepicted 13. inThein FigureFigure numerical 13.13. TheThe evaluation numericalnumerical results evaluationevaluation of this linearization resultsresults ofof thisthis approach linearizationlinearization is depicted approachapproach in Table isis depicteddepicted2 using the inin adjacent TableTable 22 usingchannelusing thethe power adjacentadjacent ratio channelchannel (ACPR) powerpower and error ratioratio vector(ACPR)(ACPR) magnitude anandd errorerror vector(EVM)vector magnitude magnitudefor GaAs and (EVM)(EVM) GaN forfor SSPA. GaAsGaAs Finally, andand GaNGaN the SSPA.obtainedSSPA. Finally,Finally, linearization thethe obtainedobtained performance linearizationlinearization of the enhanced perforperformancemance DPD of ofmodel thethe enhanced enhancedachieved anDPDDPD adequate modelmodel improvement achievedachieved anan adequateinadequate time and improvementimprovement frequency domains. inin timetime and and frequencyfrequency domains.domains.

FigureFigure 13.13. SpectrumSpectrum ofof aa dual-banddual-band LTELTE outputoutput signalsignal wiwithoutwithoutthout DPDDPD inin thethe redred tracetrace andand withwith DPDDPD inin thethe blackblack trace.trace.

Table 2. DPD model performance in adjacent channel power ratio (ACPR) and error vector magnitude TableTable 2.2. DPDDPD modelmodel performanceperformance inin adjacentadjacent channechannell powerpower ratioratio (ACPR)(ACPR) andand errorerror vectorvector (EVM) for linearizing both GaAs and GaN power amplifers (PAs). magnitudemagnitude (EVM)(EVM) forfor linearizinglinearizing bothboth GaAsGaAs andand GaNGaN powerpower amplifiersamplifiers (PAs).(PAs). GaAs GaN Evaluation Measure GaAsGaAs GaNGaN EvaluationEvaluation MeasureMeasurewithout DPD with DPD without DPD with DPD withoutwithout DPDDPD with with DPDDPD without without DPDDPD with with DPDDPD ACPR in Upper ACPRACPR inin 38.73/ 39.35 52.10/ 52.81 41.86/ 42.55 56.13/ 57.41 band/Lower band (dBc) − −−−38.73/38.73/ − −−52.10/52.10/− −−−41.86/41.86/− −−56.13/56.13/− − UpperUpper band/band/ EVM (%) 9.12− −39.3539.35 −1.86−52.8152.81 −−42.5542.558.76 −−57.4157.41 1.54 LowerLower bandband (dBc)(dBc) EVMEVM (%)(%) 9.12 9.12 1.861.86 8.768.76 1.541.54

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6. Conclusions This paper analyzed the Saleh model mathematically using polynomial expansion for modeling the odd-order IMD and described the Saleh weakness for modeling SSPAs. Therefore, a new model enhancement was proposed to improve the model accuracy for the Saleh AM/AM model when deployed for SSPA technologies in communication systems. In addition, the polynomial expansion of the enhanced Saleh model consists of both odd-order and even-order nonlinear terms for characterizing the PA nonlinearities in the AM/AM conversion. The Hammerstein-based model structure was adopted to quantify the dynamic effect in the AM/AM conversion. A least-squares approach was employed for calculating all the parameters of the enhanced Saleh model. The evaluation results showed an accuracy improvement of the enhanced Saleh model to refect the measured AM/AM characteristics of SSPAs over a wide range of the PA input magnitude. A linearization approach using DPD was developed and evaluated for mitigating the nonlinear distortion in SSPAs.

Author Contributions: Conceptualization, H.A.; Methodology, H.A. and F.L.; Software, H.A.; Supervision, F.L.; Writing—original draft, H.A.; Writing—review & editing, H.A. and F.L. All authors have read and agreed to the published version of the manuscript. Funding: Publication of this article is funded by Portland State University’s open access fund. Acknowledgments: The authors would like to acknowledge James Morris from the Department of electrical and computer engineering at Portland State University for contributions on proofreading this article. Conficts of Interest: The authors declare no confict of interest.

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