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 Open Access
Theoretical analysis of BER performance of nonlinearly amplified FBMC/OQAM and OFDM signals
 Hanen Bouhadda^{1, 2}Email author,
 Hmaied Shaiek^{1},
 Daniel Roviras^{1},
 Rafik Zayani^{2},
 Yahia Medjahdi^{3} and
 Ridha Bouallegue^{2}
https://doi.org/10.1186/16876180201460
© Bouhadda et al.; licensee Springer. 2014
 Received: 29 November 2013
 Accepted: 17 April 2014
 Published: 5 May 2014
Abstract
In this paper, we introduce an analytical study of the impact of highpower amplifier (HPA) nonlinear distortion (NLD) on the bit error rate (BER) of multicarrier techniques. Two schemes of multicarrier modulations are considered in this work: the classical orthogonal frequency division multiplexing (OFDM) and the filter bankbased multicarrier using offset quadrature amplitude modulation (FBMC/OQAM), including different HPA models. According to Bussgang’s theorem, the inband NLD is modeled as a complex gain in addition to an independent noise term for a Gaussian input signal. The BER performance of OFDM and FBMC/OQAM modulations, transmitting over additive white Gaussian noise (AWGN) and Rayleigh fading channels, is theoretically investigated and compared to simulation results. For simple HPA models, such as the soft envelope limiter, it is easy to compute the BER theoretical expression. However, for other HPA models or for real measured HPA, BER derivation is generally intractable. In this paper, we propose a general method based on a polynomial fitting of the HPA characteristics and we give theoretical expressions for the BER for any HPA model.
Keywords
 FBMC/OQAM
 OFDM
 HPA
 Nonlinear distortion
 BER
1 Introduction
Nowadays, 4G systems such as 3GPPLTE are using the cyclic prefix (CP)based orthogonal frequency division multiplexing (OFDM) modulation offering better robustness to multipath channel effects. However, the use of CP and the high side lobes of the rectangular pulse shape spectrum induce a loss of the spectral efficiency.
Filter bankbased multicarrier/offset quadrature amplitude modulation (FBMC/OQAM) modulations are potential promising candidates for next generation systems [1] as well as 5G systems [2]. Indeed, the good frequency localization of the prototype filters [3–5] used in FBMC/OQAM offers to this latter the robustness to several impairments such as the timing misalignment between users [6].
It is well known that multicarrier signals are constructed by a sum of N independent streams transmitted over N orthogonal subcarriers. Considering high values of N and according to the central limit theorem [7], the superposition of these independent streams leads to a complex Gaussian multicarrier signal. For this reason, OFDM and FBMC/OQAM exhibit large peaktoaverage power ratios (PAPR) [8–10], i.e., large fluctuations in their signal envelope, making both modulations very sensitive to nonlinear distortion (NLD) caused by a highpower amplifier (HPA).
The main objective of this paper is to study the bit error rate (BER) performance in the presence of memoryless NL HPA for both OFDM and FBMC/OQAM systems under additive white Gaussian noise (AWGN) and Rayleigh fading channels. A theoretical characterization of NLD effects on OFDM systems has been proposed in [11], where the authors focused on the impact of the nonlinear amplitude distortions induced by three HPA models: the soft envelope limiter (SEL), the solid state power amplifier (SSPA), and the traveling wave tube amplifier (TWTA). In this paper, the authors, proposed a theoretical characterization of the NLD parameters. We note that, except for the SEL HPA model, the investigation presented in [11] gives only semianalytical results which could not be easily extended to measured HPA. Other contributions [12, 13] used the results presented in [11] to study the effect of HPA on multipleinput multipleoutput (MIMO) transmit diversity systems. The impact of the HPA on outofband spectral regrowth was also studied for the OFDM system [14] as well as the FBMC/OQAM case [9, 15].
Our aim is to evaluate the impact of the inband distortions on OFDM and FBMC/OQAM modulations when a memoryless NL HPA is used. It is worth noting that, to the best of our knowledge, this is supposed to be the first paper that investigates the theoretical BER performance of the nonlinearly amplified FBMC/OQAM systems. In another way, this work extends the results presented in [11] to any modeled or measured HPA. The polynomial fitting of the amplitudeamplitude modulation (AM/AM) and amplitudephase modulation (AM/PM) conversion characteristics allows complete theoretical characterization of the nonlinear distortion parameters for both multicarrier modulation schemes.
We will first demonstrate that FBMC/OQAM signals are more sensitive to phase rotation than OFDM ones. Indeed, if no phase correction is done, or in the presence of phase estimation errors, the intrinsic interference in FBMC/OQAM will increase the error probability compared to the OFDM case. In the case of a perfect phase correction at the receiver side, both techniques exhibit the same performance. Analytical closedform expressions of the BER are established based on polynomial decomposition of the HPA NL characteristics. We notice that the proposed method can be applied for any memoryless HPA model and even for real measured ones. In order to validate the obtained BER expressions, various comparisons are made with respect to simulation results.
The rest of this paper is organized as follows: Section 2 describes the system model with brief introduction to OFDM and FBMC/OQAM modulations in addition to some elementary information on commonly used memoryless HPA models, exhibiting AM/AM and AM/PM distortions. In Sections 3 and 4, we present the theoretical model and the estimation of NLD parameters. Further, we develop a theoretical analysis of the BER for both OFDM and FBMC/OQAM systems in Section 5. The obtained BER expressions are then evaluated through various simulation results, which are presented in Section 6. Finally, Section 7 gives the conclusion of this work.
2 System model
2.1 Introduction to OFDM and FBMC/OQAM
2.1.1 OFDM
where

N is the number of subcarriers,

T is the OFDM symbol period,

c_{m,n} is a complexvalued symbol transmitted on the m th subcarrier and at the instant nT, and

f(t) is a rectangular time window, defined by$f\left(t\right)=\left\{\begin{array}{c}\frac{1}{\sqrt{T}}\phantom{\rule{1em}{0ex}}t\in \left[0,T\right]\\ 0\phantom{\rule{1em}{0ex}}\text{elsewhere}\end{array}\right..$
Considering high values of N and according to the central limit theorem [7], the IFFT block transforms a set of independent complex random variables to a set of complex Gaussian random ones.
where

${\hat{c}}_{{m}_{0},{n}_{0}}$ is the received symbol, and

〈.,.〉 stands for the inner product.
2.1.2 FBMC/OQAM
where

a_{m,n} is a real symbol transmitted on the m th subcarrier and at the instant nT,

h(t) is the prototype filter impulse response, and

φ_{m,n} is the phase term which is given by$\begin{array}{l}{\phi}_{m,n}=\frac{\pi}{2}(m+n)\mathrm{\pi mn.}\end{array}$
where ${\gamma}_{{m}_{0},{n}_{0}}^{\ast}\left(t\right)$ is the complex conjugate of ${\gamma}_{{m}_{0},{n}_{0}}\left(t\right)$.
Transmultiplexer impulse response
n_{0}−3  n_{0}−2  n_{0}−1  n _{0}  n_{0}+1  n_{0}+2  n_{0}+3  

m_{0}−1  0.043j  0.125j  0.206j  0.239j  0.206j  0.125j  0.043j 
m _{0}  −0.067j  0  −0.564j  1  0.564j  0  0.067j 
m_{0}+1  0.043j  −0.125j  0.206j  −0.239j  −0.206j  −0.125j  0.043j 
2.2 HPA model
A HPA model or a real measured one can be entirely described by its input/output or transfer function characteristics. The AM/AM and AM/PM characteristics indicate the relationship between, respectively, the modulus and the phase variation of the output signal as functions of the modulus of the input one.
where

$\rho \left(t\right)=\sqrt{\leftx\right(t){}^{2}+y\left(t\right){}^{2}}$ is the signal input modulus, and

$\phi \left(t\right)=arctan\left(\frac{y\left(t\right)}{x\left(t\right)}\right)$ is the signal input phase.
where

F_{ a }(ρ) is the AM/AM characteristic of the HPA,

F_{ p }(ρ) is the AM/PM characteristic of the HPA, and

S(ρ)=F_{ a }(ρ) exp(j F_{ p }(ρ)) is the complex soft envelop of the amplified signal u(t).
where

h_{ c }(t) is the channel impulse response, and

⊗ stands for the convolution product.
In our analysis, we will consider some memoryless HPA models that are commonly used in the literature.
2.2.1 Soft envelope limiter (SEL)
where A_{sat} is the HPA input saturation level.
2.2.2 Solid state power amplifier (SSPA)
where v is a smoothness factor that controls the transition from the linear region to the saturation region, (v>0). This HPA model assumes a linear performance for low amplitudes of the input signal. Then, a transition towards a constant saturated output is observed. When v→∞, the Rapp model converges towards the SEL.
2.2.3 Traveling wave tube amplifier (TWTA)
where φ_{0} controls the maximum phase distortion introduced by this HPA model.
The AM/AM and AM/PM characteristics cause distortions on the constellation scheme and spectral regrowth, degrading then the system performance.
where σ^{2} is the variance of the input signal (mean input signal power).
3 Nonlinear distortion modeling
where

d(t) is a zero mean noise, which is uncorrelated from i(t).

K(t) is a complex gain with modulus K and phase ϕ_{ K }(t).
where $\mathbb{E}$ is the expectation operator. We recall that S(ρ)=F_{ a }(ρ) exp(j F_{ p }(ρ)) is the complex soft envelop of the amplified signal u(t).
Comparison between estimated values of K and ${\sigma}_{d}^{2}$ for both nonlinearly amplified OFDM and FBMC/OQAM signals
OFDM  FBMC/OQAM  Error  

N  K  ${\sigma}_{d}^{2}$  K  ${\sigma}_{d}^{2}$  K  ${\sigma}_{d}^{2}$ 
IBO = 4 dB  
4  0.5796+0.1137j  1.6392×10^{−2}  0.5792+0.1137j  1.6272×10^{−2}  1.6000×10^{−7}  1.4400×10^{−7} 
64  0.5912+0.1107j  1.1338×10^{−2}  0.5918+0.1107j  1.1339×10^{−2}  3.6000×10^{−7}  1.0000×10^{−12} 
1,024  0.5916+0.1106j  1.1339×10^{−2}  0.5923+0.1104j  1.1339×10^{−2}  5.3000×10^{−7}  0.0000 
IBO = 8 dB  
4  0.7662+0.0868j  7.0879×10^{−3}  0.7677+0.0858j  6.9633×10^{−3}  3.2500×10^{−7}  1.5525×10^{−8} 
64  0.7728+0.0830j  2.4249×10^{−3}  0.7728+0.0831j  2.4256×10^{−3}  1.0000×10^{−8}  4.9000×10^{−13} 
1,024  0.7727+0.0830j  2.4256×10^{−3}  0.7728+0.0830j  2.4258×10^{−3}  1.0000×10^{−8}  4.0000×10^{−14} 
4 Analytical computation of K and ${\sigma}_{d}^{2}$
Nevertheless, for more complicated expressions of S(ρ), such as (12) and (13) (SSPA and TWTA models), the derivation of analytical expressions for the parameters K and ${\sigma}_{d}^{2}$ is intractable. In [11], no closedform expression for K and ${\sigma}_{d}^{2}$ is given for SSPA and TWTA HPA models. In order to simplify the computation and obtain analytical expressions for K and ${\sigma}_{d}^{2}$ for any HPA model, we propose a polynomial approximation of S(ρ). By doing this, we will be able to analytically compute the NLD parameters of Equations 20 and 21 for any HPA conversion characteristics after polynomial fitting.
4.1 Proposed method
where

L is the polynomial order, and

a_{ n } are the complex coefficients of the polynomial approximation.
The complex valued polynomial coefficients a_{ n }, n=1..L can be easily obtained by using a classical least square (LS) method [27].
4.2 Analytical computation of K and ${\sigma}_{d}^{2}$using polynomial approximation
where ℜ [.] stands for the real part.
The above theoretical expressions of K and ${\sigma}_{d}^{2}$ involve the computation of the expectation of ρ^{ n } (n is a positive integer). This expectation is equivalent to calculate the n th derivation of the momentgenerating function (MGF).
where

n is a positive integer, and

M(t) is the MGF given by$M\left(t\right)={e}^{\mathrm{\rho t}}.$(30)
A generic expression for the computation of $\mathbb{E}\left[\phantom{\rule{0.3em}{0ex}}{\rho}^{n}\right]$ is given in [27]. It is expressed as follows:

For odd values of n, we have$\begin{array}{ll}\mathbb{E}\left[{\rho}^{n}\right]& ={\left.\frac{{\partial}^{n}M\left(t\right)}{\partial {t}^{n}}\right}_{t=0}\\ =\sqrt{\frac{\pi}{2}}{\sigma}^{n}\prod _{i=0}^{\frac{n1}{2}}(2i+1).\end{array}$(31)

For even values of n, we have$\begin{array}{ll}\mathbb{E}\left[{\rho}^{n}\right]& ={\left.\frac{{\partial}^{n}M\left(t\right)}{\partial {t}^{n}}\right}_{t=0}\\ ={\left(\sqrt{2}\sigma \right)}^{n}\left(\frac{n}{2}\right)!\end{array}$(32)
where ! stands for the factorial operator.
4.2.1 Validation of the analytical expressions of K and ${\sigma}_{d}^{2}$
Comparison between numerical and theoretical values of K and ${\sigma}_{d}^{2}$
Simulation  Theoretical  Error  

IBO (dB)  φ _{0}  K (Equation 20)  ${\sigma}_{d}^{2}$ (Equation 21)  K (Equation 33)  ${\sigma}_{d}^{2}$ (Equation 34)  K  ${\sigma}_{d}^{2}$ 
SSPA  
4    0.7699  4.4759×10^{−3}  0.7690  4.4431×10^{−3}  6.7240×10^{−7}  9.5481×10^{−10} 
6    0.8307  1.8950×10^{−3}  0.8297  1.8722×10^{−3}  1.1025×10^{−6}  7.0560×10^{−11} 
8    0.8798  7.0742×10^{−4}  0.8785  7.0583×10^{−4}  1.6129×10^{−6}  1.6384×10^{−12} 
TWTA  
4  0  0.6042  1.0330×10^{−2}  0.6036  1.0317×10^{−2}  1.9360×10^{−7}  1.0609×10^{−8} 
6  0  0.6976  4.9589×10^{−3}  0.6969  4.9654×10^{−3}  4.9000×10^{−7}  1.2532×10^{−9} 
8  0  0.7784  2.1228×10^{−3}  0.7775  2.1030×10^{−3}  7.9210×10^{−7}  1.9600×10^{−12} 
4  π/6  0.5917+0.1106j  1.1339×10^{−2}  0.5904+0.1068j  1.1271×10^{−2}  1.6590×10^{−5}  5.3290×10^{−9} 
6  π/6  0.6887+0.0995j  5.5901×10^{−3}  0.6870+0.0948j  5.5812×10^{−3}  2.4324×10^{−5}  2.3104×10^{−10} 
8  π/6  0.7727+0.0830j  2.4194×10^{−3}  0.7727+0.0830j  2.4345×10^{−3}  3.5527×10^{−5}  2.2801×10^{−10} 
4.2.2 Influence of polynomial approximation order
Impact of the polynomial order approximation on the estimated values of K and ${\sigma}_{d}^{2}$
FBMC/OQAM  Error  

L  K  ${\sigma}_{d}^{2}$  K  ${\sigma}_{d}^{2}$ 
5  0.7976+0.1570j  3.1941×10^{−3}  6.0985×10^{−3}  5.9259×10^{−7} 
7  0.8248+0.0713j  3.8676×10^{−3}  2.8651×10^{−3}  2.0831×10^{−6} 
10  0.7728+0.0830j  2.4345×10^{−3}  3.5527×10^{−5}  2.2801×10^{−10} 
15  0.7728+0.0830j  2.4345×10^{−3}  3.5527×10^{−5}  2.2801×10^{−10} 
20  0.7728+0.0830j  2.4345×10^{−3}  3.5527×10^{−5}  2.2801×10^{−10} 
30  0.7728+0.0830j  2.4345×10^{−3}  3.5527×10^{−5}  2.2801×10^{−10} 
100  0.7728+0.0830j  2.4345×10^{−3}  3.5527×10^{−5}  2.2801×10^{−10} 
TWTA model  0.7728+0.0830j  2.4345×10^{−3} 
5 Theoretical performance analysis
Looking at Equation 36, it is clear that the effect of the NL factor K will be taken into account during frequency equalization at the receiver side (h_{ c }(t) and K will be estimated jointly). Nevertheless, in order to stress the impact of the phase rotation on both OFDM and FBMC/OQAM performances, we will present in Section 5.2 an analysis of the BER when the channel is an AWGN one and when no correction is made for the NL factor K. Within the same section, we will evaluate theoretical BER in an AWGN channel with correction of the phase rotation related to the factor K. Section 5.3 is dedicated to the BER analysis of OFDM and FBMC/OQAM modulations in the case of the Rayleigh channel.
5.1 Sensitivity of OFDM and FBMC/OQAM to phase error
For simplicity reasons, we will conduct our analysis for 4QAM modulated symbols. The extension of this analysis to MAM (M>4) is straightforward.
5.1.1 OFDM case
where ${a}_{m,n}^{I}$ and ${a}_{m,n}^{Q}$ denote, respectively, the inphase and the quadrature components of the transmitted complex symbol.
5.1.2 FBMC/OQAM case
With this modulation technique, the transmission of a real symbol on subcarrier m_{0} generates a pure imaginary intrinsic interference ${u}_{{m}_{0},{n}_{0}}$.
However, in the presence of a phase offset (ϕ_{ k }), this interference is no longer imaginary. Consequently, by taking the real part of the received signal, we obtain a part of the useful signal distorted by the interference signal (${u}_{{m}_{0},{n}_{0}}$).
u_{m,n} is given by Equation 7, and it corresponds to all possible combinations of adjacent symbols.
5.2 Performance analysis in the case of an AWGN channel
where

$u=\frac{{d}_{\text{min}}}{2}$ is the decision distance (d_{min}, being the minimum distance),

pdf(u) is the probability density function of u, and

N_{0}/2 is the power spectral density of the additive noise.
where

${\sigma}_{w}^{2}$ is the variance of the AWGN,

${\sigma}_{d}^{2}$ is the variance of the NL distortion d(t), and

T is the symbol duration.
where

M is the alphabet of the modulation,

E_{ b a v g } is the average energy per bit normalized to 1, and

N_{0}/2 is the power spectral density of the additive noise.
5.3 BER analysis with phase rotation correction in the case of a Rayleigh channel
where $\Omega =\mathbb{E}\left[{\alpha}^{2}\right]$ is the average fading power.
where γ_{ c }=K^{2}E_{ b } and E_{ b } is the energy per bit.
Derivation of the pdf of γ
Lemma 1.
where g(γ)=σ_{ w }^{2}γ/(γ_{ c }−σ_{ d }^{2}γ).
6 Simulation results
In this section, we present numerical results illustrating the impact of memoryless HPA nonlinearity on the performance of FBMC/OQAM and OFDM systems under AWGN and Rayleigh fading channels. In this work, we have considered FBMC/OQAM and OFDM systems with N=64 subcarriers transmitting MQAM modulated symbols.
The BER is computed by averaging on 5×10^{7} randomly generated FBMC/OQAM and OFDM symbols. We will investigate the cases where transmission is achieved through AWGN and Rayleigh fading channels. For both OFDM and FBMC/OQAM systems, we have considered three scenarios. In the first one, a SEL HPA model is used. In the second scenario, a TWT HPA model with only AM/AM distortion is considered. In the last scenario, a TWT HPA model is used, exhibiting both AM/AM and AM/PM distortions. In all the simulations, the curve referred by ‘linear’ in the legend corresponds to the case when the power amplifier is perfectly linear.
6.1 Impact of phase error on OFDM and FBMC/OQAM
The gap between the BER of the two modulation techniques can be explained by the intrinsic interference term introduced by the FBMC/OQAM. As an example, for IBO=4 dB, φ_{0}=π/3, and a BER=5×10^{−3}, FBMC/OQAM modulation shows a loss of ≈7 dB in SNR compared to the OFDMbased one. Then, when considering a selective frequency channel, an error in estimating the phase rotation of the channel will have a bigger impact in FBMC/OQAM than in OFDM due its higher sensitivity.
6.2 BER analysis in the case of an AWGN channel
6.3 BER analysis in the case of a Rayleigh channel
7 Conclusion
In this paper, we have studied the impact of inband nonlinear distortions caused by memoryless HPA on both OFDM and FBMC/OQAM systems. This study is valid for any measured or modeled HPA, exhibiting amplitude distortion (AM/AM) and phase distortion (AM/PM) over the multicarrier modulated signal. A theoretical approach was proposed to evaluate the BER performance for both OFDM and FBMC/OQAM systems. This approach is based on modeling the inband nonlinear distortion with a complex gain and an uncorrelated additive white Gaussian noise, given by Bussgang’s theorem. The theoretical determination of the NLD parameters is related to the HPA model used. For simple HPA models, the analytical expressions for the NLD parameters can be easily established. However, for more complicated HPA models exhibiting amplitude and phase distortions, the task is more complicated. The idea proposed in this paper is based on polynomial approximation of any HPA model at sufficient order. This makes possible the theoretical analysis for any measured or modeled HPA.
When only the amplitude of the modulated signals is distorted by the HPA, OFDM and FBMC/OQAM show the same performances in terms of BER. Simulations and theoretical results are shown to be in agreement for various IBO values. However, the FBMC/OQAM system is shown to be more sensitive to phase distortions than the OFDM one. The sensitivity of the FBMC/OQAM system to phase distortion is directly related to the intrinsic interference term introduced by this modulation. This sensitivity cannot be seen as a limitation to FBMC modulation, since the phase error is practically taken into account during the channel estimation/equalization process.
Declarations
Acknowledgements
This work is supported by the European project EMPhAtiC (ICT 318362) and the PHCUtique C3 project (code 12G1414).
Authors’ Affiliations
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