- Open Access
Theoretical analysis of BER performance of nonlinearly amplified FBMC/OQAM and OFDM signals
© Bouhadda et al.; licensee Springer. 2014
- Received: 29 November 2013
- Accepted: 17 April 2014
- Published: 5 May 2014
In this paper, we introduce an analytical study of the impact of high-power 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 bank-based multicarrier using offset quadrature amplitude modulation (FBMC/OQAM), including different HPA models. According to Bussgang’s theorem, the in-band 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.
- Nonlinear distortion
Nowadays, 4G systems such as 3GPP-LTE 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 bank-based multicarrier/offset quadrature amplitude modulation (FBMC/OQAM) modulations are potential promising candidates for next generation systems  as well as 5G systems . 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 .
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 , the superposition of these independent streams leads to a complex Gaussian multicarrier signal. For this reason, OFDM and FBMC/OQAM exhibit large peak-to-average 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 high-power 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 , 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  gives only semi-analytical results which could not be easily extended to measured HPA. Other contributions [12, 13] used the results presented in  to study the effect of HPA on multiple-input multiple-output (MIMO) transmit diversity systems. The impact of the HPA on out-of-band spectral regrowth was also studied for the OFDM system  as well as the FBMC/OQAM case [9, 15].
Our aim is to evaluate the impact of the in-band 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  to any modeled or measured HPA. The polynomial fitting of the amplitude-amplitude modulation (AM/AM) and amplitude-phase 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 closed-form 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.
2.1 Introduction to OFDM and FBMC/OQAM
N is the number of subcarriers,
T is the OFDM symbol period,
cm,n is a complex-valued symbol transmitted on the m th subcarrier and at the instant nT, and
f(t) is a rectangular time window, defined by
Considering high values of N and according to the central limit theorem , the IFFT block transforms a set of independent complex random variables to a set of complex Gaussian random ones.
is the received symbol, and
〈.,.〉 stands for the inner product.
am,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
φm,n is the phase term which is given by
where is the complex conjugate of .
Transmultiplexer impulse response
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.
is the signal input modulus, and
is the signal input phase.
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).
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 Asat 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 φ0 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 σ2 is the variance of the input signal (mean input signal power).
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 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 for both nonlinearly amplified OFDM and FBMC/OQAM signals
IBO = 4 dB
IBO = 8 dB
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 is intractable. In , no closed-form expression for K and is given for SSPA and TWTA HPA models. In order to simplify the computation and obtain analytical expressions for K and 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
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 .
4.2 Analytical computation of K and using polynomial approximation
where ℜ [.] stands for the real part.
The above theoretical expressions of K and 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 moment-generating function (MGF).
n is a positive integer, and
M(t) is the MGF given by(30)
A generic expression for the computation of is given in . It is expressed as follows:
For odd values of n, we have(31)
For even values of n, we have(32)
where ! stands for the factorial operator.
4.2.1 Validation of the analytical expressions of K and
Comparison between numerical and theoretical values of K and
K (Equation 20)
K (Equation 33)
4.2.2 Influence of polynomial approximation order
Impact of the polynomial order approximation on the estimated values of K and
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 M-AM (M>4) is straightforward.
5.1.1 OFDM case
where and denote, respectively, the in-phase 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 m0 generates a pure imaginary intrinsic interference .
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 ().
um,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
is the decision distance (dmin, being the minimum distance),
pdf(u) is the probability density function of u, and
N0/2 is the power spectral density of the additive noise.
is the variance of the AWGN,
is the variance of the NL distortion d(t), and
T is the symbol duration.
M is the alphabet of the modulation,
E b a v g is the average energy per bit normalized to 1, and
N0/2 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 is the average fading power.
where γ c =|K|2E b and E b is the energy per bit.
Derivation of the pdf of γ
where g(γ)=σ w 2γ/(γ c −σ d 2γ).
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 M-QAM modulated symbols.
The BER is computed by averaging on 5×107 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, φ0=π/3, and a BER=5×10−3, FBMC/OQAM modulation shows a loss of ≈7 dB in SNR compared to the OFDM-based 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
In this paper, we have studied the impact of in-band 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 in-band 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.
This work is supported by the European project EMPhAtiC (ICT- 318362) and the PHC-Utique C3 project (code 12G1414).
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