- Research
- Open Access
Peak reduction in OFDM using second-order cone programming relaxation
- Marko Beko^{1, 2}Email author,
- Rui Dinis^{3, 4} and
- Ramo Šendelj^{5}
https://doi.org/10.1186/1687-6180-2014-130
© Beko et al.; licensee Springer. 2014
- Received: 14 May 2014
- Accepted: 8 August 2014
- Published: 21 August 2014
Abstract
In this paper, we address the problem of peak-to-average power ratio (PAPR) reduction in orthogonal frequency-division multiplexing (OFDM) systems. We formulate the problem as an error vector magnitude (EVM) optimization task with constraints on PAPR and free carrier power overhead (FCPO). This problem, which is known to be NP hard, is shown to be approximated by a second-order cone programming (SOCP) problem using a sequential convex programming approach, making it much easier to handle. This approach can be extended to the more general problem when PAPR, EVM, and FCPO are constrained simultaneously. Our performance results show the effectiveness of the proposed approach, which allows good performance with lower computational complexity and infeasibility rate than state-of-the-art PAPR-reduction convex approaches. Moreover, in the case when all the three system parameters are constrained simultaneously, the proposed approach outperforms the convex approaches in terms of infeasibility rate, PAPR, and bit error rate (BER) performances.
Keywords
- Error vector magnitude
- Free carrier power overhead
- Orthogonal frequency-division multiplexing
- Peak-to-average power ratio
- Semidefinite programming
- Second-order cone programming
1 Introduction
Orthogonal frequency-division multiplexing (OFDM) [1] are suitable for high rate transmission over severely time-dispersive channels. For this reason, they were selected for broadband wireless communication systems such as digital video broadcasting (DVB) and long-term evolution (LTE). However, OFDM signals have high envelope fluctuations and high peak-to-average power ratio (PAPR), which makes them very sensitive to nonlinear effects and leads to amplification difficulties [2]. For this reason, several techniques were proposed to reduce the PAPR of OFDM signals. We can reduce the PAPR by using specially designed codes [3], but its application is limited to very specific cases and/or a small number of subcarriers. Multiple signal representations such as partial transmit sequences (PTS) techniques [4, 5] and clipping techniques [6–8, 10] are much more flexible and suitable for OFDM signals with a large number of subcarriers. However, with PTS techniques, we might need to transmit side information, and the transmitter complexity can increase substantially. On the other hand, the nonlinear nature of clipping might lead to significant performance degradation. Tone reservation techniques (TR) [9, 11, 12] are particularly interesting for large constellations since we have only a small degradation in the power and spectral efficiency (due to nondata subcarriers).
Convex optimization has recently emerged as an efficient tool for reducing the PAPR of OFDM signals [10–16]. This can be explained in part by the fact that convex optimization methods can efficiently compute global solutions to large-scale problems in polynomial time. In [15], an iterative second-order cone programming (SOCP) approach was proposed to pursue the quasi-constant PAPR value of OFDM signals. In [16], a semidefinite relaxation (SDR) technique is employed to reduce the PAPR values of OFDM symbols. Although convex optimization approaches show advantages over the classical repeated clipping and filtering (RCF) approach [7], it is important to note that they may fail to deliver feasible solution to the PAPR problem; see [15, 16]. This is due to the fact that in [15, 16], the feasible (nonconvex) set of the original (nonconvex) PAPR problem lies within the feasible (convex) set of the relaxed convex problem. Hence, the solutions provided by the methods proposed in [15, 16] are not necessarily feasible for the original problem. Furthermore, a complex semidefinite programming problem (SDP) is solved in [16].
To overcome these drawbacks, in this paper, we propose to minimize the error vector magnitude (EVM) subject to constraints on the PAPR and free carrier power overhead (FCPO). We introduce an efficient sequential convex programming approach to solve the corresponding nonconvex problem by rewriting the nonconvex constraints as a difference of two convex functions. We show that the new approach can also be successfully applied to the case when the PAPR, EVM, and FCPO are simultaneously constrained.
The remainder of this paper is organized as follows. The OFDM symbol design for PAPR reduction is formulated as an optimization problem in Section 2, and in Section 3, we introduce our approach for solving this optimization problem. A set of performance results is presented in Section 4, and Section 5 is concerned with the conclusions of this paper.
2 Problem formulation
Let ${\mathit{c}}_{0}\in {\u2102}^{N}$ be an original OFDM frequency-domain symbol. The corresponding OFDM time-domain symbol, x, of the optimized frequency-domain symbol, c, can be obtained by inverse fast Fourier transform (IFFT) with ℓ-times oversampling, i.e., x = IFFT_{ ℓ }(c) = A c, where the matrix $\mathit{A}\in {\u2102}^{\mathrm{\ell N}\times N}$ is the first N columns of the corresponding IFFT matrix.
The OFDM subcarriers are usually divided into three disjoint sets: data subcarriers, free subcarriers, and pilot subcarriers with cardinalities d, f, and p, respectively, so that d + f + p = N, where N is the total number of subcarriers. For simplicity, we do not consider pilot subcarriers in this paper, although the results can be easily generalized to systems with pilot subcarriers. Let $\mathit{S}\in {\mathbb{R}}^{N\times N}$ be a diagonal matrix with S_{ ii } = 1 when the i th subcarrier is reserved for data transmission and S_{ ii } = 0 otherwise. We are now ready to define three parameters responsible for OFDM’s reliable performance: PAPR, EVM, and FCPO.
where ${\mathit{M}}_{i}=\mathrm{\ell N}{\mathit{A}}^{H}{\mathit{e}}_{i}{\mathit{e}}_{i}^{T}\mathit{A}$, P_{ α } = α A^{ H }A, S_{ β } = (β + 1)S and e_{ i } represents the i th column of the identity matrix I_{ ℓ N }. The EVM optimization framework (7) to (9) results in a nonconvex optimization problem since the matrices I_{ N }-S_{ β } and M_{ i }-P_{ α }, for i = 1,…,ℓ N, are indefinite; in other words, all the constraints are nonconvex [17]. As most nonconvex problems, our problem is NP hard and, thus, difficult to solve [17].
3 Optimization procedure
where ℜ{z} denotes the real part of the complex number z. The problem in (10) to (12) is obtained by linearizing the concave parts of the constraint functions in (8) to (9) around c_{(0)}. (Note that this is actually the first-order Taylor expansion). This leads to a SOCP problem that can be readily solved by CVX[18]. The method continues with linearizing the original problem around c_{(1)} and repeating this procedure until convergence is reached. A justification/motivation of this reformulation lies in the fact that the best convex approximation of a concave function is an affine function.
with (12) when c = c_{(1)}. Then, after some basic manipulations, we obtain ${\mathit{c}}_{\left(1\right)}^{H}\left({\mathit{I}}_{N}-{\mathit{S}}_{\beta}\right){\mathit{c}}_{\left(1\right)}\le 0$, i.e., c_{(1)} satisfies the constraint in (9). Thus, c_{(1)} is feasible for the original problem (7) to (9).
Next, remark that c_{(0)} is feasible for (10) to (12) (when c = c_{(0)}, the constraints (8) and (9) become identical to (11) and (12), respectively). This implies that the feasible set of (10) to (12) contains c_{(0)}. Thus, ||S(c_{(1)}-c_{0})||^{2}≤||S(c_{(0)}-c_{0})||^{2}, i.e., the objective value of the new feasible point c_{(1)} cannot be higher than the objective value of the starting point c_{(0)}. The algorithm stops when ||S(c_{(k)}-c_{0})||-||S(c_{(k+1)}-c_{0})||<10^{-4} for some k. We refer to the proposed method as ‘NEW SOCP’ from hereafter.
where ${f}_{1}={\mathit{c}}_{\left(\text{rand}\right)}^{H}{\mathit{P}}_{\alpha}{\mathit{c}}_{\left(\text{rand}\right)}$, ${f}_{2}={\mathit{c}}_{\left(\text{rand}\right)}^{H}{\mathit{S}}_{\beta}{\mathit{c}}_{\left(\text{rand}\right)}$ and c_{(rand)} is randomly generated; the real and imaginary parts of the i th entry of the vector c_{(rand)} are randomly generated in the intervals [ℜ{c_{0}_{ i }}-1,ℜ{c_{0}_{ i }} + 1] and [I{c_{0}_{ i }}-1,I{c_{0}_{ i }} + 1], respectively, where c_{0}_{ i } is the i th entry of the vector c_{0} and I{a} denotes the imaginary part of the complex number a.
Problem (13) to (15) is obtained from (10) to (12) with c_{0} = c_{(rand)} and can be solved using CVX. Problem (13) to (15) may be infeasible; however, when it is feasible, this heuristic will provide a feasible solution to problem (7) to (9) which is used as the starting point of the algorithm^{a}.
It is important to note that the existing convex methods [15, 16] may fail to deliver feasible solution to the EVM minimization problem (4) to (6); please see ([15], Sec. III-C), and ([16], Sec. IV). This is due to the fact that in [15, 16], the feasible (nonconvex) set of the original (nonconvex) problem (4) to (6) lies within the feasible (convex) set of the relaxed convex problem. Note that in our work, in sharp contrast to [15, 16], the feasible set of the relaxed convex problem is a convex subset of the original (nonconvex) feasible set. Hence, the feasibility of the new solution is guaranteed.
where EVM_{max} is the maximum allowed EVM. Note that the optimization problem (4) to (6) is different from the one in (16) to (18), since the search space for c in the former is larger than that in the latter.
We remark that the constraint (17) is convex and, consequently, the EVM problem (16) to (18) can also be addressed by the proposed method^{b}. The simulation results in Section 4 will assess the effectiveness of the new method.
4 Performance results
Extensive simulations were performed to compare the performance of the proposed algorithm with existing algorithms. Unless stated otherwise, the number of subcarriers is N=64, the number of data subcarriers is d=52 and the number of free subcarriers is f=12. The data subcarriers were generated from 16-quadrature amplitude modulation (QAM), and 64-QAM constellations and the oversampling factor was assumed to be ℓ=4. The CVX package [18] for specifying and solving convex programs was used to solve (10) to (12).
To evaluate the performance of the proposed technique and compare it with existing ones, we obtained the complementary cumulative density function (CCDF) of the PAPR, which corresponds to the probability that the PAPR of an arbitrary OFDM symbol exceeds a given threshold.
The performance of the new algorithm, denoted here by ‘NEW SOCP’, will be compared with the performance of the SDP-based approach proposed in [16], denoted here by ‘SDP WANG’, iterative SOCP-based approach presented in [15], denoted here by ‘SOCP WANG’, and RCF (four iterations) [7], denoted here by ‘RCF’.
For the sake of presentation simplicity, we divide the comparison in two cases: without EVM constraint and with EVM constraint.
4.1 Without EVM constraint
4.2 With EVM constraint
Infeasibility analysis for 16-QAM
16-QAM | EVM_{max} = 0. 04 | EVM_{max} = 0. 09 |
---|---|---|
NEW SOCP | 19.29 | 2.44 |
SOCP WANG | 40.41 | 38.40 |
Infeasibility analysis for 64-QAM
64-QAM | EVM_{max}=0. 04 | EVM_{max}=0. 09 |
---|---|---|
NEW SOCP | 1.67 | 0.20 |
SOCP WANG | 42.57 | 32.84 |
From the comparisons presented in Tables 1 and 2, it can be seen that the probability of generating infeasible solutions by the new approach is reduced when compared to SOCP WANG. Furthermore, as expected, for a given modulation scheme, this probability decreases as the preset threshold EVM_{max} increases since the search space defined by the constraints (17) to (18) becomes larger. As a final note, remark that it is an open question whether for a given set of system parameters (EVM_{max}, α, β, c_{0}, N, d, f) a feasible solution to the EVM problem actually exists.
4.3 Complexity analysis
Computational complexity analysis
NEW SOCP | SDP WANG | SOCP WANG | RCF |
---|---|---|---|
O(L N^{3.5}ℓ^{1.5}) | O(N^{4.5}ℓ^{2}) | O(L N^{3.5}ℓ^{1.5}) | O(2L ℓ N log(ℓ N)) |
5 Conclusions
We have revisited the PAPR problem in OFDM systems. We formulated the problem as a nonconvex optimization problem which is solved approximately by an efficient iterative method. The simulation results show that in the case when the EVM constraint is not active, the new method outperforms the one in [16] in terms of complexity and feasibility, whereas, in the case when a constraint on the EVM is imposed, the new method outperforms the method in [15] in terms of PAPR performance, BER performance, and feasibility. This confirms the relevance of the approach proposed herein.
Endnotes
^{a} It is not difficult to see that the solution of (13) to (15), c_{(0)}, is feasible for the original problem (7) to (9).
^{b} In [13], Aggarwal and Meng proposed to minimize the PAPR subject to constraints on the EVM and FCPO. Note that the proposed approach can be readily applied to this framework as well.
^{c} It was shown in [16] that ‘RCF’ exhibits a bad BER performance. Consequently, the BER performance of ‘RCF’ was not presented in Figures 3, 4, 7 and 8.
Declarations
Acknowledgements
This work was partially supported by Fundação para a Ciência e a Tecnologia under Projects PTDC/EEI-TEL/2990/2012–ADIN, EXPL/EEI-TEL/1582/2013–GLANC, PEst-OE/EEI/UI0066/2014 and EXPL/EEI-TEL/0969/2013–MANY2COMWIN, and Ciência 2008 Post-Doctoral Research grant. M. Beko is a collaborative member of INESC-INOV, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal.
Authors’ Affiliations
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