Optimizing cubic metric in orthogonal frequency division multiplexing systems using chaotic differential search algorithm

Abstract

Cubic metric (CM) is a more precise measure to compute the large envelope fluctuations of orthogonal frequency division multiplexing signals. In this study, we use the partial transmit sequence method to reduce the high CM of signals. Despite the dramatic CM reduction ability of the partial transmit sequence method, its computational complexity is very high because of an exhaustive search over all possible combinations of phase factors. In order to overcome the search complexity of the partial transmit sequence method, we introduce an effective and fast convergence optimizer called a chaotic differential search algorithm (CDSA). The CDSA is a population based optimization method to solve the complex, large-scale and non-linear problems. Simulation results show the superiority of the proposed CDSA compared with several counterpart algorithms in terms of search complexity and CM mitigation performance.

Introduction

Cubic metric (CM) [1] has been recently approved as a more accurate indicator than the peak-to-average power ratio (PAPR) for envelope fluctuations and estimation of the non-linear distortion in orthogonal frequency division multiplexing (OFDM) systems [2], [3]. In wireless applications, several PAPR and CM reduction approaches have been reported [4], [5]. These approaches are non-linear companding [6], clipping [7], precoding [8], partial transmit sequence (PTS) [9], tone injection and reservation [10], and selected mapping [11]. In the companding approach, a non-linear compander is applied at the transmitter side and a decompanding procedure is utilized at the receiver end to retrieve the primary signal. The clipping approach diminishes the CM of signals by clipping to a preset level. In the precoding scheme, first, a precoder matrix is applied to the frequency-domain signals. Afterward, an inverse fast Fourier transform (IFFT) is performed. In the tone injection and reservation approaches, several additional sub-carriers are employed for CM reduction. In the selected mapping, the input frames are multiplied to the some phase factor streams, and then each frame is passed to an IFFT operation. Finally, the PAPRs of all time-domain signals are calculated and the signal with minimum PAPR is selected for transmission. Using the PTS scheme, the input frame is split into several disjoint sub-frames. To make a single data frame from sub-frames three operations should be done: multiplying the sub-frames with their corresponding phase factors, applying the IFFT operation, and finally merging the sub-frames. For the data frame, to have the least amount of PAPR, the phase factors must be in optimal order. It needs to explore all combinations of phase factors to achieve an optimal permutation of phase factors. The PTS scheme is regarded as the most efficient approach due to its robust CM reduction capability [12], [13], and [14]. The original PTS usually suffers from the high computational complexity derived from the exploring of optimal phase factors and a huge number of IFFT procedures.

The original PTS is often combined with an optimization technique to overcome the search complexity issue. Recently, several evolutionary and swarm intelligence algorithms have been incorporated with the PTS technique to explore the optimal phase factors and reduce the high PAPR or CM of OFDM signals. Some of the well-known evolutionary algorithms are genetic algorithm (GA) [15], hybrid genetic algorithm (HGA) [16], backtracking search algorithm (BSA) [17], shuffled frog leaping algorithm [18], and adaptive differential evolution (ADE) algorithm [19]. In [15], the authors proposed the GA-PTS method, in which sub-optimal phase factors are modeled as a population of chromosomes. This population updates using three operators including cross-over, mutation, and selection. In [16], a hybrid version of GA is proposed to enhance the performance of the GA-PTS method. A local search strategy is embedded in HGA to efficiently find the sub-optimal permutation of phase factors. In [17], the BSA is employed to solve the search complexity shortcoming of the PTS scheme. The BSA is a powerful extension of the standard GA, which is equipped with an improved crossover and a memory to use experiences obtained in previous iterations. Some of the most popular swarm intelligence algorithms used in PTS scheme are particle swarm optimization (PSO) [20], artificial bee colony (ABC) [21], harmony search (HS) [22], tabu search (TS) [23], firefly algorithm (FF) [24], grey wolf optimization (GWO) [25], and ant colony optimization (ACO) [26]. These algorithms are multi-agent strategies, in which several agents explore the search space through cooperating or competing with each other. The existing optimization algorithms achieved encouraging results in minimizing the search complexity of the standard PTS, but their performance is far from ideal state and more improvements are required.

We propose an efficient evolutionary algorithm named chaotic differential search algorithm (CDSA). The proposed CDSA is a powerful extension of the differential search algorithm (DSA)  [27]. We combine the CDSA with PTS to solve the search complexity issue of the conventional PTS approach and decrease the CM of OFDM signals. The CDSA–PTS is a fast convergence algorithm and needs a few control parameters and explores an optimal set of phase factors. The CDSA–PTS approach is compared with the optimal PTS (OPTS), standard OFDM system, and several state-of-the-art methods including GA-PTS [15], PSO-PTS [20], ABC-PTS [21], TS-PTS [23], GWO-PTS [25]. The achieved results verify that the CDSA–PTS is a more efficient approach than its counterparts in terms of CM mitigation performance.

The rest of the paper is arranged as follows: Section 2 discusses the basic concepts and background theoretical foundation. The suggested CDSA–PTS approach is explained in Section 3. Simulation results are exhibited in Section 4. Eventually, conclusions are given in the latest section.