# PAPR Reduction Using Fireworks Search Optimization Algorithm in MIMO-OFDM Systems

Authors: Lahcen Amhaimar; Saida Ahyoud; Ali Elyaakoubi; Abdelmoumen Kaabal; Kamal Attari; Adel Asselman
Journal: Journal of Electrical and Computer Engineering
 (2018)

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## Abstract

The transceiver combination technology, of orthogonal frequency division multiplexing (OFDM) with multiple-input multiple-output (MIMO), provides a viable alternative to enhance the quality of service and simultaneously to achieve high spectral efficiency and data rate for wireless mobile communication systems. However, the high peak-to-average power ratio (PAPR) is the main concern that should be taken into consideration in the MIMO-OFDM system. Partial transmit sequences (PTSs) is a promising scheme and straightforward method, able to achieve an effective PAPR reduction performance, but it requires an exhaustive search to find the optimum phase factors, which causes high computational complexity increased with the number of subblocks. In this paper, a reduced computational complexity PTS scheme is proposed, based on a novel swarm intelligence algorithm, called fireworks algorithm (FWA). Simulation results confirmed the adequacy and the effectiveness of the proposed method which can effectively reduce the computation complexity while keeping good PAPR reduction. Moreover, it turns out from the results that the proposed PTS scheme-based FWA clearly outperforms the hottest and most important evolutionary algorithm in the literature like simulated annealing (SA), particle swarm optimization (PSO), and genetic algorithm (GA).


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## Body

## 1. Introduction

The transmission data through the use of multicarrier modulation techniques like orthogonal frequency division multiplexing (OFDM) has been considered as a good choice for data transmission over single carrier systems. The OFDM technique has various advantages and now being used in a number of wireless communication systems. Meanwhile, the multiple-input multiple-output (MIMO) with OFDM system, has recently attracted a great deal of attention due to its various advantages such as high data rate, spectral efficiency, diversity in a fading environment, and robustness to channel fading [1]. Hence, a system with OFDM modulation and multiple transmit and multiple receive antennas (MIMO-OFDM) is now becoming adopted by several applications such as digital audio broadcasting (DAB), Worldwide Interoperability for Microwave Access (WiMAX), the fourth generation of telecommunication systems (4G), digital video broadcasting (DVB), high speed WLAN standards, and many others application areas of MIMO-OFDM [2]. But besides these useful advantages, it still suffers from the high envelope fluctuations of the transmitted signal called the peak-to-average power ratio (PAPR), which decreases the efficiency of high power amplifiers (HPA), improves the complexity of nonlinear elements, and causes out-of-band radiation with degradation of bit error rate (BER).To deal with this problem, several solutions have been proposed to mitigate the high PAPR of OFDM [3, 4] and MIMO-OFDM signals [5, 6], as clipping [7], clipping and filtering [8], coding [9], tone injection [10], peak windowing [11], selected mapping [12], and partial transmit sequence (PTS) [13–15]. All these methods have their own advantages and disadvantages, but the PTS technique is still the most attractive one due to its efficiency in PAPR reduction. However, the exhaustive search complexity of finding the optimal phase combination for PTS increases exponentially with number of subblocks and to reduce the computational complexity, many evolutionary algorithms for optimization-based PTS schemes have been proposed such as genetic algorithm (GA) [16, 17], ant colony optimization (ACO) [18], simulated annealing algorithm (SA) [19], and the most well-known algorithm of particle swarm optimization (PSO) [20, 21]. In this paper, we propose a novel swarm intelligence algorithm called fireworks algorithm (FWA) [22], to reduce the PAPR with less complexity and more easy implementation, and this developed search optimization algorithm is based on the explosion process simulation of fireworks. It turns out from the results that the proposed method FWA-PTS effectively reduces PAPR of MIMO-OFDM signal and clearly outperforms the old algorithms in both global high precision of calculated solution and convergence speed.This manuscript is organized as follows. In Section2, MIMO-OFDM system model and the PAPR problem is formulated, and then the principles of PTS techniques are introduced. Section 3 describes the framework of the FWA-based PTS and mechanisms of the algorithm with some improved version, while Sections 4 and 5 are devoted to the analysis of simulation results and conclusions successively.


## 2. MIMO-OFDM System and PTS Approach

### 2.1. PAPR of the MIMO-OFDM Signal

The transmission of signal through the use of transceiver based on orthogonal frequency division multiplexing (OFDM) system is a typical technique which divides the effective spectrum channel to a number of orthogonal subchannels, and with equal bandwidth, each subchannel handles independently with its own data using individual subcarrier, and the OFDM signal is the sum of all independent subcarriers. In transmission systems with multicarrier signal, the input data of binary sequences are mapped into symbols by a modulator (PSK, QPSK, QAM, etc.). Then, the N symbolsX=X0,X1,…,XN−1T are inserted into the IFFT block to modulate each subcarrier independently and to obtain the OFDM signal in time domain x=x0,x1,…,xN−1T.The complex envelope of OFDM signal in the discrete time domain with oversampled factorL (usually used L=4) can be mathematically written as(1)xn=1N∑k=0N−1Xkej2πnk/LN,0≤n≤LN−1,where N is the number of subcarriers and Xk is the nth complex symbol carried and transmitted by the kth subcarrier.From equation (1), the signal in time domain generated by IFFT operation consists N number of independently modulated and orthogonal subcarriers with large peak values (PAPR) when added up at the output of IDFT block. The PAPR of the OFDM signal in discrete time is defined as the ratio between the maximum power and the average power of the complex OFDM signal, and it can be defined as(2)PAPRxn=maxxn2Exn2,0≤n≤LN−1,where xn is given by (1) and E⋅ denotes the expected value (average power).MIMO-OFDM is a generalized case of OFDM systems based on space time block code (STBC) [23–25] for two, three, and four antennas. The encoder signal with two transmitting antennas, using Alamouti code and an input signal X=X0,X1,…,XN−1 is written as(3)X1=X0,−X∗1,…,XN−2,−X∗N−1T,X2=X1,X∗0,…,XN−1,X∗N−2T.The signalsX1 and X2 are transmitted by antennas 1 and 2, respectively.At each antenna of MIMO-OFDM system, the peak-to-average power ratio is defined as(4)PAPRxi=maxxin2Exin2,0≤n≤NL−1,where i=1,2,…,NT number of transmitted antennas. The time domain signal at each transmit antennas can be presented as(5)xin=1N∑k=0N−1Xkiej2πnk/LN.The expression characterized the peak power variation of MIMO-OFDM systems is defined as(6)PAPRMIMO−OFDM=maxPAPRxi,i=1,…,NT,


### 2.2. PAPR Reduction by Partial Transmit Sequence

The probabilistic distortionless technique of partial transmit sequence presented in Figure1, divided an input data block X=X1,X2,…,XNT into V disjoint subblocks, represented by the vector Xv, v=1,2,…,V such that X=∑v=1VXv. The partition of subblocks was performed with a simple method, where all the used subcarriers by another block must be zero so that the sum of all the disjoint subblocks constitutes the original signal. Then, the subblocks are oversampled and transformed to time domain by LN-point IFFT (inverse fast Fourier transform) and using an optimization algorithm or conventional searching method, each subblock is multiplied by a phase factor as follows:(7)x=IFFT∑v=1Vbv⋅Xv=∑v=1Vbv⋅IFFTXv=∑v=1Vbv⋅xv,where xv is the time domain signal of each subblock v. The complex phase factor bv=ejφv,v=1,2,…,V, rotates the sequences independently to obtain the OFDM signals with the lowest PAPR possible. The phase vector is chosen within 0,2π interval, and the optimum one can be presented as(8)b1,…,bV=argminb1,…,bVmaxn=0,1,…,N−1∑v=1Vbv⋅xvn.Figure 1
Block diagram of PTS technique.In the practical application of wireless communication systems using the PTS approach, the PAPR performance will be improved as the number of subblocksV is increased and to match the optimal phase weighting sequence for each input data sequence, WV possible combinations should be checked (W number of phase factors), which needs a huge quantity of computations to analyze all possible applicant rotation phase vectors. Therefore, the computational complexity increases.


## 2.1. PAPR of the MIMO-OFDM Signal

The transmission of signal through the use of transceiver based on orthogonal frequency division multiplexing (OFDM) system is a typical technique which divides the effective spectrum channel to a number of orthogonal subchannels, and with equal bandwidth, each subchannel handles independently with its own data using individual subcarrier, and the OFDM signal is the sum of all independent subcarriers. In transmission systems with multicarrier signal, the input data of binary sequences are mapped into symbols by a modulator (PSK, QPSK, QAM, etc.). Then, the N symbolsX=X0,X1,…,XN−1T are inserted into the IFFT block to modulate each subcarrier independently and to obtain the OFDM signal in time domain x=x0,x1,…,xN−1T.The complex envelope of OFDM signal in the discrete time domain with oversampled factorL (usually used L=4) can be mathematically written as(1)xn=1N∑k=0N−1Xkej2πnk/LN,0≤n≤LN−1,where N is the number of subcarriers and Xk is the nth complex symbol carried and transmitted by the kth subcarrier.From equation (1), the signal in time domain generated by IFFT operation consists N number of independently modulated and orthogonal subcarriers with large peak values (PAPR) when added up at the output of IDFT block. The PAPR of the OFDM signal in discrete time is defined as the ratio between the maximum power and the average power of the complex OFDM signal, and it can be defined as(2)PAPRxn=maxxn2Exn2,0≤n≤LN−1,where xn is given by (1) and E⋅ denotes the expected value (average power).MIMO-OFDM is a generalized case of OFDM systems based on space time block code (STBC) [23–25] for two, three, and four antennas. The encoder signal with two transmitting antennas, using Alamouti code and an input signal X=X0,X1,…,XN−1 is written as(3)X1=X0,−X∗1,…,XN−2,−X∗N−1T,X2=X1,X∗0,…,XN−1,X∗N−2T.The signalsX1 and X2 are transmitted by antennas 1 and 2, respectively.At each antenna of MIMO-OFDM system, the peak-to-average power ratio is defined as(4)PAPRxi=maxxin2Exin2,0≤n≤NL−1,where i=1,2,…,NT number of transmitted antennas. The time domain signal at each transmit antennas can be presented as(5)xin=1N∑k=0N−1Xkiej2πnk/LN.The expression characterized the peak power variation of MIMO-OFDM systems is defined as(6)PAPRMIMO−OFDM=maxPAPRxi,i=1,…,NT,


## 2.2. PAPR Reduction by Partial Transmit Sequence

The probabilistic distortionless technique of partial transmit sequence presented in Figure1, divided an input data block X=X1,X2,…,XNT into V disjoint subblocks, represented by the vector Xv, v=1,2,…,V such that X=∑v=1VXv. The partition of subblocks was performed with a simple method, where all the used subcarriers by another block must be zero so that the sum of all the disjoint subblocks constitutes the original signal. Then, the subblocks are oversampled and transformed to time domain by LN-point IFFT (inverse fast Fourier transform) and using an optimization algorithm or conventional searching method, each subblock is multiplied by a phase factor as follows:(7)x=IFFT∑v=1Vbv⋅Xv=∑v=1Vbv⋅IFFTXv=∑v=1Vbv⋅xv,where xv is the time domain signal of each subblock v. The complex phase factor bv=ejφv,v=1,2,…,V, rotates the sequences independently to obtain the OFDM signals with the lowest PAPR possible. The phase vector is chosen within 0,2π interval, and the optimum one can be presented as(8)b1,…,bV=argminb1,…,bVmaxn=0,1,…,N−1∑v=1Vbv⋅xvn.Figure 1
Block diagram of PTS technique.In the practical application of wireless communication systems using the PTS approach, the PAPR performance will be improved as the number of subblocksV is increased and to match the optimal phase weighting sequence for each input data sequence, WV possible combinations should be checked (W number of phase factors), which needs a huge quantity of computations to analyze all possible applicant rotation phase vectors. Therefore, the computational complexity increases.


## 3. FWA Optimization-Based PTS

### 3.1. FWA

In the PTS technique, finding the optimal combination of phase factors requires exhaustive searching by full enumeration of allWV possible combinations of phase vectors. In recent years, swarm intelligence (SI) has become popular among researchers working on optimization problems. SI algorithms have advantages in solving many optimization problems. The firework algorithm (FWA) is the most recently discovered swarm intelligence [22, 26], which seems effective at problem solving and finding a good enough solution for global optimization of complex functions. In this part, we propose a novel approach based on the FWA to find the optimal phase vector combination to reduce the PAPR with less complexity compared with old algorithms. The objective function fObj is to minimize the PAPR of the transmitted OFDM signals as follows:(9)minimizefObjb,x=∑v=1Vbv⋅xv,subjecttobv=ejφv,v=1,2,…,V,0≤φv≤2π.And the bounds of the potential space are defined by0≤φv≤2π.As any probabilistic technique without downward compatibility and in order for the receiver to be able to recover the original data block, the information about the selected phase sequence should be transmitted as a side information. In conventional PTS schemes or the PTS with metaheuristic algorithm, side information about this choice needs to be explicitly transmitted along with the chosen candidate signal. If the side information (phase vector sequence) about the transmitter’s choice is received in error, then the information in the transmitted candidate signal cannot be recovered in the receiver and is completely lost. Therefore, the side information needs a particularly strong protection against transmission errors.In this proposed technique PTS-FWA, we proposed the transmission of particularly protected side information with a simple way. Our approach is based on the use of null subcarriers of each symbol OFDM to transmit the selected phase vectors in order to perform data detection in the receiver, which can easily detect the position of the transmitted phase sequence after deleting the guard interval of each symbol.FWA is an iterative algorithm which starts to run until the termination criteria are reached, and it is made up of four key parameters: explosive operator, mutation operation, mapping rule, and selection strategy.


#### 3.1.1. Explosion Operator

Based on these four steps, the realization of FWA starts by randomly generatingN fireworks in the feasible space 0,2π as the initial swarm, then the explosion operator generate sparks around N fireworks, and it can be divided further into explosion amplitude, explosion strength, and displacement operation [22]. Let us assume N fireworks with a number of dimensions d, and the number of sparks generated by each firework bi and the explosion amplitude Ai is defined as follows:(10)si=m⋅ymax−fObjbi+ε∑i=1nymax−fObjbi+ε,(11)Ai=A^⋅fObjbi−ymin+ε∑i=1nfObjbi−ymin+ε,where m and A^ are two constants to control the total number of sparks and the explosion amplitude, respectively, ymax=maxfObjbi, ymin=minfObjbi, (maximum and minimum PAPR), and ε is the machine constant.In the process of FA, the number of sparkssi, is bounded in order to avoid the overwhelming effects of splendid fireworks as follows:(12)si=rounda⋅m,ifsi<am,roundb⋅m,ifsi>am,a<b<1roundsi,otherwise,where a and b are constant parameters.


#### 3.1.2. Gaussian Mutation Operator

In order to keep the diversity of a population after the explosion, the Gaussian mutation process is performed to generate Gaussian sparks. We select arbitrarily a firework from the present population with arbitrarily dimensionZ. If we consider that bjd is the current individual with d∈Z, the Gaussian mutation applied to the firework to generate the second type of sparks is calculated by bjd=bjd∗Gaussian1,1, where the function Gaussian1,1 denotes a Gaussian distribution with mean 1 and variance 1. Although the Gaussian mutation operator improves the diversity by conducting search in a local Gaussian space around a firework, it may produce some sparks that exceed the workable space 0,2π, for that the mapping rule will be done to guarantee all the individual sparks remain in the feasible space.


#### 3.1.3. Mapping Rule

During the explosion generation of Gaussian sparks, some obtained location of generated individuals may lie out of the feasible space. Therefore, it needs to be mapped back into the potential space using the mapping rule to ensure that all generated sparks stay in the feasible space and is stated as follows:(13)bjd=Bmin,d+bjd%Bmax,d+Bmin,d.where bjd is the individual spark’s position that is found to fall out of maximum and minimum boundaries Bmax,d and Bmin,d, respectively. And % represents the mathematics modular arithmetic (remainder of division).


#### 3.1.4. Selection Strategy

At the end of each iteration, after applying the first three steps, a new population of fireworks is selected to be a part of the next generation based on selection strategy, to keep information exchange. In the process of this step, the best spark having the best location is constantly preserved for the next iterations and the remainingN−1 individuals are chosen in light of their distance to other individuals so as to keep diversity of the population.In selection operation, the distance between two individualsbi and bj is defined as(14)Rbi=∑j∈Kdbi,bj=∑j∈Kbi−bj,where K is the set of all current locations combining fireworks and both types of sparks.Then, the selection probabilitypbi to choose individuals for next generation is calculated by(15)pbi=Rbi∑j∈KRbi.As a result, the fireworks or sparks with larger distances have greater chance to be selected for next generation.


#### 3.1.5. Firework Algorithm-Based PTS

The following algorithm1, summarizes in detail the steps employed by FWA to search the optimal phase vector for PTS.Algorithm 1: Employ FWA based PTS to reduce the PAPR.
(1)
Set general input parameters, stopping criterion, phase factors, etc;
(2)
selectN locations for fireworks randomly;
(3)
computing the fitness of each seeds (PAPR);
(4)
while stop criteria is not met do
(5)
Set offn fireworks respectively at the n locations:
(6)
for each firework bi do
(7)
Calculate the number of sons that each firework should generates^i (10);
(8)
Calculate the amplitude of each individual (sparks)Ai (11);
(9)
end for
(10)
fork = 1 to m^ (number of generated sparks by Gaussian Mutation Operator) do
(11)
Generate a specific spark for a randomly selected fireworkbj;
(12)
end for
(13)
selection strategy to choose the best sparks for next generation using the selection probability given in (15);
(14)
end while


### 3.2. Comparison of FWA with Other SI Algorithms

One of the most important categories in computational intelligence community is the swarm intelligence algorithm which is an active branch of evolutionary computation. The fireworks algorithm (FWA), particle swarm optimization (PSO), and genetic algorithms (GA) are the most respected and popular SI algorithms, which are inspired by the intelligent behavior existing in nature and widely used to solve combinatorial optimization problems and real parameter optimization. The FWA algorithm has several characteristics and advantages, such as explosiveness, instantaneity, parallelism (i.e., there is no central control mechanism among fireworks), diversity, robustness (i.e., fireworks are highly independent), and flexibility (i.e., the problem does not need to have an explicit expression to be optimized by FWA), and with these advantages FWA surpasses other SI algorithms.


#### 3.2.1. Comparison between GA and FWA

The comparison between FWA and GA leads to see that the FWA and the genetic algorithms have a lot in common, for example, both algorithms initialize a population randomly, evaluate the individuals of each iteration according to their fitness values (objective function), and perform a certain random search. In addition, FWA and GA are not guaranteed to find the optimal values. Comparing both algorithms, the mechanisms of sharing the information are quite different. In the fireworks algorithm, a distributed information sharing mechanism is used, while the number of sparks and the explosion amplitudes are determined by the fitness values of fireworks which are located in different areas. However, in the GA, chromosomes share information with each other so the entire population moves relatively homogeneously in the feasible space.The strategy of selection in FWA sounds like the selection operator in GA, but they are different. A spark which has less similar sparks is selected with a higher probability, on the contrary, a spark which has more similar sparks is chosen with a lower chance. Hence, the sparks with lower fitness values have the chance to be selected such that the diversity of the sparks is ensured. Comparing this strategy with the GA, the selection operator is based on the roulette to determine which individual is to be selected by the fitness values of the individuals. But the diversity of the population is not guaranteed compared with FWA. In addition, the fireworks are always selected from different areas and can hardly stay together due to the immune-based selection. Yet, the FWA has more mechanisms to avoid premature compared to genetic algorithm. Finally, There is no crossover operator in the FWA, and the mutation operator in the GA is totally different from that in the fireworks algorithm.


#### 3.2.2. Comparison between PSO and FWA

The particle swarm optimization is one of the popular SI algorithms which is inspired by the social behavior of bird flocking, and compared with the FWA, we can conclude that both algorithms have much in common. They adopt random initial populations, evaluate the objective functions, and perform the search based on the fitness values. Also, all the algorithms are not guaranteed to find the optimal solution. In PSO, there is no mutation operation, while there is both displacement operation and Gaussian mutation in FWA. Hence, the diversity of the population is not guaranteed in PSO compared with FWA. Besides, FWA utilizes the idea of immune concentration to keep the diversity of the population, whereas the idea is not contained in PSO.Furthermore, the mechanism of sharing the information is different. FWA uses a distributed information sharing mechanism, so as to determine the number of sparks and explosion amplitude by the fitness values of each spark in different regions. It also needs to maintain the best firework throughout the iterative process. But on the other hand, with PSO, only gbest gives information to other particles, which is one way of information delivery, as the search process follows the information about the best particle which limits optimization process.


### 3.3. Enhanced Fireworks Algorithm

Enhanced fireworks algorithm (EFWA) [27], is an improved extension version of the FWA, and it was proposed to tackle some of the limitations inherent in the original algorithm. The results of FWA decline when being applied on some functions which have a distance between their optimum and origin of the search space (shifted functions). The main factors causing this behavior are the Gaussian sparks and mapping operator. Moreover, the fireworks algorithm suffers from runtime disadvantage, and compared with other optimization algorithms, the FWA has a significant computational cost per iteration, leading to high power consumption. Comparing the two algorithms, EFWA outperforms FWA in terms of results stability and time of convergence. EFWA based on five major improvements is summarized as follows [27].


#### 3.3.1. New Minimal Explosion Amplitude Operator

The explosion amplitudeAi (11) in FWA shows that the fireworks having the best fitness value will have a very small explosion amplitude (close to 0). If this operator is close to 0, the explosion sparks will be located at (almost) the same location as the firework itself. As a result, the location of these sparks cannot be improved. In order to overcome this situation, EFWA binds the explosion amplitude of each firework using a minimal explosion amplitude strategy as follows:(16)Aik=Amink,ifAik<Amink,Aik,otherwise,where Amin, the minimal explosion amplitude, is calculated in each iteration according to [27].


#### 3.3.2. A New Explosion Sparks Operator

In the explosion sparks of firework algorithm, the offset displacement is similar, and it is just computed once for all selected dimensions and in each selection step, the same value is added to the location of selected dimensions. However, this operation leads to bad local search ability. To avoid this bad behavior of generating explosion sparks, EFWA proposed a new process to compute a new and different offset displacement for each explosion sparkssi [27].


#### 3.3.3. A New Mapping Strategy

To avoid the problem of mapping operator in the traditional algorithm of firework, a new uniform random mapping strategy is proposed in EFWA for the out-of-band sparks which exceed the search space0,2π. The following operator maps each spark si to another location with uniform distribution within the search space:(17)Bik=Bmink+rand∗Bmaxk−Bmink,where rand is a function of uniformly distributed random numbers.


#### 3.3.4. Gaussian Sparks Operator

After the explosion, the Gaussian mutation operator is employed to generate another type of spark using an improved operator for generating Gaussian sparks in order to avoid FWA mutation disadvantages. The operator is computed by [26, 27](18)Bik=Bik+Bbk−Bik∗e,e=Gaussian0,1,where Bb is the best firework (phase factor) position found so far for each dimension and Gaussian0,1 is a random selection from a Gaussian distribution with mean 0 and variance 1.


#### 3.3.5. Selection Operator

The selection operator is among the most critical key which aims to keep diversity for the next explosion generation. In fireworks algorithm, the selection strategy is processed based on the distance between two individuals, which favors the selection of fireworks or sparks in less crowded regions of the feasible space ((14) and (15)). However, it has the downside of being responsible of most of the runtime within optimization process. To speed the selection process, EFWA applies a simple selection method, which is referred to as the elitism-random selection method [28]. In this selection procedure, the optima of the set will be chosen first. Then, the other individuals are selected randomly. The comparison between two algorithms FWA and EFWA indicated that the new operator diminishes the runtime largely in EFWA.


### 3.4. Dynamic Search in FWA

One of the developed versions of enhanced fireworks algorithm (EFWA) is a fireworks algorithm-based dynamic search mechanism which enhances the results of the optimization research and also deals some limitations of EFWA. The explosion amplitude is the key operator which is used to control locality and globality of search based on the quality of the fireworks and also the present number of iterations. As a result, a firework with smaller fitness value (i.e., small explosion amplitude) will perform local search while the global search produces a smaller population (i.e., higher explosion amplitude) performed by fireworks with a fitness value significantly higher. Therefore, in [29], the authors propose a dynamic search fireworks algorithm (dynFWA) to tackle EFWA limitations, using an appropriate strategy applied on the firework at the present best position with changing the explosion amplitude dynamically. Moreover, the authors improve the efficiency of dynFWA by removing the Gaussian sparks operator compared with EFWA without a loss in accuracy.


### 3.5. Adaptive Fireworks Algorithm

Several versions of the fireworks algorithm are developed to ensure the quality of optimization. EFWA and dyFWA are two examples of this improvement. However, these algorithms remain to be improved in many aspects. The adaptive fireworks algorithm (AFWA) [30], is an improved version of the recently developed EFWA where the amplitude operator is replaced by new adaptive amplitude so as to enhance the computational mechanism of explosion amplitude. The amplitude is a critical factor influencing the performance of EFWA, and for that reason the AFWA tries to control it precisely, in order to compute the explosion amplitude operator of the best firework using the old sparks with an adaptive method. And in the next generation, we use the retain information from the previous generation to calculate the amplitude of the best firework according to the algorithm process described in [26, 30].


## 3.1. FWA

In the PTS technique, finding the optimal combination of phase factors requires exhaustive searching by full enumeration of allWV possible combinations of phase vectors. In recent years, swarm intelligence (SI) has become popular among researchers working on optimization problems. SI algorithms have advantages in solving many optimization problems. The firework algorithm (FWA) is the most recently discovered swarm intelligence [22, 26], which seems effective at problem solving and finding a good enough solution for global optimization of complex functions. In this part, we propose a novel approach based on the FWA to find the optimal phase vector combination to reduce the PAPR with less complexity compared with old algorithms. The objective function fObj is to minimize the PAPR of the transmitted OFDM signals as follows:(9)minimizefObjb,x=∑v=1Vbv⋅xv,subjecttobv=ejφv,v=1,2,…,V,0≤φv≤2π.And the bounds of the potential space are defined by0≤φv≤2π.As any probabilistic technique without downward compatibility and in order for the receiver to be able to recover the original data block, the information about the selected phase sequence should be transmitted as a side information. In conventional PTS schemes or the PTS with metaheuristic algorithm, side information about this choice needs to be explicitly transmitted along with the chosen candidate signal. If the side information (phase vector sequence) about the transmitter’s choice is received in error, then the information in the transmitted candidate signal cannot be recovered in the receiver and is completely lost. Therefore, the side information needs a particularly strong protection against transmission errors.In this proposed technique PTS-FWA, we proposed the transmission of particularly protected side information with a simple way. Our approach is based on the use of null subcarriers of each symbol OFDM to transmit the selected phase vectors in order to perform data detection in the receiver, which can easily detect the position of the transmitted phase sequence after deleting the guard interval of each symbol.FWA is an iterative algorithm which starts to run until the termination criteria are reached, and it is made up of four key parameters: explosive operator, mutation operation, mapping rule, and selection strategy.


### 3.1.1. Explosion Operator

Based on these four steps, the realization of FWA starts by randomly generatingN fireworks in the feasible space 0,2π as the initial swarm, then the explosion operator generate sparks around N fireworks, and it can be divided further into explosion amplitude, explosion strength, and displacement operation [22]. Let us assume N fireworks with a number of dimensions d, and the number of sparks generated by each firework bi and the explosion amplitude Ai is defined as follows:(10)si=m⋅ymax−fObjbi+ε∑i=1nymax−fObjbi+ε,(11)Ai=A^⋅fObjbi−ymin+ε∑i=1nfObjbi−ymin+ε,where m and A^ are two constants to control the total number of sparks and the explosion amplitude, respectively, ymax=maxfObjbi, ymin=minfObjbi, (maximum and minimum PAPR), and ε is the machine constant.In the process of FA, the number of sparkssi, is bounded in order to avoid the overwhelming effects of splendid fireworks as follows:(12)si=rounda⋅m,ifsi<am,roundb⋅m,ifsi>am,a<b<1roundsi,otherwise,where a and b are constant parameters.


### 3.1.2. Gaussian Mutation Operator

In order to keep the diversity of a population after the explosion, the Gaussian mutation process is performed to generate Gaussian sparks. We select arbitrarily a firework from the present population with arbitrarily dimensionZ. If we consider that bjd is the current individual with d∈Z, the Gaussian mutation applied to the firework to generate the second type of sparks is calculated by bjd=bjd∗Gaussian1,1, where the function Gaussian1,1 denotes a Gaussian distribution with mean 1 and variance 1. Although the Gaussian mutation operator improves the diversity by conducting search in a local Gaussian space around a firework, it may produce some sparks that exceed the workable space 0,2π, for that the mapping rule will be done to guarantee all the individual sparks remain in the feasible space.


### 3.1.3. Mapping Rule

During the explosion generation of Gaussian sparks, some obtained location of generated individuals may lie out of the feasible space. Therefore, it needs to be mapped back into the potential space using the mapping rule to ensure that all generated sparks stay in the feasible space and is stated as follows:(13)bjd=Bmin,d+bjd%Bmax,d+Bmin,d.where bjd is the individual spark’s position that is found to fall out of maximum and minimum boundaries Bmax,d and Bmin,d, respectively. And % represents the mathematics modular arithmetic (remainder of division).


### 3.1.4. Selection Strategy

At the end of each iteration, after applying the first three steps, a new population of fireworks is selected to be a part of the next generation based on selection strategy, to keep information exchange. In the process of this step, the best spark having the best location is constantly preserved for the next iterations and the remainingN−1 individuals are chosen in light of their distance to other individuals so as to keep diversity of the population.In selection operation, the distance between two individualsbi and bj is defined as(14)Rbi=∑j∈Kdbi,bj=∑j∈Kbi−bj,where K is the set of all current locations combining fireworks and both types of sparks.Then, the selection probabilitypbi to choose individuals for next generation is calculated by(15)pbi=Rbi∑j∈KRbi.As a result, the fireworks or sparks with larger distances have greater chance to be selected for next generation.


### 3.1.5. Firework Algorithm-Based PTS

The following algorithm1, summarizes in detail the steps employed by FWA to search the optimal phase vector for PTS.Algorithm 1: Employ FWA based PTS to reduce the PAPR.
(1)
Set general input parameters, stopping criterion, phase factors, etc;
(2)
selectN locations for fireworks randomly;
(3)
computing the fitness of each seeds (PAPR);
(4)
while stop criteria is not met do
(5)
Set offn fireworks respectively at the n locations:
(6)
for each firework bi do
(7)
Calculate the number of sons that each firework should generates^i (10);
(8)
Calculate the amplitude of each individual (sparks)Ai (11);
(9)
end for
(10)
fork = 1 to m^ (number of generated sparks by Gaussian Mutation Operator) do
(11)
Generate a specific spark for a randomly selected fireworkbj;
(12)
end for
(13)
selection strategy to choose the best sparks for next generation using the selection probability given in (15);
(14)
end while


## 3.1.1. Explosion Operator

Based on these four steps, the realization of FWA starts by randomly generatingN fireworks in the feasible space 0,2π as the initial swarm, then the explosion operator generate sparks around N fireworks, and it can be divided further into explosion amplitude, explosion strength, and displacement operation [22]. Let us assume N fireworks with a number of dimensions d, and the number of sparks generated by each firework bi and the explosion amplitude Ai is defined as follows:(10)si=m⋅ymax−fObjbi+ε∑i=1nymax−fObjbi+ε,(11)Ai=A^⋅fObjbi−ymin+ε∑i=1nfObjbi−ymin+ε,where m and A^ are two constants to control the total number of sparks and the explosion amplitude, respectively, ymax=maxfObjbi, ymin=minfObjbi, (maximum and minimum PAPR), and ε is the machine constant.In the process of FA, the number of sparkssi, is bounded in order to avoid the overwhelming effects of splendid fireworks as follows:(12)si=rounda⋅m,ifsi<am,roundb⋅m,ifsi>am,a<b<1roundsi,otherwise,where a and b are constant parameters.


## 3.1.2. Gaussian Mutation Operator

In order to keep the diversity of a population after the explosion, the Gaussian mutation process is performed to generate Gaussian sparks. We select arbitrarily a firework from the present population with arbitrarily dimensionZ. If we consider that bjd is the current individual with d∈Z, the Gaussian mutation applied to the firework to generate the second type of sparks is calculated by bjd=bjd∗Gaussian1,1, where the function Gaussian1,1 denotes a Gaussian distribution with mean 1 and variance 1. Although the Gaussian mutation operator improves the diversity by conducting search in a local Gaussian space around a firework, it may produce some sparks that exceed the workable space 0,2π, for that the mapping rule will be done to guarantee all the individual sparks remain in the feasible space.


## 3.1.3. Mapping Rule

During the explosion generation of Gaussian sparks, some obtained location of generated individuals may lie out of the feasible space. Therefore, it needs to be mapped back into the potential space using the mapping rule to ensure that all generated sparks stay in the feasible space and is stated as follows:(13)bjd=Bmin,d+bjd%Bmax,d+Bmin,d.where bjd is the individual spark’s position that is found to fall out of maximum and minimum boundaries Bmax,d and Bmin,d, respectively. And % represents the mathematics modular arithmetic (remainder of division).


## 3.1.4. Selection Strategy

At the end of each iteration, after applying the first three steps, a new population of fireworks is selected to be a part of the next generation based on selection strategy, to keep information exchange. In the process of this step, the best spark having the best location is constantly preserved for the next iterations and the remainingN−1 individuals are chosen in light of their distance to other individuals so as to keep diversity of the population.In selection operation, the distance between two individualsbi and bj is defined as(14)Rbi=∑j∈Kdbi,bj=∑j∈Kbi−bj,where K is the set of all current locations combining fireworks and both types of sparks.Then, the selection probabilitypbi to choose individuals for next generation is calculated by(15)pbi=Rbi∑j∈KRbi.As a result, the fireworks or sparks with larger distances have greater chance to be selected for next generation.


## 3.1.5. Firework Algorithm-Based PTS

The following algorithm1, summarizes in detail the steps employed by FWA to search the optimal phase vector for PTS.Algorithm 1: Employ FWA based PTS to reduce the PAPR.
(1)
Set general input parameters, stopping criterion, phase factors, etc;
(2)
selectN locations for fireworks randomly;
(3)
computing the fitness of each seeds (PAPR);
(4)
while stop criteria is not met do
(5)
Set offn fireworks respectively at the n locations:
(6)
for each firework bi do
(7)
Calculate the number of sons that each firework should generates^i (10);
(8)
Calculate the amplitude of each individual (sparks)Ai (11);
(9)
end for
(10)
fork = 1 to m^ (number of generated sparks by Gaussian Mutation Operator) do
(11)
Generate a specific spark for a randomly selected fireworkbj;
(12)
end for
(13)
selection strategy to choose the best sparks for next generation using the selection probability given in (15);
(14)
end while


## 3.2. Comparison of FWA with Other SI Algorithms

One of the most important categories in computational intelligence community is the swarm intelligence algorithm which is an active branch of evolutionary computation. The fireworks algorithm (FWA), particle swarm optimization (PSO), and genetic algorithms (GA) are the most respected and popular SI algorithms, which are inspired by the intelligent behavior existing in nature and widely used to solve combinatorial optimization problems and real parameter optimization. The FWA algorithm has several characteristics and advantages, such as explosiveness, instantaneity, parallelism (i.e., there is no central control mechanism among fireworks), diversity, robustness (i.e., fireworks are highly independent), and flexibility (i.e., the problem does not need to have an explicit expression to be optimized by FWA), and with these advantages FWA surpasses other SI algorithms.


### 3.2.1. Comparison between GA and FWA

The comparison between FWA and GA leads to see that the FWA and the genetic algorithms have a lot in common, for example, both algorithms initialize a population randomly, evaluate the individuals of each iteration according to their fitness values (objective function), and perform a certain random search. In addition, FWA and GA are not guaranteed to find the optimal values. Comparing both algorithms, the mechanisms of sharing the information are quite different. In the fireworks algorithm, a distributed information sharing mechanism is used, while the number of sparks and the explosion amplitudes are determined by the fitness values of fireworks which are located in different areas. However, in the GA, chromosomes share information with each other so the entire population moves relatively homogeneously in the feasible space.The strategy of selection in FWA sounds like the selection operator in GA, but they are different. A spark which has less similar sparks is selected with a higher probability, on the contrary, a spark which has more similar sparks is chosen with a lower chance. Hence, the sparks with lower fitness values have the chance to be selected such that the diversity of the sparks is ensured. Comparing this strategy with the GA, the selection operator is based on the roulette to determine which individual is to be selected by the fitness values of the individuals. But the diversity of the population is not guaranteed compared with FWA. In addition, the fireworks are always selected from different areas and can hardly stay together due to the immune-based selection. Yet, the FWA has more mechanisms to avoid premature compared to genetic algorithm. Finally, There is no crossover operator in the FWA, and the mutation operator in the GA is totally different from that in the fireworks algorithm.


### 3.2.2. Comparison between PSO and FWA

The particle swarm optimization is one of the popular SI algorithms which is inspired by the social behavior of bird flocking, and compared with the FWA, we can conclude that both algorithms have much in common. They adopt random initial populations, evaluate the objective functions, and perform the search based on the fitness values. Also, all the algorithms are not guaranteed to find the optimal solution. In PSO, there is no mutation operation, while there is both displacement operation and Gaussian mutation in FWA. Hence, the diversity of the population is not guaranteed in PSO compared with FWA. Besides, FWA utilizes the idea of immune concentration to keep the diversity of the population, whereas the idea is not contained in PSO.Furthermore, the mechanism of sharing the information is different. FWA uses a distributed information sharing mechanism, so as to determine the number of sparks and explosion amplitude by the fitness values of each spark in different regions. It also needs to maintain the best firework throughout the iterative process. But on the other hand, with PSO, only gbest gives information to other particles, which is one way of information delivery, as the search process follows the information about the best particle which limits optimization process.


## 3.2.1. Comparison between GA and FWA

The comparison between FWA and GA leads to see that the FWA and the genetic algorithms have a lot in common, for example, both algorithms initialize a population randomly, evaluate the individuals of each iteration according to their fitness values (objective function), and perform a certain random search. In addition, FWA and GA are not guaranteed to find the optimal values. Comparing both algorithms, the mechanisms of sharing the information are quite different. In the fireworks algorithm, a distributed information sharing mechanism is used, while the number of sparks and the explosion amplitudes are determined by the fitness values of fireworks which are located in different areas. However, in the GA, chromosomes share information with each other so the entire population moves relatively homogeneously in the feasible space.The strategy of selection in FWA sounds like the selection operator in GA, but they are different. A spark which has less similar sparks is selected with a higher probability, on the contrary, a spark which has more similar sparks is chosen with a lower chance. Hence, the sparks with lower fitness values have the chance to be selected such that the diversity of the sparks is ensured. Comparing this strategy with the GA, the selection operator is based on the roulette to determine which individual is to be selected by the fitness values of the individuals. But the diversity of the population is not guaranteed compared with FWA. In addition, the fireworks are always selected from different areas and can hardly stay together due to the immune-based selection. Yet, the FWA has more mechanisms to avoid premature compared to genetic algorithm. Finally, There is no crossover operator in the FWA, and the mutation operator in the GA is totally different from that in the fireworks algorithm.


## 3.2.2. Comparison between PSO and FWA

The particle swarm optimization is one of the popular SI algorithms which is inspired by the social behavior of bird flocking, and compared with the FWA, we can conclude that both algorithms have much in common. They adopt random initial populations, evaluate the objective functions, and perform the search based on the fitness values. Also, all the algorithms are not guaranteed to find the optimal solution. In PSO, there is no mutation operation, while there is both displacement operation and Gaussian mutation in FWA. Hence, the diversity of the population is not guaranteed in PSO compared with FWA. Besides, FWA utilizes the idea of immune concentration to keep the diversity of the population, whereas the idea is not contained in PSO.Furthermore, the mechanism of sharing the information is different. FWA uses a distributed information sharing mechanism, so as to determine the number of sparks and explosion amplitude by the fitness values of each spark in different regions. It also needs to maintain the best firework throughout the iterative process. But on the other hand, with PSO, only gbest gives information to other particles, which is one way of information delivery, as the search process follows the information about the best particle which limits optimization process.


## 3.3. Enhanced Fireworks Algorithm

Enhanced fireworks algorithm (EFWA) [27], is an improved extension version of the FWA, and it was proposed to tackle some of the limitations inherent in the original algorithm. The results of FWA decline when being applied on some functions which have a distance between their optimum and origin of the search space (shifted functions). The main factors causing this behavior are the Gaussian sparks and mapping operator. Moreover, the fireworks algorithm suffers from runtime disadvantage, and compared with other optimization algorithms, the FWA has a significant computational cost per iteration, leading to high power consumption. Comparing the two algorithms, EFWA outperforms FWA in terms of results stability and time of convergence. EFWA based on five major improvements is summarized as follows [27].


### 3.3.1. New Minimal Explosion Amplitude Operator

The explosion amplitudeAi (11) in FWA shows that the fireworks having the best fitness value will have a very small explosion amplitude (close to 0). If this operator is close to 0, the explosion sparks will be located at (almost) the same location as the firework itself. As a result, the location of these sparks cannot be improved. In order to overcome this situation, EFWA binds the explosion amplitude of each firework using a minimal explosion amplitude strategy as follows:(16)Aik=Amink,ifAik<Amink,Aik,otherwise,where Amin, the minimal explosion amplitude, is calculated in each iteration according to [27].


### 3.3.2. A New Explosion Sparks Operator

In the explosion sparks of firework algorithm, the offset displacement is similar, and it is just computed once for all selected dimensions and in each selection step, the same value is added to the location of selected dimensions. However, this operation leads to bad local search ability. To avoid this bad behavior of generating explosion sparks, EFWA proposed a new process to compute a new and different offset displacement for each explosion sparkssi [27].


### 3.3.3. A New Mapping Strategy

To avoid the problem of mapping operator in the traditional algorithm of firework, a new uniform random mapping strategy is proposed in EFWA for the out-of-band sparks which exceed the search space0,2π. The following operator maps each spark si to another location with uniform distribution within the search space:(17)Bik=Bmink+rand∗Bmaxk−Bmink,where rand is a function of uniformly distributed random numbers.


### 3.3.4. Gaussian Sparks Operator

After the explosion, the Gaussian mutation operator is employed to generate another type of spark using an improved operator for generating Gaussian sparks in order to avoid FWA mutation disadvantages. The operator is computed by [26, 27](18)Bik=Bik+Bbk−Bik∗e,e=Gaussian0,1,where Bb is the best firework (phase factor) position found so far for each dimension and Gaussian0,1 is a random selection from a Gaussian distribution with mean 0 and variance 1.


### 3.3.5. Selection Operator

The selection operator is among the most critical key which aims to keep diversity for the next explosion generation. In fireworks algorithm, the selection strategy is processed based on the distance between two individuals, which favors the selection of fireworks or sparks in less crowded regions of the feasible space ((14) and (15)). However, it has the downside of being responsible of most of the runtime within optimization process. To speed the selection process, EFWA applies a simple selection method, which is referred to as the elitism-random selection method [28]. In this selection procedure, the optima of the set will be chosen first. Then, the other individuals are selected randomly. The comparison between two algorithms FWA and EFWA indicated that the new operator diminishes the runtime largely in EFWA.


## 3.3.1. New Minimal Explosion Amplitude Operator

The explosion amplitudeAi (11) in FWA shows that the fireworks having the best fitness value will have a very small explosion amplitude (close to 0). If this operator is close to 0, the explosion sparks will be located at (almost) the same location as the firework itself. As a result, the location of these sparks cannot be improved. In order to overcome this situation, EFWA binds the explosion amplitude of each firework using a minimal explosion amplitude strategy as follows:(16)Aik=Amink,ifAik<Amink,Aik,otherwise,where Amin, the minimal explosion amplitude, is calculated in each iteration according to [27].


## 3.3.2. A New Explosion Sparks Operator

In the explosion sparks of firework algorithm, the offset displacement is similar, and it is just computed once for all selected dimensions and in each selection step, the same value is added to the location of selected dimensions. However, this operation leads to bad local search ability. To avoid this bad behavior of generating explosion sparks, EFWA proposed a new process to compute a new and different offset displacement for each explosion sparkssi [27].


## 3.3.3. A New Mapping Strategy

To avoid the problem of mapping operator in the traditional algorithm of firework, a new uniform random mapping strategy is proposed in EFWA for the out-of-band sparks which exceed the search space0,2π. The following operator maps each spark si to another location with uniform distribution within the search space:(17)Bik=Bmink+rand∗Bmaxk−Bmink,where rand is a function of uniformly distributed random numbers.


## 3.3.4. Gaussian Sparks Operator

After the explosion, the Gaussian mutation operator is employed to generate another type of spark using an improved operator for generating Gaussian sparks in order to avoid FWA mutation disadvantages. The operator is computed by [26, 27](18)Bik=Bik+Bbk−Bik∗e,e=Gaussian0,1,where Bb is the best firework (phase factor) position found so far for each dimension and Gaussian0,1 is a random selection from a Gaussian distribution with mean 0 and variance 1.


## 3.3.5. Selection Operator

The selection operator is among the most critical key which aims to keep diversity for the next explosion generation. In fireworks algorithm, the selection strategy is processed based on the distance between two individuals, which favors the selection of fireworks or sparks in less crowded regions of the feasible space ((14) and (15)). However, it has the downside of being responsible of most of the runtime within optimization process. To speed the selection process, EFWA applies a simple selection method, which is referred to as the elitism-random selection method [28]. In this selection procedure, the optima of the set will be chosen first. Then, the other individuals are selected randomly. The comparison between two algorithms FWA and EFWA indicated that the new operator diminishes the runtime largely in EFWA.


## 3.4. Dynamic Search in FWA

One of the developed versions of enhanced fireworks algorithm (EFWA) is a fireworks algorithm-based dynamic search mechanism which enhances the results of the optimization research and also deals some limitations of EFWA. The explosion amplitude is the key operator which is used to control locality and globality of search based on the quality of the fireworks and also the present number of iterations. As a result, a firework with smaller fitness value (i.e., small explosion amplitude) will perform local search while the global search produces a smaller population (i.e., higher explosion amplitude) performed by fireworks with a fitness value significantly higher. Therefore, in [29], the authors propose a dynamic search fireworks algorithm (dynFWA) to tackle EFWA limitations, using an appropriate strategy applied on the firework at the present best position with changing the explosion amplitude dynamically. Moreover, the authors improve the efficiency of dynFWA by removing the Gaussian sparks operator compared with EFWA without a loss in accuracy.


## 3.5. Adaptive Fireworks Algorithm

Several versions of the fireworks algorithm are developed to ensure the quality of optimization. EFWA and dyFWA are two examples of this improvement. However, these algorithms remain to be improved in many aspects. The adaptive fireworks algorithm (AFWA) [30], is an improved version of the recently developed EFWA where the amplitude operator is replaced by new adaptive amplitude so as to enhance the computational mechanism of explosion amplitude. The amplitude is a critical factor influencing the performance of EFWA, and for that reason the AFWA tries to control it precisely, in order to compute the explosion amplitude operator of the best firework using the old sparks with an adaptive method. And in the next generation, we use the retain information from the previous generation to calculate the amplitude of the best firework according to the algorithm process described in [26, 30].


## 4. Simulation Results

### 4.1. Parameters

In this section, numerous computer simulations have been examined based on IEEE 802.11a (wireless LAN) and IEEE 802.16e (WiMAX) standards to verify the performance of PTS-OFDM with FWA searching the optimal combination of phase factors for PAPR reduction. The IEEE 802.16e standard form includes 256 subcarriers (IFFT size), in which the data are carried by 192 subcarriers, 8 subcarriers are reserved as a pilot for channel estimation by the receiver, and the rest are used as guard band subcarriers.Similarly, the WLAN design has 52 subcarriers, which includes 48 data, 4 pilots, and 12 null subcarriers. For simulation, 104 random OFDM symbols have been generated and QPSK modulation is employed, and L = 4 is the oversampling factor chosen and the communication channel was AWGN models. The phase weighting factors W = 0,2π have been used, and subblocks V = 2; 4; 6; 8 are chosen. For the FWA, the parameters were chosen by some preliminary experiments, and as described in [22], the FWA worked quite well at the setting: m^=5, A^=40, N = 5, m = 50, a = 0.04, and b = 0.8. These parameters are applied in all the comparison. Firstly, the simulation of MIMO-OFDM system is carried out using two transmit antennas 2Tx and one receiver antenna (2 × 1), and then the simulation generalized to the case of four transmit and one receiver antenna (4 × 1). Table 1 documented all the parameters used in this simulation.Table 1
Simulation parameters.
Parameter
802.11a
802.16e
FFT size
64
256
User carriers
52
200
Pilot carriers
4
8
Number of null/guard band subcarriers
12
56
Cyclic prefix or guard time
1/4, 1/8, 1/16, 1/32
Modulation
QPSK, 3/4
Oversampling factor
L = 4
Number of subblocks
V = 2, 4, 6, 8
Phase factors
W = 0,2π
Channel type
AWGN
Population sizeN
5
Gaussian mutation
5
Number of sparks (m)
50
Parametersa and b
0.04 and 0.8
m
^
and
A
^
5 and 40
Channel bandwidth (MHz)
3.5


### 4.2. PAPR Reduction Performance

In this section, we compare the PAPR performance of the PTS-based FWA with the most popular algorithms used to search the phase factor as like standard particle swarm optimization (SPSO) [21], genetic algorithm (GA), simulated annealing (SA), conventional PTS, and SLM methods in terms of both optimization accuracy for PAPR reduction and convergence speed.The CCDF curves of the simulations results are given in Figure2. The OFDM system was simulated with 64 subcarriers, in which 4 subblocks are employed, and the PTS, SLM, and GA used W = 4 {1 −1 j −j} phase weighting factors to optimize the PAPR of the modulated OFDM symbol, while other algorithms chose randomly the four phases within the interval W = [0, 2π). It can be seen that the proposed scheme (PTS-FWA) yields the best performance while the SLM and GA-PTS yields the worst performance for the PAPR reduction of OFDM systems. For example, for the probability of 10−3, the PAPRs are 4 dB, 4.421 dB, 4.948 dB, 5.226 dB, 5.879 dB, 7.034 dB, and 10.66 dB for the FWA, SPSO, SA, PTS, GA, SLM scheme, and original OFDM signals, respectively.Figure 2
CCDFs comparison with different searching algorithms and conventional PTS.Figure3 shows the simulated results of the fireworks algorithm-assisted PTS technique and the recently improved version of FWA, in comparison against normal OFDM and different PAPR reduction approaches. The PTS scheme based on the EFWA, dynFWA, and AFWA algorithms are given to compare their PAPR reduction performances in OFDM systems. It is clear from the figure that the FWA and all developed version here can all effectively reduce the PAPR in OFDM systems. However, their PAPR reduction performances and convergence speed are different. In general, EFWA and dynFWA show a few improvements over conventional FWA. For CCDF = 10−3, we have 3.942 dB and 3.979 dB for EFWA and dynFWA, respectively, while AFWA has 4.283 dB compared with FWA (4 dB). In addition, the PAPR reduction based-new version of FWA shows a good improvement in terms of computational cost.Figure 3
PAPR reduction comparison-based improved version of FWA.Besides the PAPR reduction performances of the novel swarm intelligence algorithm, convergence speed is quite essential parameter for any optimization algorithm. Figure4 depicts the convergence curves of the proposed schemes in comparison with GA and SPSO in order to validate the convergence speed of the FWA. The simulation is carried out for random OFDM symbol with 10 independent runs and 3000 iterations. From these results, we can conclude that the four methods of FWA have a much faster speed convergence than the SPSO and the GA, while dynFWA and AFWA were a very promising schemes due to their efficiency and simplicity.Figure 4
Convergence curves of the FWA, EFWA, dynFWA, AFWA, SPSO, and GA algorithms-based PTS for one symbol OFDM.Figure5 shows the influence of different numbers of subblocks V on PAPR performance for a QPSK-OFDM system and 104 randomly OFDM symbols using the dynamic search fireworks algorithm, which is characterized by a good simulation runtime without losing optimization accuracy. It is seen in Figure 5 that the PAPR performance in OFDM systems-based dynFWA enhances as the number of subblocks increases with V = 2, 4, 6, and 8. Therefore, the system with the bigger number of subblocks performs the better PAPR.Figure 5
PAPR performance of a QPSK/OFDM system with dynFWA-based PTS technique when the number of subblocks varies.Finally, Figure6 depicts the PAPR reduction performance for WiMAX OFDM systems with AFWA. For randomly generated QPSK symbols, both CCDF graphs of original and optimized signals have been illustrated, and space time block coding (STBC) is used in order to diminish the computational complexity at the receiver side and achieve more diversity gain at the transmit side, where the test starts with two transmit antennas-based Alamouti scheme [23] (2 × 1) and finished by using four transmit antennas (4 × 1) [24, 25]. With a half-rate orthogonal STBC encoding and 104 OFDM symbol, it can be observed that PAPR of MIMO OFDM with AFWA was ≈5.5 dB and ≈5.812 dB at CCDF = 10−3 compared with original PAPR ≈11.25 dB and ≈11.76 dB for WiMAX with two and four transmit antennas, respectively.Figure 6
PAPR reduction performances in MIMO-OFDM-based WiMAX standard.


### 4.3. Complexity Comparison

Table2 shows the computational complexity to find the phase factors and the reduced PAPR values of the various PTS schemes at CCDF = 10−3. These proposed techniques are diversified in terms of reducing the PAPR as well as computational complexity. It must be kept in mind that the complexity of a PAPR reduction system increases linearly with the number of iterations of such an algorithm. There will therefore be a trade-off between reducing the PAPR and the complexity of the system.Table 2
Computational complexity of the different PTS schemes for CCDF = 10−3.
Methods
Number of searches
PAPR (dB)
Original
0
10.66
PTS
W
V = 44 = 256
5.226
PTS-GA
P × G = 10 × 10 = 100
5.879
PTS-SPSO
S × Iter = 100 × 2 = 200
4.421
PTS-FWA
Iter × (N + m + m^) = 2 × (5 + 50 + 5) = 120
4In the original OFDM signal, the PAPR value is calculated before the PTS optimization and therefore its search number is 0. Optimum PTS considers all the phase factors and thus it needsWV = 256 iterations. For GA-PTS, the search complexity is proportional to P × G = 100, where P is the maximum size of the population and G is the generation. In PTS-PSO, the search number is equal to S × Iter = 200, where Iter is the maximum iteration number and S is the size of particle swarm. Finally, in the FWA-PTS, approximate (n + m + m) function evaluations are done in each generation. Suppose the optimum of a function can be found in Iter generations, then we can deduce that the complexity of the FWA is O (Iter × (n + m + m)) = 120.Table3 depicts the optimization accuracy of the proposed algorithms on one symbol OFDM in comparison with SPSO and GA. It can be seen that the proposed schemes clearly outperform both the SPSO and GA. In addition, the fireworks algorithm and its improved versions can find optimal solutions in less than 500 function evaluations.Table 3
Performance evaluation by the FWA, EFWA, dynFWA, AFWA, SPSO, and GA on one symbol OFDM over 10 independent runs.
Methods
Function evaluations
PAPR (dB)
Original
—
10.66
SPSO
2000
4.055
GA
2000
4.447
FWA
500
2.839
EFWA
500
3.015
dynFWA
500
3.021
AFWA
500
2.9


### 4.4. Spectrum Performance

One of the major problems for wireless communications is the RF sidelobe generation [31]. Out-of-band radiations have decisive effects on adjacent channel interference (ACI) performances and reduce the spectrum efficiency, which are of great importance especially for multiuser and RF wireless communications systems. The transmission under ideal conditions leads to no spectral spread, while the transmission of original OFDM with height PAPR via a nonlinear device produces considerable out-of-band power when approaching the saturation zone (6, 7, 8, and 9 dB backoff from saturation point). Our approach consists of finding an alternative representation of the OFDM signal with minimum PAPR based on fireworks algorithm with dynamic search mechanism (dynFWA), and then the output signal will be amplified after digital predistortion (DPD) process. This efficient approach suggested to reduce the PAPR and to compensate the high power amplifiers (HPA), by making a combination between the proposed scheme (PTS-dynFWA) of PAPR reduction and digital predistortion (DPD).Figure7 shows the signal of spectrum performance before and after passing through the nonlinear power amplifier model for conventional OFDM system and OFDM system with the proposed approach (PAPR reduction method PTS-dynFWA and DPD). The results are obtained for two values of the HPA backoff, 7 dB and 1 dB from saturation point.Figure 7
Power Spectral Density comparison performance of an OFDM signal without and with the proposed method. (a, b) 7 dB from saturation point (moderate nonlinearity) and (c, d) 1 dB from saturation point (severe nonlinearity). (a) Without PTS-dynFWA-DPD, (b) with PTS-dynFWA-DPD, (c) without PTS-dynFWA-DPD, and (d) with PTS-dynFWA-DPD.
(a)
(b)
(c)
(d)Figure7(b) shows the frequency spectra with 7 db backoff below the input power that causes amplifier saturation. It is clearly seen that the proposed method (PTS-dynFWA with DPD) has the ability to suppress the out-of-band distortion completely. Compared to the case (a) where no PAPR reduction and no predistortion are used, Figures 7(c) and 7(d) show the spectrum with severe nonlinearity (1 db from saturation point). Comparing the two spectra (Figure 7(c)), we can view that the transmission of original OFDM signal via a HPA device produces considerable out-of-band power. However, the proposed method was able to reduce the spectral regrowth around each of the two bands.


## 4.1. Parameters

In this section, numerous computer simulations have been examined based on IEEE 802.11a (wireless LAN) and IEEE 802.16e (WiMAX) standards to verify the performance of PTS-OFDM with FWA searching the optimal combination of phase factors for PAPR reduction. The IEEE 802.16e standard form includes 256 subcarriers (IFFT size), in which the data are carried by 192 subcarriers, 8 subcarriers are reserved as a pilot for channel estimation by the receiver, and the rest are used as guard band subcarriers.Similarly, the WLAN design has 52 subcarriers, which includes 48 data, 4 pilots, and 12 null subcarriers. For simulation, 104 random OFDM symbols have been generated and QPSK modulation is employed, and L = 4 is the oversampling factor chosen and the communication channel was AWGN models. The phase weighting factors W = 0,2π have been used, and subblocks V = 2; 4; 6; 8 are chosen. For the FWA, the parameters were chosen by some preliminary experiments, and as described in [22], the FWA worked quite well at the setting: m^=5, A^=40, N = 5, m = 50, a = 0.04, and b = 0.8. These parameters are applied in all the comparison. Firstly, the simulation of MIMO-OFDM system is carried out using two transmit antennas 2Tx and one receiver antenna (2 × 1), and then the simulation generalized to the case of four transmit and one receiver antenna (4 × 1). Table 1 documented all the parameters used in this simulation.Table 1
Simulation parameters.
Parameter
802.11a
802.16e
FFT size
64
256
User carriers
52
200
Pilot carriers
4
8
Number of null/guard band subcarriers
12
56
Cyclic prefix or guard time
1/4, 1/8, 1/16, 1/32
Modulation
QPSK, 3/4
Oversampling factor
L = 4
Number of subblocks
V = 2, 4, 6, 8
Phase factors
W = 0,2π
Channel type
AWGN
Population sizeN
5
Gaussian mutation
5
Number of sparks (m)
50
Parametersa and b
0.04 and 0.8
m
^
and
A
^
5 and 40
Channel bandwidth (MHz)
3.5


## 4.2. PAPR Reduction Performance

In this section, we compare the PAPR performance of the PTS-based FWA with the most popular algorithms used to search the phase factor as like standard particle swarm optimization (SPSO) [21], genetic algorithm (GA), simulated annealing (SA), conventional PTS, and SLM methods in terms of both optimization accuracy for PAPR reduction and convergence speed.The CCDF curves of the simulations results are given in Figure2. The OFDM system was simulated with 64 subcarriers, in which 4 subblocks are employed, and the PTS, SLM, and GA used W = 4 {1 −1 j −j} phase weighting factors to optimize the PAPR of the modulated OFDM symbol, while other algorithms chose randomly the four phases within the interval W = [0, 2π). It can be seen that the proposed scheme (PTS-FWA) yields the best performance while the SLM and GA-PTS yields the worst performance for the PAPR reduction of OFDM systems. For example, for the probability of 10−3, the PAPRs are 4 dB, 4.421 dB, 4.948 dB, 5.226 dB, 5.879 dB, 7.034 dB, and 10.66 dB for the FWA, SPSO, SA, PTS, GA, SLM scheme, and original OFDM signals, respectively.Figure 2
CCDFs comparison with different searching algorithms and conventional PTS.Figure3 shows the simulated results of the fireworks algorithm-assisted PTS technique and the recently improved version of FWA, in comparison against normal OFDM and different PAPR reduction approaches. The PTS scheme based on the EFWA, dynFWA, and AFWA algorithms are given to compare their PAPR reduction performances in OFDM systems. It is clear from the figure that the FWA and all developed version here can all effectively reduce the PAPR in OFDM systems. However, their PAPR reduction performances and convergence speed are different. In general, EFWA and dynFWA show a few improvements over conventional FWA. For CCDF = 10−3, we have 3.942 dB and 3.979 dB for EFWA and dynFWA, respectively, while AFWA has 4.283 dB compared with FWA (4 dB). In addition, the PAPR reduction based-new version of FWA shows a good improvement in terms of computational cost.Figure 3
PAPR reduction comparison-based improved version of FWA.Besides the PAPR reduction performances of the novel swarm intelligence algorithm, convergence speed is quite essential parameter for any optimization algorithm. Figure4 depicts the convergence curves of the proposed schemes in comparison with GA and SPSO in order to validate the convergence speed of the FWA. The simulation is carried out for random OFDM symbol with 10 independent runs and 3000 iterations. From these results, we can conclude that the four methods of FWA have a much faster speed convergence than the SPSO and the GA, while dynFWA and AFWA were a very promising schemes due to their efficiency and simplicity.Figure 4
Convergence curves of the FWA, EFWA, dynFWA, AFWA, SPSO, and GA algorithms-based PTS for one symbol OFDM.Figure5 shows the influence of different numbers of subblocks V on PAPR performance for a QPSK-OFDM system and 104 randomly OFDM symbols using the dynamic search fireworks algorithm, which is characterized by a good simulation runtime without losing optimization accuracy. It is seen in Figure 5 that the PAPR performance in OFDM systems-based dynFWA enhances as the number of subblocks increases with V = 2, 4, 6, and 8. Therefore, the system with the bigger number of subblocks performs the better PAPR.Figure 5
PAPR performance of a QPSK/OFDM system with dynFWA-based PTS technique when the number of subblocks varies.Finally, Figure6 depicts the PAPR reduction performance for WiMAX OFDM systems with AFWA. For randomly generated QPSK symbols, both CCDF graphs of original and optimized signals have been illustrated, and space time block coding (STBC) is used in order to diminish the computational complexity at the receiver side and achieve more diversity gain at the transmit side, where the test starts with two transmit antennas-based Alamouti scheme [23] (2 × 1) and finished by using four transmit antennas (4 × 1) [24, 25]. With a half-rate orthogonal STBC encoding and 104 OFDM symbol, it can be observed that PAPR of MIMO OFDM with AFWA was ≈5.5 dB and ≈5.812 dB at CCDF = 10−3 compared with original PAPR ≈11.25 dB and ≈11.76 dB for WiMAX with two and four transmit antennas, respectively.Figure 6
PAPR reduction performances in MIMO-OFDM-based WiMAX standard.


## 4.3. Complexity Comparison

Table2 shows the computational complexity to find the phase factors and the reduced PAPR values of the various PTS schemes at CCDF = 10−3. These proposed techniques are diversified in terms of reducing the PAPR as well as computational complexity. It must be kept in mind that the complexity of a PAPR reduction system increases linearly with the number of iterations of such an algorithm. There will therefore be a trade-off between reducing the PAPR and the complexity of the system.Table 2
Computational complexity of the different PTS schemes for CCDF = 10−3.
Methods
Number of searches
PAPR (dB)
Original
0
10.66
PTS
W
V = 44 = 256
5.226
PTS-GA
P × G = 10 × 10 = 100
5.879
PTS-SPSO
S × Iter = 100 × 2 = 200
4.421
PTS-FWA
Iter × (N + m + m^) = 2 × (5 + 50 + 5) = 120
4In the original OFDM signal, the PAPR value is calculated before the PTS optimization and therefore its search number is 0. Optimum PTS considers all the phase factors and thus it needsWV = 256 iterations. For GA-PTS, the search complexity is proportional to P × G = 100, where P is the maximum size of the population and G is the generation. In PTS-PSO, the search number is equal to S × Iter = 200, where Iter is the maximum iteration number and S is the size of particle swarm. Finally, in the FWA-PTS, approximate (n + m + m) function evaluations are done in each generation. Suppose the optimum of a function can be found in Iter generations, then we can deduce that the complexity of the FWA is O (Iter × (n + m + m)) = 120.Table3 depicts the optimization accuracy of the proposed algorithms on one symbol OFDM in comparison with SPSO and GA. It can be seen that the proposed schemes clearly outperform both the SPSO and GA. In addition, the fireworks algorithm and its improved versions can find optimal solutions in less than 500 function evaluations.Table 3
Performance evaluation by the FWA, EFWA, dynFWA, AFWA, SPSO, and GA on one symbol OFDM over 10 independent runs.
Methods
Function evaluations
PAPR (dB)
Original
—
10.66
SPSO
2000
4.055
GA
2000
4.447
FWA
500
2.839
EFWA
500
3.015
dynFWA
500
3.021
AFWA
500
2.9


## 4.4. Spectrum Performance

One of the major problems for wireless communications is the RF sidelobe generation [31]. Out-of-band radiations have decisive effects on adjacent channel interference (ACI) performances and reduce the spectrum efficiency, which are of great importance especially for multiuser and RF wireless communications systems. The transmission under ideal conditions leads to no spectral spread, while the transmission of original OFDM with height PAPR via a nonlinear device produces considerable out-of-band power when approaching the saturation zone (6, 7, 8, and 9 dB backoff from saturation point). Our approach consists of finding an alternative representation of the OFDM signal with minimum PAPR based on fireworks algorithm with dynamic search mechanism (dynFWA), and then the output signal will be amplified after digital predistortion (DPD) process. This efficient approach suggested to reduce the PAPR and to compensate the high power amplifiers (HPA), by making a combination between the proposed scheme (PTS-dynFWA) of PAPR reduction and digital predistortion (DPD).Figure7 shows the signal of spectrum performance before and after passing through the nonlinear power amplifier model for conventional OFDM system and OFDM system with the proposed approach (PAPR reduction method PTS-dynFWA and DPD). The results are obtained for two values of the HPA backoff, 7 dB and 1 dB from saturation point.Figure 7
Power Spectral Density comparison performance of an OFDM signal without and with the proposed method. (a, b) 7 dB from saturation point (moderate nonlinearity) and (c, d) 1 dB from saturation point (severe nonlinearity). (a) Without PTS-dynFWA-DPD, (b) with PTS-dynFWA-DPD, (c) without PTS-dynFWA-DPD, and (d) with PTS-dynFWA-DPD.
(a)
(b)
(c)
(d)Figure7(b) shows the frequency spectra with 7 db backoff below the input power that causes amplifier saturation. It is clearly seen that the proposed method (PTS-dynFWA with DPD) has the ability to suppress the out-of-band distortion completely. Compared to the case (a) where no PAPR reduction and no predistortion are used, Figures 7(c) and 7(d) show the spectrum with severe nonlinearity (1 db from saturation point). Comparing the two spectra (Figure 7(c)), we can view that the transmission of original OFDM signal via a HPA device produces considerable out-of-band power. However, the proposed method was able to reduce the spectral regrowth around each of the two bands.


## 5. Conclusion

In this paper, we propose a FWA-based PTS technique to reduce the PAPR of MIMO-OFDM system with low computational complexity. The FWA-PTS is compared with GA-PTS, SA-PTS, SPSO-PTS, SLM, and conventional PTS, in terms of CCDF. The simulation results show that the proposed method has a promising performance in both optimization accuracy of PAPR and convergence speed over conventional schemes and algorithms. Moreover, in order to eliminate the drawbacks of conventional FWA, we have presented in this paper some improved version of FWA, like EFWA, dynFWA, and AFWA. Simulation results showed that the improved versions significantly improve the results in term of PAPR reduction and computation cost over conventional FWA, and clearly outperform SA, GA, and SPSO without loss in optimization accuracy. Besides that, the results analyzed of each method alone come to the conclusion that dynFWA-based PTS and AFWA-based PTS were very successful and have good prospects.



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*Source: 3075890-2018-09-03.xml*