PAPR reduction of OFDM signals using PTS: a real-valued genetic approach
© Lain et al; licensee Springer. 2011
Received: 16 May 2011
Accepted: 11 October 2011
Published: 11 October 2011
The partial transmit sequences (PTS) scheme achieves an excellent peak-to-average power ratio (PAPR) reduction performance of orthogonal frequency division multiplexing (OFDM) signals at the cost of exhaustively searching all possible rotation phase combinations, resulting in high computational complexity. Several researchers have proposed using binary-coded genetic algorithms (BGA) PTS to reduce both the PAPR and computational load. To improve the PAPR statistics of OFDM signals further while still reducing the computational complexity, this paper proposes a new PTS using the real-valued genetic algorithm (RVGA). By defining a cost function based on the amount of PAPR, PTS can be formulated as an optimization problem over a multidimensional real space and solved by implementing the RVGA method. The simulation results show that the performance of the proposed RVGA PTS, along with an extinction and immigration strategy, provides approximately the same PAPR statistic as the exhaustive PTS scheme, while maintaining a low computational load.
Orthogonal frequency division multiplexing (OFDM) is an attractive technique for achieving high-bit-rate wireless communication  and has been applied extensively to digital transmission, such as in wireless local area networks and digital video and audio broadcasting systems. Moreover, OFDM has been regarded as a promising transmission technique for next generation wireless mobile communication. However, due to its multicarrier nature, one of the major drawbacks in OFDM systems is the high PAPR, causing high out-of-band radiation when OFDM signals are passed through a radio frequency power amplifier. A number of approaches have been proposed to solve the PAPR problem in OFDM . Among these methods, the PTS is one of the most attractive schemes because of high-quality PAPR reduction performance with no restrictions to the number of subcarriers . In the PTS scheme, the input symbols are partitioned into several disjoint subblocks. Inverse fast Fourier transform (IFFT) is applied to each disjoint subblock, and each corresponding time-domain signal is multiplied by a rotation phase. The objective of the PTS scheme is to select the rotation phases such that the PAPR of the combined time-domain signal is minimized. Increasing exponentially with the number of subblocks and the number of the rotation phases that can be chosen, the searching complexity to find the optimal phases becomes intractable and impractical.
To reduce the computational complexity for searching rotation phases in PTS, various suboptimal methods that achieve significant reduction in complexity were presented in [4–11]. Owing to an intensive improvement of circuit design for genetic algorithms (GAs) in recent years [12, 13], PTS based on GAs not only has moderate PAPR reduction performance but also shows potential for practical implementation among these methods. The GA has proved to be a robust, domain-independent mechanism for numeric and symbolic optimization. With the trend of GA hardware becoming more popular and low-priced, the PTS based on GA may provide a practical and economical approach toward solving the difficulty of high PAPR in OFDM systems. Previous studies have demonstrated that the BGA PTS achieves a moderate PAPR reduction in discrete domains [7–9]. However, rotation phases involved in this phase-searching problem are real-valued radians. This prompts consideration of a novel implementation of PTS to reduce the PAPR based on a real-valued genetic algorithm (RVGA) method. In the proposed RVGA method, a cost function related to the amount of PAPR is first defined. The cost function is then translated into a real-valued parameter optimization problem, which can be solved effectively by the RVGA. The simulation results show that the performance of the proposed RVGA PTS along with an extinction and immigration strategy provides a PAPR statistic approaching that of the exhaustive PTS while maintaining a low computational load.
The rest of this paper is organized as follows. Section 2 presents a description of the OFDM system and formulates the PTS PAPR reduction problem as a combinatorial optimization problem over a multidimensional real space. Section 3 describes how to solve this problem using the RVGA method along with an extinction and immigration strategy. Section 4 describes the simulative results and discussion. Finally, conclusions are drawn in Section 5.
2 System model and problem formulation
2.1 OFDM systems and PAPR definition
where E[.] denotes expectation operation.
2.2 Formulation of OFDM with PTS
where Φ = [ϕ1ϕ2 ⋯ ϕ M ]. Here, the objective of the PTS scheme is to design a rotation phase vector Φ that minimizes the PAPR. PAPR reduction with the PTS technique is related to the problem of minimizing max|x' (Φ)| subject to 0 ≤ ϕ m ≤ 2π, m = 1, 2,..., M, and however, it is equivalent to an exhaustive search for a combinatorial optimization problem, which requires an enormous amount of computations to search all over possible candidate rotation phase vectors.
3 The real-valued genetic algorithm PTS
3.1 RVGA PTS
By translating the phase-searching problem of the PTS into a real-valued parameter optimization, this study proposes using the RVGA to find a rotation phase vector to reduce PAPR. This study associates every rotation phase vector using a chromosome to apply the RVGA to the PTS PAPR reduction problem. The following delineates the steps involved in the RVGA PTS.
where ϕ hi , ϕ lo , and u are the highest value in the variable range, the lowest value in the variable range, and a uniformly distributed random variable in [0,1]. In the PTS scheme, the values of ϕ lo and ϕ hi are set at 0 and 2π, respectively. Given an initial population of P chromosomes, the full matrix of P × M random rotation phases is generated.
Step 1--Evaluation and Selection: In each generation, the cost values are computed for each of the P chromosomes by substituting the corresponding rotation phase vector Φ into the cost function of max|x'(Φ)|. Thereafter, the T chromosomes with the lowest cost values are chosen for a mating pool, from which two chromosomes are selected according to a roulette wheel selection for the next crossover step .
Step 2--Crossover: Crossover is a recombination operation that combines subparts of two parent chromosomes to exchange the genetic material between chromosomes. A crossover probability p c controls the degree of crossover. A 1 × M sequence, often referred to as a crossover mask, is constructed, consisting of 1s generated with crossover probability p c and 0s generated with probability (1- p c ). When the elements in the crossover mask are 1s, the genes of the two parent chromosomes in the corresponding positions will be mixed with each other, where if they are 0s, the corresponding genes will be unchanged. Suppose Φ1 and Φ2 are two parents selected, the i th element in the crossover mask is 1, and ϕ1,iand ϕ2,iare the i th genes in Φ1 and Φ2, respectively. The i th genes in the next generation of Φ1 and Φ2 are r ϕ1,i+ (1 - r)ϕ2,iand r ϕ2,i+ (1 - r)ϕ1,i, respectively, where r is a uniformly distributed random variable in [0,1]. This crossover operation will repeat until the number of the new population size reaches P.
Step 3--Mutation: To explore more regions within the solution space, mutation should be adopted in the RVGA method . This study constructs a 1 × P mutation mask sequence, consisting of 1s generated with the mutation probability p m and 0s generated with probability (1 - p m ), for all chromosomes in each generation. When the elements in the mutation mask are 1s, the genes of the chromosome in the corresponding positions will change. However, if they are 0s, the corresponding genes will remain unchanged. Supposing the i th element ϕ i in rotation phase vector Φ is selected for mutation, (7) can easily be used to regenerate ϕ i .
Step 4--Elitism: According to the costs evaluated by max|x'(Φ)|, this study places the T chromosomes with the lowest costs into the mating pool. This ensures that each generation retains better chromosomes.
Step 5--Repeat/End: Repeat steps 1-4 until the number of generations is G. Finally, the chromosome with the lowest cost is selected to be the rotation phase factor in the PTS scheme.
3.2 Modified RVGA PTS
3.3 Modified RVGA PTS with extinction and immigration
Conventionally, the GA suffers from close breeding. As the number of chromosomes in the mating pool associated with smaller costs grows exponentially, after some generations, the T parent chromosomes chosen to mate are eventually almost identical. If two parents are identical, their children will also be identical and no new information will be disseminated. This study adopts the strategy of Extinction and Immigration (EI) to react against the aforementioned problems . By operations of extinction and immigration, the strategy of EI functions like a particular time varying mutation probability in which p m is close to 1 at the beginning of each new era and then gets smaller for the remaining generations.
Extinction eliminates all of the chromosomes in the current generation except for the chromosome corresponding to the minimum cost. Immigration randomly generates (P - 1) chromosomes to propagate the population (a mass immigration). (T - 1) chromosomes associated with the least costs among these immigrants are then selected as the parents. Together with the surviving chromosome, these are allowed to mate as usual to form the next generation. Generally, there are two cases when extinction and immigration will occur. One is the case when all of the T parents are the same, and the other is the case when no further decrease in the cost values has been reached. This study adopts the second case to determine when to execute the strategy of extinction and immigration.
4 Numerical results
Summaries of simulation parameters
Subcarriers number (K)
Subblock number (M)
Number of phases (W)
Oversampling factor (f s )
Population size (P)
Crossover probability (p c )
Mutation probability (p m )
Comparison between rotation phase-searching schemes
The number of enumerations
28, (M - 1) × W
4,480, C r M-1W r I
4,000, P × G
4,000, P × G
This paper presents an RVGA method that was used to obtain the rotation phase vector for the PTS technique to reduce the PAPR of OFDM signals. Simulations were conducted and show that the performance of the proposed MRVGA-EI PTS provided almost the same PAPR statistics as that of the optimal exhaustive PTS, while maintaining a low computational load. With the trend that GA hardware is becoming more popular and low-priced, the proposed MRVGA-EI PTS provides a practical and economical approach toward solving the difficulty of high PAPR in OFDM systems.
This work was supported by National Science Council of Taiwan under Contract NSC98-2221-E-224-019-MY3.
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