Peak to Average Power Reduction Using Amplitude and Sign Adjustment Masoud Sharif, Cedric Florens, Maryam Fazel, and Babak Hassibi Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125 Abstract—In this paper, we propose a method to reduce the peak to mean envelope power ratio (PMEPR) of multicarrier signals by modifying the constellation. For MPSK constellations, we minimize the maximum of the multicarrier signal over the sign and amplitude of each subcarrier. In order to find an efficient solution to the aforementioned non-convex optimization problem, we present a suboptimal solution by first optimizing over the signs using the result of [1], and then optimizing over the amplitudes given the signs. We prove that the minimization of the maximum of a multicarrier signal over the amplitude of each subcarrier can be written as a convex optimization problem with linear matrix inequality constraints. We also generalize the idea to other constellations such as 16QAM. Simulation results show that by an average power increase of 0.21 db and not sending information over the sign of each subcarrier, PMEPR can be decreased by 5.1 db for a system with 128 subcarriers. I. I NTRODUCTION High peak to mean envelope power ratio (PMEPR) of mul- ticarrier signal is one of the major obstacles in implementing OFDM, xDSL, and other broadband multicarrier systems. The occurrence of the large peaks in the signal seriously hampers the efficiency of the power amplifier. Over the years, different schemes have been proposed for PMEPR reduction such as coding, deliberate clipping, selective mapping (SLM), reserved carriers, and tone injection [2], [3], [4], [5], [6], [7]. In all these schemes, there is always a trade off between PMEPR and other parameters in the systems, including coding rate, average power, signal distortion, and bandwidth. Methods like coding usually give a worst case guarantee on the PMEPR, on the other hand, there are other methods such as SLM that improve the probability distribution of PMEPR, i.e. reduce the probability of encountering large PMEPR. Recently, in [1], an algorithm has been proposed to choose the sign of each subcarrier in order to reduce the PMEPR. In this paper, we further generalize this idea and adjust the sign and amplitude of each subcarrier. The price to adjust the amplitude of the subcarrier is a slight increase in the average power. Even though the optimization over the signs is not a convex optimization problem, we show that the amplitude This work was supported in part by the National Science Foundation under grant no. CCR-0133818, by the office of Naval Research under grant no. N00014-02-1-0578, and by Caltech’s Lee Center for Advanced Networking. optimization can be written as a convex optimization problem using the Bounded Real Lemma [8]. This enables us to effi- ciently solve the problem and add more practical constraints to the problem like limiting the amplitude of each subcarrier in order to bound the peak to average in frequency domain. Our approach can be considered as a method to refine the constellation for PMEPR reductions. Other methods to shape the constellation have appeared in [9] and [7] to reduce the maximum of the samples of the multicarrier signal. In [7], extending the number of constellation points is proposed, however, in [9] outer points in the constellation are allowed to move within margin-preserving constraints. In this paper, we consider a different constellation modification and we further show that reducing the peak of the continuous multicarrier signal by optimizing the amplitude of the subcarriers is a convex optimization problem. In our approach we first reduce the peak by optimizing over the signs of the multicarrier signal which is not a convex problem. Simulation results show that the PMEPR can be signifi- cantly reduced by using just 0.21 db (i.e. 5%) average power increase. More specifically, for a system with 128 subcarriers, and considering the peaks with probability less than 10 -2 as negligible, PMEPR is reduced from 10.3 to 3.1, i.e. 5.1 db PMEPR improvement. The paper is organized as follows: Section 2 introduces our notations and the statement of the problem, and furthermore reviews the sign optimization algorithm. Section 3 deals with amplitude optimization and proves that it is a convex problem using bounded real lemma. Simulations results are presented in Section 4 and Section 5 concludes the paper. II. DEFINITIONS AND PROBLEM STATEMENT In this paper, we consider a normalized multicarrier signal s C (θ) that consists of n subcarriers. More specifically, s C (θ)= n  i=1 c i e jθi , (1) where C =(c 1 ,...,c n ) is the modulating vector, c i ’s are chosen from some constellations like MPSK or 16QAM, and θ denotes time. Clearly, if c i ’s are chosen from BPSK constellation and they add up coherently, s C (θ) will have a large peak of order n. Therefore it is of great practical interest 0-7803-8533-0/04/$20.00 (c) 2004 IEEE IEEE Communications Society 837