Directed Selected Mapping for Peak-to-Average Power Ratio Reduction in Single-Antenna OFDM

The high peak-to-average power ratio of the transmit signal is a serious drawback of multicarrier transmission. To cope with this problem, peak power reduction schemes have been developed. One popular method is selected mapping (SLM), where several alternative representations are generated from the information carrying signal and that one which exhibits the lowest peak power is selected for transmission. Increasing the number of alternative candidates, i.e., spending higher computational complexity, better peak power reduction can be achieved. In this paper, a novel approach of SLM is presented, which considers a number of consecutive OFDM frames, a hyper frame, jointly. A trade-off between complexity and introduced delay is possible. Choosing an appropriate combination of number of assessed candidates and hyper frame lengths significant gains over conventional SLM can be achieved. I. I NTRODUCTION AND SYSTEM MODEL In wireless communication systems the application of multicarrier modulation is common for digital transmission over frequency-selective channels. In particular, o rth gonal frequency-division multiplexing (OFDM) is a very popular implementation. One serious drawback of OFDM is the high p eak-toaverage power r atio (PAR) of its transmit signal. Due to non-linear amplifiers in the transmitter a high PAR leads to signal clipping which causes transmission errors and, even worse, out-of-band radiation. To overcome these issues it is necessary to control the transmit signal in order to reduce its PAR. For many years, peak-power reduction algorithms for single-antenna systems (also denoted as s ingle-input/singleoutput (SISO) systems) have been an important field of research. Various methods have been developed, e.g., partial transmit sequences (PTS) [1], active constellation extension [2], or tone reservation [3], [4]; for an overview see [5]. One of the most popular techniques for PAR reduction is selected mapping (SLM) [6]. SLM and PTS are both based on multiple signal representation. Recently, the application of OFDM together with multiple transmit antennas (often denoted as MIMO OFDM) has been discussed. In this context an extension of SLM, namelydirected SLM (dSLM) was proposed [7], [8] which is able to utilize the additional degree of freedom offered by multiple transmit antennas. Thereby, significant gains in PAR reduction compared to a straightforward application of SLM to MIMO systems can be achieved. In this paper we apply the ideas of [7], [8] to single-antenna systems. This work was supported by Deutsche Forschungsgemeinschaft (DFG) within the framework TakeOFDM under grant FI 982/1-1. Throughout this paper, we consider a discrete-time OFDM system model. The binary data stream is mapped (mapping M) to (complex-valued) data symbols taken from an arbitraryM -QAM or M -PSK constellation. A frequency-domain OFDM frame is constructed of D data symbols and denoted by the vector A = [Ad]. This frequency-domain OFDM frameA is transformed into a time-domain OFDM framea = [ak] via an inverse discrete Fourier transform (IDFT) of lengthD, defined byak = 1/ √ D ∑D−1 d=0 Ad · e, k = 0, . ,D − 1. Assuming statistical independent frequency-domain symbols Ad, due to their superposition within the IDFT, the elements of the time-domain vector are almost Gaussian distributed. As performance measure for PAR reduction we resort to thecomplementary cumulative distribution function ccdf def =Pr{PAR > PARth}, which gives the probability that the PAR of the current OFDM frame exceeds a certain thresholdPARth. Thereby we consider the PAR of the discrete-time signal, i.e., PAR def = max∀k=0,. ,D−1 |ak| E{|ak|2} . (1) This measure is common in literature and sufficient for a fair comparison of different PAR reduction techniques. We also do not consider the cyclic prefix, pulse shaping, or modulation to radio frequency. Assuming the time-domain samples ak to be Gaussian distributed, the ccdf of the orig inal OFDM scheme is given by [6] ccdforig Gauss = 1− (1− e−PARth)D . (2) In Section II, a brief review of different SLM approaches is given. Section III introduces the new dSLM approach for single-antenna OFDM and in Section IV analytic expressions for the ccdf are derived. In Section V, performance results of the different SLM approaches are compared; Section VI draws some conclusions. II. REVIEW OF STATE-OF-THE-ART SLM A. Original SLM Approach The idea of SLM is to generate multiple, say U , different representations from the information carrying frequencydomain OFDM frameA. TheseU signal representations may be obtained by an element-wise multiplication of A by U different, randomly selected phase vectors P . According to [6] the elements ofP may be chosen from the set {±1,±j}, which gives good results in PAR reduction and does not affect the receiver sided synchronisation. Next, the U different signal representations are transformed into time domain and that one exhibiting the lowest PAR is selected for transmission. Thus, a total computational complexity ofU IDFTs andU metric calculations (1) is necessary. In order to decode the signal correctly, the receiver must be informed which one of the U possible signal representations was transmitted. Assuming both, transmitter and receiver are aware of a codebook of possible phase vectors it is sufficient to transmit the index of the actual used phase vector. The side information is represented by log2(U) bits. In [9] a variant of SLM was proposed, which does not require the transmission of explicit side information. In the top most subfigure of Fig. 1 a delay profile of SLM is depicted. For any computational calculations, e.g., calculation of IDFT or PAR, an infinite processing speed is assumed. The procedure of calculating the best time-domain samples from the original amplitude coefficients is done as follows. First, the frequency-domain OFDM frames (white) are generated by grouping D amplitude coefficients into one OFDM frame. Next, the OFDM frame is passed to the SLM algorithm. For each OFDM frame U alternative signal representations are assessed (illustrated by thin red lines). After the SLM algorithm the best time-domain OFDM frame (gray) is available and can be transmitted symbolby-symbol. Thus, an inherent delay of one OFDM frame is present. Assuming again Gaussian distributed time-domain samples, the ccdf of SLM is given by [6] ccdfSLM Gauss = (1− (1− e−PARth)D)U . (3) B. Ordinary and Directed SLM for Multi-Antenna OFDM For future communication systems the use of multiple transmit antennas is envisaged in order to increase capacity [10]. The combination of OFDM with so called multipleinput/multiple-output systems is often denoted as MIMO OFDM. An application of SLM to MIMO OFDM, calledordinary SLM, was first proposed in [11]. It is just a simple, independent implementation of the original SLM technique to each antenna. Performance is now governed by the worstcase PAR, i.e., the worst PAR over all transmit antennas. Due to this fact, PAR reduction gets a more tough task in MIMO communication and the PAR performance gets worse with an increasing number of transmit antennas. In [7], a novel SLM technique, called irected SLM (dSLM), was proposed for MIMO OFDM. This technique considers the OFDM frames over all transmit antennas jointly. After calculating an initial PAR of all transmit antennas this method only treats that transmit antenna exhibiting the worst PAR. A pseudocode description of this algorithm is given in [7, Fig. 1]. Demanding fixed computational complexity, for each OFDM frame only on averageU candidates are tested; more computational effort is spent for antennas with higher PAR, less for that with already lower PAR. (n on − ov er la pp in g)