# ORTHOGONAL frequency division multiplexing (OFDM)

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2 336 IEEE TRANSACTIONS ON BROADCASTING, VOL. 50, NO. 3, SEPTEMBER 2004 Fig. 1. Block diagram of OFDM transmitter with the SLM technique. II. PAPR OF AN OFDM SIGNAL Let us denote the data block of length as a vector where is equal to the number of subcarriers and denotes transpose. The duration of a data symbol in is. Each data symbol in modulates one of a set of subcarriers,. The subcarriers are chosen to be orthogonal, that is,, where and is the duration of an OFDM data block. The complex envelope of the transmitted OFDM signal The PAPR of the transmitted signal in (1) is defined as (1) (2) III. MODIFIED SELECTED MAPPING TECHNIQUE FOR CODED OFDM SIGNAL A. Review of SLM Technique To begin with, we briefly review the ordinary SLM technique [9]. Block diagram of the SLM technique is shown in Fig. 1. At first, input data is partitioned into a data block of length. Then the OFDM data block is multiplied element by element with phase sequences,, to make the phase rotated OFDM data blocks where,. All phase rotated OFDM data blocks represent the same information as the unmodified OFDM data block provided that the phase sequence is known. To include unmodified OFDM data block in the set of the phase rotated OFDM data blocks, we may set the first phase sequence as all one vector of length. After applying the SLM technique to, (1) becomes In the remaining part of the paper, an approximation will be made in that only equidistant samples of (1) will be considered where is an integer which is larger than or equal to 1. This -times oversampled time-domain signal samples are represented as a vector and obtained as It can be seen that the sequence can be interpreted as the inverse discrete Fourier transform (IDFT) of the OFDM data block with zero padding. It is well known that PAPR of the continuous-time OFDM signal cannot be obtained precisely by the use of the Nyquist rate sampling, which corresponds to the case of. It is shown in [14] that can provide sufficiently accurate PAPR results. The PAPR computed from the -times oversampled time-domain signal samples where denotes expectation. (3) (4) (5) PAPR is calculated for phase rotated OFDM data blocks by using (3) and (4). Among the phase rotated OFDM data blocks, one with the lowest PAPR is selected and transmitted. The information about the selected phase sequence should be transmitted to the receiver as side information. At the receiver, reverse operation should be performed to recover the unmodified OFDM data block. SLM technique needs IDFT operations for each OFDM data block and the number of required side information bits is where denotes the smallest integer which does not exceed. The phase sequences are selected in a way such that the phase rotated OFDM data blocks are sufficiently different. In the ordinary SLM technique, there is no restriction on the construction of the phase sequences. However, we set a structural limitation on the phase sequences for the modified SLM technique described below. B. Channel Coding for OFDM Various channel coding techniques, such as block code, convolutional code, or Turbo code can be used for OFDM. Here, we consider a simple block code. A linear block code is a nonempty set of -tuples over, called codeword, such

3 IEEE TRANSACTIONS ON BROADCASTING, VOL. 50, NO. 3, SEPTEMBER Fig. 2. Block diagram of the modified SLM technique. (a) Transmitter. (b) Receiver. that the sum of two codewords is a codeword, and the product of any codeword with a field element is a codeword [12], [13]. Let be the dimension of. The code is referred to as an code where is the block length of the code and is the dimension of code. When we are dealing with the linear block code, it is convenient to use systematic encoder. A systematic encoder for an block code is one that maps each dataword (consists of data symbols) into a codeword with the data symbols unmodified in the first symbols of the codeword. The remaining symbols are called as the check symbols. We may use a systematic convolutional code, such as rate 1/2 systematic convolutional code in which half of the symbols are data symbols and the rest half of the symbols are check symbols. It should be noted that channel coding is used for error correction, not for PAPR reduction in this paper. When the code rate is, data rate is reduced by a factor of due to the channel coding. C. Modified SLM Technique for Coded OFDM Signal Fig. 2 shows the block diagram of the modified SLM technique. At first, input data is mapped into -ary symbols and then processed by a rate code over. After channel code encoding, data symbols and check symbols are separately mapped to -ary symbols and are grouped into blocks of length. The number of codewords required to make an OFDM data block of length After collecting codewords, the data symbols and check symbols are arranged appropriately among subcarriers. subcarriers are used to transmit data symbols and the rest subcarriers are used for check symbols. Fig. 3 shows an example of such subcarrier arrangement when,,, and. In this example, 2 codewords of length 5 make an OFDM data block (6) Fig. 3. Example of subcarrier arrangement for N =10 with (5; 3) block code. of length 10. The first 3 subcarriers are used to transmit the first dataword. The next 2 subcarriers are used to transmit the check symbols of the first codeword. The second codeword is mapped to the rest 5 subcarriers in a similar way. The specific carrier arrangement may vary according to system requirements. Note again that the check symbols are transmitted using separate subcarriers. After making an OFDM data block, we need to construct phase rotated OFDM data blocks for the SLM technique. Special attention should be paid to the design of the phase sequences used in the proposed technique. In the ordinary SLM technique, there is no restriction on the design of phase sequences as long as they are sufficiently different. In the proposed technique, however, we set the element of phase sequences, which correspond to the positions of data symbols, as one. In other words, the phase factors have a limitation on the positions of subcarriers in which the phase factors can have arbitrary phase. Hence, all phase rotated OFDM blocks have same values in the positions of data symbols. In Fig. 3, these positions are 0, 1, 2, 5, 6, and 7. Other positions of the phase sequences may have arbitrary phase. An example of the set of phase sequences for,, with block code when is shown in Fig. 4. As in the ordinary SLM technique,

5 IEEE TRANSACTIONS ON BROADCASTING, VOL. 50, NO. 3, SEPTEMBER By extensive simulations, we have found that is a good approximation for 4-times oversampled OFDM signals. Next, we derive an approximate expression for the PAPR of the SLM technique. If the phase rotated OFDM data blocks from different branches in the SLM technique is independent and uncorrelated, the complementary CDF (CCDF) of the PAPR of an OFDM signal after applying SLM technique with branches is derived as (13) where denotes the PAPR of the SLM technique with branches and denotes the PAPR of the th branch in the SLM technique. Equation (13) can be used for SLM technique with no structural limitation on the phase sequences. In the modified SLM technique, however, we lost some of the randomness in the SLM technique from the restriction on the structure of the phase sequences. We may represent the PAPR of the modified SLM technique for coded OFDM signal as (14) where denotes the PAPR of the SLM technique with branches for coded OFDM signal with code rate and (between 0 and 1) represents the amount of loss in randomness. The CCDF of the modified SLM technique can be approximated by (14) with properly chosen. IV. SIMULATION RESULTS AND DISCUSSIONS We use computer simulations to evaluate the performance of the proposed PAPR reduction technique. To approximate the effect of nonlinear power amplifier in the transmitter, we adopt Rapp s model for amplitude conversion [15]. The relation between amplitude of the normalized input signal and amplitude of the normalized output signal of the nonlinear power amplifier (15) where is a parameter that represent the nonlinear characteristic of the power amplifier. The power amplifier approaches linear amplifier as gets larger. We choose which is a good approximation of a general power amplifier [15]. The phase conversion of the power amplifier is neglected in this paper. Fig. 5 shows the input-output relation curve of the Rapp s power amplifier model when. The input signal is normalized by a normalization factor to appropriately fit the input signal into the desired range in the input-output relation curve. The normalized output signal is processed back into original scale before Fig. 5. p =3. Input output relation curve of the Rapp s power amplifier model when normalization. The amount of nonlinear distortion depends on the output back-off (OBO) which is defined as (16) where is the output power at the saturation point and the average power of the output signal. As performance measures, we use the CCDF of the PAPR and symbol error rate (SER) in a Rayleigh fading channel. Modulation is QPSK for all subcarriers. Assume that each subcarrier experience identically independently distributed Rayleigh fading with additive white Gaussian noise. We set the oversampling factor. Asa block coding scheme, we adopt a Reed-Solomon (RS) code over that has minimum distance of 9 and can correct up to 4 error in each codeword. We limit the value of the elements of the phase sequences in. Fig. 6 shows the PAPR of the proposed technique (Prop.) and ordinary SLM technique (Ord.). Fig. 6(a) shows the PAPR for with, 16, 32. It is shown in Fig. 6(a) that the unmodified OFDM signal (designated as Unmodified in the legend) has a PAPR which exceeds 10.9 db for less than 0.1% of the OFDM data blocks for. In this case, we say that the 0.1% PAPR of the unmodified OFDM signal is 10.9 db. When, the 0.1% PAPR of the ordinary SLM technique and that of the proposed technique is 8.10 db and 7.45 db, respectively. When, the 0.1% PAPR of the ordinary SLM technique and that of the proposed technique is 7.90 db and 7.35 db, respectively. When, the 0.1% PAPR of the ordinary SLM technique and that of the proposed technique is 7.60 db and 7.10 db, respectively. The performance difference between the proposed technique and the ordinary SLM technique is due to the fact that, in the proposed technique, the phase sequences have a limitation on their structure and thus the improvement of PAPR provided by the proposed technique is not so much as the that from the ordinary SLM technique. In fact, the difference in PAPR improvement between the proposed technique and the ordinary SLM technique is closely related to the value of, or the code rate. If code rate is close to 0, there is little difference between the CCDF of the proposed technique

6 340 IEEE TRANSACTIONS ON BROADCASTING, VOL. 50, NO. 3, SEPTEMBER 2004 Fig. 7. PAPR of the modified SLM technique (simulation). (a) CCDFs of PAPR for N = 120. (b) CCDFs of PAPR for N = 240. Fig. 6. SER of the modified SLM technique for N = 240. (a) SER for OBO = 3 db. (b) SER for OBO = 6 db. (c) SER for OBO = 10 db. and that of the ordinary SLM technique. On the contrary, the higher the code rate is, the larger the difference is. But the difference remains quite small when compared with the difference between the proposed technique and unmodified OFDM signal if the code rate is lower than about a half. Fig. 6(b) shows the PAPR for with, 16, 32. The trends are the similar for. Fig. 7 plots the approximate expression for the CCDF of the ordinary SLM technique and that for the modified SLM technique. The approximate expression for the CCDF of the unmodified OFDM signal is also plotted in Fig. 7 for comparison. Fig. 7(a) shows the CCDF of PAPR for and Fig. 7(b) shows that for. The CCDF of the unmodified OFDM signal is plotted by using (12) with. The CCDF of the ordinary SLM technique is plotted by using (13) with and that of the modified SLM technique is plotted by using (14) with and, 0.75, 0.70 for, 16, 32, respectively. When we compare the CCDF plot of unmodified OFDM signal in Fig. 7 with the simulation results in Fig. 6, we can see that the approximate expression is quite accurate for when the oversampling factor is 4. We can also see that the approximate expression closely matches the simulation results for both the ordinary SLM technique and modified SLM technique with properly selected parameters and. The approximate expression can therefore be used to predict the PAPR statistic of the OFDM signal after applying SLM technique without simulations. Fig. 8 shows the SER of the proposed technique for. SER of uncoded OFDM signal is also shown in Fig. 8 for comparison. In the legend, Prop. refers to the proposed technique

7 IEEE TRANSACTIONS ON BROADCASTING, VOL. 50, NO. 3, SEPTEMBER modified PTS technique is proposed for coded OFDM signal. Some simulation results are provided to validate the technique; but analysis is not given for the PAPR statistic of an OFDM signal after applying the technique. It might be interesting to compare the proposed technique and the technique in [16] under a same framework. V. CONCLUSIONS In this paper, we proposed a modified SLM technique for the PAPR reduction of coded OFDM signal. By appropriately embedding the phase sequence information on the check symbols of the coded OFDM data block, we can achieve both PAPR reduction from the SLM technique and error performance improvement from the channel coding with no loss in data rate. We also derived approximate expression for the distribution of PAPR of modified SLM technique. It is shown that the approximate expression matches quite well with the simulation results with properly chosen parameters. REFERENCES Fig. 8. PAPR of the modified SLM technique (analysis). (a) CCDFs of PAPR for N =120. (b) CCDFs of PAPR for N =240. and Perf. refers to the case when the phase sequence estimation at the receiver is perfect. Fig. 8(a) (c) show the SER for,,, respectively. At first, we can see that SER performance is better for larger OBO values since the amount of nonlinear distortion is less for larger OBO. When the phase sequence estimation is perfect, the SER of is slightly better than that for since PAPR statistic is better for. On the contrary, it is shown that SER of the proposed technique is slightly better for than for. This is due to that the minimum Euclidean distance among the set of phase sequences is smaller for, resulting in a larger error in the estimation of the phase sequence at the receiver. It is also shown that when the phase sequence is estimated from the received OFDM data block in the proposed technique, there is some performance degradation when compared with the perfect estimation case. However, the performance degradation is not severe when compared with SER of the uncoded case. It can be concluded that most of the channel coding gain can be achieved by using the proposed technique. During the final revision of this manuscript, we have found that a similar study was done independently in [16]. In [16], a [1] L. J. Cimini Jr, Analysis and simulation of a digital mobile channel using orthogonal frequency division multplexing, IEEE Trans. Commun., vol. 33, pp , July [2] A. E. Jones, T. A. Wilkinson, and S. K. Barton, Block coding scheme for reduction of peak to mean envelope power ratio of multicarrier transmission scheme, Electron. Lett., vol. 30, no. 25, pp , Dec [3] M. J. E. Golay, Complementary series, IEEE Trans. Inform. Theory, vol. 7, pp , Apr [4] S. Boyd, Multitone signals with low crest factor, IEEE Trans. Circuits Syst., vol. 33, pp , Oct [5] B. M. Popovic, Synthesis of power efficient multitone signals with flat amplitude spectrum, IEEE Trans. Commun., vol. 39, pp , July [6] J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences and reed-muller codes, IEEE Trans. Inform. Theory, vol. 45, pp , Nov [7] V. Tarokh and H. Jafarkhani, On the computation and reduction of the peak-to-average power ratio in multicarrier communications, IEEE Trans. Commun., vol. 48, pp , Jan [8] S. H. Muller and J. B. Huber, OFDM with reduce peak-to-average power ratio by optimum combination of partial transmit sequences, Electron. Lett., vol. 33, no. 5, pp , Feb [9] R. W. Bauml, R. F. H. Fisher, and J. B. Huber, Reducing the peak-toaverage power ratio of multicarrier modulation by selected mapping, Electron. Lett., vol. 32, no. 22, pp , Oct [10] A. D. S. Jayalath and C. Tellambura, Reducing the peak-to-average power ratio of orthogonal frequency division multiplexing signal through bit or symbol interleaving, Electron. Lett., vol. 36, no. 13, pp , June [11] E. Lawrey and C. J. Kikkert, Peak to average power ratio reduction of OFDM signals using peak reduction carriers, in Proc. ISSPA 99, Brisbane, Australia, Aug. 1999, pp [12] R. E. Blahut, Algebraic Codes for Data Transmission. Cambridge, U.K.: Cambridge University Press, [13] S. B. Wicker, Error Control Systems for Digital Communication and Storage. Englewood Cliffs, NJ: Prentice Hall, [14] C. Tellambura, Computation of the continuous-time PAR of an OFDM signal with BPSK subcarriers, IEEE Commun. Lett., vol. 5, pp , May [15] R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications. Boston, MA: Artech House, [16] O. Muta and Y. Akaiwa, A peak power reduction scheme with phase-control of clustered parity-carriers for a systematic block-coded OFDM signal, in Proc. IEEE VTC 2003-Fall, Orlando, FL, Oct. 2003, pp

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