Joint Channel Estimation and Peak-to-Average Power Reduction in Coherent OFDM: A Novel Approach M. Julia Fernhndez-Getino Garcia, Ove Edforst, JosC M. Piez-Borrallo Univ. Polittcnica de Madrid, Dpto. de SeAales, Sist. y Radiocom., 28040 Madrid, Spain tLund University, Dept. of Applied Electronics, SE-221 00 Lund, Sweden E-mail:
[email protected]
Abstract
Several alternative solutions have been proposed in the literature. One of these approaches, and the simplest, is to deliberately clip the OFDM signal before amplification. However, clipping is a nonlinear process and may cause even greater distortion [2], [3], [4]. The second approach, with two subcategories, are distortionless methods based on the insertion of a treatment block. In this case, the reduction of the PAR can be attained by way of introducing little redundancy (as Selective Scrambling [SI, SLM [6], PTS [7] [SI) or by a joint block coding and modulation scheme with Golay and Reed-Muller codes (e.g., [9], [lo], [ll]). However, it should be taken into account that coherent wireless OFDM systems commonly use 2D-PSAM [ 121, with pilot symbols spreaded out throughout the timefrequency grid and primarily provided for channel estimation, but also feasible for synchronization [13]. In this paper, we propose a novel approach for PAR reduction consisting of finding the more adequate values for pilot symbols to build peak power optimized sequences. In Section 2, the discrete-time signal model is presented. Section 3 introduces the pilots-aided power reduction scheme, and in section 4 several suboptimal approaches are proposed. Finally, two methods are described in section 5, providing a joint analysis for channel estimation and PAR reduction. In section 6 simulation results are presented. and section 7 discusses and concludes the paper.
In this papel; a new peak-to-average power reduction approachfor OFDM has been addressed. Two-Dimensional Pilot-Symbol Assisted Modulation (2D-PSAM) is employed in coherent OFDM for channel estimation and it is based on inserting known symbols spreaded out throughout the 2 0 time-frequency grid. We show that these scattered symbols can also be employed to perform distortionless peak power reduction, with a negligible loss in data-rate. To reduce additional system complexity and side information, several suboptimal strategies are proposed. A comparison between two schemes with different approaches to the joint design of pilot symbols for both channel estimation and peak power reduction, has also been carried out'.
1. Introduction Orthogonal Frequency Division Multiplexing (OFDM) is an attractive technique for mitigating the effects of multipath channels, but a major obstacle shows up since the multiplex signal has a large Peak-to-Average power Ratio (PAR), exhibiting occasionally very high peaks. When passed through a nonlinear device, such as a High Power Amplifier (HPA), the signal may suffer significant spectral spreading, in-band distortion and more critical undesired out-of-band radiation. The distortion introduced by the HPA is worsened in the OFDM systems by the relative long-tailed distribution (Rayleigh) of the signal amplitude [l], [2]. The conventional solutions to this problem are to use a linear amplifier or to backoff the operating point of a nonlinear amplifier. This leads to power inefficiency and expensive transmitters so that it is highly desirable to reduce the PAR. With the increased interest in OFDM, overcoming this problem is a very active area of research.
2. OFDM signal model OFDM consists of multiplexing in frequency N separate subcarriers with a uniform frequency spacing, and this can be efficiently implemented by using and IDFTDFT operation. Then, the signal in time-domain for lth OFDMsymbol, q[n] can be written as:
Work supported by Projects 07T/0032/2000& TIC2000-1395-C02-02
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where Sl( k ) are the complex symbols in frequency-domain at kth subcarrier frequency. In coherent OFDM systems, it is usual the insertion of pilot symbols in the twodimensional time-frequency grid to estimate the channel. Let’s consider that Ith OFDM-symbol consists of N subcarriers, of which N p are pilot symbols and we will denote as T the set of sub-carriers indexes where a pilot is transmitted, which will be usually uniformly spreaded, and then,
be represented as Is1[n]lejarg{sf[nl), where the argument of
sl[n]is uniformly distributed between 0 and 27r and the magnitude of sl[n]is Rayleigh distributed. (It should be noted that the maximal amplitude of the transmitted signal is N , for an N-subcarrier system. Even though the distribution of the output is close to Gaussian, it is in fact amplitude limited). The discrete-time PAR associated with the Ith OFDM symbol is defined as,
where Pl ( k ) is the pilot-symbol transmitted in the kth subcarrier and Xl ( k ) is the complex data symbol, belonging to a certain constellation. The transmitted discrete-time symbol can be then separated in two parts, one containing the ( N - N p ) data symbols, and the other containing the N p pilot-symbols, and it can be modelled as in Eq. (3), where pl [n]refers to the pilots part and ,q[n], contains the datasymbols. Sl
I.[
= PlI.[
+ a I.[
where E { . } denotes expected value. The well-known Crest Factor-CF, denoted with is preferred by some authors to the PAR; both are simply related since x is the square of
e,
3. Pilots-Aided PAR Reduction This novel approach consists of finding the more adequate values for pilot symbols, primarily provided for channel estimation, to build peak power optimized sequences. The optimal sequence, 51, fulfils kl < xl,and will be found by means of a careful selection of the pilots,
(3)
These expresssions can be written more compactly by using matrix notation. For vectors, it will assumed bold letters for the time-domain representation and capital bold letters for the frequency-domain; matrices are also denoted with capital bold letters. The‘frequency-domain N x 1vector Sl = [ S ~ ( O..., ) , s ~ ( N- 1)ITcomprises all carrier amplitudes associated with Ith OFDM symbol interval and it is transformed into time domain, using an N point IDFT, resulting in a discrete-time representation of the transmit signal, given by s1 = [sl[O], ...,sl[N - 1]]*,where sz = I D F T { S l } denotes this transform relationship. Then, the signal in time-domain can be written as S I = pl zl. If we represent the IDFT operation with a diagonal N x Nmatrix denoted with W, we can write,
Sl
=w(xl+ P 1 ) =
WXl +WPl
+ WPl
(8)
(9)
(4)
Given the definition of x, the criterium to determine f‘l will be the minimization of the maximum of the sequence sl [n], so the cost function is (max{sl[n]}). An optimal approach can be given, e.g., with a nonlinear optimization algorithm, as the Nelder-Mead simplex (direct search) method [ 141; however, an optimal scheme burdens the system with a high computational complexity and, also, requires the transmission to the receiver of a lot of side information about pilot values to perform channel estimation. Thus, to reduce complexity and keep side information to a minimum, several suboptimal schemes has been proposed.
where N x 1-vectors Xl and Pl are defined as follows,
Assuming that the sent complex symbols 5’1 ( k ) are independent and identically distributed (i.i.d.), if N is sufficiently large ( N >_ 32, is claimed to be sufficient), the distribution of the output signal sl [n]are near Gaussian distributed random variables, due to the Central Limit Theorem. Then, the probability density function (pdf) of sl[n]can be approximated by the normal distribution. The complex sl[n] can
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= WSl = WXl
The aim is finding the optimum pilot values, Fl, so that they mininize the PAR, PI + min,g,, x l . They will be dependent on the data, so 1’1 = A(Xl), where A ( . )denotes a generalized function. Since we want to maintain constant power at any subcarrier, we will impose the constraint of optimizing only the phases of the pilots values, as shown in Eq. (9).
+
s1= WSl
e.
4. Suboptimal approaches Instead of using the optimal values for the pilots, we propose different approaches consisting of limiting to a small
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number, the possible values of the pilots. The idea is employing a finite set, s,composed of M possible pilot vectors, which can be obtained from two different ways, as described in the following subsections. Then, the number of bits necessary for the transmission of M possible pilotvectors is B = log, M . A choice for the transmission of this side information can be the reservation of the first B sub-carriers, each devoted per bit, employing an FSK modulation. A simple and secure strategy consists of sending a tone to transmit a ’1’ and an empty sub-carrier for the ’0’; thus, detecting the energy at the subcarrier is enough to receive the appropriate transmitted bit.
4.1. Gravity centers of optimal approaches
Figure 1. Suboptimal approximation with an 8-PSK constellation, whose complex-valued positions are like gravity centers for the optimal values.
Considering N p pilot symbols per OFDM-symbol, a first and intuitive approach will be finding the optimal values with an optimization algorithm and then reducing them to discrete values; then, the optimal phases are translated to the closest phase belonging to an m-PSK constellation, whose symbols can be named ’gravity centers’. If m is high, the pilot values are close to the original optimal values and the degradation is negligible. However, if m decreases, the accuracy of the values employed is worse compared to the optimal ones, and the PAR reduction obtained is smaller. In this case, the number of possible pilot-vectors, M is given by the combinations of the m possible symbols, M = m N P , and therefore
value at k E
s = {Pl,P2,.. . , P P , . . . , P M }
(1 1)
It has been found that there is always a pilot-vector in the set which drastically reduces the PAR. Also, as far as a small but reasonable value of M is considered, increasing this value means no great improvement in the PAR reduction; a slight refinement in the PAR reduction requires an excesive increase in the side information to be transmitted and therefore, M should be kept as low as possible but still an excellent PAR reduction will be obtained. For example, M = 4 drives to B = 2, so side information is severely decreased and this proposal seems to be an attractive choice to reduce PAR.
A constellation close to optimal values would be 8-PSK (m = S), shown in Fig. 1, but the number of bits required to transmit the side information is considerably high. On the other hand, considering a BPSK ( m = 2) constellation, significantly reduces the benefits of PAR reduction. Considering a system with N = 32 and pilot spacing in frequency domain N f = 4, so N p = 8, this strategy drives to the need of B = 8,9,10 bits to code the side information, when considering respectively BPSK, QPSK and 8-PSK.
5. Comparison of methods At this point, one question arises inherently to this proposal. Instead of considering pilot values which simultaneously perform channel estimation and PAR reduction, a second possibility would be considering two kinds of pilots tones: known pilots as classically, to perform channel estimation, and some pilot tones to perform PAR reduction, whose values would be disregarded at the receiver since they are not required for channel estimation purposes. Then, these methods are described. Method I: This method requires N p pilots to perform simultaneously PAR reduction and channel estimation, chosen from a set of size M . The side information generated by these pilots must be known at the receiver, so we need a number of subcarriers equal to the number of bits to code that information, B = log, M . Therefore, the number of non-data sub-carriers is given by N p B.
4.2. Predefined set of values Another possibility, with the aim of reducing side information, consists of choosing a small set S of predefined pilot vectors P p , randomly chosen and orthogonal between them. In this case, the transmitted pilot-vector, PL,will be found trialing the M possibilities predefined in the set, and it will provide the lowest PAR compared to the other pilot-vectors in the set. Possible values are only the elements of the set, Eq. (1 l), where each element P p is a frequency-domain pilot-vector of N x 1dimensions defined T
+
0 Pf . . . 0 P/ 0 . . . PgP01 , being P/ a pilot
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Method 11: In this case, there are two types of pilots, of which Q are known at the receiver and devoted for channel estimation, and R are pilots for PAR reduction, whose values are not used for other purpose than peak power reduction. Since their values have not to be known at the receiver, side information has not to be transmitted; and also, optimal values can be used to a obtain maximum benefit. The technique of reserving certain tones to include optimum values which minimize the PAR has been previously named Tone Reservation [15]. Performance of both methods in terms of PAR reduction can be compared by imposing the same loss in data-rate. The condition of having the same number of non-data subcarriers is, as stated in the following expression,
happen rarely, and if happens, the channel conditions will be so awful that even detection will be wrong in most cases and communication under these conditions makes no sense.
6. Simulation results The scenario is a QPSK scheme, with N = 32 - 64 subcarriers. The pilot density is such that every N f subcarriers, a pilot-symbol is inserted. Figure 2 shows peaks reduction after performing an exhautive search on the set and selecting the optimum pilots; minimizing the maximum of the sequence is the cost function to reduce PAR in the OFDMsymbol. Large amplitude spikes arise very rarely, due to statistical averaging. 0.35
Under this assumption, these proposals present similar performance for the systems considered; simulation results are shown in Section 6. However, different advantages and drawbacks can be considered about these proposals.
- CE pilots - - Optimal pilolr
0.3 -
Computational complexity in Method I1 is higher since optimization will be carried out to obtain a maximum benefit from the tones only devoted for PAR reduction. In Method I, the operation of trying a small predefined finite set and choosing the one with best results is much simpler and faster than the optimal solution. 0
Typically, in a coherent system, channel parameters determine spacing of the pilot-symbols in frequencydomain to guarantee a reliable estimation of the channel’s frequency response. Thus, if a number of N p pilots are required for channel estimation, it is clearly an advantage in Method I using all these pilots to carry out PAR reduction and the extra pilots required to select the element of the set will always be very small. These extra pilots would be the sub-carriers that could be used for PAR reduction in Method 11, to maintain a fixed data rate in both schemes. This consideration is significantly notorious in the case of applications where a large N is considered. From a different point of view, in Method I, the same pilots for PAR reduction can also be used for channel estimation. This technique requires a joint design of parameters for both funcionalities, with a trade-off among performance, complexity, and reliability. There is a drawback in Method I, since the receiver could be unable to acquire the pilot values for channel estimation, due to unsuccessful channel conditions; Method 11, however, will never experience this, since they are known at the receiver. This situation will
10
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30
50
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0
N
Figure 2. Peak reduction with pilot symbols selected from an orthogonal set. Arrows mark spikes reduction when selecting the best pilot-vector. N = 64, Nf = 8. Power reduction results for a system with N = 32 and N f = 4 are shown in Table 1. An M = 4 set has been em-
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ployed, what means a low computational load at the trans2 dB is mitter. It can be observed that a reduction of attained when considering the set, compared to the classical choice of all ones pilot symbols. OFDM-symbols with higher PAR are the ones whose peaks should be more desirably reduced. So, it can also be analyzed the reduction obtained when consideringjust the OFDM-symbols exceeding a certain threshold; if > 7 dB, the reduction is 3 dB and the reduction increases to 4 dB if > 8 dB. A comparison for the two methods proposed in section 5 has been carried out considering N = 32 and N j = 4, as in previous results. To maintain the same data-rate, the following parameters has been chosen. In method I, N p = 8 and B = 2 ( M = 4) are considered; and in method 11,
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11 Ones I Out. 1 8-PSK I OPSK I BPSK 1 Set(M=4) 1
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7
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I
I
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4.88
I
5.41
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[3] X. Li and L.J. Cimini, Jr., ”Effects of Clipping and Filtering on the Performance of OFDM’. IEEE Commun. Letters, Vol. 2, no. 5, pp. 131-133, May 1998.
,
5.12
Table 1. Mean PAR in [dB] of OFDM-symbols with different choices for pilot symbols, either optimal (Opt.) or suboptimal (gravity centers (&PSK,QPSK,BPSK) or predefined set).
[4] J. Rinne and M. Renfors, ’The Behaviour of Orthogonal Frequency Division Multiplexing Signals in an Amplitude Limiting Channel”. Proc. of IEEE Int. Con$ on Communications, ICC’94, Vol. 1, pp. 381-385, May 1-5 1994. [5] P. Van Eetvelt, G. Wade and M. Tomlinson, ”Peak to Average Power Reduction for OFDM Schemes by Selective Scrambling”. IEE Electron. Letters, Vol. 32, no. 21, pp. 1963-1964, 10th Oct. 1996.
values are Q = 8 and R = 2. For these methods, the mean PAR are 5.02 dB and 5.33 dB, respectively; these results show a similar reduction of 2 dB, when no threshold is considered. Several considerations must be outlined upon the experimental analysis. Firstly, a significantly small number of pilots is required to attain a reasonable PAR reduction. Secondly, given a certain numer of pilots to reduce the PAR, the reduction is independent of the positions of the pilots. Finally, if we do not use pilot symbols for PAR reduction and just only for channel estimation is still important to consider this aspect jointly; pseudo-random sequences must be employed for the pilot values, since the mean PAR is much lower than using sequences of ones.
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[6] R.W. Bauml, R.F.H. Fischer and J.B. Huber, ”Reducing the Peak-to-Average Power Ratio of Multicarrier Modulation by Selected Mapping”. IEE Electron. Letters, Vol. 32, no. 22, pp. 2056-2057, 24th Oct. 1996. [7] S.H. Muller and J.B. Huber, ”OFDM with Reduced Peak-toAverage Power Ratio by Optimum Combination of Partial Transmit Sequences”. IEE Electron. Letters, Vol. 33, no. 5, pp. 368-369, 27th Feb. 1997. [8] L.J. Cimini, Jr., and N.R. Sollenberger, ”Peak-to-Average Power Ratio Reduction of an OFDM Signal Using Partial Transmit Sequences”. IEEE Commun. Letters, Vol. 4, no. 3, pp. 86-88, March 2000. [9] T.A. Wilkinson and A.E. Jones, ”Minimisation of the Peak to Mean Envelope Power Ratio of Multicarrier Transmission Schemes by Block Coding”. Proc. of IEEE Vehic. Technol. Con$ VTC’95, Vol. 2, pp. 825-829. July 25-28 1995.
7. Conclusions The proposed technique makes more efficient use of the embedded signalling introduced primarily for channel estimation and it has been proven feasible to reduce the peak power in practical systems. To reduce the side information, an efficient suboptimal proposal consists of defining finite sets of pilot-vectors, and possible values are only the elements of this set. Also, this significantly lowers the additional system complexity. Simulation results have proven that this flexible scheme provides good results with very low side information, but it has also been compared with schemes considering pilot symbols separately for channel estimation and PAR reduction. Further research should be carried out to extend this analysis to systems with large N , since early results seems to be very promising.
[lo] J.A. Davis and J. Jedwab, ”Peak-to-Mean Power Control in
OFDM, Golay Complementary Sequences and Reed-Muller Codes”. IEEE Trans. on Inform. Theory, Vol. 45, no. 7, pp. 2397-2417, NOV.1999. [ 111 V. Tarokh and H. Jafarkhani, ”On the Computation and Re-
duction of the Peak-to-AveragePower Ratio in Multicarrier Communications”.IEEE Trans. on Commun., Vol. 48, no. 1, pp. 37-44, Jan. 2000. [12] P. Hoher, S. Kaiser and P. Robertson, ”WO-Dimensional Pilot-Symbol-Aided Channel Estimation by Wiener Filtering”. Proc. of IEEE Int. Con$ on Acoustics, Speech and Sig. Proc., ICASSP’97, Vol3, pp. 1845-1848, Munich, Germany. April 21-24 1997. [13] M.J. Fernindez-Getino Garcia, 0. Edfors and J.M. P6ezBorrallo, ”Joint 2D-Pilot-Symbol-Assisted-Modulation and Decision-Directed Frequency Synchronization Schemes for Coherent OFDM”. Proc. of IEEE Int. Con$ on Acoustics, Speech and Sig. Proc., ICASSP’OO, Vol 5, pp. 2961-2964, Istanbul, Turkey. June 5-9 2000.
References [l] E. Costa, M. Midrio and S. Pupolin, ”Impact of Amplifier Nonlinearities on OFDM Transmission System Performance”. IEEE Commun. Letters, Vol. 3, no. 2, pp. 37-39, Feb. 1999.
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[15] J. Tellado-Mourelo, Peak to Average Power Reduction for Multicarrier Modulation, Ph.D. Thesis, Stanford University, Stanford, CA, Sept. 1999.
[2] M.-G. Di Benedetto and P. Mandarini, ”An Application of MMSE Predistortion to OFDM Systems”. IEEE Trans. on Commun., Vol. 44, no. 11, pp. 1417-1420, Nov. 1996.
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