Peak Power Reduction Using a Unitary Rotation in Multiple Transmit

Peak Power Reduction Using a Unitary Rotation in Multiple Transmit Antennas Heechoon Lee, Daniel N. Liu, Weijun Zhu and Michael P. Fitz Department of Electrical Engineering University of California Los Angeles Email: {coet, daniell, zhuw and fitz}@ee.ucla.edu

Abstract— This paper proposes a new peak-to-average power ratio (PAPR) reduction scheme in multiple transmit antenna environments. By applying a unitary rotation, overall PAPR of the multiple transmit antenna system is reduced. This scheme doesn’t require any side information to decode the signal in the receiver, enabling throughput-lossless PAPR reduction. Furthermore, there is no increase in the complexity of the receiver.

I. I NTRODUCTION Multiple-input multiple output (MIMO) radio systems have seen a great deal of attention since Telatar [1], Foschini and Gans [2] showed that exploiting the multiple transmit/receive antenna increases the outage capacity. For high data rate wireless wideband applications MIMO combined with orthogonal frequency division multiplexing (OFDM) is being considered in a large number of current technology applications. While OFDM has a great advantage of having simple equalization, it has an inherent drawback of high peak-to-average power ratio (PAPR). Researchers have extensively examined PAPR reduction in single-input single-output (SISO) systems. Clipping [3] is the simplest way to limit the maximum magnitude of transmit signals. Clipping causes distortion resulting in increased bit error rate (BER) and out-of-band spectral radiation. Selective mapping (SLM) [4] and partial transmit sequence (PTS) [5] transmit the lowest peak power signal among several candidates. These methods, however, require redundant bits to decode the information bits in the receiver. Tone reservation [6], [7] reduces the PAPR using a small set of pre-allocated tones, that wastes the bandwidth. Nonbijective constellation [7], [8] changes the constellations to combat large signal peaks. It doesn’t need any redundant information, but results in increased average power and high complexity. PAPR reduction using coding [9], [10] was also proposed, but it sacrifices the data rate. All aforementioned schemes involve some compromises between bandwidth, peak power and complexity. So far none of the SISO PAPR reduction schemes dominates the others. PAPR reduction in MIMO case has rarely been considered yet. If the PAPR reduction schemes in SISO are directly applied to MIMO, the complexity and redundancy increase with the number of transmit antennas, which is not desired. One approach has been taken to solve the PAPR problem in MIMO case. In [11], the MIMO-SLM is applied through overall transmit antennas, i.e., the sequence with the lowest

0-7803-8938-7/05/$20.00 (C) 2005 IEEE

PAPR over all transmit antennas is selected instead of applying individual SLM to each transmit antenna. Using the MIMOSLM, redundancy can be reduced or the reliability of the side information can be increased at the sacrifice of reduced PAPR reduction gain. In this paper a new method of PAPR reduction in multiple transmit antenna environments is proposed. By applying a unitary rotation through transmit antennas, the maximum peak power over all transmit antennas can be decreased. The unitary rotation plays a role in dispersing the maximum power in one transmit antenna into the other transmit antennas. This scheme is not limited to an OFDM system, but it can be combined with any system using multiple transmit antennas. The unitary rotation can be made transparent at the receiver so that no loss in bandwidth is needed to specify which transformation was used at the transmitter. Hence this technique offers improved PAPR without any loss in bandwidth efficiency. The rest of the paper is organized as follows. Section II introduces the system model and PAPR problem formulation. Section III-A presents a new PAPR reduction scheme in multiple transmit antenna environments, which is called a unitary PAPR reduction. An example packet structure for the unitary PAPR reduction is described in Section III-B. Simulation results are provided in Section IV. Systematic generation of unitary matrix for 2 transmit antenna is introduced in Section V. Finally, Section VI gives concluding remarks. II. S YSTEM M ODEL The MIMO OFDM system with Lt transmit and Lr receive antennas and D subcarriers is considered. Dd data subcarriers among the D subcarriers are used for data transmission. For all the examples considered in the paper D = 64 and Dd = 52 are considered as is often the case with IEEE 802.11x systems. The OFDM signal transmitted from the ith transmit antenna is given as Xi (t) =

Nf Dd



Xi (l, k)u(l, t − (k − 1)Ts )

(1)

k=1 l=1

where u(l, t−(k−1)Ts ) is the subcarrier pulse shape and Ts is the OFDM symbol time. It should be noted that the subcarrier pulse is typically a time-limited pulse of a complex sinusoid.

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The PAPR of the MIMO OFDM system can be defined as maxi,t |Xi (t)|2 PAPR = Ei,t [|Xi (t)|2 ]

OFDM Mod

(2)

Bit Source

III. P EAK P OWER R EDUCTION A. Unitary Rotation Since all the existing SISO PAPR reductions involve some drawbacks, it is very advantageous to reduce PAPR over all transmit antennas together, in place of separate SISO reduction in each transmit antenna. Most PAPR reduction schemes in SISO systems try to eliminate the occurrences of the peak by controlling the frequency domain signals before the IFFT. The idea of unitary PAPR reduction exploits the characteristics of the multiple antennas. Applying a relevant unitary rotation to the multiple streams of the time domain signals, the peak power in one transmit antenna can be dispersed into signals in the other transmit antennas. That is, instead of avoiding peak generation, unitary PAPR reduction trims down the peaks by spreading out the energy over all antennas. As a result, the peak power over all the transmit antennas can be reduced. In the operation of the MIMO OFDM system the unitary rotation can be absorbed into an effective channel. For example if U represents the unitary rotation then the received signal is given as 
(m, n) = H(m, n)U Es X(m,

(m, n) (4) Y n) + N  ˜

= H(m, n) Es X(m, n) + N (m, n) ˜ where H(m, n) is denoted the effective channel matrix. The effective channel matrix can be retrieved from a standard channel estimator. There is a tradeoff between achieving good PAPR and achieving good channel estimation. The best PAPR would be achieved by applying a unitary PAPR reduction to each OFDM symbol. In that case, the channel estimator should work at every symbol time independently, since the effective channel experiences the abrupt change at each symbol time. The unitary matrix for PAPR reduction needs to be constant during the period of channel estimation and channel estimation

OFDM Mod

Pilot Insertion

Unitary PAPR Reduction

OFDM Mod

where E[·] denotes the expectation (over antennas and time). There is a relatively simple model for the output of the MIMO OFDM matched filters. Assuming that a cyclic prefix of an appropriate length is used, then the matched filter outputs for the mth symbol and the nth subcarrier are given as an Lr ×1 vector 

(m, n)
(m, n) = H(m, n) Es X(m, n) + N (3) Y

where Es is the energy per transmitted symbol; Hji (m, n) is the complex path gain from transmit antenna i to receive
antenna j at time mT on subcarrier n; X(m, n) is the Lt × 1 vector of symbols transmitted at symbol time m and subcarrier
(m, n) is the additive white Gaussian noise vector of n; N size Lr × 1. The noise samples are independent samples of circularly symmetric zero-mean complex Gaussian random variables with variance N0 /2 per dimension.

STBC Encoder

Mapper

A/D

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Channel Estimator

A/D

OFDM Demod

Channel Estimator

A/D

OFDM Demod

Channel Estimator

STBC Decoder

Demapper

D/A

HPA

D/A

HPA

D/A

HPA

Bit Sink

Fig. 1. Overall block diagram of MIMO OFDM using unitary PAPR reduction.

can be improved by smoothing over multiple symbols. In addition, if a space-time block code (STBC) is used, the unitary matrix also should be constant for the duration of space-time block length if the low complexity decoders for block codes are to be used. In this paper, a set of example short packet structures containing pilot symbols and STBCs are presented that can use the unitary PAPR reduction technique. The STBC of Lt transmit antenna and Nf time duration is employed [12]. The packet length of the presented short packet structure is Nf OFDM symbols, i.e., one STBC block per subcarrier composes one packet. The detailed packet structure is described in Section III-B. In order to reduce the continuous time PAPR, oversampling should be used when sampling the transmit signals [13]. In this paper, oversampling rate L = 5 is used. The resulting packet length is LNf (D + Dg ) samples per each transmit antenna, where Dg = 16 is the cyclic prefix length. The cyclic prefix is included in the system, but it doesn’t affect the PAPR performance, given that it is just a cyclic extension of original signal. The optimal unitary matrix for PAPR reduction can be written as (5) U o = arg min[max{U X}] U

where X represents a Lt × LNf (D + Dg ) matrix of one transmitted short packet signals and max{U X} denotes the maximum magnitude element of matrix U X. That is, the unitary PAPR reduction reduces the peak power using the unitary matrix that minimizes the maximum peak power over all transmit antennas through the packet. The block diagram of the MIMO OFDM system adopting the unitary PAPR reduction is depicted in Fig. 1. The unitary PAPR reduction is located just before the D/A converter and high power amplifier (HPA). By reducing peak power using the unitary rotation, the distortion due to the nonlinearity of HPA can be decreased. As shown in Fig. 1, the unitary PAPR reduction doesn’t require any additional processing in the receiver. The original channel estimator can retrieve the channel information (the effective channel matrix) including

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the unitary matrix generated in the transmitter. In addition, this scheme doesn’t have any redundant information to transmit, resulting in throughput-lossless PAPR reduction. Furthermore, it can be combined with any multiple antenna systems regardless of modulation or coding structure. It is obvious that the longer packet length degrades the performance of the PAPR reduction. Thus, this scheme is advantageous with a short packet structure. Long packets would require multiple transformations to get effective PAPR reduction. In this case the redundant information bits representing the unitary matrices would be transmitted or separate channel estimation per each transformation is required. An algorithm to find the optimal unitary matrix is a tough optimization problem. In this paper, the unitary matrix is generally determined by a random search, but a constructive method for 2 transmit antenna case is introduced in Section V. A systematic generation of unitary matrix for general structure would be a future goal. B. Packet Structure Channel state information (CSI) is required for the OFDM receiver to perform coherent detection, or diversity combining in MIMO environments. In practice, a channel estimate is acquired in the receiver using a known pilot symbol (PS) [14], [15]. The channel estimates are obtained by the interpolation from the observations at the PS locations. Performance with finite samples in time and frequency is aided by having more pilot symbols at the edges of the packet. The simplest and most efficient way to estimate the CSI from MIMO environment is to use the orthogonal modulation on each of the transmit antennas [15]. Denote the observation of pilot value at the receiver by xp and the channel at the position in data symbol (DS) by c. Due to the orthogonality, channel estimates can be obtained by   −1 xp ≡ Wxp , (6) ˆ c = E cxH p cov (xp ) where cov (xp ) is the covariance matrix of xp , W is Wiener filter coefficients and superscript H denotes a complex conjugate transpose. Short packets are very important for MAC layer communication; RTS, CTS and TCP/IP ACKs are the examples of short packet. Even though they include little information compared to data packet, but their reliability is critical to the overall performance of network. The short packet structures for the MIMO OFDM system is designed: Lt = Nf = 2 and Lt = Nf = 4 cases are described in Table I. Since channel estimation takes larger percentage of bandwidth in short packet, data and pilot are mixed together instead of preamble-based structure. It should be noted that unitary PAPR reduction can also be applied to preamble-based structure. Based on the structure in Table I, randomized pilots are designed to exhibit low PAPR so that the PAPR of transmit signal is not dominated by pilots. These particular PS placements in the frequency-time grid enables the system to track a frequency roll across the short packet. It is compliant to IEEE 802.11x system so 52 effective

TABLE I S HORT PACKET STRUCTURE .

2 Tx

Pilot Subcarrier Data Subcarrier

Pilot Subcarrier

4 Tx Data Subcarrier

-26, -25, -24, -22, -20, -18, -16, -14, -12, -10, -8, -6, -4, -2, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26 -23, -21, -19, -17, -15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 -26, -25, -24, -23, -21, -19, -17, -15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 24, 25, 26 -22, -20, -18, -16, -14, -12, -10, -8, -6, -4, -2, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22

subcarriers are used out of 64 subcarriers. As per the 802.11a standard the DC subcarrier is not used. In some situations, the optimum interpolation filter from this sampling of the noisy channel response is a linear filter whose tap coefficients are a function of particular channel statistics [14], [15]. Therefore, the Wiener filter coefficients can be pre-computed and the resulting channel estimator is an open loop structure which has no acquisition time. IV. P ERFORMANCE The performance of the proposed unitary PAPR reduction is provided via computer simulation. The use of 2 × 2 MIMO OFDM system with Alamouti STBC [16] and 4 × 4 system with super-orthogonal STBC [12] are considered. Alamouti STBC employs a 64 QAM and super-orthogonal STBC employs a 64 QAM with 2 additional bits. This PAPR reduction scheme is compatible with more powerful space-time modulations but this paper considers simple block codes to keep the focus on the achieved PAPR reduction and performance improvement. First, the performance of unitary PAPR reduction is evaluated in terms of complementary cumulative distribution function (CCDF) of PAPR. In simulation, a sub-optimal unitary matrix is chosen among 200 randomly generated unitary matrices. The CCDF of PAPR in 2 and 4 transmit antennas are depicted in Fig. 2. It can be seen that about 2 and 3 dB of gains are observed at CCDF = 10−6 in 2 and 4 transmit antennas, respectively. These gains are relatively small compared to SISO techniques, but it should be emphasized that these gains can be obtained without sacrificing spectral efficiency at all and it can be used in conjunction with SISO PAPR techniques in the MIMO case. To evaluate the overall system performance using unitary PAPR reduction, RAPP nonlinear amplifier model [17] is used for the HPA. The RAPP model is developed for a solid-state power amplifier and it is recommended to use in IEEE 802.11 TGn comparison criteria document [18]. The RAPP model is given by Vin (7) Vout =   2p 1/2p in | 1 + |V Vsat

where Vin and Vout is the input and output magnitude of the amplifier, Vsat is the saturation magnitude and p is the

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(b) 4 transmit antennas with super-orthogonal STBC [12] using 64 QAM Fig. 2.

16

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PAPR0

Fig. 3. Performance of unitary PAPR reduction with RAPP model [17] with p = 3.

Complementary cumulative distribution function of PAPR.

smoothness factor. p = 3 is used in the simulation [18]. The backoff of the amplifier (Pbo ) is defined as the input power backoff from the full saturation, i.e.,   Ptx Pbo = −10 log10 (8) Psat where Ptx is the average power per transmit antenna and Psat is the saturation power of the amplifier. The BER performance is evaluated in Fig. 3 with respect to different Pbo as a function of Eb /N0 per receive antenna. Bypass channel is used with additive white Gaussian noise to measure the BER. ‘NoAmp:PCSI’ curve denotes the BER without HPA using perfect CSI and ‘NoAmp:PSAM’ denotes the BER without HPA using estimated CSI. All other curves shown in Fig. 3 use the estimated CSI. Apparently, the lower Pbo is, the bigger gain is obtained in both cases. The unitary PAPR reduction enables to use the HPA with lower Pbo requirement. Furthermore, it is also seen that high throughput cannot be achieved without PAPR reduction due to high error floor when the HPA with low Pbo is deployed in the transmitter. This nonlinearity effect from the HPA is more dominant in 2 transmit antenna due to lower space diversity.

V. S YSTEMATIC I MPLEMENTATION For practical implementation a constructive method to find the optimal/sub-optimal matrix is essential. This section provides a simple gradient search method to generate a suboptimal unitary matrix for 2 transmit antenna short packet structure. Since PAPR is independent of the phase of transmit signals, a 2 × 2 unitary matrix can be parameterized as √

1 − r2 ejθ r √ (9) U (r, θ) = r − 1 − r2 e−jθ

where 0 ≤ r ≤ 1 and 0 ≤ θ ≤ 2π. The PAPR reduction problem in (5) can be rewritten as U o = arg min[ U

max

n∈[0,LNf D−1]

U
xn ∞ ]

(10)

where
xn is the transmit signal vector at time instance n. Maximization over all n in (10) is numerically very expensive. Intuitively peak occurrences might depend mostly on when 
xn 2 is large. Therefore, maximization over all n can be reduced into maximization over reduced set of n. Random searches using only Nc time samples corresponding to Nc largest 
xn 2 were evaluated with different Nc ’s. Fig. 4 shows

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CCDF Degradation (dB) at PAPR CCDF = 10

−5

1

VI. C ONCLUSIONS

Random search Gradient search

0.9

In this paper, a new PAPR reduction scheme for the multiple transmit antenna environments was proposed. PAPR can be reduced by a unitary rotation and it achieves the throughputlossless PAPR reduction. Furthermore, this doesn’t require any processing in the receiver. The drawback of this scheme is that it is only advantageous in short packet structure for the costless channel estimation. Finally, a constructive method to find the optimal/quasi-optimal unitary matrix for general structure is an open research problem.

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Fig. 4. Comparison of PAPR degradation between random search and gradient search with respect to the number of optimization points.

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Fig. 5. An example of object function (maxn∈{Nc }
U
xn
∞ ) with Nc = 16.

the PAPR degradation in terms of CCDF at CCDF = 10−5 compared from the random search over entire packet. Using only 16 out of 640 (L = 5, D = 64 and Nf = 2) samples, the degradation less than 0.1 dB can be achieved. An example of object function (maxn∈{Nc } U
xn ∞ ) with Nc = 16 is depicted in Fig. 5. Typically there exist two or three minimum valleys in the object function. Considering the use of gradient search with four fixed starting points, a sub-optimal solution can be attained. Performance degradation of gradient search is also shown in Fig. 4 against random search over entire packet, where the degradation is less than 0.02 dB when Nc = 16. Performance of gradient search is worse than that of random search when Nc = 16, because of local minima in the objective function. The gradient search method demonstrates a direction to make unitary PAPR reduction feasible, but more exact and less complex optimization technique would be expected. Furthermore, a technique to find optimal/sub-optimal unitary matrix for general structure (e.g., general number of transmit antenna) is an open research problem.

[1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” AT&T Bell Labs Internal Tech. Memo., June 1995. [2] G. J. Foschini and W. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communication, vol. 6, pp. 314–335, Mar. 1998. [3] X. Li and L. J. Cimini, Jr., “Effects of clipping and filtering on the performance of OFDM,” in Proc. IEEE Vehicular Tech. Conf., vol. 3, May 1997, pp. 1634–1638. [4] R. W. Bauml, R. F. H. Fischer, and J. B. Huber, “Reducing the peak-toaverage power ratio of multicarrier modulation by selected mapping,” IEEE Elec. Letters, vol. 32, no. 22, pp. 2056–2057, Oct. 1996. [5] S. H. Muller and J. B. Huber, “OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequence,” IEEE Elec. Letters, vol. 33, no. 5, pp. 368–369, Feb. 1997. [6] A. Gatherer and M. Polley, “Controlling clipping probability in DMT transmission,” in Proc. Asilomar Conf. Signals, Systems, and Computers, vol. 1, Nov. 1997, pp. 578–584. [7] J. Tellado, “Peak to Average Power Reduction for Multicarrier Modulation,” Ph.D. dissertation, Stanford University, September 1999. [8] B. S. Krongold and D. L. Jones, “Par reduction in OFDM via active constellation extension,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, vol. 4, Apr. 2003, pp. IV–525–IV–528. [9] R. van Nee, “Ofdm codes for peak-to-averge power reduction and error correction,” in Proc. IEEE Global Telecommunications Conf., vol. 1, Nov. 1996, pp. 740–744. [10] T. A. Wilkinson and A. E. Jones, “Minimisation of the peak to mean envelope power ratio of multicarrier transmission schemes by block coding,” in Proc. IEEE Vehicular Tech. Conf., vol. 2, July 1995, pp. 825–829. [11] Y. Lee, Y. You, W. Jeon, J. Paik, and H. Song, “Peak-to-average power ratio in MIMO-OFDM systems using selective mapping,” IEEE Commun. Letters, vol. 7, no. 12, pp. 575–577, Dec. 2003. [12] H. Lee, M. Siti, W. Zhu, and M. P. Fitz, “Super-orthogonal space-time block code using a unitary expansion,” in Proc. IEEE Vehicular Tech. Conf., Sep. 2004, to appear. [13] C. Tellambura, “Computation of the continuous-time PAR of an OFDM signal with BPSK subcarriers,” IEEE Commun. Letters, vol. 5, no. 5, pp. 185–187, May 2001. [14] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh faded channels,” IEEE Trans. Veh. Tech., vol. VT-40, pp. 686– 693, Nov. 1991. [15] J. C. Guey, M. P. Fitz, M. R. Bell, and W. Y. Kuo, “Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels,” in Proc. IEEE Vehicular Tech. Conf., Apr. 1996. [16] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [17] C. Rapp, “Effects of HPA-nonlinearity on a 4-DPSK/OFDM-signal for a digital sound broadcasting system,” in Proc. 2nd European Conf. Satellite Communications, Liege, Belgium, Oct. 1991, pp. 179–184. [18] A. P. Stephens, “IEEE 802.11 TGn comparison criteria, Tech. Rep. IEEE 802.11-03/814r30, May 2004.

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