Partial Transmit Sequence Using EVM Optimization Metric for BER Reduction in OFDM Systems

Transcription

1 2013 8th International Conference on Communications and etworking in China (CHIACOM) Partial Transmit Sequence Using EVM Optimization Metric for BER Reduction in OFDM Systems Shun Zhang, Jianhua Ge, Jun Hou, and Fengkui Gong The State Key Lab. of Integrated Service etworks, Xidian University, Xian, P. R. China Abstract One of the main drawbacks of orthogonal frequency division multiplexing (OFDM) is its high peak-to-average power ratio (PAPR). Conventional partial transmit sequence (PTS) is an efficient PAPR reduction technique. However, reducing the PAPR cannot achieve an optimal bit error rate (BER) performance when the signals are passed through a power amplifier (PA) because of nonlinear distortion. In this paper, an error vector magnitude (EVM) reduction PTS scheme is proposed to improve the BER performance in the presence of PA. By properly choosing the phase factor of the PTS, this scheme can dramatically reduce the nonlinear distortion. In addition, a simplified two-layer PTS optimization model is introduced to reduce the complexity. Simulation results show that the proposed scheme can significantly improve the BER performance while maintaining low complexity. Index Terms Orthogonal frequency division multiplexing (OFDM), peak-to-average power ratio (PAPR), error vector magnitude (EVM), solid state power amplifier (SSPA), two-layer PTS optimization model. I. ITRODUCTIO Orthogonal frequency division multiplexing (OFDM) is one of the key techniques in B3G/4G system due to its high spectral efficiency, robustness in the multipath fading and resistance to the narrowband interference. However, one of the main drawbacks of OFDM system is its high peak-to-average power ratio (PAPR). Since the linear range of the power amplifier is limited, signals with high PAPR after power amplifier (PA) can result in in-band distortion and out-band radiation. Thus, many PAPR reduction schemes have been proposed, such as clipping-filtering (CF) [1], [2], coding schemes [3], [4], tone reservation (TR) [5], [6], tone injection (TI) [7]-[11], selective mapping (SLM) [12] and partial transmit sequence (PTS) [13]-[17]. Among them, SLM and PTS are efficient and distortionless techniques. However, PAPR reduction cannot guarantee a low distortion caused by the PA. Thus, in order to improve bit error rate (BER) performance, we should minimize the distortion instead of PAPR. In [15], a peak interference-to-carrier ratio (PICR) reduction scheme is introduced to reduce the inter carrier interference (ICI). In [16], an inter-modulation distortion (IMD) reduction scheme is presented and it can also be used to minimize the distortion. However, the main limitation of these PICR and IMD schemes are high computational complexity. Therefore, in order to reduce the complexity and further reduce the nonlinear distortion, an efficient PTS scheme based on error vector magnitude (EVM) reduction is proposed. This scheme takes the EVM as the measure metric to select the optimal coefficients of the PTS. Simulation results show that the proposed scheme can achieve a better BER performance than the PAPR, PICR and IMD reduction while maintaining a low computational complexity. This paper is organized as follows. Section II reviews the OFDM system and conventional PTS scheme. Section III introduces the distortion metric based on EVM and a suboptimal searching algorithm. The performance of our proposed scheme is simulated in Section IV and Section V concludes this paper. II. OFDM SYSTEM MODEL AD COVETIOAL PTS SCHEME A. OFDM System Model In an OFDM system, since the rate of serial data stream is high, we separate it into parallel data streams X = [X 1,X 2,..., X ] T which are usually modulated with quadrature amplitude modulation (QAM) and phase-shift keying (PSK) techniques. Thus, the time domain signal is: x (t) = 1 X k e j 2πkt T 0 t<t, (1) where T is the sampling interval. The PAPR of the time domain signal x (t) is given by: max { x (t) 2} 0t<T PAPR =, (2) 2] E [ x (t) [ where E x (t) 2] denotes the expectation operation of x (t) 2. However, the signal is usually transformed into discretetime domain signal so that digital signal processing (DSP) technology can be applied to it. Therefore, the transmitted OFDM signal in the time domain x =[x 1,x 2,..., x ] T can be expressed as: x n = 1 X k e j 2πnk L n =0, 1,..., L 1, (3) where j = 1, L is the oversampling factor and X k is the frequency domain signal. The modulation operation can be replaced by IFFT to simplify the hardware implementation IEEE

2 The PAPR of this L-times oversampling time domain signal can be defined as: max { x n 2} 0nL PAPR =, (4) 2] E [ x n note that when L 4, this PAPR definition is approximately equal to that of the continuous time signal. As we state before, one advantage of the OFDM system is its resistance to ICI. In the practical OFDM system, a cyclic prefix (CP) is usually appended to x n. Thus, the transmitted symbol x with CP can be rewritten as x = [x L g,..., x L,x 0,x 1,..., x L ] T, where g is the length of CP, which is usually a quarter of the length of x n. B. The Conventional PTS Scheme In the conventional PTS scheme, the input frequency { data D is divided into M disjoint subblocks D (m),m=1, 2,..., M }, each of which has only /M nonzero elements. Then, multiply D (m) with phase factors b m and add them together. Therefore, the new data D in frequency domain can be expressed as: M D = b m D (m), (5) m=1 where b m = e jϕm (m =1,..., M). The phase angle ϕ m is uniformly distributed in [0, 2π) and usually chosen as 0 and π. Since the inverse discrete Fourier transform (IDFT) operation is linear, the time domain signals can be expressed as: M x = b m d (m), (6) m=1 where d (m) (m =1,..., M) are the IDFT operation of D (m) (m =1,..., M). By searching over the phase factors, we can get the combined signals with the lowest PAPR. Therefore, the optimization problem of the PTS method can be described as: {b 1,b 2,..., b M } best = argmin {b 1,b 2,...,b M } {PAPR(x)}. (7) III. PROPOSED EVM METRIC I THE PRESECE OF PA MODEL A. PA Model The output of the solid state power amplifier (SSPA) model with the input signal x n = s n e jθn can be expressed as: y n = A (s n ) e j[θn+ϕ(sn)]. (8) Here, assume that the amplitude modulation (AM)/phase modulation (PM) distortion is negligibly small, that is, ϕ (x n )=0. Therefore, the SSPA model only introduces AM/AM distortion, which can be expressed as: A (s n )= [ 1+ s n ( ) ] 1 2p 2p s n A o, (9) where A o is the maximum linear range output of the amplifier. The parameter p controls the smoothness of the transition from the linear region to the saturation region. A baseband equivalent polynomial model for a nonlinear power amplifier is also given as [18] K y n = α k x (n) x (n) k, (10) k=1 where α k is the coefficient of the amplifier. Since the odd order k =1and 3 produce most IMD. Thus an approximation of (10) can be rewritten as: y n α 1 x n + α 3 x n x n 2. (11) Therefore, the output signal after SSPA in an OFDM system can be written as: Y k = 1 2πnk j y n e = 1 ( α1 x n + α 3 x n x n 2) 2πnk j e k =1,...,. (12) As stated in [18], the output signal after SSPA can also be rewritten as: Y k = μx k + η k k =1,...,. (13) The gain factor μ and the distortion of SSPA η k can be written as: μ = α 1 + α 3 η k = α 3 i=0,i k where IBO =10 (IBO/10). x n 2 = α 1 + α 3 IBO, (14) X i x n 2 e j 2πn(i k), (15) B. Proposed EVM Metric In [15], a PICR-reduction scheme is proposed to improve the BER performance of nonlinearly distorted OFDM, the PICR is defined as: { } η k 2 PICR = max 0k μ 2 X k 2. (16) In [16], an IMD reduction metric is also introduced and IMD is defined as: { { IMD =max max Re {η k} 0k Re {μx k }, Im {η }} k}. Im {μx k } (17) In this section, we propose an EVM reduction metric which is called root-mean-square error vector magnitude (RMS- EVM). EVM is defined as the ratio of the square root of the error signal power and the square root of the useful signal power and it represents the amount of in-band distortion signal. The more distortion signal is, the worse the BER. Thus, it is one of 114

3 Fig. 1. Scheme of the RMS-EVM optimization in PTS. the main factors that result in BER loss at the receiver of the OFDM system. EVM may be caused by many factors such as the PA, the digital-to-analog converter (DAC), the mixer and various nonlinear signal processing methods. EVM is usually used to measure the amount of the distortion of the useful signal, which is defined as [19]: η k 2 EV M k =. (18) μx k 2 Algorithm 1 : RMS-EVM Reduction Partial Transmit Sequence Scheme 1: Divide the input data D into M disjoint subblocks, { D (m),m=1, 2,..., M } ; 2: Convert the frequency domain data D (m) into time domain data d (m) by IDFT operation; 3: Set b best m = 1(m =1,..., M), Best EV M = 100, t=0 and the number of searching times T =2 M 1 ; 4: Change t into eight binary sequences, convert 0 to 1 and 1 to -1. Select this sequence as phase factors b m (m =1,..., M). Multiply D (m) with b m, and then add them together to get x; 5: Pass x into the SSPAs based on (11) to get y and take FFT operation on y to get Y ; 6: Calculate η k with equation (13) and (14) instead of (15); 7: Calculate RMS EV M with equation (19); 8: if RMS EV M < Best EV M then 9: Best EV M = RMS EV M and b best = b; 10: end if 11: t = t +1; 12: if t>t then 13: Transmit the optimized signals; 14: else 15: t = t +1 and go to step 4; 16: end if In this paper, we focus on reducing the RMS EV M of the signal, which can be defined as: RMS EV M = 1 EV Mk 2. (19) Thus, our proposed RMS-EVM reduction method can be described as: {b 1,b 2,..., b M } best = argmin {b 1,b 2,...,b M } {RMS EV M}. (20) The block diagram of this RMS-EVM based scheme is shown in Fig. 1, and the algorithm of our proposed PTS scheme to reduce the computational complexity of RMS-EVM metric will be given in the subsection C. C. Algorithm of Proposed EVM Reduction Scheme Since minimizing the nonlinear distortion can improve the BER performance. Therefore, we can search the phase factors in the PTS scheme to find the symbol with the minimum RMS EV M. For a detailed description of this RMS-EVM based algorithm, the procedures of this algorithm can be described as Algorithm 1. The core idea of this algorithm is that taking RMS EV M value as the metric. Through the loop iteration, we search over all available phase factors to select the one with the minimum RMS EV M value after the PA as the optimal one. Since the searching complexity of the conventional PTS, or called optimal PTS (OPTS), is too high to be used in practical OFDM systems. Therefore, a low searching complexity PTS method is also proposed in this section. Although the searching complexity of ordinary suboptimal searching method is low, the performance is not good. Thus, we can combine two suboptimal searching method together as a simplified twolayer optimization model. The resulting performance is close to that of the optimal method, but the searching complexity reduces dramatically. In the first layer, we use the iterative flipping PTS (IPTS) algorithm [20]. In the second layer, we apply exhaustive searching only to the even subblocks. The procedures of this algorithm are only different from the Algorithm 1 with the number of searching times T =2 M/2 + M in step 3 and the searching method in step 4:... step 3: Set b best m =1(m =1,..., M), Best EV M = 100, t = 0 and the number of searching times T =2 M/2 + M; step 4: In the first layer, apply the iterative flipping PTS algorithm: set all the b m =1(m =1,..., M) and compute RMS EV M as the Best EV M. Then, invert the first phase factor b 1 = 1 and compute the resulting RMS EV M again. If the new RMS EV M is lower than the Best EV M, update Best EV M and retain b 1 = 1 as the final b 1. Take the same operation on the other b m until all the phase factors are flipped. In the second layer, keep the phase factors of odd subblocks constant, and then apply the exhaustive searching algorithm on the even subblocks. Finally, we obtain the input data of the PA x

4 10 1 Original PAPR PICR IMD RMS EVM 10 1 Optimal EVM RMS Suboptimal EVM RMS GA EVM RMS IPTS EVM RMS Bit error rate Bit error rate SR/dB SR/dB Fig. 2. BER Performance of PTS based on various metrics. Fig. 4. BER Performance of various RMS EVM based PTS metrics Original QAM PAPR PICR IMD RMS EVM OPTS Ë«²ãPTS IPTS CCDF(Pr[PAPR>PAPR0]) Times of FFTs PAPR0(dB) umber of Subblocks Fig. 3. Comparison of PAPR of various methods. Fig. 5. umber of FFTs of different methods. IV. SIMULATIO RESULTS To evaluate the performance of the proposed scheme, Monte Carlo simulations are performed. In the simulations, 10 5 random OFDM symbols are generated with 16QAM modulation and 256 subcarriers. The PTS parameter M =8subblocks. The transmitted signals are oversampled by a factor of 4 to get the accurate results. The phase factors b m are selected from {+1, 1}. Here, the phase factors are assumed perfectly known at the receiver. The polynomial model of SSPA is simplified as (11) with α 1 = 1 and α 3 = 0.132, which means p=3. The IBO of the SSPA is 3 db. Fig. 2 compares the BER performance over additive white Gaussian noise (AWG) channel of PTS scheme based on PAPR, IMD, PICR and RMS-EVM reduction. As shown, the performance of the proposed EVM-RMS based PTS scheme outperforms that of other methods significantly. To achieve a BER performance, the SRs of the proposed EVM- RMS, PAPR, IMD and PICR based PTS methods are 20.1 db, 21.4 db, 22.7 db and 23.8 db, respectively. That is, the proposed RMS-EVM based PTS method outperforms PAPR, IMD and PICR with 1.3 db, 2.6 db and 3.7 db, respectively. In addition, the error floors of the proposed RMS-EVM, PAPR, IMD, PICR based PTS method and the original OFDM system are , , , and 6.2, respectively. 116

5 TABLE I UMBERS OF FFTS I VARIOUS PTS METHODS WITH DIFFERET SUBBLOCKS M Methods M=4 M=8 M=16 OPTS = = = IPTS Sub OPTS = = = 144 Fig. 3 compares the complementary cumulative distribution function (CCDF) of these PTS schemes. When Pr(P AP R > P AP R 0 )=, the CCDF of PAPR, RMS- EVM, IMD, PICR based schemes and the original OFDM system are 8.2 db, 9.1 db, 9.7 db, 9.8 db and 11.9 db, respectively. From this figure, we can see that the RMS-EVM method is only worse than PAPR method and better than the other ones. It confirms that the RMS-EVM based method performs well in both PAPR reduction and BER in presence of PA. Fig. 4 compares the BER performance of RMS-EVM based PTS with various suboptimal methods over AWG channel. In this section, a simplified two-layer optimization PTS described before and the GA-assisted PTS are also simulated. In the GAassisted PTS, the numbers of individuals in one generation and the maximum generation are 4 and 6, respectively. So the times of FFTs operation is 24, which is the same as that of our proposed suboptimal PTS. The cross and variation operations are performed to the last 4 phase factors and the second phase factor, respectively. The mutation probability is 0.7. As described before, the overall searching complexity of the proposed scheme is M+2 M/2 FFTs. The searching complexity of IPTS and OPTS are M and 2 M 1 FFTs, respectively. With the growth of M, the growths of searching complexity are drawn in Fig. 5. In Fig. 5, as the number of subblocks M increases, the FFT number of IPTS, sub OPTS and OPTS increase linearly, exponentially and exponentially faster. We can see that when M 16, the number of FFTs of OPTS is so large that it can not be used in the practical OFDM system. The precise numbers of FFTs of these schemes are listed in Table I. From this table, we can see that as the number of subblocks M increases, the number of FFTs of sub OPTS does not increase so fast as OPTS. Therefore, when M 16, the sub OPTS is a good choice. V. COCLUSIO In this paper, an RMS error vector magnitude (RMS-EVM) reduction based PTS scheme is proposed to achieve a better BER in OFDM systems. By searching the PTS phase factors, this scheme can dramatically reduce the nonlinear distortion. Simulation results have proven that the proposed scheme can significantly improve the system performance with low computational complexity. ACKOWLEDGMET The work was supported in part by the ational Science and Technology Major Project of the Ministry of Science and Technology of China under Grant 2012ZX , the ational High-tech R&D Program of China (863 Program) under Grant 2012AA011701, the ational atural Science Foundation of China under Grant and the 111 Project (B08038). REFERECES [1] J. Armstrong, Peak-to-average power reduction for OFDM by repeated clipping and frequency domain filtering, Electron. Lett., vol. 38, no. 5, pp , Feb [2] L. Wang and C. 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