Iterative Correction of Clipped and Filtered Spatially Multiplexed OFDM Signals

Transcription

1 Iterative Correction of Clipped and Filtered Spatially Multiplexed OFDM Signals Steffen Bittner, Peter Zillmann and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Technische Universität Dresden, D-0062 Dresden, Germany bittner, zillmann, Abstract Peak-to-Average Power Ratio PAPR) reduction in multicarrier systems with multiple transmit and receive antennas is considered. One attractive method for reducing the PAPR is to use clipping and filtering in the digital domain at the transmitter, which results in signal distortion prior to transmission. In this work, soft clipping correction at the receiver, recently proposed in literature, is extended to multi antenna systems. It is shown that taking the clipping noise into consideration leads to a significant performance improvement. The behaviour of the clipping correction algorithm is studied by error rates as well as EXIT charts. Moreover, the low complexity of the proposed scheme makes it attractive for various kinds of multi antenna detection algorithms. I. INTRODUCTION Multiple antennas at transmitter and receiver MIMO) allow to achieve high spectral efficiency by spatially multiplexing several data streams into the same time frequency bin. For broadband wireless systems a very promising approach is to combine the MIMO concept with multicarrier techniques such as orthogonal frequency division multiplexing OFDM), leading to the MIMO-OFDM idea []. However, OFDM signals have a high dynamic range, which has a significant impact on the design of the power amplifier of a wireless transmitter. The high dynamic range requires the power amplifier to operate at high input power backoff, which decreases the efficiency. Low efficiency and power utilisation factor increase both hardware cost and power consumption. Therefore, it is desirable to reduce the Peak-to-Average Power Ratio PAPR) of the transmit signal prior to transmission. Nonlinear distortion of the transmit signal creates undesired out-of-band power and loss in error performance. In literature, a variety of different PAPR reduction methods are known. An attractive method, which also preserves the spectral mask, is clipping of the discrete time transmit signal in combination with oversampling and digital filtering [2] [3]. Clipping and filtering CF) provides good PAPR reduction with comparably low computational complexity at the transmitter, especially for OFDM systems with a large number of active subcarriers. However, at the receiver the distortion noise generated by clipping has to be compensated. In this paper, we study PAPR reduction in MIMO-OFDM systems. Due to CF the orthogonality among the subcarriers is lost. Hence, a joint equalisation would be the optimum detection strategy. However, a maximum-likelihood detection is computationally infeasible even in single antenna systems Source Fig.. Coder Π Interleaver S/P QAM ifft Limiter FFT Filter ifft CP QAM ifft Limiter FFT Filter ifft CP MIMO-OFDM transmission chain with clipping and filtering SISO) [4]. In this work an iterative decision-feedback algorithms is proposed, which offers a promising trade-off between performance and complexity. Incorporating soft information likelihood values) of the transmitted bits into the equalisation process, in order to improve the convergence behaviour of the algorithm, has been studied for SISO systems in [5]. The issue addressed in this paper is an extension of PAPR reduction to MIMO, which is only rarely discussed in literature. The idea presented in this paper is to include the scaling of the transmitted data due to the clipping process in the channel, resulting in an effective channel. Based on this effective channel, the proposed iterative correction method is suitable for low complexity, near optimal MIMO detection schemes e.g. linear MMSE) as well as for powerful schemes e.g. sphere detection, LISS, m-algorithm [6]). In a coded system, a general expression of the remaining distortion noise based on the soft information of the transmitted bits is of importance. This problem will also be addressed in this paper. Due to the superposition of the individual carriers, the distortion noise is almost Gaussian distributed. Thus an analysis of the equalisation process in an EXIT chart [7] is applicable as already introduced in [8] for a single antenna systems. The remainder of this paper is organised as follows. In Sec. II the MIMO-OFDM model extended by clipping and filtering at the transmitter is introduced. In Sec. III we give a detailed description of the iterative receiver processing including clipping compensation and linear minimum mean squared error LMMSE) MIMO detection. Performance for a HiperLAN A channel and EXIT transfer curves are presented in Sec. IV. Finally, conclusions are drawn in Sec. V. II. SYSTEM MODEL PAPR reduction and equalisation is done for each MIMO- OFDM symbol separately. According to figure, we consider MIMO-OFDM transmission with N transmit Tx), N Rx receive Rx) antennas and N C subcarriers. Let V be a vector of information bits which are encoded by an outer code and inter-

2 leaved. The resulting code bit stream is partitioned into blocks X containing N N C M independent binary digits. Here M represents the number of bits per symbol and hence allows to distinguish between 2 M different constellation points. As part of the transmission process, every single block X is mapped onto a N C signal vector S = [S,, S l,, S NC ] T, whose components S l = [S l,, S ltx,, S ln ] T denote the N frequency domain MIMO transmit vectors for each subcarrier. Per antenna, the vector is transformed into time domain by performing an inverse fast Fourier transform IFFT), denoting the time index as n. In order to reduce the PAPR of the transmitted symbols, clipping and filtering CF) is employed per transmit antenna. In this work we deliberately clip the signal in the digital domain in order to avoid clipping in the analog domain. Additionally, a predistortion of the analog power amplifier up to the saturation point is assumed. The complete digital) clipping can hence be modelled by a soft limiter gn)) with clipping level A: z tx n) = gs tx n)) s tx n) = Ae j s txn) 0 s tx n) A s tx n) A where s tx n) denotes the phase angle of the discrete time signal s tx n). A filtering stage is applied on the FFT transformed signal, setting the guard carriers to zero and restoring the pilot symbols. Therefore, any out-of-band radiation is discarded. Although this will lead to a peak regrowth, however, the spectral mask can now be maintained. An additional oversampling would improve the PAPR reduction [2], however, due to simplicity we will not consider additional oversampling in this paper. Afterwards the signal is transformed back into time domain and denoted by z f,tx n). The characteristic of the clipping device is determined by the input power backoff IBO) of the nonlinear clipping device, defined per transmit antenna as: ) IBO tx = A 2 /P tx 2) where P tx represents the transmit power per transmit antenna. Before transmission a cyclic prefix CP) of length N G is added. This is assumed to be longer than the channel impulse response, to avoid possible intersymbol interference caused by frequency selective channels. The time domain signal at receive antenna rx can be expressed as: y rx n) = N tx= z f,tx n) h rx,tx n)/ β f + ξ rx n), 3) where stands for convolution and ξ rx n) represents additive white Gaussian noise with variance σ 2 n. In this notation, h rx,tx n) describes the time domain channel impulse response between transmit antenna tx and receive antenna rx. We applied a power adaption of the transmitted signal represented We use uppercase letters to describe frequency domain and lowercase letters for time domain signals. by the factor / β f in order to be comparable with unclipped transmission. After removing the CP, a discrete Fourier transform is performed per antenna, transforming the received signal back into frequency domain. The overall transmission chain including Fourier transforms is given by the following vector matrix notation: Y = F I Rx )Ψ h/ ) β f ΘF I )Z f + ξ. 4) The symbol denotes the Kronecker Product, I is the identity matrix and F represents the N C N C Fourier matrix. The cyclic prefix is added by multiplication with the matrix Θ and removed by the multiplication with the matrix Ψ. The frequency correspondence of the clipped and filtered transmit signal is represented by Z f. Furthermore, the channel is given by the matrix h of channel impulse responses. Performing standard matrix manipulation the received signal in the frequency domain can be written as: Y = H/ β f Z f + η 5) where we applied the fact that the circular block matrix ΨhΘ can be diagonalised by the IFFT and FFT operation resulting in the N C N Rx N C N block diagonal matrix H which is the frequency domain representation of the channel matrix h. The main tasks are: to solve the spatial multiplexing detection problem and to correct the clipping effects done at the transmitter. III. ITERATIVE CLIPPING CORRECTION A. Bussgang-Theorem Applying the Bussgang-Theorem [9], the output of the nonlinear device in the frequency domain can be written as: Z = αs + D. 6) Note that the distortion noise D is uncorrelated with S and the scaling factor α which is important for the detector filter design as will be shown later. The theorem holds only for strict sense stationary S which is in general not the case. However, it has been found to be an accurate model also for wide sense stationary signals [0] and therefore will be used throughout this paper. Clipping effects the transmit power reflected by the scaling factor β. The power budget of the process can thus be written as: P Z = βp S = α 2 P S + P D 7) where P Z is the power of the clipped signal, P S is the power of the unclipped signal and P D represents the distortion power. The parameters α and β depend on the nonlinearity of the clipping device and can be computed as given in []. The filtering of Z only affects the distortion noise and has no impact on the active subcarriers, resulting in a new scaling factor β f. Thus the output of the clipping and filtering unit can be written as: Z f = αs + D f 8)

3 Fig LMMSE Filter Detector Π Clipping & Filtering SISO Decoder Π Soft Modulation Receiver structure with iterative symbol reconstruction which results in the following power budget Sink P Zf = β f P S = α 2 P S +P Df P Df = β f α 2 )P S. 9) Note, that Due to clipping the orthogonality among the symbols is lost, which is represented by the data depending distortion noise D f, which is statistically dependent on S. B. Signal Detection The optimal MIMO-OFDM detector finds the symbol vector S that has been sent with the highest a-posteriori probability APP) conditioned on the received vector Y Ŝ = arg max P[S Y]. 0) S S However, the computation of the APP would be computationally infeasible. Work on a reduced complexity APP approximation for the SISO case has been done in [4], [2]. In order to reduce the complexity, the APP problem is divided into 2 subtasks which results in a suboptimal representation of the APP task. Firstly the MIMO-OFDM detection problem is replaced by a subcarrier based MIMO detection. Secondly a soft symbol based clipping correction method is applied resulting in a modified received symbol Ỹ l. Here we extend the ideas presented in [3], [4] and [5]. The aim of the MIMO detector is now to provide soft information for each bit based on Ỹ l to the outer decoder. In the following we drop the subcarrier index l to increase readability. Figure 2 depicts the proposed receiver structure. The received signal per subcarrier is given by: Y = H Z f + η = H αs + D f ) + η. ) βf βf The modified received symbol Ỹ is computed by subtracting an estimate of the distortion noise it from the received symbol: Ỹ = Y H/ β f D f. 2) The distortion noise term itself is based on remodulated soft symbols which can be determined by evaluating the likelihood values of the decoded bits [5]: D f = α S + D f ) α S = Z f α S. 3) Note that in the first iteration the soft symbols are equal to zero, thus no distortion noise is subtracted from the received symbols. Thus, the full distortion noise power has to be included in the equalisation process. Using Bayes theorem and under the assumption of statistically independent bits, the L-value soft information) of a certain bit based on the modified received signal Ỹ is defined as: X X LX tx,m Ỹ) = ln tx,m+ pỹ X) P [X] X X tx,m pỹ X) P [X]. 4) The conditioned probability density is given by the complex Gaussian distribution: ) pỹ X) = π Rx detφ νν ) exp Ỹ HS) H Φ ννỹ HS) 5) where Φ νν represents the total noise covariance matrix. Here ν comprises the noise terms D f and η. Using the so called max-log approximation, the L-values from the detector can be approximated by a difference of two maximum operation: LX tx,m Ỹ) max ln pỹ X) ) N + ln X X + tx= max X X ln pỹ X) ) N + ln M m= M tx= m= } P [X tx,m] } P [X tx,m]. 6) For the implementation of Eq. 6) a lot of MIMO detection strategies are known. In this work we will not concentrate on more advanced detection schemes such as sphere detection, LISS detection or m-algorithm [6]. However, the proposed clipping correction algorithm is also applicable for these schemes by applying them on the effective channel H = α H. 7) βf A low complexity scheme is an adapted linear MMSE equalisation, which minimises the mean squared error arg min W E S WỸ 2 }. 8) The resulting filter matrix is given as [ α α W =Φ SS H H 2 HΦ SS H H + βf β f H ) ] Φ Df D β f E D f D H f } H H +Φ ηη 9) f }} Φ D f D f where Φ SS represents the covariance matrix of the transmitted symbols per subcarrier and Φ Df D f represents the covariance matrix of the distortion noise per subcarrier. During the derivation of the filter matrix, the fact was applied that the distortion noise and the symbols S are uncorrelated. Furthermore, we assumed that ED f D H f } E D f D H f }. The novelty of the proposed filter is that the remaining distortion covariance matrix Φ D f D f is included in the filter in addition to the AWGN covariance matrix Φ ηη resulting in Φ νν. During the iterations, the distortion covariance is constantly updated. For that we use a more general definition of the variance of the estimated symbols as given in [6]. E S tx S tx} = σ 2 S tx ϑ tx ), 20)

4 BER Iteration Iteration 2 Iteration 3 Iteration 4 Fig. 3 shows the iterative performance results in terms of bit error rate BER) for 6-QAM modulated system and IBO tx = 0dB per antenna. Note, one iteration means one equalisation and one decoding step. Applying no compensation in the first iteration results in a performance loss of roughly 4dB compared to the linear transmission. The most significant improvement is achieved by the second iteration. Here the gap to linear transmission could be narrowed to approximately db. Performing further iterations allows to get even closer to the clipping free transmission. However, the SNR improvement during the iterations is getting lower. The impact of clipping and filtering gets even worse if higher order modulation schemes are used. Fig. 4 shows the SNR [db] 0 0 Fig. 3. BER performance per iteration, 6-QAM, IBO tx = 0dB per antenna with σs 2 tx as transmit power per antenna, ϑ tx = Var S tx } as soft symbol variance and ϑ = diag ϑ tx ). A variance ϑ tx = 0 corresponds to the case that the soft symbol perfectly matches with one constellation point. On the other hand ϑ tx = represents a soft symbol which is located in the origin of the constellation diagram. Alternatively one can say that the power of the soft symbol is directly related to its variance. The expectation E D f D H f } can now be written as: E D f D H f } = Φ Df D f ϑ. 2) using the power budget given in Eq. 9). The MMSE approach leads to the fact that the filter is not unbiased. This problem can be solved by modifying the filter by a diagonal matrix that restores unit gain, leading to a new filter matrix given as: ) W UB = diag diag W H) W. 22) Using an iterative system in a coded environment, the knowledge of the SINR at each antenna is important for the computation of the L-values. For each antenna the SINR can be computed via the diagonal element of the covariance matrix at the filter output: γ tx = [Φ SS + H H Φ D f D f + Φ ηη ) H) Φ SS ] tx,tx IV. NUMERICAL RESULTS. 23) The performance of the iterative clipping correction in a MIMO-OFDM environment was tested by simulating a 4 4 MIMO IEEE 802.a system with N C = 64 carriers, where 4 carriers are reserved for pilots and 2 for guard carriers. For the channel we simulated for each antenna link an uncorrelated realisation of an HiperLAN A model. For channel coding we used a rate /2 convolutional code with generator polynomial G = [33, 7] 8. FER Iteration Iteration 2 Iteration 3 Iteration 4 Iteration SNR [db] Fig. 4. FER performance per Iteration, 64-QAM, IBO tx = 0dB per Antenna coded frame error rate FER) for 64-QAM transmission and again IBO tx = 0dB per antenna. Performing no correction results in an unacceptable performance loss. Gaining in performance by the second iteration, there is still a loss of roughly 3.5dB compared to the linear clipping free transmission. However, the SNR loss can be further reduced by performing more iterations, resulting in a loss of less than db after the 4th iteration. The improvement of performing a 5th or any further iterations is only marginal compared to the 4th one. An alternative to running extensive computer simulations in order to study the behaviour of an iterative correction algorithm is the use of so called EXIT charts [7]. EXIT charts reflect the convergent behaviour of a system and thus are only valid for infinity long block lengths. Nevertheless, even based on shorter block length they allow a good approximation of the system properties. Knowing the transfer characteristic of the equaliser, one is able to match an appropriate decoder to this characteristic which will be the focus of future studies. Fig. 5 shows transfer curves of the linear transmission diamond markers), linear MMSE based equalisation circular markers) and the decoder for a memory 6 convolutional code no markers). The vertical axis represents the extrinsic information of the equaliser detector) corresponding to the a-priori

5 I A, Dec ; I E, Det SNR: 26dB SNR: 8dB the behaviour of an iterative clipping compensation, which is done besides to linear MMSE equalisation. It has been shown that including the distortion noise in the equalisation step leads to promising performance results. Even for low number of subcarriers the loss in SNR compared to linear transmission is less than 0.5dB. Finally, an analysis of the transfer information using EXIT charts has been done, allowing a system evaluation without performing long computer simulations. Further studies will focus on more advanced detection schemes as well as on code optimisation. 0.2 Memory 6 CC 0. Clipping I E, Dec ; I A, Det Fig. 5. Detector Decoder EXIT chart behaviour, 64-QAM, IBO tx = 0dB per Antenna information of the decoder, while the horizontal axis represents the a-priori information of the equaliser corresponding to the extrinsic information of the decoder. The exchange of information is stopped as soon as an intersection of the detector transfer curve with the decoder transfer curve occurs. At this point any further iteration will not improve the performance of the system. Using linear transmission without clipping together with a 64-QAM gray coded system setup and linear MMSE filtering at the receiver results in an almost horizontal transfer curve. The only gain in extrinsic information is due to the fact that a higher order QAM modulation scheme is used. On the other hand, performing clipping at the transmitter results in a significant loss in extrinsic information if no correction is done at the receiver. Applying the proposed clipping correction algorithm in addition with increasing a-priori information leads to a gain in extrinsic information and hence to a performance improvement of the system. The increase of extrinsic information can nicely be seen by the slope of the curves with circular markers. Furthermore, one can see that providing the equaliser with perfect a-priori information I A,Det = ), the extrinsic information provided from the equaliser almost matches with the extrinsic information of linear transmission. Finally, one can roughly predict the behaviour of the system using EXIT charts. In order to illustrate this statement we compare the extrinsic information of linear transmission at SNR = 8dB with the extrinsic information of clipping performing no correction at SNR = 26dB. Both extrinsic information are approximately 0.55bit. This implies that both systems should roughly result in the same performance behaviour as can be seen in Fig. 4. REFERENCES [] H. Yang, A road to future broadband wireless access: MIMO-OFDM- Based air interface, IEEE Communications Magazine, vol. 43, no., pp , [2] H. Ochiai and H. Imai, Performance Analysis of Deliberately Clipped OFDM Signals, IEEE Transactions on Communications, vol. 50, no., pp. 89 0, [3] J. 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Cioffi, Maximum-Likelihood Detection of Nonlinearly Distorted Multicarrier Symbols by Iterative Decoding, IEEE Transactions on Communications, vol. 5, no. 2, pp , [4] H. Chen and A.M. Haimovich, Iterative Estimation and Cancellation of Clipping Noise for OFDM, IEEE Communications Letters, vol. 7, no. 7, pp , [5] S. Bittner, E. Zimmermann, and G. Fettweis, Low Complexity Soft Interference Cancellation for MIMO-Systems, in IEEE Vehicular Technology Conference VTC), [6] R. Wohlgenannt, K. Kansanen, D. Tujkovic, and T. Matsumoto, Outage-based LDPC Code Design for SC/MMSE Turbo-Equalization, in IEEE Vehicular Technology Conference VTC), V. CONCLUSIONS Looking at advanced multi antenna detection schemes it is common to perform detector-decoder iterations in order to improve the system performance. In this work we discussed