Diversity-Enabled and Power-Efficient Transceiver Designs for Peak-Power-Limited SIMO-OFDM Systems
Orthogonal frequency division multiplexing (OFDM) has been widely adopted for high data rate wireless transmissions. By deploying multiple receiving antennas, single-input multiple-output- (SIMO-) OFDM can further enhance the performance with spatial diversity. However, due to the large dynamic range of OFDM signals and the nonlinear nature of analog components, it is pragmatic to model the transmitter with a peak-power constraint. A natural question to ask is whether SIMO-OFDM transmissions can still enjoy the antenna diversity in this case. In this paper, the effect of the peak-power limit on the error performance of uncoded SIMO-OFDM systems is studied. In the case that the receiver has no information about the transmitter nonlinearity, we show that full antenna diversity can still be collected by carefully designing the transmitters, while the receiver performs a maximum ratio combining (MRC) method which is implemented the same as that in the average power constrained case. On the other hand, when the receiver has perfect knowledge of the peak-power-limited transmitter nonlinearity, zero-forcing (ZF) equalizer is able to collect full antenna diversity. In addition, an iterative method on joint MRC and clipping mitigation is proposed to achieve high performance with low complexity.
KeywordsOrthogonal Frequency Division Multiplex Fading Channel Orthogonal Frequency Division Multiplex System Orthogonal Frequency Division Multiplex Symbol Maximal Ratio Combine
Orthogonal frequency division multiplexing (OFDM) has been adopted by various modern communication standards because of its high spectral efficiency and low complexity in combating frequency-selective fading effects [1, 2]. Equipped with multiple antennas, OFDM systems can further enhance the performance by collecting spatial diversity . Thus, multiple-input multiple-output (MIMO) OFDM transmission has been adopted by several communication standards and becomes a strong candidate for future cellular systems .
However, OFDM experiences certain implementation challenges due to the large dynamic range of its signal waveforms, which is usually measured by the peak-to-average power ratio (PAR) . Large PAR values may lead to low power efficiency or severe nonlinear distortions which decrease system performance. It is possible to back-off (i.e., scale down) the waveform so that distortions are less likely, but this comes at the cost of the reduced transmission power efficiency. Conversely, although the signal power can be boosted by reducing the amount of back-off, nonlinear distortions will inevitably be increased. Power efficiency and nonlinear distortions are thus conflicting metrics that must be balanced. There has been extensive research on improving the transmission power efficiency with constraints on nonlinear distortions in single-input single-output (SISO) OFDM channels (see e.g., ).
In light of the power efficiency and nonlinear distortion considerations, the error performance of OFDM systems with peak-power-limited power amplifier (PA) should be investigated. To quantify the error performance of wireless transmissions over fading channels, two parameters are usually used: diversity order and coding gain (see e.g., [7, 8]). The diversity order describes how fast the error probability decays with signal-to-noise ratio (SNR), while the coding gain measures the error performance gap among different schemes when they have the same diversity. Thus, diversity-enabled transceivers have well-appreciated merits. For single-antenna OFDM systems with clipping at the transmitter, the approximated symbol error rate (SER) has been derived for maximum-likelihood sequence detection (MLSD) . The results show that the clipping nonlinearity leads to a certain (may not be full) multipath diversity order over frequency-selective Rayleigh fading channels. However, MLSD requires near exponential complexity to collect some diversity gain. When the number of subcarriers is large which is usually the case in current standards, the complexity of MLSD is prohibitive. In such a case, the OFDM system loses its advantage as a simple equalizer which may reduce its practical applicability.
In this paper, we are interested in low-complexity diversity-enabled transceiver design over peak-power-limited channels. Instead of the multipath diversity, we focus on the antenna diversity from multiple antennas deployed at the receiver (i.e., single-input multiple-output (SIMO) channels). When OFDM signals are linearly transmitted, linear equalizers are sufficient to collect the antenna diversity by optimally combining the multiple faded replicas of the same information bearing signal [10, 11]. However, to the best of our knowledge, the question of whether and how the peak-power-limited SIMO-OFDM system can still enjoy antenna diversity with linear equalizers has not been addressed in the literature. A few iterative methods to reconstruct the clipped OFDM signals in multiple-antenna systems have been proposed in [12, 13, 14] based on the assumption that the receiver knows the transmitter nonlinearity. However, the diversity gain has not been quantitatively analyzed.
This paper focuses on error performance analysis for SIMO-OFDM systems over peak-power-limited channels. Several low-complexity transceiver designs are proposed to collect the antenna diversity and near maximum-likelihood (ML) SER performance is achieved.
The rest of the paper is organized as follows. The OFDM system and SIMO channel models are described in Section 2. In Section 3, the diversity combining methods for linear SIMO channels are briefly reviewed. The transceiver designs over the peak-power-limited SIMO channels are mainly discussed in Sections 4 and 5 based on different a priori information requirements. Numerical results are shown in Section 6. Finally, conclusions are drawn in Section 7.
Throughout this paper, we use lower-case and upper-case bold face letters for column vectors and matrices, respectively. Their elements are denoted in italic with subindices. * denotes conjugate, T transpose, and H Hermitian. Let blackboard bold letters represent number sets, then Open image in new window stands for an Open image in new window matrix whose elements belong to a number set Open image in new window . In particular, we use Open image in new window to represent the set of all complex numbers. Open image in new window stands for the Open image in new window th norm of vector Open image in new window . Open image in new window is an Open image in new window -by-1 vector with all zero entries and Open image in new window is an Open image in new window -by- Open image in new window identity matrix. Open image in new window denotes a diagonal matrix with vector Open image in new window on its diagonal and Open image in new window stands for the trace of a matrix. Additionally, Open image in new window is used for the expectation over a random variable Open image in new window .
2. System Model
where Open image in new window is the Open image in new window oversampling FFT matrix formed by retaining only the first Open image in new window rows of a full FFT matrix whose Open image in new window th entry is Open image in new window . Since this FFT operation is unitary, we have Open image in new window .
where Open image in new window denotes the phase of a complex variable Open image in new window . Without loss of generality, unit gain is assumed for the PA linear region. The input back-off (IBO) is defined as Open image in new window . Clipping occurs when Open image in new window .
Notice that, by digital clipping and filtering methods, out-of-band spectral regrowth can be constrained according to certain spectral mask or totally eliminated [15, 16]. In this case, the following analysis still holds valid and the proposed methods can be modified accordingly by treating the clipping and filtering as a deterministic nonlinear process.
where Open image in new window denotes the OFDM symbol received on the Open image in new window th antenna. Open image in new window , Open image in new window , and Open image in new window ( Open image in new window ) is the channel frequency response of the Open image in new window th in-band subcarrier on the Open image in new window th receiving antenna. In addition, Open image in new window and Open image in new window where Open image in new window ( Open image in new window ) consists of the circularly complex white Gaussian noise with variance Open image in new window .
In this paper, we study the symbol error rate (SER) in peak-power-limited SIMO fading channels. First, we give some definitions.
Definition 1 (PSNR).
Definition 2 (Diversity gain).
Unlike linear channels, the SER and the diversity gain are defined in terms of PSNR in peak-power-limited channels. For certain transmitters with a given Open image in new window , the diversity gain describes how fast the SER decays with decreasing channel noise power.
3. Diversity Combining in Linear SIMO Channels
For linear SIMO channels, several diversity combining techniques are available to achieve the antenna diversity , for example, maximal ratio combining (MRC) and selective combining (SC). Before discussing the peak-power-limited case, we briefly review the MRC method for the linear SIMO-OFDM channel.
where Open image in new window is the Open image in new window th column of Open image in new window , that is, Open image in new window . The corresponding received SNR is Open image in new window . In the end, the decision Open image in new window is obtained by hard decoding on Open image in new window , denoted as Open image in new window
that is, MRC collects full antenna diversity. From the existing literature, however, it is not clear yet whether (and if so, how) full antenna diversity can be achieved in the presence of the peak-power constraint. We address this open question in the following sections.
4. Transparent Receivers: A Statistical Model
By "transparent" we mean that the receivers have no information about the transmitter nonlinearities. In this case, no receiver-side cooperation is expected. The nonlinear distortion noise has to be dealt with in the same way as the uncorrelated Gaussian channel noise. Therefore, the following statistical model is introduced at first to quantify the clipping noise.
Definition 3 (Statistical model).
where Open image in new window is the equivalent channel frequency response. The frequency-domain nonlinear distortion noise can be found as Open image in new window with power Open image in new window .
To maximize the SNDR, the MRC weights are given in the following proposition.
For transparent receivers that have no information about the transmitter nonlinearity, the optimal MRC weights are given by Open image in new window whose Open image in new window th column is Open image in new window , where Open image in new window ( Open image in new window ).
See Appendix .
At first, it appears that the transparent receiver has to know Open image in new window in order to acquire Open image in new window , which is inconsistent with the "transparent" definition. In fact, for OFDM systems with embedded pilot subcarriers, since the pilot signals are also attenuated by Open image in new window , Open image in new window is the effective channel response which is acquired by channel estimation at the receiver. Therefore, transparent receivers do not need to know Open image in new window aforehand and the SNDR-maximizing combining weights can be used to achieve the best error performance.
Unlike the linear case, using the optimal MRC weights at the receiver may not guarantee full antenna diversity. The necessary and sufficient condition for achieving the antenna diversity gain with transparent receivers is given as follows.
For OFDM transmitters with a fixed peak-power limit, the transparent receiver is able to achieve full antenna diversity if and only if the distortion noise vanishes as the PSNR increases.
See Appendix .
Example 1 (Constant clipping).
When a constant IBO is maintained, clipping occurs if the PAR of an OFDM symbol exceeds the IBO. It implies that Open image in new window and Open image in new window for the statistical model in (12). Therefore, no antenna diversity can be achieved with transparent receivers. In fact, as indicated in Appendix , error floor should be observed.
Example 2 (Piece-wise linear scaling).
Because Open image in new window so that clipping never occurs, we have Open image in new window and Open image in new window in (3) and (4). The symbol-wise gain will not affect the demodulation because it is essentially a part of the channel and can be recovered by receivers with pilot-aided channel estimation.
which is inversely proportional to the harmonic mean of the PAR. Still, low power efficiency and small coding gain may result due to the large PAR of OFDM signals. A number of distortionless methods have been proposed to reduce the PAR of OFDM signals, for example, coding , selected mapping , and tone reservation . They can be combined with PWLS and improve the coding gain at the cost of implementation complexity, spectral efficiency and/or receiver-side cooperation.
Example 3 (Optimal clipping).
where Open image in new window is the SER for Open image in new window -ary QAM constellations and Open image in new window [7, page 278]. When the OFDM waveform is approximated as a complex Gaussian random variable, a numerical method to solve for Open image in new window can be found in .
Unlike PWLS which is trying to avoid any clipping, the optimum clipping method maximizes the SNDR in (18) for a given PSNR. In the high PSNR region, a large Open image in new window is yielded in which case Open image in new window and Open image in new window . Thus, full antenna diversity is sustained according to Proposition 2. On the other hand, in the low PSNR region, some distortion is introduced to achieve a more desired tradeoff with the increase in signal power so that the error performance is optimized. Therefore, the optimal clipping method can achieve a better coding gain while maintaining the full antenna diversity for transparent receivers.
5. Transmitter Nonlinearity Known at the Receiver: A Deterministic Model
Instead of a random process, the clipping distortion, based on the PA model in (3), is a deterministic function of the data. When the receiver knows or estimates a priori the transmitter nonlinearity, it can exploit the deterministic nature of the clipping process for better performance . In this case, receiver-side cooperation can be adopted to achieve antenna diversity with nondiminishing distortion noise at the transmitter. The corresponding system diagram is given in Figure 1(b).
In order to design the receiver-side cooperation, we first establish a deterministic model to characterize the clipping process.
Definition 4 (Deterministic model).
Although clipping was also shown to enable certain multipath diversity in frequency-selective fading channels , we focus on antenna diversity in this paper. In addition, the SER performance for clipping and filtering oversampled OFDM signals was shown to be well approximated by that of the Nyquist sampling in SISO fading channels . This approximation remains for the SIMO channel case. Therefore, the average SER for general SIMO fading channels can be approximated by (22), which is referred as the MLSD bound in accordance with . Again, full antenna diversity can be verified similar to (B.4) in Appendix .
where Open image in new window . In the following, we first quantify the diversity order collected by the ZF equalizer when Open image in new window is known. Then, an iterative method will be proposed to jointly estimate both Open image in new window and Open image in new window and realize the ZF equalizer in the absence of a priori knowledge about Open image in new window .
For clipped OFDM signals transmitted through SIMO fading channels with Open image in new window receiving antennas, if the receiver has perfect knowledge of the Open image in new window given in (21), the diversity order collected by the ZF equalizer is Open image in new window .
See Appendix .
Proposition 3 states that ZF equalizers can achieve full antenna diversity if the clipping-based matrix Open image in new window is known or can be estimated at the receiver. It also indicates that in frequency-selective fading channels, ZF equalizers are not able to collect any multipath diversity. It is the compromise that low-complexity solutions have to make. The same fact was previously observed in  without proof. It is also worth mentioning that, unlike the linear case in Section 3, MRC is no longer the same as the ZF equalizer in the presence of clipping.
where Open image in new window denotes the estimate for the corresponding variable and the superscript Open image in new window stands for the iteration index. As the initialization, Open image in new window .
We refer to it as the joint MRC and clipping mitigation method. Its complexity is dominated by one pair of Open image in new window operations per iteration and on the order of Open image in new window .
Open image in new window is decreasing quickly, especially in the high PSNR region, which will be shown in Section 6. As a result, the joint estimation method can empirically approach the ideal case of ZF equalizers. Acting as the receiver-side cooperation as plotted in Figure 1(b), it can collect full antenna diversity with constant clipping at the transmitter.
Two more remarks about the use of the joint MRC and clipping mitigation method are now in order.
where Open image in new window denotes the simulated average SER performance for the joint MRC and clipping mitigation method.
The proposed method can be regarded as an extension to the iterative quasi-ML clipping estimation method , which was designed for SISO-OFDM systems. However, the quasi-ML clipping estimation method provides poor error performance in fading channels, which will be shown in Section 6. The main reason is that the subcarriers with deep fadings will have low received SNR and large error probabilities. The clipping estimation then propagates the errors and yields degraded estimation for both clipping noise and data. In SIMO fading channels, multiple receptions over independently faded channels not only provide the diversity gain for the data error performance, but also achieve better estimation for the clipping noise. The proposed joint MRC and clipping mitigation method thus exploits this benefit. In Section 6, we will show that the SER performance gets close to the MLSD bound within five iterations even for very small IBOs.
In summary, the proposed joint MRC and clipping mitigation method can provide the near-MLSD error performance. It requires the knowledge about the transmitter nonlinearity as well as receiver-side modifications. Compared with the transparent receiver, the extra complexity is on the order of Open image in new window , which is far less than the complexity of MLSD. From the transmitter perspective, the joint MRC and clipping mitigation method has lower complexity than PWLS and optimal clipping schemes. In addition, it can achieve better coding gain, which will be shown in the following section.
6. Simulation Results
For all simulations in this section, the uncoded OFDM system has Open image in new window subcarriers and uses 16-QAM modulation. Unless otherwise specified, frequency-selective Rayleigh fading channel with two taps and Open image in new window receiving antennas are assumed. Since the antenna diversity is focused in this paper, the results are independent with the number of channel taps as long as the total average gain of these taps stays the same. In addition, ideal channel estimation is assumed so that Open image in new window is known at the receivers.
First, the ideal case with Open image in new window dB but linear PA (i.e., no clipping, thus Open image in new window and Open image in new window ) is plotted as a benchmark in Figure 2. Although only constant-envelope modulations (rather than OFDM) may actually achieve this error performance in practice, it gives an SER lower bound for this channel. For OFDM, by setting Open image in new window and assuming no clipping happens, Monte Carlo simulation gives the SER curve for this ideal case. The curve agrees well with the theoretical MLSD bound in (22) with Open image in new window dB and Open image in new window .
Using the transparent receivers with the MRC weights given in Proposition 1, three transmitter schemes are also compared in Figure 2, namely, the constant clipping, the PWLS, and the optimal clipping approaches. As expected in Section 4, no antenna diversity can be obtained with the constant clipping method. In fact, the SER reaches an error floor that is determined by the clipping threshold. The PWLS-based transceiver can provide full antenna diversity but poor coding gain. Compared to the case with ideal linear PA, the PSNR degradation Open image in new window is more than 9 dB in the simulated system, as shown in Figure 2. On the other hand, the optimal clipping method achieves about 3 dB coding gain better than PWLS.
In this paper, we have examined the antenna diversity gain in the peak-power-limited SIMO-OFDM system. The main conclusion is that full antenna diversity can be achieved for the transparent receiver by intelligently choosing the transmission method: PWLS and optimal clipping achieve diversity, while a constant back-off clipping does not. To achieve full antenna diversity, the MRC coefficients are derived for the peak-power-limited channel and can be obtained in the same way with those in the average-power-constrained linear channel. Additionally, we showed that for systems where the receiver has perfect knowledge of the transmitter nonlinearity, antenna diversity can be achieved with low-complexity linear equalizers. The joint MRC and clipping mitigation method is also proposed to employ the multiple antennas to better estimate both the clipping noise and the data. To extend the results to coded multiantenna OFDM systems is a part of our future work.
A. Proof of Proposition 1
Obviously, Open image in new window satisfies (A.2). In addition, these weights are channel-normalizing (i.e., Open image in new window ) as well as orthogonal to the channels of other subcarriers (i.e., Open image in new window , Open image in new window ). Therefore, Open image in new window with Open image in new window gives the optimal MRC weights and the transparent receiver can decode according to Open image in new window .
B. Proof of Proposition 2
As mentioned in Section 4, Open image in new window . In addition, Open image in new window . Therefore, Open image in new window is the necessary condition for the limit of SNDR in (B.2) to go to infinity, as well as for the transparent receiver to collect antenna diversity.
where Open image in new window .
Therefore, for a fixed Open image in new window , the necessary and sufficient condition for the transparent receiver to collect full antenna diversity is that the distortion noise power vanishes as the PSNR increases.
C. Proof of Proposition 3
This work was supported in part by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011.
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