to deliver just-acceptable error rate (JAER) performance. Error rate models for two popular MIMO detection algorithms are derived given the channel matrix.
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Transactions Papers Efficient Channel-Adaptive MIMO Detection Using Just-Acceptable Error Rate I-Wei Lai, Student Member, IEEE, Gerd Ascheid, Senior Member, IEEE, Heinrich Meyr, Fellow, IEEE, and Tzi-Dar Chiueh, Senior Member, IEEE
Abstract—This paper proposes a new concept of multipleinput multiple-output (MIMO) detection, aiming at minimizing the average computational cost. The detection methods are adapted according to the estimated channel state information to deliver just-acceptable error rate (JAER) performance. Error rate models for two popular MIMO detection algorithms are derived given the channel matrix. From these models, a channeladaptive-MIMO (CA-MIMO) receiver with detector-switching strategies is proposed. Simulation results demonstrate that the proposed CA-MIMO detector meets the JAER criterion efficiently. Compared with the sophisticated Sphere Search (SS) MIMO detector, the average saving at moderate signal-to-noise ratio (SNR) is around 58% to 72%, depending on different modulation alphabets. At high SNR, the CA-MIMO detector almost always switch to Zero-Forcing (ZF) detection, where the complexity is several orders lower than the SS detector. Index Terms—MIMO detection, OFDM, sphere search, zeroforcing, error probability, maximum likelihood, just-acceptable error rate (JAER).
I. I NTRODUCTION
W
ITH growing user demand for ubiquitous and broadband connection to the internet, future wireless communication system must provide higher data rate and wider coverage with limited spectrum resource. Thus, a top priority for wireless communication engineering is to develop reliable wireless systems with high spectral efficiency. Multiple-input multiple-output (MIMO) technology has attracted a great deal of attention and was adopted in several recent wireless standards, such as IEEE 802.11n, IEEE 802.16e, and 3GPP Long-Term Evolution (LTE). It exploits multiple antennas at both the transmitting and the receiving ends to achieve
Manuscript received July 28, 2009; revised April 15, 2010 and September 3, 2010; accepted September 20, 2010. The associate editor coordinating the review of this paper and approving it for publication was M. Uysal. This work was supported in part by the National Science Council, Taiwan, under Grant no. NSC 99-2219-E-002 -018, the Ministry of Education, Taiwan, under Excellent Research Projects of National Taiwan University, 99R80304, and the UMIC Research Centre, RWTH Aachen University. The authors are with the Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, Taiwan; and the Institute for Integrated Signal Processing System, RWTH Aachen University, Aachen, Germany. This paper was presented in part at IEEE VTC 2009-Spring, Barcelona, Spain. Digital Object Identifier 10.1109/TWC.2010.101810.091129
different objectives. Among them, spatial multiplexing can multiply attainable data rates by transmitting independent data streams from the transmit antennas. Over the years, several MIMO detection algorithms for spatially-multiplexed signals have been developed. An optimal solution to detecting signals is exhaustive search (ES) according to the Maximum Likelihood (ML) criterion. Despite minimizing error rate, this approach suffers from enormous computational complexity, especially in wireless communication systems with large number of antennas. On the other hand, Ordered Successive Interference Cancellation (OSIC) [1], Minimum Mean-Squared Error (MMSE), and Zero-Forcing (ZF) detection [2] all demand relatively lower complexity and therefore are more popular in actual implementations. Unfortunately, these solutions, though computationally efficient, may perform poorly in channels with low signal-tonoise ratio (SNR) because they fail to fully reconstruct the signal for final decision as the channel gets noisier. To achieve satisfactory error rate in these situations, reduced-complexity search methods that perform almost as well as the ES have been proposed. Among these improved search-based methods, Sphere Search (SS) [3] effectively transforms the ES into a tree-based search with extensive pruning of unlikely branches. There are three major types of SS: depth-first search, breadthfirst search, and best-first search [4], [5], [6], [7]. All these three types of searches have been shown to be capable of efficient MIMO detection at low SNR, though their complexity is still huge when compared with that of OSIC, MMSE, and ZF MIMO detectors. Many current wireless standards specify minimum error rates to guarantee quality of service (QoS) for their users. The rationale is that setting a criterion on minimum error rate performance mandates all standard-compatible receivers to provide acceptable system performance. For example, in DVB-T/H a coded bit error rate (BER) after the inner FEC decoder of 2 × 10−4 leads to no discernible disturbance (after the outer decoder)when viewing video programs (Quasi-Error Free, QEF). Likewise, in the IEEE 802.16e standard, 10−6 is specified as the minimum required coded BER. Another example is the IEEE 802.11a/g/n wireless LAN standard, wherein the receiver packet error rate (PER) is required to be
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less than 10%. In this context, reaching performance beyond the required minimum error rate seems not only unnecessary but also wasteful. Toward this end, it is natural to employ only low-complexity MIMO detection solutions, such as ZF detection or OSIC, under “good” channel scenarios as they achieve acceptable performance with less computational complexity. On the other hand, for “poor” channel conditions, the searchbased MIMO detectors seem to be indispensable. Mobile communication receivers experience fluctuating channel conditions, where the SNR and channel responses vary all the time. To fully exploit channel variations that vacillate between “good” conditions and “poor” conditions, we propose a new concept called channel-adaptive-MIMO (CAMIMO) detection with “just-acceptable error rate (JAER)” performance. In this adaptive receiver, a proper MIMO detection method is adopted on the fly according to the estimated channel conditions and the acceptable error rate. Specifically, given a few MIMO detection methods with different levels of detection performance, computational complexity, and transmission parameters, the receiver always adopts the lowest-complexity method that provides just low enough error rates. The proposed CA-MIMO detection can operate in a wide range of channels with minimum cost (e.g., power consumption) since the JAER criterion translates to substantial computational complexity reduction during good channel conditions. Similar concept of on-line switching MIMO detection based on the condition number have been proposed in [8], [9], [10]. Nevertheless, these works aim at switching to lowercomplexity algorithms with negligible error rate performance loss, while we propose to adopt the lowest-complexity algorithm that meets the JAER criterion, resulting in more complexity saving. The above concept, though intuitive and quite effective, still leaves a few hurdles to overcome. For instance, the relationship between MIMO detection methods and their respective achievable error rates must be established. Besides, for successful deployment the receiver must detect the channel condition and set up a workable detector-switching strategy in all operating environments. In case of actual hardware realization, the MIMO detection methods should preferably comprise similar mathematical operations to enable hardware reuse and thus reduce the overhead of multiple detector hardware. In this paper, the CA-MIMO detection with the JAER criterion is implemented in a wireless baseband receiver conforming to the IEEE 802.11n standard using MIMO orthogonal frequency division multiplexing (OFDM) technology. Several features of the proposed CA-MIMO detection are listed below ∙ Just Acceptable Error Rate (JAER) concept, enabling low-complexity adaptive MIMO detection, ∙ Theoretical error rate models for two MIMO detection algorithms allowing smooth and timely MIMO detector switching according to estimated channel condition, ∙ Derivation of an uncoded BER threshold based on the coded BER target such that once the proposed soft-output CA-MIMO meets the uncoded threshold, the coded JAER is guaranteed, ∙ A fine-grained MIMO detector switching strategy that consider subcarriers with drastically different fading levels and balance their BER performance,
Delta lattice scheme that simplifies the complexity of onthe-fly error rate prediction, and ∙ Up to 72% computational complexity saving at moderate SNR (for 64QAM) and very low complexity ZF detection at high SNR while still meeting the error rate requirement. The rest of the paper is organized as follows: in Section II the MIMO-OFDM signal model and the proposed 802.11n receiver are described. Section III derives the mathematical expression of the error rate performance for ZF and ES detection. Based on these two error rate models and the derived uncoded BER threshold, a detector-switching strategy is introduced in Section IV. Numerical simulations in Section V verify the feasibility and efficiency of the proposed scheme. Finally, discussions about further application of CA-MIMO and the JAER concept and conclusions are presented in Section VI. ∙
II. S YSTEM M ODEL AND CA-MIMO D ETECTION A. System Model The IEEE 802.11n wireless LAN standard is one of the first systems adopting the MIMO-OFDM technology. This powerful combination of MIMO with OFDM effectively enhances the achievable data rate and spectral efficiency by utilizing both the frequency and space diversity. A simplified 𝑀𝑡 × 𝑀𝑟 MIMO-OFDM communication system with 𝑀𝑡 transmit antennas and 𝑀𝑟 receive antennas transmitting over a multipath channel is illustrated in Fig. 1. Information bits are encoded with a rate 𝑟 error-correcting code. The encoded bit stream is then interleaved and mapped onto an 𝑀 -ary constellation set 𝜒 with a mapping rule ℳ. After modulation via the inverse fast Fourier transform (IFFT) and the guard interval (GI) insertion, the resulting 𝑀𝑡 symbols are simultaneously transmitted. At the receiver side, we assume that both the normalized delay and Doppler spread are small, and the synchronization is perfect so that the intersymbol interference (ISI) and intercarrier interference (ICI) can be ignored. When the subcarrier spacing of an OFDM system is narrower than the coherent bandwidth of the channel, one then deals with a flat channel scenario in each subcarrier [2]. Therefore, the received frequency-domain baseband signal, i.e., signal after the fast Fourier transform (FFT) and the GI removal, in a MIMO-OFDM receiver is given by y𝑘 = H𝑘 x𝑘 + n𝑘 , where y𝑘 , H𝑘 , x𝑘 , and n𝑘 are the received MIMO-OFDM signal, the channel frequency response matrix, the transmitted MIMO signal, and the noise vector at the 𝑘th subcarrier of the current OFDM symbol, respectively. The complex additive noise n ∈{𝐶 𝑀𝑟 ×1 } is assumed to be zero mean with covariance matrix E nn𝐻 = 𝜎𝑛2 I𝑀𝑟 = 𝑁0 I𝑀𝑟 , where (⋅)𝐻 denotes the Hermitian (conjugate transpose) operation, and I𝑀𝑟 is the identity matrix of size 𝑀𝑟 . The{ transmitted symbol vector x } is spatially independent, i.e., E xx𝐻 = 𝜎𝑥2 I𝑀𝑡 . We focus on MIMO detection with estimated CSI (instead of perfect CSI) but ignore other non-ideal effects, such as carrier frequency offset, symbol timing uncertainty, IQ imbalance, DC offset and phase noise [2]. The remaining part of this section will introduce the proposed channel estimator,
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noise-plus-interference power estimator, and MIMO detectors with different levels of detection performance and complexity. We assume a quasi-static channel model, i.e. the channel impulse responses are identical over some time interval, e.g., one 802.11n packet period. The channel estimator employs the high throughput long training fields (HT-LTFs) to extract channel frequency responses by the Least Square (LS) method [2]. A linear MMSE (LMMSE) filter with 2𝐿 + 1 taps is cascaded to smooth the LS channel estimates. It should be emphasized that for the filter with infinite taps, the estimation error is always smaller after smoothing [12]. However, the implementation loss (finite tap [13]) in practical scenarios restricts such smoothing gain to low-to-moderate SNR cases. The noiseplus-interference power estimator calculates the additive noise and the interference caused by channel estimation error using ˆ 𝑘 is the estimated channel matrix at the 𝑘th subcarrier where H ˆ 𝑘 is the ˆ 𝑘 = H𝑘 − H of the current OFDM symbol, and E channel estimation error. Then, we can estimate the power of noise plus interference by averaging 𝜀𝑘 over all subcarriers in the HT-LTFs ∑
1 𝑁HT-LTF𝐾
𝐾 ∑
𝜀𝑘 𝜀𝐻 𝑘 ,
(1)
all symbols in 𝑘=1 HT-LTF
where 𝑁HT-LTF is the number of HT-LTFs, and 𝐾 is the number of used subcarriers in one HT-LTF. Since the data symbol power and HT-LTF preamble power are both 𝜎𝑥2 , Eq. (1) also represents the estimation of the power of noise plus interference at data subcarriers 𝐸{𝒫ˆ𝜀 I𝑀𝑟 } = 𝐸{
1 𝑁HT-LTF𝐾
= (𝑁0 +
∑
𝐾 ∑
Fig. 2.
Details of the CA-MIMO detector.
B. CA-MIMO Detection
ˆ 𝑘 x𝑘 = E𝑘 x𝑘 + n𝑘 , 𝜀𝑘 = y𝑘 − H
𝒫ˆ𝜀 I𝑀𝑟 =
˜
𝜀𝑘 𝜀𝐻 𝑘 }
all symbols in 𝑘=1 HT-LTF 𝑀𝑡 𝜎𝑥2 𝜎𝑒2 )I𝑀𝑟 ,
where 𝜎𝑒2 is the variance of the channel estimation error.
As an illustration of the proposed soft-output channeladaptive MIMO (CA-MIMO) detection, we adopt two wellknown algorithms, ZF (lowest complexity, worst performance) and SS with unified tree search [11] (very high complexity, best performance). A detector-switching unit first predicts the error-rate performance of different MIMO detection methods according to the estimated channel matrix and power of noise plus interference. The lowest-complexity detection method meeting the JAER criterion is then determined (the details of the detector-switching strategy will be explained in Section VI). The proposed CA-MIMO detector is depicted in Fig. 2. The detector-switching unit informs the CA-MIMO detector the adopted detection method for each stream using a flag vector F, where 𝐹𝑙 = {ZF, SS} and 𝑙 ∈ [1, 𝑀𝑡 ]. In other words, each stream is detected by either the ZF detection or SS. Therefore, in a 4 × 4 MIMO system, the proposed CA-MIMO detector switches among four different schemes: all ZF, (2, 2)ZF/SS, (1, 3)ZF/SS, and all SS, where (𝑎, 𝑀𝑡 − 𝑎)ZF/SS refers to the detection method that 𝑎 streams are detected by ZF, and the other 𝑀𝑡 − 𝑎 streams are collectively detected by SS. In
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the second and the third schemes, the order of ZF detection and SS detection is decided based on the predicted uncoded BER of the two detection methods. Note that the streams detected by the SS method is considered as a group as they have to be jointly detected, while the streams using the ZF method can be detected individually and hence their order is determined in this phase. Then, the order of the streams inside the SS group is sorted based on the predicted uncoded BER under ZF assumption, similar to the conventional sorted QR decomposition (QRD) [14]. With all streams sorted, a 𝑀𝑡 × 𝑀𝑟 sorted QRD is perˆ and formed. The resulting signals, i.e., the triangular matrix R the received symbol vector times the Hermitian of the unitary ˆ 𝐻 y are fed to the two MIMO detectors with in˜=Q matrix y terference cancellation (IC). Note that the inter-group interference cancellation (IC) between two detection methods should be treated carefully. In the proposed CA-MIMO detector, softvalued symbols are adopted in IC to reduce error propagation. The detection is executed from the stream corresponding to the ˆ and proceeds upwards. Finally, based bottom-most layer of R on the max-log-MAP criterion [15], [11], the soft-output loglikelihood ratios (LLRs) are computed, reordered, and fed into the de-interleaver and the channel decoder. III. E RROR R ATE M ODELS FOR MIMO D ETECTOR S WITCHING To select an adequate detection method, the CA-MIMO receiver has to establish the relationship between the error rate performance and the channel condition for different MIMO detection methods. For convolutional codes, a tight upper bound on the coded BER 𝑃𝑐𝑏 is given by [16] 𝑃𝑐𝑏 ≤
∞ 1 ∑ 𝑊𝐼 (𝑑)𝑓 (𝑑, ℳ, 𝜒), 𝑘𝑐
(2)
𝑑=𝑑min
where 𝑘𝑐 denotes the number of information bits used per encoding operation, 𝑊𝐼 (𝑑) is the total input weight of error events, 𝑓 (𝑑, ℳ, 𝜒) is the pairwise error probability (PEP), which depends on the mapping rule, constellation set, and the Hamming distance between two error events, and 𝑑min is the minimum Hamming distance of the convolutional code. As the PEP with 𝑑 = 1 is simply the uncoded BER, the coded BER can be linked to the uncoded BER by the PEP. The PEP can be evaluated by integrating 𝜓(𝑠), the average of the bilateral Laplace transform of the probability density function (pdf) of the metric difference Λ(𝒙, 𝒛) [17] )𝑑 ∫ 𝜈+𝑗∞ ( 𝑑𝑠 1 , (3) 𝜓(𝑠) 𝑓 (𝑑, ℳ, 𝜒) = 2𝜋𝑗 𝜈−𝑗∞ 𝑠 where 𝜈 is chosen such that the integral converges. 𝜓(𝑠) is defined as 𝜓(𝑠) =
𝑀𝑡 ∑ 𝐼 ∑ 1 ∑ ∑ 1 𝐼𝑀𝑡 2𝐼𝑀𝑡 x 𝑖=1 𝑙=1
𝑏=0
∑
z 𝑥𝑙 ∈𝜒𝑖𝑏 𝑧𝑙 ∈𝜒𝑖∼𝑏
ΦΛ(𝒙,𝒛) (𝑠). (4)
Herein, ΦΛ(𝒙,𝒛) (𝑠) is the bilateral Laplace transform of the pdf of the metric difference Λ(𝒙, 𝒛) Λ(𝒙, 𝒛) = log 𝑝(y∣x) − log 𝑝(y∣z),
(5)
where x is the transmit symbol vector and z represents the erroneously-detected symbol vector. The 𝑙th entry of the transmitted symbol 𝑥𝑙 belongs to the subset 𝜒𝑖𝑏 whose labels have the binary value 𝑏 ∈ [0, 1] at the 𝑖th bit position with 𝑖 ∈ [1, 𝐼] and 𝐼 = log2 (𝑀 ). Note that 𝑝(y∣x) and 𝑝(y∣z) in Eq. (5) depend on the adopted detection algorithm. In the following subsections, we analyze the PEP and the uncoded BER of the ZF detection and ES in preparation for the detector-switching strategy. The uncoded BER is of our interest because the online switching is performed on a stream basis such that the uncoded error rate is the direct metric of adaptation rather than coded one. An uncoded threshold is also derived based on a given coded BER value and is proved that as long as the CA-MIMO detector meets such uncoded threshold, the coded JAER criterion is satisfied.
A. Error Rate Analysis for ZF Detection In a MIMO receiver with ZF MIMO detection, it is proven that the expurgated bound for a single-input single-output (SISO) system [17] can be directly applied [15]. Dropping the subcarrier index 𝑘 and the OFDM symbol index 𝑗 without loss of generality, the ZF detector recovers the transmitted data symbol vector with the estimated channel matrix ˆ 𝐻 Ex + G ˆ 𝐻n ˆ 𝐻y = x + G ˆ=G x where
ˆ −1 H ˆ𝐻 ˆ 𝐻 = (H ˆ 𝐻 H) G ˆ E = H − H.
Following the orthogonal projection theorem, which the LMMSE channel estimator enjoys, we have ˆ 𝐻 } = 0. 𝐸{HE The metric difference of the 𝑙th stream in the ZF detection is expressed as [18] { 𝐻 } ˆZF ˆ 𝑙 (n + Ex)Δ∗𝑙 + 𝛽ˆ𝑙ZF ∣Δ𝑙 ∣2 , (6) ΛZF 𝑙 (x, z) = 2𝛽𝑙 Re g where 𝛽ˆ𝑙ZF is denoted as the inverse of the squared norm of ˆ 𝑙𝐻 ; Δ𝑙 is the 𝑙th row vector of the pseudoinverse matrix g the difference between 𝑥𝑙 and 𝑧𝑙 . Elements in n and Ex are mutually-independent, zero-mean complex Gaussian random variables. Consequently, ΛZF 𝑙 (x, z) is a Gaussian random variable with its mean value and variance given by ˜ 𝑙 ∣2 𝜎𝑥2 , 𝜇ΛZF = 𝛽ˆ𝑙ZF ∣Δ ˜ 𝑙 ∣2 𝜎 2 (𝑀𝑡 𝜎 2 𝜎 2 + 𝑁0 ), 𝜎 2 ZF = 2𝛽ˆZF ∣Δ Λ
𝑙
𝑥
𝑥 𝑒
where ˜ 𝑙 = Δ𝑙 . Δ 𝜎𝑥
(7)
The bilateral Laplace transform is then derived as follows ( )) ( ZF ˜ 2 2 2 2 2 ˆ ΦΛZF (𝑠)=exp 𝛽 ∣ Δ ∣ 𝜎 +𝑀 𝜎 𝜎 )𝑠 −𝑠 . (𝑁 𝑙 0 𝑡 𝑙 𝑥 𝑥 𝑒 𝑙 (𝒙,𝒛) Averaging over different transmitted symbol 𝑥𝑙 , bit value 𝑏, and bit position 𝑖, the average bilateral Laplace transform
LAI et al.: EFFICIENT CHANNEL-ADAPTIVE MIMO DETECTION USING JUST-ACCEPTABLE ERROR RATE
for 𝑙th stream is
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B. Uncoded JAER Threshold Derivation and Proof
𝐼 1 1 ∑∑ ∑ ZF 𝜓ex,𝑙 (𝑠) = 𝐼 ΦΛZF (𝑠), 𝑙 (𝒙,𝒛) 𝐼2 𝑖=1 𝑖
(8)
𝑏=0 𝑥𝑙 ∈𝜒𝑏
where “ex” is the abbreviation for ”expurgated”, meaning that only the most-likely erroneous event is considered for ZF detection case, and thus the summation of z is removed and the summation of x is reduced to 𝑥𝑙 when compared with Eq. (4). In the following error rate evaluation, the expurgated PEP will be used to approximate the true PEP in Eq. (3). Dropping the stream index 𝑙 without loss of generality, we apply a numerical technique called Gauss-Chebyshev quadratures [19] to approximate the integral in Eq. (3) ∫ 𝜈+𝑗∞ 𝑑𝑠 1 [𝜓 ZF (𝑠)]𝑑 𝑓 ZF (𝑑, ℳ, 𝜒) = 2𝜋𝑗 𝜈−𝑗∞ ex 𝑠 𝑤 ∑ } { ZF } { ZF 1 ≈ Re [𝜓ex (𝑠𝑡 )]𝑑 + 𝜏 Im [𝜓ex (𝑠𝑡 )]𝑑 + 𝜉, 2𝑤 𝑡=1 where 𝑠𝑡 = 𝜈 + 𝑗𝜈𝜏𝑡 , 𝜏𝑡 = tan((2𝑡 − 1)𝜋/2𝑤), and 𝜈 is the ZF real part of the value that minimizes 𝜓ex (𝑠). The numerical approximation error 𝜉 is negligible as 𝑤 approaches infinity. Herein, 𝑤 = 32 is sufficient in our case. Inserting 𝑠𝑡 with 𝜈 = 1/2(𝑁0 + 𝑀𝑡 𝜎𝑥2 𝜎𝑒2 ) and using (8), we get the expression of the PEP of ZF detection as 𝑓 ZF (𝑑, ℳ, 𝜒) ≈ ⎛ )⎞𝑑 ( 𝐼 ∑ 𝑤 1 ∑ ZF ˜ 2 2 2 ˆ ∑ ∑ −𝛽 ∣Δ∣ 𝜎𝑥 (1 + 𝜏𝑡 ) ⎠ 1 ⎝1 exp . 2𝑤 𝑡=1 𝐼2𝐼 𝑖=1 4(𝑁0 + 𝑀𝑡 𝜎𝑥2 𝜎𝑒2 ) 𝑖 𝑏=0 𝑥∈𝜒𝑏
The uncoded BER distance to 1
ZF 𝑃𝑢𝑏
(9)
is acquired by setting the Hamming
ZF = 𝑓 ZF (1, ℳ, 𝜒). 𝑃𝑢𝑏
Clearly, the error rate performance is closely related to the instantaneous signal-to-interference-plus-noise ratio (iSINR) of ZF detection defined as follows 𝛽ˆZF 𝜎𝑥2 . (10) 𝛾 ZF = 𝑁0 + 𝑀𝑡 𝜎𝑥2 𝜎𝑒2 Given ℳ and 𝜒, the uncoded BER is monotonically decreasing with respect to 𝛾 ZF . Consequently, we can build a lookZF up table (LUT) storing the relationship between 𝛾 ZF and 𝑃𝑢𝑏 such that the uncoded BER can be easily evaluated with the estimated iSINR 𝛽ˆZF 𝜎𝑥2 , (11) 𝛾ˆ ZF = 𝒫ˆ𝜀 where the denominator is calculated by Eq. (1) and 𝐸{𝒫ˆ𝜀 } = 𝑁0 + 𝑀𝑡 𝜎𝑥2 𝜎𝑒2 , as mentioned before. The coded BER of ZF detection can also be implemented by the LUTs with Eq. (2). In other words, a LUT for certain (ℳ, 𝜒) containing the relationships between ZF ) can be constructed beforehand to facilitate the error (𝛾 ZF , 𝑃𝑢𝑏 rate prediction.
It should be emphasized that the under the assumption that applying ZF detection with all streams and subcarriers experiencing the same iSINR 𝛾 ZF , the corresponding coded BER ZF can be derived so that we can make another LUT for (𝑃𝑢𝑏 , ZF 𝑃𝑐𝑏 ). This uncoded and coded BER relationship is identical to a SISO system operated in an additive white Gaussian noise (AWGN) channel with effective iSINR 𝛾 AWGN = 𝛾 ZF for every subcarrier. On the other hand, CA-MIMO can be regards as another SISO system suffering different effective iSINR 𝛾ˇ for each subcarrier depending on the channel conditions and the adopted detection algorithms. Now, we will prove that given the same uncoded BER, the coded BER of the system operates in AWGN channel serves as upper bound, i.e., AWGN AWGN 𝑃ˇ𝑐𝑏 < 𝑃𝑐𝑏 , if 𝑃ˇ𝑢𝑏 = 𝑃𝑢𝑏 .
(12)
Note that as the metric difference in Eq. (6) is a Gaussian random variable given 𝛽ˆZF , the relationship between iSINR and the uncoded BER can alternatively expressed by the Qfunction (√ 2 ) 𝐼 ∑ 1 ∑ ∑ 𝜇ΛZF 1 ZF = 𝐼 𝑄 𝑃𝑢𝑏 2 𝐼2 𝑖=1 𝜎Λ ZF 𝑏=0 𝑥∈𝜒𝑖𝑏 √ ( 𝐼 1 ˜ 2) 𝛾ˆ ZF ∣Δ∣ 1 ∑∑ ∑ = 𝐼 𝑄 (13) 𝐼2 𝑖=1 2 𝑖 𝑏=0 𝑥∈𝜒𝑏
Without loss of generality and for clearness, we use the SISO systems with BPSK to simplify the following proof. Assuming the systems experiencing 𝑁 ∑ different effective iSINR {ˇ 𝛾 } with 𝑁 weight factors {𝑤} so that 𝑘=1 𝑤𝑘 = 1. Applying Eq. (13), the uncoded BER of BPSK SISO system is expressed as (√ ) 𝑁 ∑ 𝛾ˇ𝑘 ˇ 𝑃𝑢𝑏 = 𝑤𝑘 𝑄 , (14) 2 𝑘=1
where the triple summation and the constant of 1/𝐼2𝐼 is removed because of the BPSK modulation. For the system AWGN operated in AWGN channel with 𝑃𝑢𝑏 = 𝑃ˇ𝑢𝑏 , the effective AWGN is as follows iSINR 𝛾 ) (∑ 𝑁 AWGN −1 =𝑔 𝑤𝑘 𝑔(ˇ 𝛾𝑘 ) (15) 𝛾 𝑘=1
with
(√ ) ( ) ∫ 𝑥 𝑥 1 𝜋/2 𝑔(𝑥) = 𝑄 exp − = 𝑑𝜃, 2 𝜋 0 4 sin2 𝜃
(16)
the so-called Craig representation for 𝑄(𝑥), 𝑥 ≥ 0. Then, we can derive the corresponding PEP based on the same procedure in Section III-A. Instead of applying the numerical Gauss-Chebyshev quadrature, the Chernoff bound is adopted to provide a cleaner relationship among parameters ( ) 𝑑𝛾 AWGN AWGN,𝑐ℎ AWGN (𝑑, ℳ, 𝜒) ≈ min 𝜓 (𝑠) = exp − , 𝑓 𝑠>0 4 (17) where 𝜓 AWGN (𝑠) is the bilateral Laplace transform of the metric difference of the systems with AWGN channel. On
78
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 1, JANUARY 2011
the other hand, as the metric difference of the systems with 𝑁 iSINR contains many independent Gaussian random variables, the Chernoff bound of the PEP is presented by ( (∑ ))𝑑 𝑁 𝛾ˇ𝑘 ˇ 𝑓ˇ𝑐ℎ (𝑑, ℳ, 𝜒) ≈ min 𝜓(𝑠) 𝑤𝑘 exp − . = 𝑠>0 4 𝑘=1 (18) As the coded BER is proportional to the PEP, we change the coded relation Eq. (12) to the PEP relation 𝑓ˇ𝑐ℎ < 𝑓 AWGN,𝑐ℎ ,
𝑘=1
where note that 𝑔(−4 log(⋅)/𝑑) is a monotonic increasing function so that the for 𝑔(𝑎) < 𝑔(𝑏), 𝑎 < 𝑏. To prove Eq. (20), we first expand them by Eq. (16). The left-hand side in Eq. (20) thus becomes )) ( (∑ 𝑁 𝛾ˇ𝑘 ) 𝑤𝑘 exp(− 𝑔 − 4 log 4 𝑘=1 ) ( (∑ ∫ 𝑁 1 1 𝜋/2 𝛾ˇ𝑘 = ) 𝑑𝜃 exp 𝑤 exp(− log 𝑘 𝜋 0 4 sin2 𝜃 𝑘=1 ) 1 ( 𝑁 ∫ 1 𝜋/2 ∑ 𝛾ˇ𝑘 sin2 𝜃 = 𝑤𝑘 exp(− ) 𝑑𝜃, (21) 𝜋 0 4 𝑘=1
while the right-hand side is ( ) ∫ 𝑁 𝑁 ∑ ∑ 𝑤𝑘 𝜋/2 𝛾ˇ𝑘 𝑤𝑘 𝑔(ˇ 𝛾𝑘 ) = exp − 𝑑𝜃 𝜋 0 4 sin2 𝜃 𝑘=1 𝑘=1 ) 1 ( ∫ 𝑁 𝛾ˇ𝑘 sin2 𝜃 1 𝜋/2 ∑ 𝑤𝑘 exp(− ) 𝑑𝜃. (22) = 𝜋 0 4 𝑘=1
1
Defining the variable 𝜂 = exp(− 𝛾ˇ4 ) and 𝑞(𝑥) = 𝑥 sin2 𝜃 , a convex function for 0 ≤ 𝜃 ≤ 𝜋/2, we can apply the finite form of Jensens’ inequality ) ∑ (∑ 𝑁 𝑁 𝑤𝑘 𝜂𝑘 ≤ 𝑤𝑘 𝑞(𝜂𝑘 ), 𝑞 𝑘=1
ˆ ∥2 − ∥ y − Hx ˆ ∥2 ΛES (𝒙, 𝒛) =∥ y − Hz { } ˆ ˆ ∥2 , = 2 Re (n + Ex)𝐻 (HΔ) + ∥ HΔ where Δ = x − z is the difference vector between the transmitted vector and the erroneously-detected one. ˆ ΛES (𝒙, 𝒛) is a Gaussian random variable with mean Given H, and variance given by ˆ Δ∥ ˜ 2 𝜎𝑥2 , 𝜇ΛES = ∥H ˆ Δ∥ ˜ 2 𝜎 2 (𝑀𝑡 𝜎 2 𝜎 2 + 𝑁0 ) 𝜎 2 ES = 2∥H
(19)
given the same uncoded BER. Taking 𝑔(−4 log(⋅)/𝑑) on both Eq. (17) and Eq. (18) and replacing 𝛾 AWGN by Eq. (15), Eq. (19) is reformulated as )) ∑ ( (∑ 𝑁 𝑁 𝛾ˇ𝑘 < 𝑤𝑘 exp(− ) 𝑤𝑘 𝑔(ˇ 𝛾𝑘 ), (20) 𝑔 − 4 log 4 𝑘=1
be defined as
𝑘=1
∑𝑁 where the denominators are removed because 𝑘=1 𝑤𝑘 = 1. Therefore, Eq. (21) is smaller than Eq. (22) since every point in the region of the integration is smaller such that Eq. (12) is proved.
C. Error Rate Analysis for Exhaustive Search For the ES, since all data streams are detected simultaneously, the expurgated bound as in the ZF case is no longer applicable [20]. While considering all the error events will introduce an impractical union upper bound, we utilize the sphere search to find the relevant error events. Let ΛES (𝒙, 𝒛)
𝑥
Λ
𝑥 𝑒
˜ defined in Eq. (7). The with the normalized delta vector Δ bilateral Laplace transform then takes the form of ( ( )) ˆ Δ∥ ˜ 2 𝜎 2 (𝑁0 + 𝑀𝑡 𝜎 2 𝜎 2 )𝑠2 − 𝑠 . ΦΛES (𝑠) = exp ∥ H (𝒙,𝒛) 𝑥 𝑥 𝑒 𝑙 Similarly, by applying the Gauss-Chebyshev quadratures, we obtain the approximated PEP of ES detection 𝑓 ES (𝑑, ℳ, 𝜒) ≈
𝑤 ∑ 1 2𝑤(𝑀𝑡 𝐼2𝐼𝑀𝑡 )𝑑 𝑡=1
⎛ ⎞ ) 𝑑 ( 𝑀 𝐼 1 𝑡 2 2 2 ˆ Δ∥ ˜ 𝜎 (1 + 𝜏 ) ⎟ −∥H ⎜∑ ∑∑ ∑ ∑ 𝑥 𝑡 exp ⎝ ⎠ . 2 𝜎2 ) 4(𝑁 + 𝑀 𝜎 0 𝑡 𝑥 𝑒 x z 𝑖=1 𝑙=1
𝑏=0
𝑥𝑙 ∈𝜒𝑖𝑏 𝑧𝑙 ∈𝜒𝑖∼𝑏
(23)
Observing this equation, the error rate performance of ES is related to the square norm of the product of the estimated ˆ Δ∥ ˜ 2, channel matrix and the normalized error distance ∥H ˜ belongs to a finite set 𝜁 𝑀𝑡 . The SS, with the nature where Δ of expurgating the unlikely branches, is borrowed from the detection problem to reduce the set to { } 𝑀𝑡 ˜ ∈ 𝜁 𝑀𝑡 ∥H ˆ Δ∥ ˜ 2≤𝑅 , 𝜁𝑅 = Δ where 𝑅 is heuristically determined. The uncoded BER of ES is thus modified to ( ( )) 𝑤 ∑ ˆ Δ∥ ˜ 2 𝜎 2 (1+𝜏 2 ) 1 ∑ −∥H 𝑥 𝑡 ES ˜ 𝑃𝑢𝑏 ≈ 𝐶(Δ) exp , 2𝑤 𝑡=1 4(𝑁0 +𝑀𝑡 𝜎𝑥2 𝜎𝑒2 ) 𝑀 𝑡 ˜ Δ∈𝜁 𝑅
(24)
˜ is the number of Δ ˜ occurred in the quintuwhere 𝐶(Δ) ple summation in Eq. (23) divided by 𝑀𝑡 𝐼2𝑀𝑡 𝐼 . Similar to Eq. (10) and Eq. (11), the denominator is the power of interference plus noise and thus we can use its estimate 𝒫ˆ𝜀 calculated by Eq. (1). Given the modulation alphabet 𝜒 and ˜ can be calculated beforehand. the mapping rule ℳ, 𝐶(Δ) Note that we change the order of the summations so that the term inside the bracket can be calculated for each Δ. Since the improbable error events are pruned by the SS, the error rate model is more accurate and the computation complexity is reduced. IV. S TRATEGY FOR MIMO D ETECTOR S WITCHING Armed with the above error rate models for two MIMO detection algorithms, we propose a strategy for switching MIMO detection methods in a MIMO-OFDM receiver with
LAI et al.: EFFICIENT CHANNEL-ADAPTIVE MIMO DETECTION USING JUST-ACCEPTABLE ERROR RATE
a view to meeting the JAER criterion. As the channel models in the 802.11n standard are assumed to be quasi-static in one packet period, the switching strategy only operates at the beginning of the packet. We first transform the coded BER target to the uncoded BER target by finding 𝛾thZF , the required iSINR of ZF detection for the coded BER threshold, with ZF ). Then, the the LUT containing the information of (𝛾 ZF , 𝑃𝑐𝑏 uncoded BER threshold 𝑃𝑢𝑏,th is evaluated through another ZF ZF , 𝑃𝑐𝑏 ). Note that as LUT storing the information of (𝑃𝑢𝑏 mentioned in Section III-B, this uncoded threshold equals the uncoded BER of the system operated in AWGN channel with ZF . The objective is therefore to satisfy the given coded BER 𝑃𝑐𝑏 𝑃𝑢𝑏,𝜇 ≤ 𝑃𝑢𝑏,th ,
(25)
where 𝑃𝑢𝑏,𝜇 is the arithmetic mean (average) of the uncoded BER over all streams of all data subcarriers, calculated by the Gauss-Chebyshev quadratures in (9) and (24). Note that for certain channel models with infinite interleaver, the coded BER and uncoded BER has one-to-one mapping relationship when all the symbols are detected by the same algorithm such as ZF or ES. However, though the proposed system adopts various detection methods for different subcarriers, the simulation results in the next section demonstrate that once (25) is fulfilled with the proposed switching mechanism, the coded BER is smaller than and close to the target value. We propose to satisfy this constraint by switching the detection method of one stream in a certain subcarrier at a time. Initially, signals in all subcarriers are assumed to be detected by the ZF method. For the 𝑙th stream at the 𝑘th subcarrier, the ZF given in Eq. (11), is estimated iSINR of the ZF detector, 𝛾ˆ𝑙,𝑘 ˆ derived according to the outputs from the channel estimator H ZF ˆ and the noise-plus-interference power estimator 𝒫𝜀 . 𝑃𝑢𝑏,𝑙,𝑘 is ZF ). If the average of these then found through the LUT (𝛾 ZF , 𝑃𝑢𝑏 ZF 𝑃𝑢𝑏,𝑙,𝑘 meets the criterion indicated in Eq. (25), the ZF detector can be applied in all streams at all subcarriers. Otherwise, more powerful MIMO detection methods must be applied to some or even all subcarriers. Overall, there are four candidates of MIMO detection methods, ZF, (2, 2)ZF/SS, (1, 3)ZF/SS, and SS. For convenience, we index the above four candidate detectors from 1 to 4. The switching strategy finds a certain stream in a certain subcarrier and switches its detector to a more powerful one and then updates the average predicted uncoded BER 𝑃𝑢𝑏,𝜇 with the predicted uncoded BER of the new detector from Eq. (24). This process is repeated until the JAER requirement specified in Eq. (25) is satisfied. It should be noted that the predicted uncoded BERs are higher than actual values in low iSINR cases, which undermines the accuracy of the average predicted BER and the switching strategy may choose MIMO detection methods more powerful than necessary. We mitigate this problem by clipping the predicted uncoded BER as soon as it exceeds 0.5. 1 ∑ ˜ 𝑃𝑢𝑏,𝑙,𝑘 𝑃𝑢𝑏,𝜇 = ∣𝐷∣ (𝑙,𝑘)∈𝐷
with
{
𝑃˜𝑢𝑏,𝑙,𝑘 =
79
0.5
ZF ES 𝑃𝑢𝑏,𝑙,𝑘 or 𝑃𝑢𝑏,𝑙,𝑘 > 0.5
ZF ES or 𝑃𝑢𝑏,𝑙,𝑘 𝑃𝑢𝑏,𝑙,𝑘
otherwise
,
where 𝐷 is the set of all streams of all data subcarriers. In the following subsections, two switching strategies will be introduced. A. Switching Strategy In practice, the data subcarriers that experience deep fading and/or highly-correlated channel matrix dominate the overall error rate performance. Therefore, the switching strategy attacks the stream with the maximum error rate and switches that stream to a more powerful detector unless that subcarrier is already detected by SS, i.e., (𝑙𝑎 , 𝑘𝑎 ) = arg 𝐷′ =
{
max
(𝑙,𝑘) (𝑙,𝑘)∈𝐷′ , 𝑚
