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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

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Exact and Closed-Form Error Performance Analysis for Hard MMSE-SIC Detection in MIMO Systems Peng Liu, Student Member, IEEE, and Il-Min Kim, Senior Member, IEEE

Abstract—We investigate the exact error performance of hard minimum mean-squared error (MMSE) detection possibly with successive interference cancellation (SIC) in multiple-input multiple-output (MIMO) systems. To facilitate the analysis, we start with an exact bit-error rate (BER) analysis for a general system with decision statistic, 𝑧 = 𝑎𝑥 + 𝑢, where 𝑎 > 0, 𝑥 is the transmitted signal, and 𝑢 is an arbitrarily distributed noise component which is possibly dependent on the signal component 𝑥. For this general system, we derive the exact and closed-form BER expressions for 𝑀 -ary pulse amplitude modulation (PAM) and arbitrary rectangular quadrature amplitude modulation (QAM), which include the well-known BER result of [22] as a special case. Furthermore, by formulating the MIMO MMSE decision statistics in the same form as 𝑧 in the general system, we derive the exact and closed-form instantaneous BER and symbolerror rate (SER) expressions for MIMO hard MMSE detection with/without SIC employing PAM and QAM. Finally, the validity of our derived error probability expressions is verified through extensive Monte Carlo simulations and the results reported in the literature. Index Terms—Bit-error rate (BER), minimum mean-squared error (MMSE), multiple-input multiple-output (MIMO), spatial multiplexing (SM), symbol-error rate (SER).

I. I NTRODUCTION

M

ULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO) communication has emerged as one of the most promising techniques in wireless communications due to its great potential to improve system reliability and increase channel capacity [1]. Two typical approaches in the MIMO systems are to provide diversity gain as in space-time coding (STC) or to allow spatial multiplexing (SM). While STC systems are capable of improving system reliability through coding across space and/or time, SM systems are capable of increasing data transmission rate through spatial multiplexing. In this paper, we focus on the SM systems only. For SM systems with flat fading channels, linear minimum mean-squared error (MMSE) receiver has been widely adopted in hard detection for low computational complexity [2]–[7]. Paper approved by X. Wang, the Editor for Multiuser Detection and Equalization of the IEEE Communications Society. Manuscript received March 22, 2010; revised February 3, 2011. This paper was presented in part at IEEE GLOBECOM 2010 [25], Miami, Florida, USA, December 2010, and at ACM ICUIMC 2011 [26], Seoul, Korea, February 2011. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), and by the Ubiquitous Computing and Network (UCN) Project, Knowledge and Economy Frontier R&D Program of the Ministry of Knowledge Economy (MKE) in Korea, as a result of UCN’s subproject 11C3-C2-11T. The authors are with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, Ontario, K7L 3N6, Canada (e-mail: {peng.liu, ilmin.kim}@queensu.ca). Digital Object Identifier 10.1109/TCOMM.2011.060911.100166

Recently, the linear MMSE receiver has also been developed for soft detection in both flat fading [8] and frequency-selective fading channels [9], [10]. For analytical tractability of error performance analysis, however, most of the works in this context have focused on the error analysis for hard MMSE detection [11]–[15], [18]–[20].1 The error performance of hard MMSE detection without successive interference cancellation (SIC) has received much attention. Most of the works have been devoted to the approximate error analysis for simplicity. A well-known approximation is the so-called Gaussian approximation, which models the interference-plus-noise at the MMSE filter output as a Gaussian random variable [11], [12]. With the Gaussian approximation, Poor and Verd´u derived approximate instantaneous (conditional on the channel realization) bit-error rate (BER) for binary phase-shift keying (BPSK) [13]. Also, Burnashev and Poor derived a number of useful bounds on the instantaneous BER for BPSK using Gaussian approximation in conjunction with Chernoff bound [14]. Furthermore, Gaussian approximation has been employed for average symbolerror rate (SER) analysis for square 𝑀 -ary (i.e., 𝑀 = 22𝑘 for any integer 𝑘) quadrature amplitude modulation (𝑀 -QAM) [15] and 𝑀 -ary phase-shift keying (𝑀 -PSK) [16].2 While Gaussian approximation can significantly facilitate the error analysis, its accuracy deteriorates for practical scenarios, e.g., small number of antennas or users, as admitted by many works [14], [18], [19]. Other approximations such as Gamma approximation and generalized Gamma approximation have been used [19]. The basic idea is to approximate the distribution of the output signal-to-interference-plus-noise ratio (SINR) of the MMSE filter using the target distribution (either Gamma distribution or generalized Gamma distribution) with matched moments up to the third-order. Taking this approach, the authors of [19] derived the approximate BER; however, the results were only limited to a binary signaling. The exact error analysis is important because it can be used as a performance benchmark in practical system designs and to test the tightness of various bounds. In the literature, however, very limited results have been reported on the exact 1 With an exception of [8], the BER of soft MMSE detection for a convolutional coded MIMO system was studied. However, a number of approximations were employed in the analysis of [8], which led to loose BER approximation for low signal-to-noise ratio (SNR) regions. Moreover, the powerful technique of successive interference cancellation (SIC) which can substantially improve the error performance was not considered in [8]. In this paper, however, we will focus on the exact error performance analysis for hard MMSE detection with SIC. 2 The Gaussian approximation was implied in the analysis of [16] starting with [17, eq. (15)], which implicitly assumed that the interference-plus-noise at the MMSE filter output was modeled as a Gaussian random variable.

c 2011 IEEE 0090-6778/11$25.00 ⃝

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error analysis for MMSE detection without SIC. Specifically, an exact and closed-form instantaneous BER expression of BPSK was derived for MMSE detection without SIC [20]. However, the analysis of [20] only applies to a binary signaling and its generalization to higher constellations is by no means straightforward. To the best of our knowledge, the exact BER and SER expressions for hard MMSE detection without SIC employing general 𝑀 -ary pulse amplitude modulation (𝑀 PAM) or arbitrary rectangular 𝐼 × 𝐽-QAM have not been derived in the literature. For MMSE detection with SIC, the exact error analysis becomes more difficult due to the error propagation. Specifically, imperfect SIC due to erroneously detected symbols causes error propagation to the detection of other symbols, which in turn causes correlation across all symbol detections. Due to such correlation, the exact error analysis becomes very challenging. To the best of our knowledge, there has been no work analyzing the exact BER or SER for MMSE detection with SIC.3 Motivated by the above discussions, this paper focuses on the exact BER and SER analyses for MIMO hard MMSE detection with or without SIC for general 𝑀 -PAM and arbitrary rectangular 𝐼 × 𝐽-QAM signallings. To facilitate the analysis, we start with an exact BER analysis for a general system with decision statistic, 𝑧 = 𝑎𝑥 + 𝑢, where 𝑎 > 0 is a constant, and the noise component 𝑢 is arbitrarily distributed and possibly correlated with the transmitted signal 𝑥. For this general system, we derive the exact and closed-form BER expressions for both 𝑀 -PAM and 𝐼 × 𝐽-QAM. We also show that the well-known BER result in [22] can be considered as a special case of our result. Furthermore, we apply our general analysis to the error analysis for hard MMSE detection with/without SIC. The exact and closedform instantaneous BER and SER expressions for hard MMSE detection with/without SIC employing 𝑀 -PAM and 𝐼 × 𝐽QAM are derived for the first time in the literature, to the best of our knowledge. Finally, the validity of the analytical results is verified through extensive Monte Carlo simulations and also supported by the reported results in the literature. The remainder of this paper is organized as follows. We describe the system model and hard MMSE detection in Section II. In Section III, we present preliminary results on the MMSE decision statistic. In Section IV, we study the BER of a general system with arbitrarily distributed noise component which is possibly correlated with the signal. In Section V, the error performance of hard MMSE detection with/without SIC is studied. Section VI presents some numerical results and Section VII concludes this paper. Notation: Bold letters denote vectors or matrices. Also, 𝑰𝑛 denotes an 𝑛 × 𝑛 identity matrix; (⋅)𝑇 , the transpose; (⋅)† , the Hermitian; and 𝔼[⋅], the expectation. We use 𝐴 := 𝐵 to denote that 𝐴, by definition, equals 𝐵 and we use 𝐴 =: 𝐵 to denote that 𝐵, by definition, equals 𝐴. For any matrix 3 In

[21], the authors analyzed the exact average SER of a QR decomposition-based decision feedback detection for MIMO Rayleigh fading channels with square 𝑀 -QAM. However, this detection is essentially a zero-forcing (ZF)-based detection with successive interference cancellation (ZF-SIC), not the MMSE detection with SIC considered in our paper. Furthermore, it is well-known that ZF-SIC detection achieves poorer error performance than MMSE detection with SIC.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

𝑿 = [𝒙1 , 𝒙2 , ⋅ ⋅ ⋅ , 𝒙𝑛 ] with column vectors 𝒙𝑙 , 𝑿⟨−𝑙⟩ denotes a matrix obtained by removing 𝒙𝑙 from 𝑿, 𝑿(−𝑙) := [𝒙1 , 𝒙2 , ⋅ ⋅ ⋅ , 𝒙𝑙−1 ], and 𝑿(+𝑙) := [𝒙𝑙 , 𝒙𝑙+1 , ⋅ ⋅ ⋅ , 𝒙𝑛 ]; also, for any column vector 𝒚 = [𝑦1 , 𝑦2 , ⋅ ⋅ ⋅ , 𝑦𝑛 ]𝑇 , 𝒚⟨−𝑙⟩ denotes a vector obtained by removing 𝑦𝑙 from 𝒚, 𝒚(−𝑙) := [𝑦1 , 𝑦2 , ⋅ ⋅ ⋅ , 𝑦𝑙−1 ]𝑇 , and 𝒚(+𝑙) := [𝑦𝑙 , 𝑦𝑙+1 , ⋅ ⋅ ⋅ , 𝑦𝑛 ]𝑇 , where 𝑙 = 1, 2, ⋅ ⋅ ⋅ , 𝑛.4 For any complex-valued variable 𝑧, we use [𝑧]1 , or interchangeably, ℜ(𝑧), to denote the real part of 𝑧 and we use [𝑧]2 , or interchangeably, ℑ(𝑧), to denote the imaginary part of 𝑧. For random variables 𝑋 and 𝑌 , 𝑓𝑋 (𝑥) and 𝐹𝑋 (𝑥) denote the probability density function (PDF) and cumulative distribution function (CDF) of 𝑋, respectively; 𝑓𝑌 ∣𝑋=𝑥 (𝑦) and 𝐹𝑌 ∣𝑋=𝑥 (𝑦) denote the conditional PDF and conditional CDF of 𝑌 given 𝑋 = 𝑥, respectively; Pr(⋅∣⋅) denotes the conditional probability; and 𝒬{ denotes the Gaussian Q𝑇 function. For any set 𝑅, 𝑅𝑚 } := [𝑥1 , 𝑥2 , ⋅ ⋅ ⋅ , 𝑥𝑚 ] : 𝑥1 ∈ 𝑅, 𝑥2 ∈ 𝑅, ⋅ ⋅ ⋅ , 𝑥𝑚 ∈ 𝑅 denotes a set of all possible length-𝑚 column vectors whose elements are drawn from 𝑅. Furthermore, ⌊⋅⌋ denotes the floor function and ∣∣ ⋅ ∣∣ denotes the Euclidean vector norm. Finally, 𝒏 ∼ 𝒞𝒩 (𝝁, 𝑹) denotes a circularly symmetric complex Gaussian random vector with mean 𝝁 and covariance 𝑹. II. S YSTEM D ESCRIPTION AND H ARD MMSE D ETECTION We now describe the hard MMSE detection without SIC and with SIC, which are referred to as MMSE-non-SIC and MMSE-SIC, respectively, throughout the paper. A. System Model Consider an SM MIMO system with 𝑁𝑡 transmit and 𝑁𝑟 receive antennas (𝑁𝑟 ≥ 𝑁𝑡 ). The transmitted and received signals are related as 𝒓 = 𝑯𝒙 + 𝒏,

(1)

where 𝒙 := [𝑥1 , 𝑥2 , ⋅ ⋅ ⋅ , 𝑥𝑁𝑡 ]𝑇 is the transmitted signal vector, 𝒓 := [𝑟1 , 𝑟2 , ⋅ ⋅ ⋅ , 𝑟𝑁𝑟 ]𝑇 is the received signal vector, 𝒏 := [𝑛1 , 𝑛2 , ⋅ ⋅ ⋅ , 𝑛𝑁𝑟 ]𝑇 is the additive white Gaussian noise (AWGN), and 𝑯 = [𝒉1 , 𝒉2 , ⋅ ⋅ ⋅ , 𝒉𝑁𝑡 ] is an 𝑁𝑟 × 𝑁𝑡 channel matrix with each column vector 𝒉𝑖 representing the channels from the 𝑖th transmit antenna to] all receive antennas. [ [ ] We adopt the normalization, 𝔼 𝒏𝒏𝐻 = 𝑰𝑁𝑟 and 𝔼 𝒙𝒙𝐻 =: 𝜌 𝑁𝑡 𝑰𝑁𝑡 =: 𝛾𝑰𝑁𝑡 , as in [7, Chap. 9.4.2] to ensure that 𝜌 denotes the SNR at each receive antenna, and 𝛾 := 𝑁𝜌𝑡 denotes the SNR per receive antenna per symbol. Each transmitted symbol 𝑥𝑖 is drawn from any 𝑀 -PAM or rectangular 𝐼 × 𝐽QAM constellation ℭ, i.e., 𝑥𝑖 ∈ ℭ. As in [22], we denote by 2𝑑 the minimum Euclidean distance in the constellation √ 2 − 1) for 𝑀 -PAM and 𝑑 := 3𝛾/(𝑀 space ℭ, where 𝑑 := √ 3𝛾/(𝐼 2 + 𝐽 2 − 2) for 𝐼 × 𝐽-QAM.5 Then each real-valued signal 𝑥𝑖 can be uniquely represented by an integer-valued signal 𝑠𝑖 satisfying 𝑥𝑖 = 𝑑𝑠𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 . For 𝑀 -PAM, 4 For notational consistence, we let the index 𝑙 in the subscript of 𝑿 (−𝑙) and 𝒚(−𝑙) start with 𝑙 = 1; but one should note that 𝑿(−1) actually becomes a null matrix and 𝒚(−1) a null vector. 5 These choices of 𝑑 ensure that the average symbol power in the constellation space ℭ is normalized to 𝛾, that is, 𝔼[∣𝑥𝑖 ∣2 ] = 𝛾. Moreover, it is easy to see that 𝑑 in 𝐼 × 𝐽-QAM reduces to 𝑑 in 𝑀 -PAM when setting 𝐼 = 𝑀 and 𝐽 = 1.

LIU and KIM: EXACT AND CLOSED-FORM ERROR PERFORMANCE ANALYSIS FOR HARD MMSE-SIC DETECTION IN MIMO SYSTEMS

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TABLE I G RAY C ODE B IT M APPING FOR 𝑀 -PAM W ITH 𝑀 = 2, 4, 8, AND 16 symbol (𝑠) 2-PAM 4-PAM 8-PAM 16-PAM

−15 −13 −11

−9

−7

−5

−3

−1 1 3 5 7 9 11 13 15 1 0 10 11 01 00 100 101 111 110 010 011 001 000 1000 1001 1011 1010 1110 1111 1101 1100 0100 0101 0111 0110 0010 0011 0001 0000

𝑠𝑖 ∈ 𝒮𝑀 , where 𝒮𝑀 := {±1, ±2, ⋅ ⋅ ⋅ , ±(𝑀 − 1)}. For 𝐼 × 𝐽QAM, the in-phase (I) component ℜ[𝑠𝑖 ] and the quadrature (Q) component ℑ[𝑠𝑖 ] are drawn independently from 𝒮𝐼 and 𝒮𝐽 , respectively. For ease of exposition, we denote by 𝔛 the integer-valued space of 𝑠𝑖 ; that is, 𝔛 := ℭ/𝑑. Furthermore, we denote by 𝑥 ˆ𝑖 (or 𝑠ˆ𝑖 ) the hard decision estimate for 𝑥𝑖 (or 𝑠𝑖 ), which satisfies 𝑥ˆ𝑖 = 𝑑𝑠ˆ𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 . Also, we ˆ := [ˆ 𝑥1 , 𝑥 denote 𝒔 := [𝑠1 , 𝑠2 , ⋅ ⋅ ⋅ , 𝑠𝑁𝑡 ]𝑇 , 𝒙 ˆ2 , ⋅ ⋅ ⋅ , 𝑥 ˆ𝑁𝑡 ]𝑇 , and 𝑇 ˆ = 𝑑ˆ 𝒔. In 𝒔ˆ := [ˆ 𝑠1 , 𝑠ˆ2 , ⋅ ⋅ ⋅ , 𝑠ˆ𝑁𝑡 ] , and hence, 𝒙 = 𝑑𝒔 and 𝒙 this paper, we adopt the Gray code bit mapping as depicted in Table I and assume that the channel matrix 𝑯 is known at the receiver, but unknown at the transmitter. B. Hard MMSE Detection We note that the performance of hard MMSE-SIC detection can be improved via a proper detection ordering method subject to the instantaneous channel realizations. For instance, a detection order which maximizes the instantaneous output SINR of the MMSE filter at each stage substantially improves the error performance compared to any fixed detection order [6], [7]. In case any specific detection ordering is used in the MMSE-SIC detection, the input-output relation of (1) along with the transmitted signal 𝒙, the received signal 𝒓, and the channel matrix 𝑯 need to be rearranged such that the actual detection is ordered from 𝑖 = 1 to 𝑖 = 𝑁𝑡 . Hence, without loss of generality, we assume that the transmit symbols, 𝑥𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 , are sequentially detected at the receiver within 𝑁𝑡 stages in the order from 𝑖 = 1 to 𝑖 = 𝑁𝑡 , given that 𝒓, 𝑯, and 𝒙 have been rearranged subject to the detection ordering method. Therefore, the analysis presented in this paper is valid for arbitrary detection ordering methods including an optimum one.6 Following the above discussion, at stage 𝑖, the 𝑖th symbol 𝑥𝑖 is detected. It is very well-known that, at stage 𝑖 the standard MMSE filter 𝒈𝑖 is given by [7]: ⎧ ( )−1  ⎨ 𝒉† 𝑯𝑯 † + 1 𝑰𝑁𝑟 , for MMSE-non-SIC, 𝑖 𝛾 𝒈𝑖 = ( )−1 †  ⎩ 𝒉†𝑖 𝑯(+𝑖) 𝑯(+𝑖) + 𝛾1 𝑰𝑁𝑟 , for MMSE-SIC, (2) for 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 . At the receiver, the MMSE decision statistic 𝑧𝑖 is obtained through the following MMSE filtering, 𝑧𝑖 = 𝒈𝑖 𝒓𝑖 , where 𝒓𝑖 is the (residual) signal vector at stage 𝑖 and it is given by { 𝒓, for MMSE-non-SIC, (3) 𝒓𝑖 = ˆ (−𝑖) , for MMSE-SIC. 𝒓 − 𝑯(−𝑖) 𝒙 6 The optimum detection ordering requires exhaustive search of all possible orders of the symbol detection [7]. Thus, a suboptimal scheme such as the one maximizing the output SINR of the MMSE filter which has much lower complexity may be used in practice.

Form (3), it is not hard to see that erroneously predetected symbol(s) (ahead of stage 𝑖) cannot be perfectly canceled out from the received signal 𝒓, which in turn adds residual error to the signal 𝒓𝑖 at stage 𝑖. In this case, error propagation is inevitable and it results in performance degradation in the detection of other symbols. Such error propagation is a critical issue and it will be explicitly addressed in our error analysis. After some manipulations, the MMSE decision statistic 𝑧𝑖 can be rewritten as 𝑧𝑖 = 𝑎𝑖 𝑥𝑖 + 𝑢𝑖 ,

(4)

where the coefficient 𝑎𝑖 of the signal component is given by ⎧ ( )−1  ⎨ 𝒉† 𝑯𝑯 † + 1 𝑰𝑁𝑟 𝒉𝑖 , for MMSE-non-SIC, 𝑖 𝛾 𝑎𝑖 = ( )−1 †  ⎩ 𝒉†𝑖 𝑯(+𝑖) 𝑯(+𝑖) + 𝛾1 𝑰𝑁𝑟 𝒉𝑖 , for MMSE-SIC. (5) Furthermore, 𝑢𝑖 denotes the noise component composed of interference and AWGN, and it is given by { 𝑑𝒈𝑖 𝑯⟨−𝑖⟩ 𝒔⟨−𝑖⟩ + 𝒈𝑖 𝒏, for MMSE-non-SIC, (6) 𝑢𝑖 = for MMSE-SIC, 𝑑𝒈𝑖 퓗𝑖 퓢 𝑖 + 𝒈𝑖 𝒏, where 퓗𝑖 := [𝑯⟨−𝑖⟩ , −𝑯(−𝑖) ] and 퓢 𝑖 := [𝒔𝑇⟨−𝑖⟩ , 𝒔ˆ𝑇(−𝑖) ]𝑇 . Note that the interference associated with 𝒔⟨−𝑖⟩ for MMSEnon-SIC contains all transmitted symbols except 𝑠𝑖 , whereas the interference associated with 퓢 𝑖 for MMSE-SIC contains the predetected symbols 𝒔ˆ(−𝑖) as well as all transmitted symbols except 𝑠𝑖 . Finally, the hard MMSE decision on 𝑥𝑖 is made by projecting the MMSE decision statistic 𝑧𝑖 of (4) onto the nearest constellation point in ℭ (see [2, eqs. (9.1.29) and (9.1.41)], [3, eq. (6.26)], [4, eq. (4)], [5, Chap. 6.1], [6, Chap. 7.4], [7, Chap. 9.5], and numerous other publications): 𝑥 ˆ𝑖 = Prjℭ (𝑧𝑖 ),

(7)

for 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 , where Prjℭ (⋅) denotes the projection onto constellation ℭ. The hard decision rule of (7) results in an error when 𝑥 ˆ𝑖 ∕= 𝑥𝑖 . In Section VI, we will test the performance of this hard MMSE detection through both Monte Carlo simulations and our derived analytical expressions, which agree exactly. In addition, the obtained results are further confirmed by the simulation results reported in the literature [5], [6]. III. P RELIMINARY R ESULTS ON THE MMSE D ECISION S TATISTIC In this section, we derive some preliminary results on the MMSE decision statistic, 𝑧𝑖 = 𝑎𝑖 𝑥𝑖 + 𝑢𝑖 . The results are summarized in the following two lemmas.

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

Lemma 1: For the MMSE decision statistic 𝑧𝑖 = 𝑎𝑖 𝑥𝑖 + 𝑢𝑖 with 𝑥𝑖 = 𝑑𝑠𝑖 , the noise component 𝑢𝑖 is non-Gaussian distributed for both MMSE-non-SIC and MMSE-SIC. Moreover, 𝑢𝑖 is independent of the signal component 𝑠𝑖 (or 𝑥𝑖 ) for MMSE-non-SIC, and 𝑢𝑖 is dependent on 𝑠𝑖 (or 𝑥𝑖 ) for MMSESIC. Proof: Referring to (6), the non-Gaussianity of the noise component 𝑢𝑖 is due to the non-Gaussian interference, which is associated with 𝒔⟨−𝑖⟩ for MMSE-non-SIC and with 퓢 𝑖 for MMSE-SIC. Furthermore, in MMSE-non-SIC with 𝑢𝑖 = 𝑑𝒈𝑖 𝑯⟨−𝑖⟩ 𝒔⟨−𝑖⟩ + 𝒈𝑖 𝒏, both the interference component associated with 𝒔⟨−𝑖⟩ and the AWGN 𝒏 are independent of 𝑠𝑖 , and hence, 𝑢𝑖 is independent of 𝑠𝑖 . In MMSE-SIC with 𝑢𝑖 = 𝑑𝒈𝑖 퓗𝑖 퓢 𝑖 + 𝒈𝑖 𝒏, however, the interference associated with 퓢 𝑖 contains the predetected symbols 𝒔ˆ𝑇(−𝑖) , which are dependent on 𝑠𝑖 because 𝑠𝑖 was the interference in the detection of 𝒔ˆ𝑇(−𝑖) . Hence, 𝑢𝑖 is dependent on 𝑠𝑖 . Lemma 2: The constant 𝑎𝑖 of (5) satisfies 0 < 𝑎𝑖 < 1 for both MMSE-non-SIC and MMSE-SIC. 1 is true for MMSEProof: We first show that ( 0 < 𝑎𝑖 < ) −1

SIC. Rewrite 𝑎𝑖 as 𝑎𝑖 = 𝒉†𝑖 𝑹𝑖 + 𝒉𝑖 𝒉†𝑖 𝒉𝑖 , where { † 𝑯(+𝑖+1) 𝑯(+𝑖+1) + 𝛾1 𝑰𝑁𝑟 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 − 1, 𝑹𝑖 = 1 𝑖 = 𝑁𝑡 . 𝛾 𝑰𝑁𝑟 ,

= Using the Matrix Inversion Lemma yields 𝑎𝑖 ( )−1 𝒉†𝑖 𝑹−1 † † 𝑖 𝒉𝑖 𝒉𝑖 𝑹𝑖 + 𝒉𝑖 𝒉𝑖 𝒉𝑖 = 1+𝒉† 𝑹−1 𝒉 . Since 𝑹𝑖 is a 𝑖

𝑖

𝑖

positive definite matrix, 𝒉†𝑖 𝑹−1 > 0. Therefore, 𝑖 𝒉𝑖 𝒉†𝑖 𝑹−1 𝑖 𝒉𝑖 0 < 𝑎𝑖 = 1+𝒉† 𝑹−1 𝒉 < 1. For MMSE-non-SIC, we 𝑖 𝑖 𝑖 ( )−1 𝒉𝑖 , where can similarly rewrite 𝑎𝑖 = 𝒉†𝑖 𝑹𝑖 + 𝒉𝑖 𝒉†𝑖

for a system involving a decision statistic of the same form as 𝑧𝑖 in the next section. Then such general analysis will be applied to the error analysis for hard MMSE detection. IV. G ENERAL E RROR A NALYSIS The error analysis for MMSE-non-SIC and MMSE-SIC involves analysis of 𝑁𝑡 decision statistics, 𝑧𝑖 = 𝑎𝑖 𝑥𝑖 + 𝑢𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 , which are essentially of the same form. Since the analysis of all 𝑁𝑡 decision statistics are also essentially the same, in this section, we will focus on a single decision statistic and conduct a general error analysis. Specifically, we consider a decision statistic of the same form as 𝑧𝑖 ; but the considered decision statistic is more general in the sense that the coefficient of the signal component and the noise component are defined to be more general than in 𝑧𝑖 . We first define a general system as follows. Definition 1: A general system is defined to be a communication system whose decision statistic 𝑧 and the transmitted signal 𝑥, 𝑥 ∈ ℭ, are related as 𝑧 = 𝑎𝑥 + 𝑢 and the hard decision on 𝑥 is obtained via 𝑥 ˆ = Prjℭ (𝑧), where 𝑎 > 0 is any constant and 𝑢 is the noise component which can be arbitrarily distributed and possibly dependent on 𝑥.8 Recall that 2𝑑 is the minimum Euclidean distance in the constellation space ℭ. Therefore, for each real-valued signal 𝑥 ∈ ℭ, there exists a unique integer-valued signal 𝑠 ∈ 𝔛 satisfying 𝑥 = 𝑑𝑠. Henceforth, we will use only the integervalue signal 𝑠 (instead of 𝑥) to describe the BER results for ease of exposition. In the following, we present the exact BER for the general system with 𝑀 -PAM and 𝐼 × 𝐽-QAM. Theorem 1: The BER of the general system with 𝐼 × 𝐽QAM is given by ⎛ ⎞ log2 𝐼 log2 𝐽 ∑ ∑ 1 𝐵 ⎝ = 𝑃1𝐵 (𝑘) + 𝑃2𝐵 (𝑘)⎠ , (8) 𝑃𝐼×𝐽-QAM log2 (𝐼𝐽)

† 𝑹𝑖 = 𝑯⟨−𝑖⟩ 𝑯⟨−𝑖⟩ + 𝛾1 𝑰𝑁𝑟 . Then a similar proof applies. From the above two lemmas, we can derive some important implications, which are described in the following. Remark 1 (non-Gaussianity and dependence): Lemma 1 implies that the error analysis of hard MMSE detection suffers from two major challenges: the non-Gaussianity of the noise component 𝑢𝑖 and the dependence of the noise component 𝑢𝑖 on the signal 𝑠𝑖 . The former is a common challenge for both MMSE-SIC and MMSE-non-SIC, whereas the latter is unique to MMSE-SIC. Remark 2 (non-unity constant): Lemma 2 implies that there exists another challenge in the error analysis of hard MMSE detection. Specifically, a power mismatch is caused by the non-unity constant 𝑎𝑖 ; that is, the average power of the signal component 𝑎𝑖 𝑥𝑖 in the MMSE decision statistic 𝑧𝑖 is not identical to the average signal power of 𝑥𝑖 . This power mismatch will have additional impact on decision errors when the MMSE decision statistic is projected onto a constellation ℭ with mismatched power via (7).7 For this reason, the effect of the non-unity 𝑎𝑖 needs to be judiciously addressed in the error analysis. Due to the aforementioned properties, the exact error analysis for hard MMSE detection is indeed challenging. To facilitate the analysis, we first conduct a general error analysis

where 𝑃 𝐵 (𝑘) = 𝜙1 (𝑀, 1, 𝑘).

7 Note that in many other hard detection rules such as zero-forcing (ZF), matched filtering, and maximum ratio combining, the obtained decision statistic has the identical average signal power as the transmitted signal.

8 Note that, in this definition 𝑎 can be any positive constant, not necessarily given by (5). Also, 𝑢 can have any distribution, not necessarily determined by (6), and thus, the PDF of 𝑢 can be possibly a non-even function.

𝑘=1

𝑘=1

where 𝑃𝛼𝐵 (𝑘) = 𝜙𝛼 (𝐼, 𝐽, 𝑘), 𝛼 = 1, 2. Furthermore, 𝜙𝛼 (𝐼, 𝐽, 𝑘), 𝛼 = 1, 2, are defined at the top of the next page, where the function 𝜔(⋅, ⋅) is defined as 𝜔(𝑚, 𝑦) := 2𝑚−1−𝑦; the coefficients 𝒜𝑦 (𝑘, 𝑗), ℬ𝑦 (𝑘, 𝑗), 𝒞𝑦 (𝑘, 𝑗), 𝒟𝑦 (𝑘) are presented in Table II; and 𝐼𝛼 is defined as follows: 𝐼𝛼 = 𝐼 if 𝛼 = 1 and 𝐼𝛼 = 𝐽 if 𝛼 = 2. Proof: See Appendix A. Using the above theorem, therefore, one can obtain the exact BER in closed-form for the general system with 𝐼 × 𝐽-QAM, as long as the conditional CDFs, 𝐹[𝑢]𝛼 ∣𝑠 (⋅), 𝛼 = 1, 2, are given. In the following, as a direct consequence of Theorem 1, we obtain the exact BER for the general system with 𝑀 PAM Corollary 1: The BER of the general system employing 𝑀 -PAM is given by 𝐵 𝑃𝑀-PAM

log2 𝑀 ∑ 1 = 𝑃 𝐵 (𝑘), log2 𝑀

(10)

𝑘=1

LIU and KIM: EXACT AND CLOSED-FORM ERROR PERFORMANCE ANALYSIS FOR HARD MMSE-SIC DETECTION IN MIMO SYSTEMS

2467

( 𝒟𝐼𝛼 (𝑘) 𝐼3−𝛼 ( ( ∑ 1 ∑ 𝒜𝐼𝛼 (𝑘, 𝑗) 1 − 𝐹[𝑢]𝛼 ∣[𝑠]𝛼 =−𝒞𝐼𝛼 (𝑘,𝑗),[𝑠]3−𝛼 =𝜔(𝑚,𝐼3−𝛼 ) 𝑑 ℬ𝐼𝛼 (𝑘, 𝑗) + (𝑎 − 1) 𝐼𝐽 𝑗=1 𝑚=1 ) ) ( ) )) ( ×𝒞𝐼𝛼 (𝑘, 𝑗) + 𝐹[𝑢]𝛼 ∣[𝑠]𝛼 =𝒞𝐼𝛼 (𝑘,𝑗),[𝑠]3−𝛼 =−𝜔(𝑚,𝐼3−𝛼 ) − 𝑑 ℬ𝐼𝛼 (𝑘, 𝑗) + (𝑎 − 1)𝒞𝐼𝛼 (𝑘, 𝑗) .

𝜙𝛼 (𝐼, 𝐽, 𝑘) :=

(9)

TABLE II 𝒜𝑦 (𝑘, 𝑗), ℬ𝑦 (𝑘, 𝑗), 𝒞𝑦 (𝑘, 𝑗), AND 𝒟𝑦 (𝑘), 𝑘 = 1, 2, ⋅ ⋅ ⋅ , log2 𝑦, 𝑗 = 1, 2 ⋅ ⋅ ⋅ , 𝒟𝑦 (𝑘), FOR 𝑦 = 2, 4, 8, AND 16 𝑦 2 4 8

16

𝑘 1 1 2 1 2 3 1 2 3

4

𝒜𝑦 (𝑘, 𝑗) 1 1, 1 1, 1, 1, −1 1, 1, 1, 1 1, 1, 1, 1, 1, 1, −1, −1 1, 1, 1, 1, 1, 1, 1, −1, −1, −1, −1, −1, 1, 1, 1, −1 1, 1, 1, 1, 1, 1, 1, 1 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, −1, −1, −1, −1 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, −1, −1, −1, −1, −1, −1, −1, −1, −1, −1, 1, 1, 1, 1, 1, 1, −1, −1 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, −1, −1, −1, −1, −1, −1, −1, −1, −1, −1, −1, −1, −1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, −1, −1, −1, −1, −1, −1, −1, −1, −1, 1, 1, 1, 1, 1, 1, 1, −1, −1, −1, −1, −1, 1, 1, 1, −1

ℬ𝑦 (𝑘, 𝑗) 1 1, 3 1, 1, 3, 5 1, 3, 5, 7 1, 1, 3, 3, 5, 7, 9, 11 1, 1, 1, 1, 3, 3, 3, 5, 5, 5, 7, 7, 9, 9, 11, 13 1, 3, 5, 7, 9, 11, 13, 15 1, 1, 3, 3, 5, 5, 7, 7, 9, 11, 13, 15, 17, 19, 21, 23 1, 1, 1, 1, 3, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 11, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 23, 25, 27 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 15, 15, 15, 15, 17, 17, 17, 17, 19, 19, 19, 21, 21, 21, 23, 23, 25, 25, 27, 29

Proof: Setting 𝐼 = 𝑀 and 𝐽 = 1 in Theorem 1 yields the result. The BER expressions in Theorem 1 and Corollary 1 are valid for the general system. We now discuss several special cases of the general system, which have attracted particular research interest in the literature. In many practical systems, the PDF of the noise component is modeled by an even function. For instance, the additive noise in wireless communications is well-modeled as a zero mean Gaussian distribution [23] and the additive noise in image processing is often modeled as a zero mean Laplacian distribution [24]. For noise with even PDF, an important property is that the CDF and the complement CDF (CCDF) are symmetrical. If this property holds for the general system, the BER expressions of Theorem 1 and Corollary 1 can be simplified. Specifically, we have the following result. Corollary 2: Assume that the conditional PDFs of [𝑢]𝛼 , 𝛼 = 1, 2, given the knowledge of 𝑠, are even, i.e., 𝑓[𝑢]𝛼 ∣𝑠=𝑐 (𝑧) = 𝑓[𝑢]𝛼 ∣𝑠=−𝑐 (−𝑧) for any 𝑐 ∈ 𝔛; or equivalently, the conditional CDFs of [𝑢]𝛼 given the knowledge of 𝑠 satisfy the symmetry, 1 − 𝐹[𝑢]𝛼 ∣𝑠=𝑐 (𝑧) = 𝐹[𝑢]𝛼 ∣𝑠=−𝑐 (−𝑧).9 Then 9 In this conditional argument, we do not explicitly define the real and imaginary parts of 𝑠; but one should note that the signal component 𝑠 in the conditional argument can still be either real or complex, depending on the constellation ℭ.

𝒞𝑦 (𝑘, 𝑗) 1 1, 3 3, −1, 1, 3 1, 3, 5, 7 −3, 5, −1, 7, 1, 3, 5, 7 −5, −1, 3, 7, −3, 1, 5, −1, 3, 7, 1, 5, 3, 7, 5, 7 1, 3, 5, 7, 9, 11, 13, 15 −7, 9, −5, 11, −3, 13, −1, 15, 1, 3, 5, 7, 9, 11, 13, 15 −11, −3, 5, 13, −9, −1, 7, 15, −7, 1, 9, −5, 3, 11, −3, 5, 13, −1, 7, 15, 1, 9, 3, 11, 5, 13, 7, 15, 9, 11, 13, 15 −13, −9, −5, −1, 3, 7, 11, 15, −11, −7, −3, 1, 5, 9, 13, −9, −5, −1, 3, 7, 11, 15, −7, −3, 1, 5, 9, 13, −5, −1, 3, 7, 11, 15, −3, 1, 5, 9, 13, −1, 3, 7, 11, 15, 1, 5, 9, 13, 3, 7, 11, 15, 5, 9, 13, 7, 11, 15, 9, 13, 11, 15, 13, 15

𝒟𝑦 (𝑘) 1 2 4 4 8 16 8 16 32

64

𝜙𝛼 (𝐼, 𝐽, 𝑘), 𝛼 = 1, 2, of (9) can be simplified as 𝜙𝛼 (𝐼, 𝐽, 𝑘) =

𝒟𝐼𝛼 (𝑘) 𝐼3−𝛼 ∑ 2 ∑ 𝒜𝐼 (𝑘, 𝑗) 𝐼𝐽 𝑗=1 𝑚=1 𝛼

( ( ( × 1 − 𝐹[𝑢]𝛼 ∣[𝑠]𝛼 =−𝒞𝐼𝛼 (𝑘,𝑗),[𝑠]3−𝛼 =𝜔(𝑚,𝐼3−𝛼 ) 𝑑 ℬ𝐼𝛼 (𝑘, 𝑗) ) )) + (𝑎 − 1)𝒞𝐼𝛼 (𝑘, 𝑗) . (11) Proof: Applying 1 − 𝐹[𝑢]𝛼 ∣𝑠=𝑐 (𝑧) = 𝐹[𝑢]𝛼 ∣𝑠=−𝑐 (−𝑧) in (9) yields the result. So far, our attention has been devoted to a (still) general case in the sense that the noise component 𝑢 and the signal component 𝑥 are possibly dependent. Here, we consider a special but typical case in which 𝑢 is independent of 𝑥. Corollary 3: If 𝑢 is independent of 𝑥 and the PDFs of [𝑢]𝛼 , 𝛼 = 1, 2, are even functions, then 𝜙𝛼 (𝐼, 𝐽, 𝑘), 𝛼 = 1, 2, of (9) or (11) can be simplified as 𝒟𝐼𝛼 (𝑘) 2 ∑ 𝒜𝐼𝛼 (𝑘, 𝑗) 𝜙𝛼 (𝐼, 𝐽, 𝑘) = 𝐼𝛼 𝑗=1 ( ) (12) ( ( )) × 1 − 𝐹[𝑢]𝛼 𝑑 ℬ𝐼𝛼 (𝑘, 𝑗) + (𝑎 − 1)𝒞𝐼𝛼 (𝑘, 𝑗) .

Proof: Removing the conditional arguments in (11) yields the result.

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

Note that, in Lemma 1 we showed that the noise component is independent of the signal component for MMSE-non-SIC and they are dependent for MMSE-SIC. Furthermore, it will be demonstrated in the next section that the (conditional) PDFs of the noise component for both MMSE-non-SIC and MMSESIC are even functions. In the error analysis of hard MMSE detection, therefore, it is necessary to use (12) for MMSEnon-SIC and use (11) for MMSE-SIC. Before conducting the actual error analysis for hard MMSE detection, we consider two further special but practically important scenarios in the following. We first consider the BER of the general system given that 𝑎 = 1 and 𝑢 is independent of 𝑥. Corollary 4: If 𝑎 = 1 and 𝑢 is independent ℬ𝑦 (𝑘, 𝑗) and of 𝑥, the coefficients 𝒜𝑦 (𝑘, 𝑗), 𝒟𝑦 (𝑘) reduce to simple closed-forms captured by 𝐴𝑦 (𝑘, 𝑗), 𝐵𝑦 (𝑘, 𝑗), ⌊ and 𝐷𝑦⌋(𝑘, 𝑗), respectively, where (𝑗−1)2𝑘−1 ( ⌋) ⌊ 𝑘−1 𝑦 1 𝐴𝑦 (𝑘, 𝑗) = (−1) 2𝑘−1 − (𝑗−1)2 , + 𝑦 2 𝐵𝑦 (𝑘, 𝑗) = 2𝑗 − 1, and 𝐷𝑦 (𝑘) = (1 − 2−𝑘 )𝑦. Also, 𝜙𝛼 (𝐼, 𝐽, 𝑘) of (9) can be simply rewritten as 𝐷𝐼 (𝑘)

𝛼 1 ∑ 𝐴𝐼𝛼 (𝑘, 𝑗) 𝐼𝛼 𝑗=1 ( ( ( ) )) × 1 − 𝐹[𝑢]𝛼 𝑑𝐵𝐼𝛼 (𝑘, 𝑗) + 𝐹[𝑢]𝛼 − 𝑑𝐵𝐼𝛼 (𝑘, 𝑗) .

𝜙𝛼 (𝐼, 𝐽, 𝑘) =

(13) Proof: See Appendix B. The above corollary provides the BER for practically important cases. For instance, for the Laplacian additive noise channel in image processing [24], the BER result of Corollary 4 can be certainly used. On the other hand, the well-known BER result of [22] is not applicable since it is only valid for AWGN channels. Finally, if we consider the most typical scenario in the literature by imposing 𝑢 ∼ 𝒞𝒩 (0, 𝑁0 ) in Corollary 4, the CDF 𝐹[𝑢]𝛼 will be replaced by the Gaussian Q-function, and then, our expression 𝜙1 (𝐼, 1, 𝑘) reduces to [22, eq. (9)]. Therefore, Corollary 4 is a generalization of the work of [22] for AWGN channels. In the next section, we will apply the general analysis presented in this section to the error analysis for hard MMSE detection. V. E RROR P ERFORMANCE A NALYSIS FOR H ARD MMSE DETECTION

As demonstrated in Section IV, it is essential to derive the conditional CDF 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 (𝑧) and the unconditional CDF 𝐹[𝑢𝑖 ]𝛼 (𝑧), 𝛼 = 1, 2, of 𝑢𝑖 in order to have explicit closedform expressions for the BER of hard MMSE detectors. In this section, therefore, we first conduct a (conditional and unconditional) CDF analysis of 𝑢𝑖 . Then the derived CDFs are used to obtain the BER for hard MMSE detection. A. CDF Analysis In the following theorem, we now derive the conditional CDFs of 𝑢𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 , for MMSE-SIC.

Theorem 2: For MMSE-SIC with 𝑀 -PAM, 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 (𝑧) = 𝜓𝑖,𝛼 (𝑀, 1, 𝑠𝑖 , 𝑧); For MMSE-SIC with 𝐼 × 𝐽-QAM, 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 (𝑧) = 𝜓𝑖,𝛼 (𝐼, 𝐽, 𝑠𝑖 , 𝑧), 𝛼 = 1, 2, where 𝜓𝑖,𝛼 (𝐼, 𝐽, 𝑠𝑖 , 𝑧) is defined as follows: ( ) ∑ 𝑑[ℒ𝑖 ]𝛼 − 𝑧 1 𝜓𝑖,𝛼 (𝐼, 𝐽, 𝑠𝑖 , 𝑧) := 𝒬 (𝐼𝐽)𝑁𝑡 −1 𝜎𝑖 𝑁 +𝑖−2 퓢 𝑖 ∈𝔛

×

𝑖−1 ∏

𝑡

𝜑𝑙,1 (𝑠𝑖 , 퓢 𝑖 )𝜑𝑙,2 (𝑠𝑖 , 퓢 𝑖 ),

(14)

𝑙=1

√ for 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 .10 In (14), 𝜎𝑖 =∥ 𝒈𝑖 ∥ / 2, ℒ𝑖 = 𝒈𝑖 퓗𝑖 퓢 𝑖 , and the functions 𝜑𝑙,𝛽 (𝑠𝑖 , 퓢 𝑖 ), 𝛽 = 1, 2, 𝑙 = 1, 2, ⋅ ⋅ ⋅ , 𝑖 − 1, are defined as 𝜑𝑙,𝛽 (𝑠𝑖 , 퓢 𝑖 ) := i1 ([ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) + i2 ([ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) ( ( ) ( )) 𝑑 [ℒ𝑙 ]𝛽 − [ˆ 𝑠𝑙 ]𝛽 − 𝑎𝑙 [𝑠𝑙 ]𝛽 + 1 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) ×𝒬 − i3 ([ˆ 𝜎 ( 𝑙( ) ( )) 𝑑 ×𝒬 𝑠𝑙 ]𝛽 − 𝑎𝑙 [𝑠𝑙 ]𝛽 − 1 , (15) [ℒ𝑙 ]𝛽 − [ˆ 𝜎𝑙 where i1 (⋅, ⋅), i2 (⋅, ⋅), and i3 (⋅) are three indicator functions defined as follows: i1 (𝑥, 𝑦) = 1 if 𝑥 = 𝑦 − 1; i1 (𝑥, 𝑦) = 0 if 𝑥 ∕= 𝑦 −1; i2 (𝑥, 𝑦) = 1 if 𝑥 ∕= 𝑦 −1; i2 (𝑥, 𝑦) = 0 if 𝑥 = 𝑦 −1; i3 (𝑥, 𝑦) = 1 if 𝑥 ∕= 1 − 𝑦; and i3 (𝑥, 𝑦) = 0 if 𝑥 = 1 − 𝑦. Proof: See Appendix C. Note that the variables on the right hand side of (15) such as 𝑠𝑙 , 𝑠ˆ𝑙 , and 퓢 𝑙 (due to ℒ𝑙 = 𝒈𝑙 퓗𝑙 퓢 𝑙 ), are all determined by 퓢 𝑖 and 𝑠𝑖 for 𝑙 = 1, 2, ⋅ ⋅ ⋅ , 𝑖 − 1. Specifically, since 퓢 𝑖 = [𝒔𝑇⟨−𝑖⟩ , 𝒔ˆ𝑇(−𝑖) ]𝑇 , the variables 𝑠𝑙 ’s are all determined by 퓢 𝑖 though 𝒔⟨−𝑖⟩ ; the variables 𝑠ˆ𝑙 ’s, are all determined by 퓢 𝑖 though 𝒔ˆ(−𝑖) ; and 퓢 𝑙 ’s, are jointly determined by 퓢 𝑖 and 𝑠𝑖 . Also, setting 𝐼 = 𝑀 and 𝐽 = 1 in (14) in order to compute 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 (𝑧) for 𝑀 -PAM, it follows that 𝜑𝑙,2 (𝑠𝑖 , 퓢 𝑖 ) in (14) reduces to 1. Corollary 5: For MMSE-non-SIC detection, the MMSE filter yields independent noise component 𝑢𝑖 and signal component 𝑠𝑖 , and the CDFs of [𝑢𝑖 ]𝛼 , 𝛼 = 1, 2, are given by ( ) ∑ 1 𝑑[𝐿𝑖 ]𝛼 − 𝑧 𝒬 , (16) 𝐹[𝑢𝑖 ]𝛼 (𝑧) = 𝑁𝑡 −1 Ξ 𝜎𝑖 𝑁𝑡 −1 𝒔⟨−𝑖⟩ ∈𝔛

where 𝐿𝑖 = 𝒈𝑖 𝑯⟨−𝑖⟩ 𝒔⟨−𝑖⟩ , Ξ = 𝑀 for 𝑀 -PAM, and Ξ = 𝐼𝐽 for 𝐼 × 𝐽-QAM. Proof: From Lemma 1, 𝑢𝑖 is independent of 𝑠𝑖 for MMSE-non-SIC. Then applying Theorem 2 yields (16). In the following, we study an important property of the conditional and unconditional CDFs of [𝑢𝑖 ]𝛼 , 𝛼 = 1, 2. Theorem 3: The unconditional CDFs of 𝐹[𝑢𝑖 ]𝛼 (𝑧), 𝛼 = 1, 2, for MMSE-non-SIC in Corollary 5 satisfy the symmetry, 1 − 𝐹[𝑢𝑖 ]𝛼 (𝑧) = 𝐹[𝑢𝑖 ]𝛼 (−𝑧); the conditional CDFs of 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 (𝑧), 𝛼 = 1, 2, for MMSE-SIC in Theorem 2 satisfy the symmetry, 1 − 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 =𝑐𝑖 (𝑧) = 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 =−𝑐𝑖 (−𝑧), for any 𝑐𝑖 ∈ 𝔛. Proof: See Appendix D. Note that the symmetry of the unconditional and conditional CDFs in Theorem 3 can be used to substantially simplify the final BER expressions, as demonstrated by Corollaries 2 and 10 For

𝑖 = 1, the product

𝑖−1 ∏ 𝑙=1

(⋅) reduces to

0 ∏

(⋅) = 1.

𝑙=1

LIU and KIM: EXACT AND CLOSED-FORM ERROR PERFORMANCE ANALYSIS FOR HARD MMSE-SIC DETECTION IN MIMO SYSTEMS

Ψ𝛼 (𝐼, 𝐽, 𝑖, 𝑘) ⎧ ( ) ( ( 𝒟𝐼∑ 𝛼 (𝑘) ))  2  𝑑 ℬ 𝒜 (𝑘, 𝑗) 1 − 𝐹 (𝑘, 𝑗) + (𝑎 − 1)𝒞 (𝑘, 𝑗) ,  𝐼𝛼 𝐼𝛼 𝑖 𝐼𝛼 [𝑢𝑖 ]𝛼  𝐼𝛼  𝑗=1    ⎨ ( ( ( 𝒟𝐼∑ 𝛼 (𝑘) 𝐼3−𝛼 ∑ := 2 𝒜 (𝑘, 𝑗) 1 − 𝐹  [𝑢𝑖 ]𝛼 ∣[𝑠𝑖 ]𝛼 =−𝒞𝐼𝛼 (𝑘,𝑗),[𝑠𝑖 ]3−𝛼 =𝜔(𝑚,𝐼3−𝛼 ) 𝑑 ℬ𝐼𝛼 (𝑘, 𝑗)   𝐼𝐽 𝑗=1 𝑚=1 𝐼𝛼  )   ))   ⎩ + (𝑎𝑖 − 1)𝒞𝐼𝛼 (𝑘, 𝑗) ,

Φ𝛼 (𝐼, 𝐽) :=

⎧          ⎨          ⎩

2𝐼3−𝛼 (𝐼𝐽)𝑁𝑡

2 (𝐼𝐽)𝑁𝑡

𝐼𝛼 −1 ∑

(

∑

𝑚𝑖,𝛼 =1 𝒔⟨−𝑖⟩ ∈𝔛𝑁𝑡 −1 𝐼𝛼 −1 ∑

𝒬

𝐼∑ 3−𝛼

(

1 + (1 − 𝑎𝑖 )[𝑠𝑖 ]𝛼 − [𝐿𝑖 ]𝛼 (

∑

𝑚𝑖,𝛼 =1 𝑚𝑖,3−𝛼 =1 퓢 𝑖

− [ℒ𝑖 ]𝛼

𝑑 𝜎𝑖

∈𝔛𝑁𝑡 +𝑖−2

)) 𝑖−1 ∏ 𝑙=1

𝒬

𝑑 𝜎𝑖

(

3. In the next subsection, we now combine the CDF analysis in this section and the general error analysis in Section IV in order to obtain the BER expressions for both MMSE-non-SIC and MMSE-SIC. B. BER and SER Analyses of Hard MMSE Detection In Section IV, we analyzed the BER of the general system involving a single decision statistic 𝑧 = 𝑎𝑥 + 𝑢, where 𝑎 > 0 and 𝑢 is of any arbitrary distribution. In this section, we now study the error performance of the MMSE detection with 𝑁𝑡 decision statistics, 𝑧𝑖 = 𝑎𝑖 𝑥𝑖 + 𝑢𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑡 , where 𝑎𝑖 is given by (5) and 𝑢𝑖 is given by (6). It is obvious that each MMSE decision statistic 𝑧𝑖 can be seen as a special case of 𝑧 in the general system, and thus, the general analysis in Section IV can be applied to the error analysis of hard MMSE detection. In the following theorem, we derive the BER expression. Theorem 4: For hard MMSE detection employing 𝐼 × 𝐽QAM, the BER is given by ( log 𝐼 𝑁𝑡 2 ∑ ∑ 1 𝐵 𝒫1𝐵𝑖 (𝑘) 𝒫𝐼×𝐽-QAM = 𝑁𝑡 log2 (𝐼𝐽) 𝑖=1 𝑘=1 ) log2 𝐽 ∑ 𝐵𝑖 𝒫2 (𝑘) , (17) + 𝑘=1

where 𝒫𝛼𝐵𝑖 (𝑘) = Ψ𝛼 (𝐼, 𝐽, 𝑖, 𝑘), 𝛼 = 1, 2. Furthermore, the functions Ψ𝛼 (𝐼, 𝐽, 𝑖, 𝑘), 𝛼 = 1, 2, are defined in (18). Proof: Regarding each MMSE decision statistic 𝑧𝑖 as a special case of 𝑧 in the general system and using (12) for MMSE-non-SIC and (11) for MMSE-SIC, it is not hard to show the above result. By Theorem 4, one can obtain the explicit expression of the exact and closed-form BER for MMSE-non-SIC employing 𝐼 × 𝐽-QAM by substituting the unconditional CDFs 𝐹[𝑢𝑖 ]𝛼 (𝑧) of (16) into (18). Also, one can obtain the explicit expression of the exact and closed-form BER for MMSE-SIC employing 𝐼 × 𝐽-QAM by substituting the conditional CDFs 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 (𝑧)

for MMSE-non-SIC, (18)

for MMSE-SIC.

)) , for MMSE-non-SIC, (19)

1 + (1 − 𝑎𝑖 )[𝑠𝑖 ]𝛼

𝜑𝑙,1 (𝑠𝑖 , 퓢 𝑖 )𝜑𝑙,2 (𝑠𝑖 , 퓢 𝑖 ),

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for MMSE-SIC.

of (14) into (18). Furthermore, setting 𝐼 = 𝑀 and 𝐽 = 1 in these obtained BER expressions for 𝐼 × 𝐽-QAM gives the exact and closed-form BER expressions for 𝑀 -PAM in both MMSE-SIC and MMSE-non-SIC. In addition, one can also derive the SERs of MMSE-nonSIC and MMSE-SIC with 𝑀 -PAM and 𝐼 × 𝐽-QAM by taking a similar approach as in the BER analysis. In the following, we present only the final SER expressions for 𝐼 × 𝐽-QAM. Corollary 6: For hard MMSE detection employing ∑𝑁𝑡 𝑆𝑖 𝐼 × 𝐽𝑆 = 𝑁1𝑡 𝑖=1 𝒫 , where QAM, the SER is given by 𝒫𝐼×𝐽-QAM 𝒫 𝑆𝑖 denotes the SER of the 𝑖th symbol and is given by 𝒫 𝑆𝑖 = 1 − (1 − 𝒫1𝑆𝑖 )(1 − 𝒫2𝑆𝑖 ), in which, 𝒫𝛼𝑆𝑖 = Φ𝛼 (𝐼, 𝐽), 𝛼 = 1, 2. Furthermore, Φ𝛼 (𝐼, 𝐽), 𝛼 = 1, 2, are defined in (19), where [𝑠𝑖 ]𝛼 = 2𝑚𝑖,𝛼 − 1 − 𝐼𝛼 and [𝑠𝑖 ]3−𝛼 = 2𝑚𝑖,3−𝛼 − 1 − 𝐼3−𝛼 . Proof: Taking essentially the same approach as in the BER analysis yields the above result. Note that one can also obtain the explicit expressions of the exact and closed-form SER for 𝑀 -PAM in both MMSE-nonSIC and MMSE-SIC by simply setting 𝐼 = 𝑀 and 𝐽 = 1 in Corollary 6. C. BER and SER Analyses With Gaussian Approximation With Gaussian approximation, the noise term 𝑢𝑖 associated with the MMSE decision variable 𝑧𝑖 can be approximated as a zero mean complex Gaussian random variable, with the variance given by [10, eq. (15)]: Var (𝑢𝑖 ) = 𝛾𝑎𝑖 (1 − 𝑎𝑖 ). Using this approximation, we obtain the approximate BER/SER for MMSE-non-SIC and MMSE-SIC with arbitrary 𝐼 × 𝐽QAM. Specifically, the BER can be obtained by replacing Ψ𝛼 (𝐼, 𝐽, 𝑖, 𝑘) in Theorem 4 by ΨGA 𝛼 (𝐼, 𝐽, 𝑖, 𝑘), and the SER can be obtained by replacing Φ𝛼 (𝐼, 𝐽) in Corollary 6 by GA GA ΦGA 𝛼 (𝐼, 𝐽), where Ψ𝛼 (𝐼, 𝐽, 𝑖, 𝑘) and Φ𝛼 (𝐼, 𝐽) are respectively given by ΨGA 𝛼 (𝐼, 𝐽, 𝑖, 𝑘) =

𝒟𝐼𝛼 (𝑘) 2 ∑ 𝒜𝐼𝛼 (𝑘, 𝑗) 𝐼𝛼 𝑗=1

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0

0

10

10

−1

10

16−QAM

−1

10

−2

10

Average BER

Average SER

4−PAM

BPSK

−2

10

−3

10

−3

10

−4

10

−5

10 MMSE−non−SIC (Exact analysis) MMSE−non−SIC (Simulation) MMSE−SIC without ordering (Exact analysis) MMSE−SIC without ordering (Simulation) MMSE−SIC with ordering (Exact analysis) MMSE−SIC with ordering (Simulation)

−4

10

−5

10

0

5

10 15 20 SNR per receive antenna: ρ (dB)

MMSE−non−SIC (Exact analysis) MMSE−non−SIC (Simulation) MMSE−SIC without ordering (Exact analysis) MMSE−SIC without ordering (Simulation) MMSE−SIC with ordering (Exact analysis) MMSE−SIC with ordering (Simulation)

−6

10

−7

25

30

Fig. 1. Average SER for 2 × 2 MIMO system with BPSK, 4-PAM, and 16-QAM.

10

0

5

10 15 SNR per receive antenna: ρ (dB)

20

25

Fig. 3. Average BER for 4×4 MIMO system with BPSK. The curves labeled as MMSE-non-SIC (by simulation and analysis) and MMSE-SIC without ordering (by simulation and analysis) in this figure exactly overlap with the corresponding curves labeled as “MMSE” and “MMSE-IC”, respectively, in [5, Fig. 6.4].

0

10

3 × 3 MIMO (12 bps/Hz)

−1

10

−2

Average BER

10

2 × 4 MIMO (8 bps/Hz)

−3

10

2 × 3 MIMO (8 bps/Hz)

−4

10

MMSE−non−SIC (Exact analysis) MMSE−non−SIC (Simulation) MMSE−SIC without ordering (Exact analysis) MMSE−SIC without ordering (Simulation) MMSE−SIC with ordering (Exact analysis) MMSE−SIC with ordering (Simulation)

−5

10

−6

10

0

5

10 15 20 SNR per receive antenna: ρ (dB)

25

30

Fig. 2. Average BER for 2 × 3, 3 × 3, and 2 × 4 MIMO systems with 16-QAM.

(√ ) 2𝑑 (ℬ𝐼𝛼 (𝑘, 𝑗) + (𝑎𝑖 − 1) 𝒞𝐼𝛼 (𝑘, 𝑗)) √ ×𝒬 (25) 𝑎𝑖 (1 − 𝑎𝑖 ) 𝛾 (√ ) 𝐼𝛼 −1 2𝑑 (1 + (1 − 𝑎𝑖 ) [𝑠𝑖 ]𝛼 ) 2 ∑ GA √ 𝒬 Φ𝛼 (𝐼, 𝐽, 𝑖, 𝑘) = 𝐼𝛼 𝑚 =1 𝑎𝑖 (1 − 𝑎𝑖 ) 𝛾 𝑖,𝛼 (26) Furthermore, setting 𝐼 = 𝑀 and 𝐽 = 1 in (25) and (26), one can obtain the BER and SER for 𝑀 -PAM. VI. N UMERICAL R ESULTS A. Exact BER/SER Results We verify the validity of our derived BER and SER expressions for MMSE-non-SIC and MMSE-SIC by evaluating the average BER and SER. We consider flat fading with the elements of 𝑯 modeled as independent and identicallydistributed (i.i.d.) circularly symmetric complex Gaussian random variables with zero mean and unit variance. The average BER and SER can be obtained from pure Monte Carlo

simulations or computed through our derived expressions. Specifically, with our instantaneous BER/SER expressions, one can calculate the average BER/SER by generating sufficient many channel realizations and taking the mean of the corresponding instantaneous BER/SER values. We then compare our computed average BER/SER (denoted as “Exact analysis”) with the simulated average BER/SER (denoted as “Simulation”) in Figs. 1–3. Three MMSE algorithms are considered: 1) MMSE-non-SIC; 2) MMSE-SIC without ordering; and 3) MMSE-SIC with ordering.11 Various modulations and various antenna settings (denoted as 𝑁𝑡 × 𝑁𝑟 ) including 2 × 2, 2 × 3, 2 × 4, 3 × 3, and 4 × 4 are considered. All investigated cases reveal perfect agreements between our exact analysis and Monte Carlo simulations, as illustrated in Figs. 1–3. Furthermore, our curves can be backed up by the literature. For instance, the BER curves labeled as MMSE-non-SIC and MMSE-SIC without ordering in Fig. 3 exactly overlap with the corresponding curves in [5, Fig. 6.4]. B. Comparison of the Exact Analysis With Gaussian Approximation In this subsection, we evaluate the tightness of Gaussian approximation (denoted as “GA analysis”) in Figs. 4–7. From Figs. 4 and 5, we see that SER analysis based on Gaussian approximation is tight for MMSE-non-SIC, but loose for MMSE-SIC with or without ordering. Also, from Figs. 6–7 we see that BER analysis based on Gaussian approximation is tight only for MMSE-non-SIC with BPSK and QPSK. For MMSE-non-SIC with higher constellations and for MMSESIC with or without ordering, Gaussian approximation yields very loose BER approximations. 11 To test the validity of our analytical expressions, we adopt a detection ordering which maximizes the instantaneous output SINR of the MMSE filter at each stage as in [6] and [7]. Note that this ordering method is a suboptimal scheme; but it has a much lower computational complexity compared to an optimal detection ordering which requires exhaustive search over all possible ordering scenarios [7].

LIU and KIM: EXACT AND CLOSED-FORM ERROR PERFORMANCE ANALYSIS FOR HARD MMSE-SIC DETECTION IN MIMO SYSTEMS

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0

10

0

10

3 x 3 16−QAM

16−QAM

−1

10

−1

10

BPSK −2

10

−3

10

−4

10

−5

10

0

5

10 15 20 SNR per receive antenna: ρ (dB)

4 x 4 BPSK

−3

MMSE−non−SIC (Exact analysis) MMSE−non−SIC (GA analysis) MMSE−non−SIC (Simulation) MMSE−SIC without ordering (Exact analysis) MMSE−SIC without ordering (GA analysis) MMSE−SIC without ordering (Simulation) MMSE−SIC with ordering (Exact analysis) MMSE−SIC with ordering (GA analysis) MMSE−SIC with ordering (Simulation)

Average BER

Average SER

−2

10

25

30

10

−4

10

−5

10

Fig. 4. Average SER comparison: exact analysis versus Gaussian approximation, for 2 × 2 MIMO with BPSK and 16-QAM.

MMSE−non−SIC (Exact analysis) MMSE−non−SIC (GA analysis) MMSE−non−SIC (Simulation) MMSE−SIC without ordering (Exact analysis) MMSE−SIC without ordering (GA analysis) MMSE−SIC without ordering (Simulation) MMSE−SIC with ordering (Exact analysis) MMSE−SIC with ordering (GA analysis) MMSE−SIC with ordering (Simulation)

−6

10

−7

10

0

5

0

10

25

30

Fig. 6. Average BER comparison: exact analysis versus Gaussian approximation, for 4 × 4 MIMO with BPSK and 3 × 3 MIMO with 16-QAM.

−1

10

−2

10

0

10 MMSE−non−SIC (Exact analysis) MMSE−non−SIC (GA analysis) MMSE−non−SIC (Simulation) MMSE−SIC without ordering (Exact analysis) MMSE−SIC without ordering (GA analysis) MMSE−SIC without ordering (Simulation) MMSE−SIC with ordering (Exact analysis) MMSE−SIC with ordering (GA analysis) MMSE−SIC with ordering (Simulation)

−3

10

−4

10

0

5

10 15 20 SNR per receive antenna: ρ (dB)

4 x 4 16−QAM −1

10

25

Fig. 5. Average SER comparison: exact analysis versus Gaussian approximation, for 4 × 4 MIMO with 16-QAM.

4 x 4 QPSK

−2

30

10

Average BER

Average SER

10 15 20 SNR per receive antenna: ρ (dB)

−3

10

−4

10

We now provide some intuitive explanations about the above results. Since Gaussian approximation only takes into account the first and second moments (i.e., mean and variance), the expectation operators associated with the first and second moments can average out the correlation between 𝑢𝑖 and 𝑥𝑖 in the MMSE-SIC detection. Thus, for MMSE-SIC with or without ordering, Gaussian approximation yields very loose approximations for BER and SER. For MMSE-nonSIC, however, 𝑢𝑖 and 𝑥𝑖 are automatically independent since SIC is not involved; thus Gaussian approximation provides accurate SER for MMSE-non-SIC. For the BER approximation in MMSE-non-SIC, it becomes a bit more complicated. Gaussian approximation provides tight BER approximations only for MMSE-non-SIC with BPSK and QPSK. For BPSK, it is very obvious because the BER of BPSK is identical to the SER, which is accurately approximated by the Gaussian approximation. In the case of QPSK, since the I- and Qcomponents of QPSK can be seen as two independent BPSK signals, the BER of QPSK is exactly determined by the BER

MMSE−non−SIC (Exact analysis) MMSE−non−SIC (GA analysis) MMSE−non−SIC (Simulation) MMSE−SIC without ordering (Exact analysis) MMSE−SIC without ordering (GA analysis) MMSE−SIC without ordering (Simulation) MMSE−SIC with ordering (Exact analysis) MMSE−SIC with ordering (GA analysis) MMSE−SIC with ordering (Simulation)

−5

10

−6

10

0

5

10 15 20 SNR per receive antenna: ρ (dB)

25

30

Fig. 7. Average BER comparison: exact analysis versus Gaussian approximation, for 4 × 4 MIMO with QPSK and 16-QAM.

of BPSK, and thus, the BER approximation for QPSK should be tight as well. However, for MMSE-non-SIC detection with higher constellations (e.g., 𝑀 -PAM with 𝑀 ≥ 4 and 𝑀 QAM with 𝑀 > 4), each symbol is mapped into multiple (i.e., log2 𝑀 ) bits, and thereby, one symbol error can possibly result in up to log2 𝑀 bit errors, regardless of the bit-mapping

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schemes. Thus, the inaccuracy of the SER approximation due to Gaussian approximation is magnified in the BER analysis, which leads to loose BER approximation for MMSE-non-SIC with higher constellations. VII. C ONCLUSION This paper has studied the exact BER and SER for hard MMSE-non-SIC and MMSE-SIC detection with 𝑀 -PAM and arbitrary rectangular 𝐼 × 𝐽-QAM signalings, for the first time in the literature, to the best of our knowledge. Specifically, we first presented BER analysis for a general system with decision statistic, 𝑧 = 𝑎𝑥 + 𝑢, where 𝑎 > 0 is a constant and the noise component 𝑢 is arbitrarily distributed and possibly correlated with the transmitted signal 𝑥. For this general system, we derived exact and closed-form BER expressions for both 𝑀 PAM and 𝐼 × 𝐽-QAM signalings, which include the wellknown BER result in [22] as a special case. Furthermore, we applied the general analysis to the BER analysis for MIMO hard MMSE detection. In particular, we derived exact and closed-form instantaneous BER and SER expressions for both MMSE-non-SIC and MMSE-SIC detection with 𝑀 PAM and 𝐼 × 𝐽-QAM signalings. Finally, the validity of the derived expressions was verified through extensive Monte Carlo simulations and also supported by the reported results in the literature. As a further work, it is an interesting topic to extend the instantaneous BER analysis to average BER analysis for fading channels. A PPENDIX A P ROOF OF T HEOREM 1 Exploiting the separate I- and Q-components detection [22], the 𝐼 × 𝐽-QAM with 𝑥 ˆ = Prjℭ (𝑧) is equivalent to two independent PAMs: 𝐼-PAM with ℜ(ˆ 𝑥) = Prjℜ[ℭ] (ℜ(𝑧)) and 𝐽-PAM with ℑ(ˆ 𝑥) = Prjℑ[ℭ] (ℑ(𝑧)). Therefore, the BER of the general system with 𝐼 × 𝐽-QAM is the mean of the BERs of the equivalent 𝐼-PAM and 𝐽-PAM, which yields (8). As the derivations of 𝑃1𝐵 (𝑘) and 𝑃2𝐵 (𝑘) are essentially the same, in the following, we only present the derivation for 𝑃1𝐵 (𝑘), 𝑘 = 1, 2, ⋅ ⋅ ⋅ , log2 𝐼, for the equivalent 𝐼-PAM. In the equivalent 𝐼-PAM, the decision statistic is [𝑧]1 = 𝑎[𝑥]1 + [𝑢]1 , where 𝑥 = 𝑑𝑠 and 𝑢 is possibly dependent on 𝑠. First, we show that 𝑃1𝐵 (𝑘) can be written as ( ( 𝒟𝐼 (𝑘) ( 1 ∑ = 𝒜𝐼 (𝑘, 𝑗) Pr [𝑢]1 > 𝑑 ℬ𝐼 (𝑘, 𝑗) + (𝑎 − 1) 𝐼 𝑗=1 ) ( ) ( × 𝒞𝐼 (𝑘, 𝑗) [𝑠]1 = −𝒞𝐼 (𝑘, 𝑗) + Pr [𝑢]1 < −𝑑 ℬ𝐼 (𝑘, 𝑗) )) )  + (𝑎 − 1)𝒞𝐼 (𝑘, 𝑗) [𝑠]1 = 𝒞𝐼 (𝑘, 𝑗) , (A-1)

𝑃1𝐵 (𝑘)

for 𝑘 = 1, 2, ⋅ ⋅ ⋅ , log2 𝐼. The proof for (A-1) is divided into six steps. Step I: In this step, we exploit the structure of the bit sequence patterns based on the Gray mapping for the 𝐼-

…

1

… -7 - c4 Fig. 8.

0

0

0 …

1

3

5

c1

c2

c3

7 …

1

1

1

0

-5 - c3

-3 - c2

-1 -c1

c4

Bit sequence and signal space for the 1st bit.

PAM. As demonstrated in Table III, the bit sequence12 patterns based on the Gray mapping for the 𝑘th bits, 𝑘 = 1, 2, ⋅ ⋅ ⋅ , log2 𝐼, are essentially composed of two classes:13 1) 1 𝐼 0 𝐼 , that is, the left-plane bit sequence is a length2 2 𝐼 2 all-one sequence (denoted by 1 2𝐼 ) and the right-plane bit sequence is a length- 2𝐼 all-zero sequence (denoted by 0 𝐼 ); 2 2) 01 12 03 ⋅ ⋅ ⋅ 1𝐾−1 0𝐾 0𝐾 1𝐾−1 ⋅ ⋅ ⋅ 03 12 01 , that is, the leftplane is a succession of alternating zeros and ones (denoted by 01 12 03 ⋅ ⋅ ⋅ 1𝐾−1 0𝐾 ) and the right-plane is a repetition of the left-plane bit sequence in a reverse order (denoted by 0𝐾 1𝐾−1 ⋅ ⋅ ⋅ 03 12 01 ), where 0𝑛 is a length-𝑙𝑛 all-zero sequence and 1𝑛 is a length-𝑙𝑛 all-one sequence satisfying ∑ 𝐾 14 𝑛=1 𝑙𝑛 = 𝐼/2. It is obvious that the bit sequence of the 1st bit belongs to class 1) and the bit sequence of the 𝑘th (𝑘 ≥ 2) (essentially) belongs to class 2). Define 𝑒𝑘 as the event that the 𝑘th bit is detected erroneously in the 𝐼-PAM, 𝑘 = 1, 2, ⋅ ⋅ ⋅ , log2 𝐼. Then the 𝑘th bit error probability can be written as 𝑃1𝐵 (𝑘) = Pr(𝑒𝑘 ) 𝐼/2 ) ( )) 1 ∑( ( Pr 𝑒𝑘 ∣[𝑠]1 = 𝑐𝑗 + Pr 𝑒𝑘 ∣[𝑠]1 = −𝑐𝑗 , = 𝐼 𝑗=1

(A-2)

where 𝑐𝑗 = 2𝑗 − 1. The first term Pr(𝑒𝑘 ∣[𝑠]1 = 𝑐𝑗 ) of (A-2) represents the conditional error probability of the 𝑘th bit given that a right-plane signal constellation point 𝑐𝑗 (𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝐼/2) is transmitted, and the second term Pr(𝑒𝑘 ∣[𝑥]1 = −𝑐𝑗 ) of (A-2) represents the conditional error probability of the 𝑘th bit given that a left-plane signal constellation point −𝑐𝑗 is transmitted. Step II: In this step, we derive 𝑃1𝐵 (𝑘) for 𝑘 = 1. The bit sequence and the corresponding signal space for the 1st bit (𝑘 = 1) are depicted in Fig. 8. It is obvious that the errordecision region for any constellation point 𝑐𝑗 in the rightplane (the non-shaded region) is the whole left-plane (the shaded region); also, the error-decision region for any leftplane constellation point −𝑐𝑗 is the whole right-plane. Based 12 In the Gray code bit mapping for 𝐼-PAM, each signal constellation point is mapped to a Gray codeword which consists of log2 𝐼 bits, as shown in Table I. The bit sequence for the 𝑘th bit contains the 𝑘th bits in all the possible Gray codewords for the 𝐼-PAM. For example, the 4-PAM has 4 possible codewords 10, 11, 01, and 00, and hence, the bit sequence for the 1st bit is 1100 and the bit sequence for the 2nd bit is 0110. 13 Note that the Gray mapping scheme is not unique, and hence, different Gray mapping may yield different bit sequence patterns. However, for any Gray mappings, the BER analyses are essentially the same. For this reason, it is no loss of generality to focus on a particular Gray mapping scheme and the BER result holds for any other Gray mapping schemes [22]. In this paper, therefore, we adopt the same Gray mapping scheme as in [22]. 14 A bit sequence pattern, 0 1 0 ⋅ ⋅ ⋅ 0 1 2 3 𝐾−1 1𝐾 1𝐾 0𝐾−1 ⋅ ⋅ ⋅ 03 12 01 , essentially belongs to class 2) as the analysis will be the same.

LIU and KIM: EXACT AND CLOSED-FORM ERROR PERFORMANCE ANALYSIS FOR HARD MMSE-SIC DETECTION IN MIMO SYSTEMS

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TABLE III B IT S EQUENCE PATTERNS FOR E ACH B IT IN G RAY C ODE B IT M APPING bits 2-PAM 4-PAM 8-PAM 16-PAM

1st bit 10 1100 11110000 1111111100000000

G1

G2

3rd bit N/A N/A 01100110 0011110000111100

G3

0

1

1

0

0

1

1

0

-15

-13

-11

-9

-7

-5

-3

-1

S1 Fig. 9.

2nd bit N/A 0110 00111100 0000111111110000

S2

0 1

c1

4th bit N/A N/A N/A 0110011001100110

G4

G5

1

1

0

0

1

1

0

3

5

7

9

11

13

15

c2

S3

S4

A sample bit sequence and signal space for the 𝑘th (𝑘 > 1) bit.

on this observation, we can write Pr(𝑒𝑘 ∣[𝑠]1 = 𝑐𝑗 )

 = Pr([𝑧]1 − [𝑥]1 < −𝑐𝑗 𝑑[𝑠]1 = 𝑐𝑗 ), ( ( ) ) = Pr [𝑢]1 < −𝑑 𝑐𝑗 + (𝑎 − 1)𝑐𝑗 [𝑠]1 = 𝑐𝑗 ,

(A-3)

Pr(𝑒𝑘 ∣[𝑠]1 = −𝑐𝑗 ) = Pr([𝑧]1 − [𝑥]1 > 𝑐𝑗 𝑑∣[𝑠]1 = −𝑐𝑗 ), ( ) ) ( = Pr [𝑢]1 > 𝑑 𝑐𝑗 + (𝑎 − 1)𝑐𝑗 [𝑠]1 = −𝑐𝑗 ,

(A-4)

for 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝐼/2. Substituting (A-3) and (A-4) into (A-2) yields 𝑃1𝐵 (𝑘) for 𝑘 = 1: ) ) ( 1 ∑( ( Pr [𝑢]1 > 𝑑 𝑐𝑗 + (𝑎 − 1)𝑐𝑗 [𝑠]1 = −𝑐𝑗 = 𝐼 𝑗=1 ( ) )) ( + Pr [𝑢]1 < −𝑑 𝑐𝑗 + (𝑎 − 1)𝑐𝑗 [𝑠]1 = 𝑐𝑗 . (A-5) 𝐼/2

𝑃1𝐵 (1)

Therefore, (A-5) is the same as (A-1) for 𝑘 = 1, and the corresponding coefficients in (A-1) for 𝑘 = 1 are determined as follows: 𝒜𝐼 (1, 𝑗) = 1, ℬ𝐼 (1, 𝑗) = 2𝑗 −1, 𝒞𝐼 (1, 𝑗) = 2𝑗 −1, and 𝒟𝐼 (1) = 𝐼/2. Step III: In this step, we derive an initial expression of Pr(𝑒𝑘 ∣[𝑠]1 = 𝑐𝑗 ) for 𝑘 = 2, 3, ⋅ ⋅ ⋅ , log2 𝐼. As described earlier, the bit sequence for the 𝑘th (𝑘 > 1) bit is composed of alternating zeros and ones and the left-plane and right-plane bit sequences are symmetrical about the origin. Furthermore, the error-decision regions for the ones (zeros) are the decision regions for the zeros (ones). For any signal constellation point, the error-decision regions at most consist of four types: a leftmost infinite interval, a rightmost infinite interval, several finite intervals on the left, and several finite intervals on the right. As an example, Fig. 9 presents a sample bit sequence and the signal space for the 𝑘th (𝑘 > 1) bit, where the decision-regions for zeros are denoted by shaded regions G1 – G5 and the decision-regions for ones are denoted by dotted regions S1 –S4 . It is obvious that the error-decision region for constellation point 𝑐2 contains all the four types: one leftmost infinite region G1 , one rightmost infinite region G5 , two finite regions G2 and G3 on the left, and one finite region G4 on the right, whereas the error-decision region for constellation

point 𝑐1 contains only two types: two finite regions S1 and S2 on the left and two finite regions S3 and S4 on the right. Taking into account all the possible four types of errordecision regions, we can represent the error probability for the 𝑘th bit conditional on a right-plane constellation point 𝑐𝑗 , 𝑗 = 1, 2, ⋅ ⋅ ⋅ , log2 𝐼, by (A-6), where the first, second, third, and forth terms on the right-hand side account for the leftmost infinite error-decision region, the rightmost infinite errordecision region, the finite error-decision regions on the left, and the finite error-decision regions on the right, respectively. ′ can be either Furthermore, the coefficients 𝜖, 𝜖′ , 𝜋𝑚 , and 𝜋𝑚 zero if the corresponding error-decision region does not exist or one if it exists. Also, the coefficients 𝑇 (𝑘, 𝑗) and 𝑇 ′ (𝑘, 𝑗) denote the number of finite error-decision regions on the left and the number of finite error-decision regions on the right, respectively. Finally, the function Γ(𝑘, 𝑗), Γ′ (𝑘, 𝑗), 𝛿𝑚 (𝑘, 𝑗), ′ (𝑘, 𝑗), Δ𝑚 (𝑘, 𝑗), and Δ′𝑚 (𝑘, 𝑗) are all positive values 𝛿𝑚 which determine their corresponding error-decision regions. After some manipulations, (A-6) can be rewritten as (A-7). Step IV: In this step, we simplify (A-7) through reparameterization. Note that the first three terms on the right-hand side of (A-7) are essentially of the same format, and so are the last three terms. Therefore, to yield a more compact form, we introduce the following notation:

{𝑏} 𝜂𝑚

⎧ 𝜖{𝑏} ,      ⎨ 𝜋 {𝑏} , 𝑚 :=  {𝑏}   −𝜋𝑚−𝑇 {𝑏} (𝑘,𝑗) ,   ⎩

if 𝑚 = 0, if 𝑚 = 1, 2, ⋅ ⋅ ⋅ , 𝑇 {𝑏} (𝑘, 𝑗), if 𝑚 = 𝑇 {𝑏} (𝑘, 𝑗) + 1, 𝑇 {𝑏} (𝑘, 𝑗) + 2, ⋅ ⋅ ⋅ , 2𝑇 {𝑏} (𝑘, 𝑗), (A-8)

⎧ Γ{𝑏} (⋅, ⋅), if 𝑚 = 0,      ⎨ 𝛿 {𝑏} (⋅, ⋅), if 𝑚 = 1, 2, ⋅ ⋅ ⋅ , 𝑇 {𝑏} (𝑘, 𝑗), 𝑚 Λ{𝑏} 𝑚 (⋅, ⋅) :=  {𝑏}   if 𝑚 = 𝑇 {𝑏} (𝑘, 𝑗) + 1, Δ𝑚−𝑇 {𝑏} (𝑘,𝑗) (⋅, ⋅),   ⎩ {𝑏} 𝑇 (𝑘, 𝑗) + 2, ⋅ ⋅ ⋅ , 2𝑇 {𝑏} (𝑘, 𝑗), (A-9) where {𝑏} can be null indicating that the superscript does not exist, or {𝑏} can denote the superscript ′ , which was used in

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

Pr(𝑒𝑘 ∣[𝑠]1 = 𝑐𝑗 ) = 𝜖 Pr([𝑧]1 − [𝑥]1 < −Γ(𝑘, 𝑗)𝑑∣[𝑠]1 = 𝑐𝑗 ) + 𝜖′ Pr([𝑧]1 − [𝑥]1 > Γ′ (𝑘, 𝑗)𝑑∣[𝑠]1 = 𝑐𝑗 ) 𝑇 (𝑘,𝑗)

+

∑

𝜋𝑚 Pr(−Δ𝑚 (𝑘, 𝑗)𝑑 < [𝑧]1 − [𝑥]1 < −𝛿𝑚 (𝑘, 𝑗)𝑑∣[𝑠]1 = 𝑐𝑗 )

𝑚=1

(A-6)

𝑇 ′ (𝑘,𝑗)

+

∑

′ ′ 𝜋𝑚 Pr(𝛿𝑚 (𝑘, 𝑗)𝑑 < [𝑧]1 − [𝑥]1 < Δ′𝑚 (𝑘, 𝑗)𝑑∣[𝑠]1 = 𝑐𝑗 ).

𝑚=1 ′

𝑇 (𝑘,𝑗) ∑   ) ) ( ( ′ ′ Pr(𝑒𝑘 ∣[𝑠]1 = 𝑐𝑗 ) = 𝜖 Pr [𝑢]1 > (Γ′ (𝑘, 𝑗) − (𝑎 − 1)𝑐𝑗 )𝑑[𝑠]1 = 𝑐𝑗 + 𝜋𝑚 Pr [𝑢]1 > (𝛿𝑚 (𝑘, 𝑗) − (𝑎 − 1)𝑐𝑗 )𝑑[𝑠]1 = 𝑐𝑗 ′

𝑚=1 𝑇 ′ (𝑘,𝑗)

−

∑

𝑚=1 𝑇 (𝑘,𝑗)

+

  ) ( ) ( ′ 𝜋𝑚 Pr [𝑢]1 > (Δ′𝑚 (𝑘, 𝑗) − (𝑎 − 1)𝑐𝑗 )𝑑[𝑠]1 = 𝑐𝑗 + 𝜖 Pr [𝑢]1 < −(Γ(𝑘, 𝑗) + (𝑎 − 1)𝑐𝑗 )𝑑[𝑠]1 = 𝑐𝑗

∑

𝑇 (𝑘,𝑗) ∑   ) ) ( ( 𝜋𝑚 Pr [𝑢]1 < −(𝛿𝑚 (𝑘, 𝑗) + (𝑎 − 1)𝑐𝑗 )𝑑[𝑠]1 = 𝑐𝑗 − 𝜋𝑚 Pr [𝑢]1 < −(Δ𝑚 (𝑘, 𝑗) + (𝑎 − 1)𝑐𝑗 )𝑑[𝑠]1 = 𝑐𝑗 .

𝑚=1

𝑚=1

some coefficients and functions in (A-6) and (A-7).15 Then (A-7) can be simplified as Pr(𝑒𝑘 ∣[𝑠]1 = 𝑐𝑗 ) 2𝑇 ′ (𝑘,𝑗)

=

∑

 ) ( ′ 𝜂𝑚 Pr [𝑢]1 > [Λ′𝑚 (𝑘, 𝑗) − (𝑎 − 1)𝑐𝑗 ]𝑑[𝑠]1 = 𝑐𝑗

(A-7)

⎧ if 𝑚 = 0, 1, ⋅ ⋅ ⋅ , 2𝑇 (𝑘, 𝑗),  ⎨ 𝑐𝑗 , 𝜉𝑚 (𝑘, 𝑗) := −𝑐𝑗 , if 𝑚 = 2𝑇 (𝑘, 𝑗) + 1,  ⎩ 2𝑇 (𝑘, 𝑗) + 2, ⋅ ⋅ ⋅ , 2𝑇 (𝑘, 𝑗) + 2𝑇 ′(𝑘, 𝑗) + 1, (A-13)

𝑚=0 2𝑇 (𝑘,𝑗)

⎧ if 𝑚 = 0, 1, ⋅ ⋅ ⋅ , 2𝑇 (𝑘, 𝑗),   ⎨ Λ𝑚 (⋅, ⋅), ) ( + 𝜂𝑚 Pr [𝑢]1 < −[Λ𝑚 (𝑘, 𝑗) + (𝑎 − 1)𝑐𝑗 ]𝑑[𝑠]1 = 𝑐𝑗 . Θ𝑚 (⋅, ⋅) := ′ Λ (⋅, ⋅), if 𝑚 = 2𝑇 (𝑘, 𝑗) + 1,  𝑚=0 ⎩ 𝑚−2𝑇 (𝑘,𝑗)−1 2𝑇 (𝑘, 𝑗) + 2, ⋅ ⋅ ⋅ , 2𝑇 (𝑘, 𝑗) + 2𝑇 ′ (𝑘, 𝑗) + 1. (A-10) (A-14) Due to the symmetry of the bit sequence patterns in Gray Then (A-2) can be simplified as mapping, it follows that the conditional error probability given ( ( 𝐼/2 2𝑇 (𝑘,𝑗)+2𝑇 ′ (𝑘,𝑗)+1 ∑ ( 1∑ −𝑐𝑗 is transmitted can be written as 𝐵 𝜁𝑚 Pr [𝑢]1 > Θ𝑚 (𝑘, 𝑗) 𝑃1 (𝑘) = 𝐼 𝑗=1 𝑚=0 ) ( )  Pr(𝑒𝑘 ∣[𝑠]1 = −𝑐𝑗 ) + (𝑎 − 1)𝜉𝑚 (𝑘, 𝑗) 𝑑[𝑠]1 = −𝜉𝑚 (𝑘, 𝑗) + Pr [𝑢]1 2𝑇 (𝑘,𝑗) ∑  ( ) )) ( )  𝜂𝑚 Pr [𝑢]1 > (Λ𝑚 (𝑘, 𝑗) + (𝑎 − 1)𝑐𝑗 )𝑑[𝑠]1 = −𝑐𝑗 = < − Θ𝑚 (𝑘, 𝑗) + (𝑎 − 1)𝜉𝑚 (𝑘, 𝑗) 𝑑[𝑠]1 = 𝜉𝑚 (𝑘, 𝑗) . ∑

𝑚=0

2𝑇 ′ (𝑘,𝑗)

+

∑

 ( ) ′ 𝜂𝑚 Pr [𝑢]1 < −(Λ′𝑚 (𝑘, 𝑗) − (𝑎 − 1)𝑐𝑗 )𝑑[𝑠]1 = −𝑐𝑗 .

(A-15)

It can be seen that (A-15) is the same in form as (A-1) except that some coefficients are different. Step VI: In this step, we show that (A-15) is actually the Step V: In this step, we simplify Pr(𝑒𝑘 ∣[𝑠]1 = 𝑐𝑗 ) + same as (A-1) through reparameterization. Specifically, we can Pr(𝑒𝑘 ∣[𝑠]1 = −𝑐𝑗 ) in (A-2) through reparameterization. Sub- reorder Θ𝑚 (𝑘, 𝑗) in an increasing order by considering all stituting (A-10) and (A-11) into (A-2), we define 𝜁𝑚 , 𝜉𝑚 (⋅, ⋅), possible index values 𝑚 and 𝑗 for each 𝑘. Then the reordered and Θ𝑚 (⋅, ⋅) for the reparameterization as follows: new sequence of Θ𝑚 (𝑘, 𝑗) needs a new index variable in place ⎧ of two indices 𝑚 and 𝑗, and hence, the double summations if 𝑚 = 0, 1, ⋅ ⋅ ⋅ , 2𝑇 (𝑘, 𝑗),  ⎨ 𝜂𝑚 , in (A-15) reduces to a single summation represented by the ′ 𝜁𝑚 := 𝜂 , if 𝑚 = 2𝑇 (𝑘, 𝑗) + 1, new index variable. Moreover, when reordering the sequence  ⎩ 𝑚−2𝑇 (𝑘,𝑗)−1 2𝑇 (𝑘, 𝑗) + 2, ⋅ ⋅ ⋅ , 2𝑇 (𝑘, 𝑗) + 2𝑇 ′ (𝑘, 𝑗) + 1, of Θ𝑚 (𝑘, 𝑗) for each 𝑘, the parameters 𝜁𝑚 and 𝜉𝑚 (𝑘, 𝑗) (A-12) have to be reordered correspondingly. Note that all these reordered parameters are dependent on the value 𝐼 since 15 If {𝑏} is null in (A-8) and (A-9), 𝜂 the obtained BER is for 𝐼-PAM. Thus, we should define 𝑚 is defined in terms of 𝜖, 𝜋𝑚 , and 𝑇 (𝑘, 𝑗); and Λ𝑚 (⋅, ⋅) is defined in terms of Γ(⋅, ⋅), 𝛿𝑚 (⋅, ⋅), Δ𝑚 (⋅, ⋅), and all these parameters in terms of 𝐼 as well. Finally, we can ′ is defined in 𝑇 (𝑘, 𝑗). On the other hand, if {𝑏} is ′ in (A-8) and (A-9), 𝜂𝑚 ′ ′ ′ ′ ′ terms of 𝜖 , 𝜋𝑚 , and 𝑇 (𝑘, 𝑗); and Λ𝑚 (⋅, ⋅) is defined in terms of Γ (⋅, ⋅), denote the reordered sequence of Θ𝑚 (𝑘, 𝑗) by ℬ𝐼 (𝑘, 𝑗); the ′ (⋅, ⋅), Δ′ (⋅, ⋅), and 𝑇 ′ (𝑘, 𝑗). 𝛿𝑚 parameters 𝜁𝑚 and 𝜉𝑚 (𝑘, 𝑗) are denoted by 𝒜𝐼 (𝑘, 𝑗) and 𝑚 𝑚=0

(A-11)

LIU and KIM: EXACT AND CLOSED-FORM ERROR PERFORMANCE ANALYSIS FOR HARD MMSE-SIC DETECTION IN MIMO SYSTEMS

𝒞𝐼 (𝑘, 𝑗), respectively; and the total number of summations is denoted by 𝒟𝐼 (𝑘). This way we have shown that the 𝑘th (𝑘 > 1) bit BER can be written as (A-1) as well. Furthermore, we can explicitly adopt the above approach (Steps III–VI) for each 𝐼 and 𝑘, and then all the parameters 𝒜𝐼 (𝑘, 𝑗), ℬ𝐼 (𝑘, 𝑗), 𝒞𝐼 (𝑘, 𝑗), and 𝒟𝐼 (𝑘) can be determined, as shown in Table II. Finally, the BER of the 𝑘th bit, 𝑘 = 1, 2, ⋅ ⋅ ⋅ , log2 𝐼, for the equivalent 𝐼-PAM can be always written as (A-1). Applying the Total Probability Theorem to (A-1) by considering all the possibilities of [𝑠]2 , we can obtain 𝑃1𝐵 (𝑘) = 𝜙1 (𝐼, 𝐽, 𝑘), 𝑘 = 1, 2, ⋅ ⋅ ⋅ , log2 𝐼. Finally, taking essentially the same approach, we can show 𝑃2𝐵 (𝑘) = 𝜙2 (𝐼, 𝐽, 𝑘), 𝑘 = 1, 2, ⋅ ⋅ ⋅ , log2 𝐽. This completes the proof. A PPENDIX B P ROOF OF C OROLLARY 4 Starting with (9), we set 𝑎 = 1 and replace the conditional CDFs with the unconditional CDFs; then we have the following expression: ( 𝒟𝐼𝛼 (𝑘) ( ) 1 ∑ 𝒜𝐼𝛼 (𝑘, 𝑗) 1 − 𝐹[𝑢]𝛼 𝑑ℬ𝐼𝛼 (𝑘, 𝑗) 𝜙𝛼 (𝐼, 𝐽, 𝑘) = 𝐼𝛼 𝑗=1 ( )) + 𝐹[𝑢]𝛼 − 𝑑ℬ𝐼𝛼 (𝑘, 𝑗) . (B-1) It can be seen from Table II that there are some repeated blocks of odd numbers in the ℬ𝑦 (𝑘, 𝑗) column accounting for different 𝑗 values for fixed 𝐼 and 𝑘. Therefore, we may combine the terms associated with identical ℬ𝐼𝛼 (𝑘, 𝑗) in (B1), and hence, the corresponding coefficients 𝒜𝐼𝛼 (𝑘, 𝑗) add up and the total number of terms 𝒟𝐼𝛼 (𝑘) in the finite sum decreases. Finally, we represent the new obtained coefficients by 𝐴𝐼𝛼 (𝑘, 𝑗), 𝐵𝐼𝛼 (𝑘, 𝑗), and 𝐷𝐼𝛼 (𝑘), and hence, (B-1) reduces to (13). Next, the remaining task is to determine the new coefficients 𝐴𝐼𝛼 (𝑘, 𝑗), 𝐵𝐼𝛼 (𝑘, 𝑗), and 𝐷𝐼𝛼 (𝑘). From Table II, it can be verified that for each 𝑦 and 𝑘, ℬ𝑦 (𝑘, 𝑗) reduces to a sequence of continuous odd numbers if we combine the repeated blocks. Also, if we add up the numbers 𝒜𝑦 (𝑘, 𝑗) corresponding to the repeated blocks of ℬ𝑦 (𝑘, 𝑗), the 𝒜𝑦 (𝑘, 𝑗) column would reduce to the [𝑋𝐼 (𝑘)] column in [22, Table I]. Therefore, Table II in this paper reduces to [22, Table I]. To verify these observations, we provide a more rigorous proof to determine the coefficients 𝐴𝐼 (𝑘, 𝑗), 𝐵𝐼 (𝑘, 𝑗), and 𝐷𝐼 (𝑘, 𝑗) in the following. Note that we obtained (13) without adopting any assumption on the distribution of 𝑢. This suggests that the obtained coefficients 𝐴𝐼 (𝑘, 𝑗), 𝐵𝐼 (𝑘, 𝑗), and 𝐷𝐼 (𝑘, 𝑗) in (13) indeed do not depend on any particular distribution of 𝑢. Therefore, the obtained coefficients are invariant under different noise distributions. Hence, without loss of generality, we consider the Gaussian distribution, 𝑢 ∼ 𝒞𝒩 (0, 𝑁0 ), which is also the case considered in [22]. Then (13) simplifies as ( ) 𝐷𝐼𝛼 (𝑘) 𝑑𝐵𝐼𝛼 (𝑘, 𝑗) 1 ∑ √ 𝜙𝛼 (𝐼, 𝐽, 𝑘) = 𝐴𝐼𝛼 (𝑘, 𝑗)erfc , 𝐼𝛼 𝑗=1 𝑁0 (B-2) where erfc(⋅) is the complementary error function [22]. Setting 𝛼 = 1 and 𝐽 = 1 in (B-2) gives the BER of the 𝐼-PAM for

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AWGN channel, which was also presented by [22, eq. (8)]. Therefore, it is not hard to determine the coefficients 𝐴𝐼 (𝑘, 𝑗), 𝐵𝐼 (𝑘, 𝑗), and 𝐷𝐼 (𝑘) by comparing 𝜙1 (𝐼, 1, 𝑘) with the BER result of [22, eq. (8)]. Once again, it is worth noting that the Gaussian noise assumption adopted in (B-2) only specifies the explicit expression for the CDF of [𝑢]𝛼 ; however, it does not change the general forms of 𝐴𝐼 (𝑘, 𝑗), 𝐵𝐼 (𝑘, 𝑗), and 𝐷𝐼 (𝑘, 𝑗). Therefore, the above derived new coefficients are common for any noise distributions. A PPENDIX C P ROOF OF T HEOREM 2 Recall that for MMSE-SIC, 𝑢𝑖 = 𝑑𝒈𝑖 퓗𝑖 퓢 𝑖 + 𝒈𝑖 𝒏, where 퓢 𝑖 = [𝒔𝑇⟨−𝑖⟩ , 𝒔ˆ𝑇(−𝑖) ]𝑇 . Then the conditional distribution of 𝑢𝑖 given 퓢 𝑖 is circularly symmetric complex Gaussian, i.e., 𝑢𝑖 ∣퓢 𝑖 ∼ 𝒞𝒩 (𝑑ℒ𝑖 , 2𝜎𝑖2 ). In the following, we first present the derivation of 𝐹[𝑢𝑖 ]1 ∣𝑠𝑖 (𝑧) for MMSE-SIC. To proceed, we apply the Total Probability Theorem as follows: 𝐹[𝑢𝑖 ]1 ∣𝑠𝑖 (𝑧) ( ) ∑ ) ( 𝑑[ℒ𝑖 ]1 − 𝑧 𝒬 = Pr 퓢 𝑖 ∣𝑠𝑖 𝜎𝑖 퓢 𝑖 ∈𝔛𝑁𝑡 +𝑖−2 ( ) ∑ ( ) 𝑑[ℒ𝑖 ]1 − 𝑧 1 𝒬 = Pr 𝒔ˆ(−𝑖) ∣𝑠𝑖 , 𝒔⟨−𝑖⟩ 𝑁 −1 𝑡 (𝐼𝐽) 𝜎𝑖 퓢 𝑖 ∈𝔛𝑁𝑡 +𝑖−2 ( ) 𝑖−1 ∑ ) 𝑑[ℒ𝑖 ]1 − 𝑧 ∏ ( 1 = 𝒬 Pr 𝑠ˆ𝑙 ∣ˆ 𝒔(−𝑙) , 𝒔 , 𝑁 −1 𝑡 (𝐼𝐽) 𝜎𝑖 𝑁 +𝑖−2 퓢 𝑖 ∈𝔛

𝑡

𝑙=1

(C-1) ( ) where Pr 𝑠ˆ𝑙 ∣ˆ 𝒔(−𝑙) , 𝒔 is the probability of detecting 𝑠ˆ𝑙 conditional on all the transmitted symbols 𝒔 and( the predetected ) ( symbols ) 𝒔ˆ(−𝑙) . We can write Pr 𝑠ˆ𝑙 ∣ˆ 𝒔(−𝑙) , 𝒔 = Pr 𝑠ˆ𝑙 ∣𝑠𝑙 , 퓢 𝑙 and regard this probability as the conditional probability. ) pairwise ( ) We ( can further ) ( 𝑠𝑙 ]1 ∣[𝑠𝑙 ]1 , 퓢 𝑙 Pr [ˆ 𝑠𝑙 ]2 ∣[𝑠𝑙 ]2 , 퓢 𝑙 , write Pr 𝑠ˆ𝑙 ∣𝑠𝑙 , 퓢 𝑙 = Pr [ˆ owing to the separate I-component and Q-component ) ( detection. Then Pr [ˆ 𝑠𝑙 ]1 ∣[𝑠𝑙 ]1 , 퓢 𝑙 can be seen as the conditional pairwise probability (for the equivalent 𝐼-PAM ) 𝑠𝑙 ]2 ∣[𝑠𝑙 ]2 , 퓢 𝑙 can be seen with [𝑧𝑙 ]1 = 𝑎𝑙 [𝑥𝑙 ]1 +[𝑢𝑙 ]1 and Pr [ˆ as the conditional pairwise probability for the equivalent 𝐽PAM with [𝑧𝑙 ]2 = 𝑎𝑙 [𝑥𝑙 ]2 + [𝑢𝑙 ]2 , where 𝑢𝑙 = 𝑑𝒈𝑙 퓗𝑙 퓢 𝑙 + 𝒈𝑙 𝒏. It can be shown that ) ( Pr [ˆ 𝑠𝑙 ]1 ∣[𝑠𝑙 ]1 , 퓢 𝑙 ⎧ ( ) 𝐹[𝑢𝑙 ]1 ∣퓢 𝑙 𝑑([ˆ 𝑠𝑙 ]1 − 𝑎𝑙 [𝑠𝑙 ]1 + 1) , if [ˆ 𝑠𝑙 ]1 = −(𝐼 − 1),     ) (   𝑠𝑙(]1 − 𝑎𝑙 [𝑠𝑙 ]1 + 1) ⎨ 𝐹[𝑢𝑙 ]1 ∣퓢 𝑙 𝑑([ˆ ) 𝑠𝑙 ]1 − 𝑎𝑙 [𝑠𝑙 ]1 − 1) , = − 𝐹[𝑢𝑙 ]1 ∣퓢 𝑙 𝑑([ˆ   if [ˆ 𝑠𝑙 ]1 = −(𝐼 − 3), −(𝐼 − 5), ⋅ ⋅ ⋅ , 𝐼 − 3,     ( ) ⎩ 1 − 𝐹[𝑢𝑙 ]1 ∣퓢 𝑙 𝑑([ˆ 𝑠𝑙 ]1 − 𝑎𝑙 [𝑠𝑙 ]1 − 1) , if [ˆ 𝑠𝑙 ]1 = 𝐼 − 1. (C-2) ) ( Then substituting 𝐹[𝑢𝑙 ]1 ∣퓢 𝑙 (𝑧) = 𝒬 𝑑[ℒ𝜎𝑙 ]𝑙1 −𝑧 into (C-2) ( ) yields Pr [ˆ 𝑠𝑙 ]1 ∣[𝑠𝑙 ]1 , 퓢 𝑙 = 𝜑𝑙,1 (𝑠𝑖 , 퓢 𝑖 ). (Taking essentially ) the same approach, we can derive Pr [ˆ 𝑠𝑙 ]2 ∣[𝑠𝑙 ]2 , 퓢 𝑙 = 𝜑𝑙,2 (𝑠𝑖 , 퓢 𝑖 ). Therefore, (C-1) along with the obtained 𝜑𝑙,1 (𝑠𝑖 , 퓢 𝑖 ) and 𝜑𝑙,2 (𝑠𝑖 , 퓢 𝑖 ) yields the conditional CDF

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

𝐹[𝑢𝑖 ]1 ∣𝑠𝑖 (𝑧) = 𝜓𝑖,1 (𝐼, 𝐽, 𝑠𝑖 , 𝑧). Furthermore, taking essentially the same approach as in the derivation of 𝐹[𝑢𝑖 ]1 ∣𝑠𝑖 (𝑧), we can derive 𝐹[𝑢𝑖 ]2 ∣𝑠𝑖 (𝑧) = 𝜓𝑖,2 (𝐼, 𝐽, 𝑠𝑖 , 𝑧). Also, setting 𝐼 = 𝑀 and 𝐽 = 1 in the obtained CDFs for 𝐼 × 𝐽-QAM yields the CDFs for 𝑀 -PAM; that is, 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 (𝑧) = 𝜓𝑖,𝛼 (𝑀, 1, 𝑠𝑖 , 𝑧), 𝛼 = 1, 2. This completes the proof. A PPENDIX D P ROOF OF T HEOREM 3 We first show that the conditional CDFs of 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 (𝑧), 𝛼 = 1, 2, for MMSE-SIC satisfy the property 1 − 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 =𝑐𝑖 (𝑧) = 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 =−𝑐𝑖 (−𝑧) for any 𝑐𝑖 ∈ 𝔛. Note that this is equivalent to showing 𝜓𝑖,𝛼 (𝐼, 𝐽, −𝑠𝑖 , −𝑧) = 1−𝜓𝑖,𝛼(𝐼, 𝐽, 𝑠𝑖 , 𝑧). Starting from (14) we write ∑ 1 1−𝜓𝑖,𝛼 (𝐼, 𝐽, 𝑠𝑖 , 𝑧) = (𝐼𝐽)𝑁𝑡 −1 퓢 𝑖 ∈𝔛𝑁𝑡 +𝑖−2 (D-1) ( ) 𝑖−1 𝑧 − 𝑑[ℒ𝑖 ]𝛼 ∏ 𝒬 𝜑𝑙,1 (𝑠𝑖 , 퓢 𝑖 )𝜑𝑙,2 (𝑠𝑖 , 퓢 𝑖 ), 𝜎𝑖 𝑙=1 ( ) where the equality follows by the facts Pr 퓢( 𝑖 ∣𝑠𝑖 ) = ∑ 1 𝜑 (𝑠 , 퓢 𝑖 )𝜑𝑙,2 (𝑠𝑖 , 퓢 𝑖 ) and Pr 퓢 𝑖 ∣𝑠𝑖 = (𝐼𝐽)𝑁𝑡 −1 𝑙,1 𝑖 퓢 𝑖 ∈𝔛𝑁𝑡 +𝑖−2

1, as demonstrated by (C-1). Also, 𝜓𝑖,𝛼 (𝐼, 𝐽, −𝑠𝑖 , −𝑧) can be written as 𝜓𝑖,𝛼 (𝐼, 𝐽, −𝑠𝑖 , −𝑧) ( ) ∑ 1 𝑧 + 𝑑[ℒ˜𝑖 ]𝛼 = 𝒬 (𝐼𝐽)𝑁𝑡 −1 𝜎𝑖 𝑁 +𝑖−2 ˜ 𝑖 ∈𝔛 퓢

× =

1 (𝐼𝐽)𝑁𝑡 −1

𝑡

𝑖−1 ∏

˜ 𝑖 )𝜑𝑙,2 (−𝑠𝑖 , 퓢 ˜ 𝑖 ), 𝜑𝑙,1 (−𝑠𝑖 , 퓢

𝑙=1

∑

퓢 𝑖 ∈𝔛𝑁𝑡 +𝑖−2

×

𝑖−1 ∏

( 𝒬

𝑧 − 𝑑[ℒ𝑖 ]𝛼 𝜎𝑖

(D-2)

)

𝜑𝑙,1 (−𝑠𝑖 , −퓢 𝑖 )𝜑𝑙,2 (−𝑠𝑖 , −퓢 𝑖 ).

(D-3)

𝑙=1

˜ 𝑖 as the index of the finite sum to In (D-2), we use 퓢 distinguish it from the index 퓢 𝑖 of the finite sum in (D-1), ˜ 𝑖 . Furthermore, (D-2) is due to (14), and and ℒ˜𝑖 = 𝒈𝑖 퓗𝑖 퓢 ˜ 𝑖 = −퓢 𝑖 . Note that (D-3) follows by the change of variable, 퓢 the change of variable is valid because the constellation space 𝔛 is symmetrical about the origin. Then the remaining task is to show 𝜑𝑙,𝛽 (−𝑠𝑖 , −퓢 𝑖 ) = 𝜑𝑙,𝛽 (𝑠𝑖 , 퓢 𝑖 ) for 𝛽 = 1, 2. To this end, we write 𝜑𝑙,𝛽 (−𝑠𝑖 , −퓢 𝑖 ) as follows: 𝜑𝑙,𝛽 (−𝑠𝑖 , −퓢 𝑖 )

(

𝑑( [−ℒ𝑙 ]𝛽 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) + i2 ([−ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 )𝒬 =i1 ([−ˆ 𝜎𝑙 ) ( )) − [−ˆ 𝑠𝑙 ]𝛽 − 𝑎𝑙 [−𝑠𝑙 ]𝛽 + 1 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) − i3 ([−ˆ ( ( ) ( )) 𝑑 [−ℒ𝑙 ]𝛽 − [−ˆ 𝑠𝑙 ]𝛽 − 𝑎𝑙 [−𝑠𝑙 ]𝛽 − 1 ×𝒬 , (D-4) 𝜎𝑙 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) + i2 ([−ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) − i3 ([−ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) =i1 ([−ˆ ( ( ) ( )) 𝑑 [ℒ𝑙 ]𝛽 − [ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 )𝒬 𝑠𝑙 ]𝛽 − 𝑎𝑙 [𝑠𝑙 ]𝛽 + 1 + i3 ([−ˆ 𝜎𝑙

( 𝑠𝑙 ]𝛽 , 𝐼𝛽 )𝒬 − i2 ([−ˆ

) ( )) 𝑑( [ℒ𝑙 ]𝛽 − [ˆ 𝑠𝑙 ]𝛽 − 𝑎𝑙 [𝑠𝑙 ]𝛽 − 1 , 𝜎𝑙 (D-5)

where (D-4) is due to (15) and (D-5) follows by the fact 𝒬(−𝑥) = 1 − 𝒬(𝑥). Furthermore, by definitions of i1 (⋅, ⋅), i2 (⋅, ⋅), and i3 (⋅, ⋅), we have i1 ([−ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) + i2 ([−ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) − i3 ([−ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) = i1 ([ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ), i3 ([−ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) = 𝑠𝑙 ]𝛽 , 𝐼𝛽 ), and i2 ([−ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ) = i3 ([ˆ 𝑠𝑙 ]𝛽 , 𝐼𝛽 ). Hence, i2 ([ˆ 𝜑𝑙,𝛽 (−𝑠𝑖 , −퓢 𝑖 ) = 𝜑𝑙,𝛽 (𝑠𝑖 , 퓢 𝑖 ) for 𝛽 = 1, 2. Then it follows that 𝜓𝑖,𝛼 (𝐼, 𝐽, −𝑠𝑖 , −𝑧) = 1 − 𝜓𝑖,𝛼 (𝐼, 𝐽, 𝑠𝑖 , 𝑧), i.e., 1 − 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 =𝑐𝑖 (𝑧) = 𝐹[𝑢𝑖 ]𝛼 ∣𝑠𝑖 =−𝑐𝑖 (−𝑧) is true for MMSESIC. Taking essentially the same approach, we can show that 1 − 𝐹[𝑢𝑖 ]𝛼 (𝑧) = 𝐹[𝑢𝑖 ]𝛼 (−𝑧) is true for MMSE-non-SIC. This completes the proof. R EFERENCES [1] E. Telatar, “Capacity of multi-antenna Gaussian channels,” AT&T Bell Lab., Tech. Rep., June 1995. [2] E. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications. Cambridge University Press, 2003. [3] B. Vucetic and J. Yuan, Space-Time Coding. Wiley, 2003. [4] G. D. Golden, G. J. Foschini, R. A. Valenzuela, and P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture,” Electron. Lett., vol. 35, pp. 14–15, Jan. 1999. [5] T. M. Duman and A. Ghrayeb, Coding for MIMO Communication Systems. John Wiley & Sons, Inc., 2007. [6] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. Cambridge University Press, 2003. [7] H. Jafarkhani, Space-Time Coding: Theory and Practice. Cambridge University Press, 2005. [8] N. Kim and H. Park, “Bit error performance of convolutional coded MIMO system with linear MMSE receiver,” IEEE Trans. Wireless Commun., vol. 8, pp. 3420–3424, July 2009. [9] T. Ait-Idir, S. Saoudi, and N. Naja, “Space-time turbo equalization with successive interference cancellation for frequency-selective MIMO channels,” IEEE Trans. Veh. Technol., vol. 57, pp. 2766–2778, Sep. 2008. [10] H. Lee, B. Lee, and I. Lee, “Iterative detection and decoding with an improved V-BLAST for MIMO-OFDM systems,” IEEE J. Sel. Areas Commun., vol. 24, pp. 504–513, Mar. 2006. [11] Y.-C. Liang, G. Pan, and Z. D. Bai, “Asymptotic performance of MMSE receivers for large systems using random matrix theory,” IEEE Trans. Inf. Theory, vol. 53, pp. 4173–4190, Nov. 2007. [12] D. Guo, S. Verd´ u, and L. K. Rasmussen, “Asymptotic normality of linear multiuser receiver outputs,” IEEE Trans. Inf. Theory, vol. 48, pp. 3080–3095, Dec. 2002. [13] H. V. Poor and S. Verd´ u, “Probability of error in MMSE multiuser detection,” IEEE Trans. Inf. Theory, vol. 43, pp. 858–871, May 1997. [14] M. V. Burnashev and H. V. Poor, “On the probability of error in linear multiuser detection,” IEEE Trans. Inf. Theory, vol. 49, pp. 1922–1941, Aug. 2003. [15] N. Kim, Y. Lee, and H. Park, “Performance analysis of MIMO system with linear MMSE receiver,” IEEE Trans. Wireless Commun., vol. 7, pp. 4474–4478, Nov. 2008. [16] A. Zanella, M. Chiani, and M. Z. Win, “MMSE reception and successive interference cancellation for MIMO systems with high spectral efficiency,” IEEE Trans. Wireless Commun., vol. 4, pp. 1244– 1253, May 2005. [17] M. Chiani, M. Z. Win, A. Zanella, R. K. Mallik, and J. H. Winters, “Bounds and approximations for optimum combining of signals in the presence of multiple co-channel interferers and thermal noise,” IEEE Trans. Commun., vol. 51, pp. 296–307, Feb. 2003. [18] G. V. Moustakides and H. V. Poor, “On the relative error probabilities of linear multiuser detectors,” IEEE Trans. Inf. Theory, vol. 47, pp. 450–456, Jan. 2001. [19] P. Li, D. Paul, and R. Narasimhan, “On the distribution of SINR for the MMSE MIMO receiver and performance analysis,” IEEE Trans. Inf. Theory, vol. 52, pp. 271–285, Jan. 2006. [20] S. Verd´ u, Multiuser Detection. Cambridge University Press, 1998.

LIU and KIM: EXACT AND CLOSED-FORM ERROR PERFORMANCE ANALYSIS FOR HARD MMSE-SIC DETECTION IN MIMO SYSTEMS

[21] N. Prasad and M. K. Varanasi, “Analysis of decision feedback detection for MIMO Rayleigh-fading channels and the optimization of power and rate allocations,” IEEE Trans. Inf. Theory, vol. 50, pp. 1009–1025, June 2004. [22] K. Cho and D. Yoon, “On the general BER expression of one- and two-dimensional amplitude modulations,” IEEE Trans. Commun., vol. 50, pp. 1074–1080, July 2002. [23] J. G. Proakis, Digital Communications, 4th edition. McGraw-Hill, 2000. [24] E. Y. Lam and J. W. Goodman, “A mathematical analysis of the DCT coefficient distributions for images,” IEEE Trans. Image Process., vol. 9, pp. 1661–1666, Oct. 2000. [25] P. Liu and I.-M. Kim, “Exact and closed-form error performance analysis for hard MMSE detection in MIMO systems,” in Proc. IEEE Global Commun. Conf., Dec. 2010. [26] P. Liu and I.-M. Kim, “BER analysis for hard MMSE detection in MIMO systems,” in Proc. ACM ICUIMC, Feb. 2011. Peng Liu received the B.Eng. and M.Eng. degrees in electrical engineering from Xidian University, Xi’an, China, in July 2005 and July 2007, respectively. In September 2007, he joined the Wireless Information Transmission Lab (WITL) of the Department of Electrical and Computer Engineering, Queen’s University, Kingston, Canada, where he received his M.Sc. degree in August 2009. He is currently working towards the Ph.D. degree, and serving as a Research Assistant, in the WITL. His research interests include cooperative diversity networks, bidirectional communications, cognitive radios, and MIMO communication systems.

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Il-Min Kim received the B.S. degree in electronics engineering from Yonsei University, Seoul, Korea, in 1996, and the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Taejon, Korea, in 1998 and 2001, respectively. From October 2001 to August 2002, he was with the Dept. of Electrical Engineering and Computer Sciences at MIT, Cambridge, USA, and from September 2002 to June 2003, he was with the Dept. of Electrical Engineering at Harvard, Cambridge, USA, as a Postdoctoral Research Fellow. In July 2003, he joined the Dept. of Electrical and Computer Engineering at Queen’s University, Kingston, Canada, where he is currently an Associate Professor. His research interests include bidirectional communications, cooperative diversity networks, network coding, femto cells, CoMP, and cognitive radio. He is currently serving as an Editor for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS and Journal of Communications and Networks (JCN).