Efficient Near-Optimum Detectors for Large MIMO

Wireless Personal Communications manuscript No. (will be inserted by the editor)

Efficient Near-Optimum Detectors for Large MIMO Systems under Correlated Channels Ricardo Tadashi Kobayashi · Fernando Ciriaco · Taufik Abrão

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Abstract Recently, high spectral and energy efficiencies multiple antennas wireless systems under scattered environments have attracted an increasing interest due to their intrinsically benefits. This work focuses on the analysis of MIMO equalizers, improved MIMO detection techniques and their combinations, allowing a good balance between complexity and performance in Rayleigh channel environment. Primarily, the MIMO linear equalizers combined with detection techniques such as ordering (via sorted QR decomposition, SQRD), successive interference cancellation (SIC), list reduction and lattice reduction (LR) were investigated. An important aspect invariably present in practical system taken into account in the analysis has been the channel correlation effect, which under certain realistic conditions could result in a strong negative impact on the MIMO system performance. The goal of this paper consists in construct a framework on sub-optimum MIMO detection techniques, pointing out a MIMO detection architecture able to attain low or moderate complexity, suitable performance and full diversity. Keywords Lattice-reduction · Channel correlation · MIMO · Zeroforcing (ZF), Minimum-mean-squared-error (MMSE) · ordered successiveinterference-cancellation (OSIC) · Chase-list · Sphere-decoder (SD). R. T. Kobayashi Dept. of Electrical Engineering State University of Londrina, Brazil E-mail: [email protected] F. Ciriaco Dept. of Electrical Engineering State University of Londrina, Brazil E-mail: [email protected] T. Abrão Dept. of Electrical Engineering State University of Londrina, Brazil E-mail: [email protected]

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Ricardo Tadashi Kobayashi et al.

1 Introduction The use of multiple antennas in transmission and reception is a way to realize higher spectral efficient and/or reliable mobile communication systems, offering improved quality of service (QoS) through the wireless channel [4]. The pioneering work [24] proposed a V-BLAST architecture capable of providing spatial multiplexing gain and high data rate, motivating countless works around multiple antenna systems. Improvements obtained through multipleinput-multiple-output (MIMO) systems may be on transmit energy efficiency, data rate and/or symbol error rate, being determined by the antenna configuration and transmission-detection techniques as well. Balancing these improvements with the available resources in the system is a project necessity, since energy and spectrum are increasingly scarce resources. As a result, it is necessary to formulate solutions in order to obtain improved performance under low or moderate complexity constraints. Generally, these solutions brings an increment in communication capacity, or alternatively, reducing the system consumption power, the radiated power, size and weight of the wireless system, which are desirable characteristics, especially in mobile terminals (MTs). Hence, the goal of this work consists in pointing out and analyzing MIMO architectures and detectors with low or moderate complexity holding suitable performance under full diversity condition. Thus, linear MIMO equalizers combined with sub-optimal detection techniques such as ordering (via SQRD), interference cancellation (SIC), list and lattice reduction (LR) were carefully analyzed in terms of performance-complexity trade-off. Another aspect to be considered in this work is the fading channel correlation; since the physical dimensions required for the mobile devices is increasingly restricted, the distance between antennas in the same MT is reduced as well. Indeed, under MIMO systems operating over UHF frequency range, the distance between antennas required in order to achieve the uncorrelated channel condition is not so very small; hence, practical and versatile MIMO communication systems accommodating a large number of antennas and exploiting the maximal diversity gain (or alternatively maximal multiplexing gain) is challenging. A MIMO channel correlation scenario causes deleterious effects on the achievable data rate and on performance as well [14], requiring additional power to comply with target quality of service (QoS) indexes. This can be a concern, since the use of receivers capable of operating in high SNR and low phase noise present a high cost [18]. Therefore, efficient MIMO detectors able to operate under suitable bit error rate (BER) performance and power transmission constraints are of paramount interest. More recently, massive MIMO systems proved to be a promising technology for 5G systems [3], mainly for its highest spectral and energy efficiency, but mainly for being immune to additive noise for very large arrays [19]. However, large arrays bring two main problems to be addressed in this work: correlation between antennas and the signal processing complexity. The first comes from the fact that large antennas arrays must be accommodate in relatively small areas, causing correlated channels. On the other hand, the increase on

Title Suppressed Due to Excessive Length

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the processing complexity is straightforward, since large arrays require more hardware and larger number of operations due to the large number of antennas. Therefore, analyses on MIMO processing techniques is very important since practical high efficient communications systems must meet their own limitations. From the point-of-view of information detection, it is well known that MIMO systems sends data through different antennas, traveling through different paths, so the received signal is a combination of every transmit antennas signals (in each receive antenna), from which the original message is recovered. The MIMO detection problem consists primarily in decoupling the transmitted signal from a received bandpass signal sample, while the processing loading is concentrated almost on the receive side, if no coding is applied on the transmitter side. Since the mobile communication systems do not possess neither high processing capability nor abundant energy resource, due to battery size and weight constraints, the design of efficient MIMO detector is of paramount importance and an ongoing research topic. It’s well known that the maximum likelihood (ML) detection principle provides the best performance in terms of bit error rate, but its computational complexity makes this detector impractical. On the other hand, the sphere decoder (SD) is an approach representing the state-of-the-art on MIMO detection; SD results in a lower complexity than the ML for small amounts of noise but still remains quite complex under low SNR regime, and further dependent on the arrangement of the MIMO system. Besides, there are also the classical linear MIMO detectors, such as zero-forcing (ZF) and minimum mean squared error (MMSE)-based MIMO detector, which when combined with ordered successive interference cancellation (OSIC) [24, 25, 27] or OSIC with repetition [22, 23] are able to exhibit lower complexity than the ML or even SD detectors, though some degradation in symbol error performance. Furthermore, the lattice reduction (LR) [21] technique aided linear MIMO detectors provides full diversity exploitation, improving greatly the performance and, in some cases, achieving near ML-performance. The complexity on LRreduction is known to be polynomial in time, but correlated channels impact negatively on the MIMO system complexity, as the columns of the channel matrix become more and more likely. Even with lower complexity than ML, the LR-aided MIMO detectors can still show undesirable complexity with a large number of antennas and high correlation between them. Hence, it’s one of the challenges for the applicability of large-MIMO systems. Notation: (·)H , (·)T , (·)† and (·)−1 denote the matrix transpose, Hermitian, pseudo-inverse and inverse, respectively. Boldface lowercase letter represents vectors, while boldface uppercase letter denotes matrices. I and 0 denote the f repidentity and an all zero matrix, respectively. Finally, tilde superscript (·) c resents a symbol vector estimation and hat (·) superscript represents a symbol estimation after a slicer, which quantizes an estimated symbol to its nearest f = (·). c constellation point. This process will also be denoted as Q(·)

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2 System Model The transmission process of a point-to-point MIMO system composed by nT transmit antennas and nR receive antennas can be compactly described by: x = Hs + n.

(1)

Herein we have assumed overdetermined MIMO systems, i.e., nR ≥ nT , operating in spatial multiplexing mode. In other words, the detected symbol is obtained by solving system of nT unknowns with, at least, nT equations. Also, no precoding techniques were deployed. Eq. (1) represents a transmission of nT modulated symbols, snT ×1 , through a channel which gain is denoted by HnR ×nT and additive noise nnR ×1 . Each element of H represents the channel gain in its respectively path and these gains will be assumed known at the receiver side. After passing through the channel, the symbols are combined forming the received vector, xnT ×1 . Besides, a pre-processing block is responsible for the modulation and coding, if applied, while the MIMO detector recovers the sent data from the received signal, corrupted by background noise and inter-antenna interference. It’s assumed that the noise vector has a circularly-symmetric complex Gaussian distribution, n ∼ CN (0, N0 I), with variance N0 . Alternatively, the noise can be statistically represented by its covariance matrix E nnH = N0 InR . The transmitted symbols belong to the set S, which depends on the modulation order M , deployed on the transmitter side. Since each antenna’s symbol is independent and the power is divided between the antennas, the covariS ance matrix of the transmit symbols is given by E ssH = E nT InT , where ES represents the average symbol energy. Due to its spectral efficiency and performance trade-off, in this work the M -QAM modulated symbols has been deployed; M -QAM symbols are denoted by √ a complex number, which real and imaginary parts are odd and limited to ±( M − 1), i.e. [1, 10]: √ √ √ S = {a + jb | a, b ∈ {− M + 1, − M + 3, . . . , M − 1}}.

For this modulation, the average symbol energy is given by: ES =

2(M − 1) 3

(2)

Also, it will be assumed Gray coded symbols, where adjacent symbols presents only one bit difference, minimizing the bit error. In some MIMO detection procedures, both the noise power and the transmit energy parameters, or the power-signal-to-power noise ratio (SNR) must be estimated at the receiver side for a suitable detection procedure, as shown in the next sections.


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2.1 Spatially Correlated MIMO Rayleigh-Fading Channels The MIMO fading channel is properly modelled by flat Rayleigh distributions. Additionally, correlation between antennas is considered through the Kronecker’s correlation model [2, 28]: p p (3) H = RH,Rx G RH,Tx .

where G(nR × nT ) is composed by independent, identically distributed complex Gaussian elements, gij ∼ CN (0, 1), the matrices RH,Rx (nR × nR ) and RH,Tx (nT × nT ) represent the spatial channel correlation observed in the receiver and transmitter side, respectively. The elements of these two matrices are given, in terms of normalized correlation index ρ, by: ( 2 rH,Rx ij = ρ(i−j) (4) 2 rH,Tx ij = ρ(i−j) . Hence, if nT = nR both channel correlation matrices can be written as: 

    RH =     

1 ρ ρ4 .. . ρ(n−1)

2

 2 ρ ρ4 · · · ρ(n−1)  ..  1 ρ ··· .   4 . ρ 1 ··· ρ   .. .. . . ρ  . . .  4 ··· ρ ··· 1

(5)

3 MIMO Detectors This section revisits the most commonly MIMO detectors techniques available in the literature, including the maximum-likelihood (ML), sphere decoder, zero-forcing (ZF) and minimum-mean-squared-error (MMSE). Additionally, we have provide a brief discussion on the use of less conventional techniques for MIMO detection, such as ordering, interference-cancellation, list-reduction and lattice-based. It is of great importance the knowledge on each detector procedure, especially for complexity evaluation and BER performance analysis.

3.1 Maximum-Likelihood (ML) Despite of its complexity, the maximum likelihood detector provides the optimal BER performance. ML detection is performed by an exhaustive search for the closest symbol and signal reconstruction regarding the signal observation at the receiver. Considering a M −ary modulation order with nT transmit antennas, each one transmitting a distinct symbol at each time-slot system, the

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number of symbols combination is simply M nT . For example, the ML detector for 16−QAM and nT = 8 transmit antennas must test a list of over ≈ 4 billions of candidate-symbols. Considering s a candidate-vector from the S nT set, of size M nT , the candidatevector presenting the lowest distance from the received signal, and therefore the lowest error, can be expressed by: e s = argmin kx − Hsk2 . s∈S nT

(6)

3.2 Sphere Decoder (SD) In order to reduce the complexity of the ML detector, the sphere-decoder [12] searches for only the candidates contained in a sphere of radius d: d2 < kx − Hsk2

(7)

which, of course, is dependent to the signal-to-noise-ratio. If the radius is set too high, the SD complexity tends to the ML one. On the other hand, if the search radius is small enough, there will be no candidates into the hypersphere. Aiming to perform a sphere detection, eq. (1) is rewritten applying the QR decomposition on the channel matrix. Through this decomposition, we obtain an orthogonal matrix Q, where I = QH Q, and an upper triangular matrix R, both with convenient properties for the detection procedure: y = QH x = QH QRs + QH n

(8)

′

= Rs + n . Since Q is orthogonal, the statistics on the noise, n′ , remains unchanged and no noise enhancement is expected. Also, R matrix is upper triangular, enabling estimating antennas’ noise independently. With these facts, eq. (7) becomes: Considering R = [r1

d2 < ky − Rsk2

r2

(9)

r3 · · · rnT ]T , the noise norm is given by:

kn′ k2 = ky − Rsk2 =

nT X

k=1

|yk − rk s|2

(10)

Indeed, eq. (10) shows that the noise norm is the sum of each of its layer’s norm. Therefore, the noise norm can be updated as the symbols are tested in each layer, which benefits the radius criteria test by avoiding the evaluation of the estimated noise norm for every symbol combination. The structure of the detection problem in (9) allows a tree search that begins from the last antenna’s symbol to the first one, where the candidatesymbols are tested recursively and independently, differently from ML. It is noteworthy that this process still obeys the radius constraint defined in (10). By the end of the SD detection procedure, the most likely symbol-vector inside the sphere of radius d is taken as the solution.


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3.3 Zero-Forcing (ZF) The zero-forcing MIMO detector basically solves the linear system problem posed by (1) ignoring the additive noise. The solution proposed for this detector uses Moore-Penrose pseudo-inverse matrix [15]: e s = H† x = s + H† n.

(11)

Therefore, the equalization matrix is given by the pseudo-inverse of the channel matrix: WZF = (HH H)−1 HH . (12) Despite of its low complexity, when H is ill conditioned, it is also near singular, causing noise enhancement [15], given by H† n. Hence, suitable performance cannot be expected in this situation. 3.4 Minimum Mean-Squared-Error (MMSE) By taking into account the noise and the signal statistics, the MMSE detector is able to reduce the impact of the additive noise on the detection, improving the overall MIMO performance. In the MMSE approach, the minimization of the symbol error can be obtained solving the following optimization problem [20]: WMMSE = argmin E[ks − Wxk2 ]. (13) W

Solving (13), the equalization matrix for this detector is obtained as: −1 N0 WMMSE = HH H + I HH . (14) ES Hence, the solution for MMSE detector is: −1 N0 H e I HH x. s= H H+ ES

(15)

Alternatively, the MMSE detection can also be performed as: e s = H† x

= s + H† n

(16)

where the extended channel matrix and the received vector are given respectively by:   " # H x , . x= H = qN 0 0nT ×1 ES InT

Despite of being more complex than the approach given by (15), the extended matrix model is required for successive interference cancellation (SIC) and it is also recommended for lattice-reduction due to performance improvements [26].

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3.5 Vertical Bell Laboratories Layered Space-Time (VBLAST) The V-BLAST architecture, proposed in [24], is a MIMO detection architecture which performs ordered successive interference cancelation (OSIC) through a linear detector (ZF or MMSE). Under this detection strategy, BER reduction is expected, as well an computational complexity increment. Initially, the algorithm evaluates the equalization matrix, ZF or MMSE. Unlike the channel matrix, the equalization matrix will be represented as nT row vectors: W = [w1 w2 · · · wnT ]T . The detection’s order follows the minimization of kwki k, with symbol detection obtained by: sek = wki x.

(17)

x := x − sbki hki .

(18)

After passing by a slicer, the interference is reconstructed and cancelled:

where sbki is the sliced version of the symbol detected applying (17). Since the ki th antenna’s symbol is detected and its interference canceled, the ki th column of H is no longer needed, so it is zeroed. Due to this process, the channel matrix changes, so the equalization matrix needs to be evaluated again. This process is repeated until all symbols are detected. The whole VBLAST detection procedure is summarized in the pseudo-code ??. The presented algorithm uses the ZF equalizer, but the MMSE equalizer can also be used by evaluating the pseudo-inverse of the extended channel matrix H. One of VBLAST detection’s weakness is its high complexity, since it requires an update on the equalization matrix for every symbol interference cancellation. When compared with other OSIC detectors, VBLAST detection requires nT pseudo-inverse evaluations, while the sorted QR decomposition approach, as discussed in section 3.7, requires only one matrix decomposition. In [27], it’s showed that the performance gap between these detectors is very low, but it can disappear via post ordering. Therefore, this work will focus on the SQRD approach.

3.6 Successive Interference Cancellation (SIC) The successive interference cancellation can be obtained using the QR decomposition in the channel matrix H, as explained in section 3.2. Noticing for the ZF approach the QR decomposition procedure is applied on the H, while for the MMSE detector to the matrix H.


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The process represented in (9) prepares the system’s solution, considering R is upper triangular. Hence, the linear system is solved upwards by:

sei =

y i  ,    rii 1     rii

i = nT yi −

nT X

rik sˆk

k=i+1

!

(19) , i = nT − 1, . . . , 3, 2, 1.

Noticing that every symbol must pass by the slicing step before feed forward with the interference cancellation, which is applied in the next symbol detection. Indeed, the slicing step must be implemented in order to perform a properly interference cancellation and improve the system performance.

3.7 Sorted QR Decomposition (SQRD) Further performance improvement on SIC procedure can be achieved through a proper ordering [27], which avoid error propagation during interference cancellation computation. The ordering criteria is the minimization of the columns’ norm of Q, which makes the detection be proceeded from the strongest to the weakest symbol. The form of the decomposition is: HΠ = QR

(20)

where Π is a permutation matrix, used to reorder the symbols after applying the SIC detection. Note that the detection is followed as a conventional SIC, as described in (19). However, at the end of the detection process the reordering is carried out by multiplying the symbol vector by the permutation matrix. The sorted QR decomposition algorithm is represented by the pseudo-code in Algorithm ??. If lines 2 and 3 of the algorithm are ignored, it will perform a conventional QR decomposition with the Gram-Schimidt aprroach. Since these lines are not high complexity operations, the cost for ordering is practically negligible. Notice the QR decomposition on the extended matrix channel requires minor modifications on the algorithm, which are found in [25]. Applying this decomposition ensures suboptimal but near BLAST ordering performance. In order to achieve V-BLAST performance, a post sorting algorithm can be applied, as discussed in [25]. In our analysis, the use of this decomposition will be referred as ordered successive interference cancellation (OSIC).

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3.8 Chase List Algorithm Performance improvements on the previous discussed MIMO detectors can be obtained, with manageable complexity, through the Chase list (CL) algorithm. The key feature for its success is repetition, since in CL algorithm the symbol detection is a recurrent procedure. Fig.1 depicts a generic diagram for the Chase list MIMO detection. (1)

x

Last antenna’s symbol detection and list creation

sˆnT

x(1)

hnT b

Subdetector 1

ˆs(1)

x (2)

sˆnT

x(2)

hnT b

Subdetector 2

ˆs(2)

argminkHˆs(k) − xk ˆs k=1 to q

x b b b

(q) sˆnT b

(q)

x

hnT

Subdetector q

ˆs(q)

x

Fig. 1 Block diagram of Chase list detector.

Basically, in the CL algorithm the q best candidates are firstly selected aiming to detect one symbol; then the interference cancellation is performed, and the remaining symbols are finally detected (sub-detection). This process is repeated for all the candidates and then the best set is chosen using the Euclidean norm criterion, just like in the ML detector, but with reduced candidates list. Notice also that q is limited to M . To elaborate further, initially, the last antenna’s symbol is detected (no ordering criterion), which can be proceeded with the previous discussed detectors. From this, the q best candidates for the last antenna’s symbol, namely (q) (1) (2) (3) C = {cnT , cnT , cnT , ..., cnT }, are classified by using the Euclidean distance criterion. Next, the interference cancellation and the remaining symbol detection are carried out. Then the SIC is performed for each symbol candidate, from i = 1 to q, ignoring the noise effect: (i)

x(i) = x − snT hnT .

(21)

After the interference cancellation step, the remaining symbols are detected using the chosen MIMO sub-detector. Hence, with all the q symbols vectorcandidates evaluated, the solution for the Chase list detection is simply given by solving the problem: ˆs = argmin kHˆs(k) − xk2 .

(22)

k=1 to q

For the sake of simplicity, the Chase list was presented using linear MIMO detection techniques; however, the use of SIC-based detectors is desirable,


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since SIC’s first symbol detection requires only one division and interference cancellation is inherit to the process. A pseudo-code for CL-MIMO detector which applies the SIC approach is presented in Algorithm ??. In this Chase list algorithm a zero forcing SIC (ZF-SIC) detector has been deployed, but the use of other techniques, such as ordering combined with linear detector, can be applied successfully. In this sense, the use of MMSE equalizer is also possible by carrying out the QR decomposition on the extended channel matrix. Other Chase list approaches can be found in [22, 23]. As mentioned, ordering procedure can be included in CL-MIMO detection and the SQRD makes the strongest symbol to be detected first, as it becomes the last symbol of the s vector, as also described in Fig. 1. In order to apply the OSIC on the Chase list, its only needed two modifications in the Algorithm ??: a) SQRD is applied instead of the conventional one (line 2); b) Received symbol must be multiplied by the permutation matrix Π (after line 15).

3.9 Lattice Reduction (LR) aided MIMO detector One important issue found in the previous detectors is their narrow decision boundaries, which makes the detection more sensitive even under small amounts of noise. In order to avoid this problem, detection can be done in other domain followed by proper conversion, which is obtained trough the lattice reduction technique [8, 26].

The lattice reduction can be implemented through the LLL algorithm, proposed by Lenstra-Lenstra-Lovaz in [16]. Basically, the LLL algorithm decomposes the MIMO channel matrix into a new base: e = HT, H

(23)

e is a base with better properties, in terms of near-orthogonality, than where H the original H, while T is an unitary matrix, which presents two features: e has betdet(|T|) = ±1, and T ∈ {ZnR ×nT + jZnR ×nT }. As the new matrix H ter properties, the decision boundaries are enlarged and the noise amplification effect is reduced. Since the detection is done in LR domain, it’s desirable to rewrite the system model in terms of LR symbols: e = (HT)(T−1 s) + n x e + n. = Hz

(24)

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Under this modified system model, the LR symbols z can be detected using any linear MIMO equalization technique discussed previously. Herein, it will be presented the zero-forcing equalization approach: e †x e z=H e † n. =z+H

(25)

As revealed by (24) and (25), noise also corrupts the symbols on the LR domain. Hence, a proper decision must be made on the LR domain symbols, z, including shifting and scaling operations, as follows: e z − βT−1 1nT b z=2 + βT−1 1. (26) 2

where ⌊·⌉ is the round operator, 1nT represents a column vector of ones and β is a constant determined by the modulation format and order. Since the transmission scheme deployed in this work uses exclusively QAM modulation, we need set β = 1 + i. In the last step of the LR-aided MIMO detection, the symbols must be converted from the LR domain to the original signal space: b s = Tb z.

3.10 LR-aided Chase-list

(27)

In order to deploy jointly lattice reduction technique and Chase-list MIMO detection, herein SIC structure will be used similarly as proposed in [1, 5]. Firstly, the QR decomposition, or even the SQRD, is evaluated; after that, it is multiplied by the received signal, like (8), which leads to: y = QH x "

y = Rs + n #, # " #" # " n1 s1 R1 R2 y1 + = nnR snT 0 rnR nT ynR

(28)

where R1 is the composed by the columns 1 to nT − 1 and rows 1 to nR − 1 of R; R2 is formed by the last column and (nR − 1)th first rows of R; 0 is a zero row vector with nT − 1 elements; rnR nT is the last element of R. On the sequence, from eq. (19), the last estimated symbol becomes: senT = ynT /rnT nT

(29)

After the nT antenna’s symbol has been sliced, the interference cancellation is proceeded: (30) y1 := y1 − sbnT R2


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Assuming a successful interference cancellation, the system defined in (28) is reduced to: (31)

y 1 = R 1 s1 + n 1

Hence, with this new reduced system description, it is possible to apply the LR-aided method discussed previously in order to detect the remaining nT − 1 symbols. Notice that the lattice reduction in eq. (31) is required only once, since the interference cancellation procedure does not change the system. Assuming that the Chase-list is deployed, the signal processing operations defined by (29) to (31) are repeated q times, with the q best candidates for snT .

4 Performance Comparison In this section, the simulated BER performance of various MIMO detectors discussed previously have been compared. For a fair comparison between different MIMO transmission systems with different modulation order and number of antennas (even SISO systems), it will be considered a normalized SNR Eb SN R N0 = log2 M , and transmit power constraint, with power equally distributed among the nT antennas. Each MIMO detector performance was evaluated considering three modulation format and antennas arrangement: a) 64−QAM-4 × 4;

b) 16−QAM-8 × 8;

c) 4−QAM-20 × 20.

Furthermore, three antenna correlation scenarios has been considered: a) no correlation ρ = 0; b) medium correlation ρ = 0.5; c) strong correlation ρ = 0.9. Finally, it was considered perfect channel-state-information (CSI) available at the receiver side. Fig. 2 depicts the first evaluated arrangement for the BER performance comparison, with 64−QAM modulation format and 4×4 antennas. In low SNR regime, the nine analysed MIMO detectors provide very similar performance. On the other hand, the LR-aided MIMO detectors (LR-ZF, LR-MMSE, LRMMSE-OSIC and LR-CL-MMSE-OSIC) are able to achieve full diversity, as well as the ML and SD detectors. Recall that a system achieves full diversity when a 3 [dB] increasing on the transmission power implies in a BER reduction factor of 2nT in high SNR regime [2]. When small antennas array are applied, the LR-aided detector presents a near ML/SD performance. The difference between their performance in high SNR results in a small gap, but the performance curves are parallel, implying in same diversity order. The ZF and MMSE MIMO detectors tend to present the same performance in high SNRs, for the MMSE solution is approximately equal to the ZF one,

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as described in (11) and (15). Besides, Fig. 2 confirms that interference cancellation, lattice-reduction techniques and combination of them can greatly improve the MIMO detection performance. Regarding the antenna correlation effect on performance, still in Fig. 2, one can conclude that as the correlation index grows the BER performance deteriorates, specially when ρ ≥ 0.5. In high correlation scenarios non-LR-aided MIMO detectors requires a very high SNR to reach reasonable BER performance. This SNR requirement for high correlation is a concerning issue, which can be expressed in terms of undesirable low energy-efficient systems. Indeed, only LR-based and SD MIMO detectors are able to achieve simultaneously full diversity and transmission energy efficiency systems under high antenna correlation, i.e., ρ = 0.9 in Fig. 2.(c). As the number of antennas increases, the gap between SD/ML and other MIMO detector’s BER also grow, as depicted in Fig. 3 (8 × 8 antennas) and Fig. 4 (20 × 20 antennas). In these arrangements, it is easy to see that in low SNR non-LR-aided detectors present better performance, but in high SNR this behavior is inverted as LR-aided detectors present full diversity. Hence, given an array configuration with increasing number of antennas, the MIMO detectors combining lattice-reduction, interference cancellation and list-reduction techniques exhibit a better performance than the correspondent detector in previous array with small number of antennas. It is worthing to notice that for large array arrangements, in Fig. 4 with nT = nR = 20 antennas, the BER performance presents a different trend. As the correlation index grows, it takes much more power for the LR-aided MIMO detectors to outperform the non-LR ones. However, when the correlation index is very high, ρ = 0.9, a new trend is established. First, the ordered interferencecancellation is not able to deal with high correlation in high SNR. In these cases, it is recommended not using ordering in SIC detection, to avoid the inversion of the slope of the BER curve. Also, MMSE lost the diversity gain, while the LR-MMSE’s BER decreased and increased as the SNR increased. For this array configuration the ZF-based detectors is unable to deal with high correlation scenarios (ρ ≥ 0.9); this class of MIMO detectors fails in decoupling the high-correlated inter-antennas interference. Besides, the SDMIMO detector under high correlated channels has demonstrated an extremely exceeding computational cost.

4.1 BER Inversion Effect in OSIC Detectors As seemed trough previous BER performance results, OSIC detectors with SQRD approach are not effective under high correlated MIMO channel scenarios while operating with a large number of antennas. This happens mainly because of the method by which the sorted QR decomposition proposed by [25] is a modification on the Gram-Schimidt approach. Despite of being useful conceptually, the Gram-Schimidt method is numerically unstable. Hence, other

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15

20

(b) ρ = 0.5

−1

BER

10

−2

10

ZF MMSE MMSE−OSIC CL−MMSE−OSIC LR−ZF LR−MMSE LR−MMSEOSIC LR−CL−MMSE−OSIC SD CL−LR−MMSESIC

−3

10

0

5

10

15

20 25 Eb/N0 [dB]

30

35

40


18


available algorithms based on re-orthogonalization, such as Householder reflections, are able to mitigate this numerical instability. In a QR decomposition, despite of the matrix Q be called orthogonal, it is not strictly orthogonal in practice. Therefore, a residual error ξ, where kI − QH Qk2 = ξ 2 ≥ 0, is always present, but it is particularly larger when QR decomposition is obtained through Gram-Schimidt algorithm [9]. Although ξ is very small in most cases, rank deficiency or the high size of the matrix to be decomposed can increase ξ enough to forbid a proper solution of a linear system via QR decomposition. For further reading, recall to [6, 9]. As explained before, ordered SIC using the MMSE equalization can be obtained through the QR decomposition on the extended channel matrix, therefore: " " # # H Q1 (32) R. = QR = H= p Q2 N0 /ES InT It is straightforward that:

Q2 R = and:

p

N0 /ES InT

QH H = QH 1 H+ = R.

p

N0 /ES QH 2

(33)

(34)

For SIC detection, it must be derived an upper triangular system, using (33), (34) and (1): H QH Hs + QH n 1 x = Q 1 p 1 (35) s + QH = R − N0 /ES QH 2 1 n. Therfore, it must be considered the following approximation: QH 1 H = R,

(36)

which is reasonable for high SN R, also leading to: QH 1 Q1 = InT

(37)

At this point, it can be observed that H is a full rank matrix even if H is highly correlated, and in the worst case even if H is singular. However, for high values of N0 /ES and extremely high correlated channels, pthe extended channel matrix H tends to exhibit orthogonality deficiency and N0 /ES InT ≈ 0nT . When this condition is combined to the low numerical stability of the sorted QR decomposition, which is based on the Gram-Schimidt method, the error ξ may not be negligible; therefore, QH 1 Q1 6= I, which doesn’t meet the requirements for an effective ordered SIC detection. From this analysis, it can be concluded that ordered SIC via MMSE equalization becomes ineffective under high correlated channel scenarios and large arrays due to the characteristics of its sorted QR decomposition. From this perspective, V-BLAST would be recommended to perform ordered interference


19

cancellation if complexity was not a concern, since V-BLAST should perform one pseudo-inverse to detect each layer, resulting in a complexity of O(n4T ). Therefore, ordering procedure should be avoided for systems operating under these circumstances. In order to sustain these claims, Fig. 5 depicts the BER performance for large array (20 antennas) under very high correlated channels (ρ = 0.9), where ordered SIC was obtained through a) V-BLAST, and alternatively via b) SQRD procedure. Since V-BLAST is not affected by numerical unstability, it can perform ordered interference cancellation without experiencing BER performance inversion effect in high SNR region. 0

10

−1

BER

10

−2

10

−3

10

V−BLAST SQRD

−4

10

0

5

10

15

20 E /N b

25

30

35

40

0

Fig. 5 BER performance for MMSE-VBLAST and MMSE-OSIC(via SQRD) considering a 20 × 20 4−QAM system under high correlated channels (ρ = 0.9).

5 Complexity Analysis It is of paramount importance analyse the computational complexity of those MIMO detectors, since communication under mobility condition imposes restrictions on signal processing capability and power consumption. By combining complexity and BER performance analysis, it is possible to establish the best trade-off among those sub-optimum MIMO detectors. From this perspective, this section discusses the overall complexity of those sub-optimum MIMO detectors. The overall complexity has been measured as the number of flops needed to perform the complete MIMO detection process for each MIMO detector presented previously. Three and one flop(s)

20


for complex sum and product, respectively, have been considered in the flop counting [25]. Besides, with the necessary modifications, matrix operations flop counting were based in [11]. Also, the complexity on the sorted QR decomposition found in [25, 27], as well as the lower bound for the number of nodes visited by the SD in [13] have been considered in the following analysis. 5.1 LLL Complexity Despite of its BER performance, there are certain scenarios where LR-aided detectors may present a growing complexity. Numerical simulation results revealed not only a matrix size dependence on LLL’s complexity, but also a correlation index dependence. The dependence between complexity and matrix size is of course due to the number of operations evaluated on the matrix. On the other hand, we have found that an increasing correlation index leads to a near singular matrix, which makes difficult for LLL to find an orthogonal basis, increasing the computational complexity. Unfortunately, LLL’s complexity cannot be easily evaluated considering all its variables dependence. In order to circumvent this difficult, herein the LLL’s complexity has been determined numerically through a surface fitting procedure on the flop count of the LLL algorithm. We have conducted numerical experiments aiming to determine the LLL algorithm complexity dependence on the antenna correlation index and array dimension; as a result, it was established a flop count procedure for lattice reduction of square matrices taking into account increasing array size and correlation indexes. Besides, the LLL algorithm applied in these numerical simulations uses the complex approach with factor δ = 0.75, as suggested in [17]. In order to obtain a better and realistic prediction, the mean over 500 LLL computation flops count has been calculated for each MIMO system and channel configuration, including correlation index and matrix size. The best surface fitting, which can suitably describe the flop count on LLL algorithm has been numerically found as: flll (nT , ρ) = (aebρ + c)n3T

(38)

where a = 5.018×10−4 , b = 13.48 and c = 8.396. Remembering that this fitting is valid only for nT = nR arrays. As illustration, Fig. 6 depicts the collected data (identified by the “+” marker) and the corresponding surface fitting given by eq. (38). It is worth noting that the LLL computational complexity cost increases substantially under medium-high correlation index (ρ ≥ 0.5) and large-array configurations (nT = nR ≥ 15). 5.2 Complexity of the Analysed MIMO Detectors Several MIMO detectors complexity has been analysed due to the various techniques combination possibilities. Table 1 summarises the overall complex-


21

Fig. 6 LR flop count dependence on antenna correlation index and number of antennas.

ity related to the most relevant combination of sub-optimum MIMO detectors. The complexity of the ML-MIMO detector is included as reference. From Table 1, notice that SIC-based MIMO detectors are capable to offer lower complexities, since they don’t require pseudo-inverse evaluation. Besides, the aggregation of ordering procedure is preferable since it requires only 2n2T − 2nT flops, according to [25], while providing substantial performance improvement. The cost for using the Chase list may be low, but only when the number of repetitions q is lower than the number of transmit antennas; of course, holding the number of repetitions q much low impacts negatively on the CL-based MIMO detector performance. On the other hand, the LR-aided detectors show reasonable low complexity under low to medium correlation index scenarios, while full diversity is held, which makes it a promising sub-optimum MIMO transmitting scheme. Finally, ML has a prohibitive exponential complexity in any practical system configuration, while SD-based MIMO detectors present decreasing complexity as the SNR increases, but still highly complex under low-SNRs regime. Since the OSIC procedure represents a reasonable difference in terms of the final detection complexity, Fig. 7 provide a further graphically flops-complexity comparison, among the different detectors grouped as: a) linear MIMO detectors, and b) OSIC-based linear MIMO aided by lattice reduction and Chase list procedure as well. Indeed, in Fig. 7.(b) it has been considered the deployment of the ordered successive interference cancellation and a Chase list length equal to the constellation size, in this case M = q = 64.

22


Table 1 MIMO Detectors Complexity MIMO Detector ZF ZF-SIC ZF-OSIC CL-ZF-OSIC

Number of flops 4n3T + 8nR n2T − n2T + 3nR nT − nR + nT 4nR n2T + 15nR nT /2 + 3n2T /2 4nR n2T + 15nR nT /2 + 7n2T /2 − 2nT 4nR n2T + (4q + 15/2)nR nT + 4n2R + (2q + 3)n2T /2 +(2q − 1)nR + (9q − 3)nT /2 + 5M/2

LR-ZF LR-ZF-OSIC CL-LR-MMSE-OSIC

8n3T + 8nR n2T + 7n2T + 3nR nT − nR + 5nT + fLLL (nT , ρ) 4n3T + 12n2T + 4nR nT + 11nT /2 + fLLL 4n3T /3 + 4nR n2T + (4q − 9/2)n2T + (8q + 15/2)nR nT +2qnR (7/2 − 3q/2)nT /2 + +5M/2 + fLLL (nT − 1, ρ)

MMSE

12n3T + 8nR n2T + 2n2T + 3nR nT − nR

MMSE-SIC

4n3T /3 + 4nR n2T + 16n2T /3 + 6nT nR + 22nT /3

MMSE-OSIC

4n3T /3 + 4nR n2T + 19n2T /3 + 6nT nR + 16nT /3

CL-MMSE-OSIC

4n3T /3 + 4nR n2T + (2q + 13/3)n2T + (4q + 6)nT nR +2qnR + (9q/2 + 25/6)nT + 5M/2

LR-MMSE LR-MMSE-OSIC CL-LR-MMSE-OSIC

16n3T + 8nR n2T + 10n2T + 3nR nT − nR + 4nT + fLLL 4n3T + 16n2T + 4nR nT + 11nT /2 + fLLL (nT , ρ) 16n3T /3 + 4nR n2T + (10q − 11/3)n2T + (8q + 6)nR nT +2qnR + (55/6 − 37q/2)nT + 5M/2 + fLLL (nT − 1, ρ)

SD

ML

(4nR nT + 2nR )M nT

[13]

4n3T + 7n2T − nT /2 + (2nT + 2)M ηnT where and

1/2 [c2 (M 2 − 1)/6N 0

− 1/M − 1,

η= + 1] c2 = E khi k2 , ∀i ∈ [1, nT ]

−1

5.3 Impact of Real-valued System Model Representation on the Complexity The MIMO problem defined by eq. (1) can be rewritten in order to deal only with real-valued entries: xr = Hr sr (39) where the real matrix channel is given by: " # ℜ(H) −ℑ(H) Hr = ℑ(H) ℜ(H)

(40)

with the received and transmitted symbol vectors as: " # " # ℜ(x) ℜ(s) xr = , sr = . ℑ(x) ℑ(s) With this modification, the new system model representation has doubled its size, but all its elements are real-valued entries; mathematically, it means xr ∈ R2nR ×1 , sr ∈ R2nT ×1 and Hr ∈ R2nR ×2nT .


23

7

x 10

5

Flops

4 3 ZF MMSE LR−ZF LR−MMSE SD(ρ=0)

2 1

0.6 0.4 0.2 0 ρ

100

90

80

60

70

50

40

30

20

10

n

T

(a) Linear MIMO complexity 7

x 10

5

Flops

4 3 MMSE−OSIC CL−MMSE−OSIC LR−MMSE−OSIC LR−CL−MMSE−OSIC SD(ρ=0)

2 1

0.6 0.4 0.2 0 ρ

100

90

80

60

70

50

40

30

20

10

nT

(b) OSIC-based MIMO detectors Fig. 7 Complexity in terms of flops for MIMO detectors; 64−QAM and Eb /N0 = 22 [dB].

Despite the real model representation be considerably widespread in the literature, its use may lead to a higher computational complexity, specially for matrix arithmetic operations. Real arithmetic operations may requires less flops than complex ones, but the real system size is larger than the complex one. According to [17] and [8], the impact of complex-valued system model representation on the LLL algorithm complexity is a half the complexity of the real-valued representation. In terms of arithmetic operations, Table 2 shows the cost comparison between complex-valued and real-valued representation.

24


Table 2 Comparison in terms of number of operations between complex-valued and realvalued MIMO system model representation, nR = nT Operation

Real addition

Real product

Total arith. operations

H r sr

4n2T − 2nT

4n2T

8n2T − 2nT

Hs

2n2T − 2nT

4n2T

6n2T − 2nT

HT r Hr

8n3T − 4n2T

8n3T

16n3T − 4n2T

HH H

2n3T − 2n2T

4n3T

6n3T − 2n2T

Under MIMO system configurations discussed previously, the use of realvalued model may not be advantageous, since the complexity increases due to the real-valued system representation size. On the other hand, the use of the real-valued model reduce the constellation size. √ For example, the constellation size of a M −QAM modulation turns to a 2× M −PAM (in-phase and quadrature representation) when the real model is applied, but it does not lead to complexity reduction. Furthermore, the use of the real-valued system model representation brings a small BER improvements to the SIC-based MIMO detectors because operations such as detection and interference cancellation of both real and imaginary parts are carried out independently, leading to a more reliable detection, as demonstrated in [7]. However, this improvement representing a small performance BER gap is low for the price paid by the complexity growth.

6 Conclusions In this work the use of lattice reduction technique has been demonstrated a useful tool to improve the BER performance of linear sub-optimum MIMO detectors. On the other hand, the list reduction technique was capable to achieve reasonable MIMO system performances over different array dimensions and correlation indexes, but did not able to achieve full diversity. Since the use of the lattice reduction technique was able to provide large improvements on the BER performance, its combination with list reduction did not leave space for further BER reduction, even under combined large-MIMO and high antenna correlation condition. Furthermore, LR-aided MIMO detectors have demonstrated a noticeable complexity growth under high antenna correlation scenarios, since the channel matrix becomes near-singular, getting harder and harder for the LLL algorithm establish an orthogonal signal basis. It is worth to note that ordering procedure has demonstrated ineffective in large array systems operating under high correlated channel scenarios. In the extreme correlation and large arrays situation analyzed (ρ = 0.9 and 20 × 20 antennas), the LR-MMSE-OSIC and MMSE-OSIC MIMO detectors were not able to perform the symbol detection accordingly in medium- and high-SNRs regime. Finally, the comprehensive complexity analysis of dozen MIMO detectors carried out in this paper has demonstrated that LR-aided MIMO detectors


25

operating under large number of antennas and low-medium antenna correlation indexes present the best complexity-performance tradeoffs under the sub-optimum MIMO detection perspective.

Acknowledgment This work was supported in part by the National Council for Scientific and Technological Development (CNPq) of Brazil under Grants 202340/2011-2, 303426/2009-8 and in part by State University of Londrina – Paraná State Government (UEL), scholarship PROIC-2013-14.

References 1. Bai, L., Choi, J.: Low Complexity MIMO Detection. Springer (2012) 2. Biglieri, E.: MIMO Wireless Communications. Cambridge University Press (2007) 3. Boccardi, F., Heath, R., Lozano, A., Marzetta, T., Popovski, P.: Five disruptive technology directions for 5g. Communications Magazine, IEEE 52(2), 74–80 (2014). DOI 10.1109/MCOM.2014.6736746 4. Bolcskei, H.: Mimo-ofdm wireless systems: basics, perspectives, and challenges. Wireless Communications, IEEE 13(4), 31–37 (2006). DOI 10.1109/MWC.2006.1678163 5. Choi, J., Nguyen, H.: SIC-based detection with list and lattice reduction for mimo channels. IEEE Transactions on Vehicular Technology 58(7), 3786–3790 (2009) 6. Cox, A., Higham, N., for Computational Mathematics, M.C.: Stability of Householder QR Factorization for Weighted Least Squares Problems. Numerical analysis report. Manchester Centre for Computational Mathematics (1997) 7. Fischer, R.F.H., Windpassinger, C.: Real versus complex-valued equalisation in VBLAST systems. Electronics Letters 39(5), 470–471 (2003). DOI 10.1049/el:20030331 8. Gan, Y.H., Ling, C., Mow, W.H.: Complex lattice reduction algorithm for lowcomplexity full-diversity MIMO detection. IEEE Transactions on Signal Processing 57(7), 2701–2710 (2009). DOI 10.1109/TSP.2009.2016267 9. Gander, W.: Algorithms for the QR decomposition. Tech. rep., Eidgenössische Technische Hochschule, Zürich (1980). URL www.inf.ethz.ch/personal/gander/papers/ qrneu.pdf 10. Goldsmith, A.: Wireless Communications. Cambridge University Press (2005) 11. Golub, G.H., Van Loan, C.F.: Matrix Computations (3rd Ed.). Johns Hopkins University Press, Baltimore, MD, USA (1996) 12. Hassibi, B., Vikalo, H.: On the sphere-decoding algorithm i. expected complexity. IEEE Transactions on Signal Processing 53(8), 2806–2818 (2005). DOI 10.1109/TSP.2005. 850352 13. Jalden, J., Ottersten, B.: On the complexity of sphere decoding in digital communications. IEEE Transactions on Signal Processing 53(4), 1474–1484 (2005). DOI 10.1109/TSP.2005.843746 14. Kim, I.M.: Exact BER analysis of OSTBCs in spatially correlated MIMO channels. Communications, IEEE Transactions on 54(8), 1365–1373 (2006). DOI 10.1109/TCOMM.2006.878823 15. Kühn, V.: Wireless Communications over MIMO Channels - Applications to CDMA and Multiple Antenna Systems. John Wiley & Sons Ltd., Chichester, England (2006) 16. Lenstra, H., Lenstra, A., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982) 17. Ma, X., Zhang, W.: Performance analysis for MIMO systems with lattice-reduction aided linear equalization. IEEE Transactions on Communications 56(2), 309–318 (2008). DOI 10.1109/TCOMM.2008.060372

26


18. Paulraj, A., Gore, D., Nabar, R., Bolcskei, H.: An overview of MIMO communications - a key to gigabit wireless. Proceedings of the IEEE 92(2), 198–218 (2004). DOI 10.1109/JPROC.2003.821915 19. Rusek, F., Persson, D., Lau, B.K., Larsson, E., Marzetta, T., Edfors, O., Tufvesson, F.: Scaling up MIMO: Opportunities and challenges with very large arrays. Signal Processing Magazine, IEEE 30(1), 40–60 (2013). DOI 10.1109/MSP.2011.2178495 20. Szczecinski, L., Massicotte, D.: Low complexity adaptation of MIMO MMSE receivers, implementation aspects. In: Global Telecommunications Conference, 2005. GLOBECOM ’05. IEEE, vol. 4, pp. 6 pp.–2332 (2005). DOI 10.1109/GLOCOM.2005.1578079 21. Valente, R., Marinello, J.C., Abrão, T.: LR-aided MIMO detectors under correlated and imperfectly estimated channels. Wireless Personal Communications pp. 1–24 (2013). DOI 10.1007/s11277-013-1500-6. URL http://dx.doi.org/10.1007/ s11277-013-1500-6 22. Waters, D., Barry, J.: The sorted-QR chase detector for multiple-input multiple-output channels. In: Wireless Communications and Networking Conference, 2005 IEEE, vol. 1, pp. 538–543 Vol. 1 (2005). DOI 10.1109/WCNC.2005.1424558 23. Waters, D., Barry, J.: The chase family of detection algorithms for multiple-input multiple-output channels. IEEE Transactions on Signal Processing 56(2), 739–747 (2008). DOI 10.1109/TSP.2007.907904 24. Wolniansky, P., Foschini, G., Golden, G., Valenzuela, R.: V-BLAST: an architecture for realizing very high data rates over the rich-scattering wireless channel. In: International Symposium on Signals, Systems, and Electronics, 1998. ISSSE 98. 1998 URSI, pp. 295– 300 (1998). DOI 10.1109/ISSSE.1998.738086 25. Wubben, D., Bohnke, R., Kuhn, V., Kammeyer, K.D.: Mmse extension of V-BLAST based on sorted QR decomposition. In: Vehicular Technology Conference, 2003. VTC 2003-Fall. 2003 IEEE 58th, vol. 1, pp. 508–512 Vol.1 (2003) 26. Wubben, D., Bohnke, R., Kuhn, V., Kammeyer, K.D.: Near-maximum-likelihood detection of MIMO systems using MMSE-based lattice reduction. In: IEEE International Conference on Communications, vol. 2, pp. 798–802 Vol.2 (2004). DOI 10.1109/ICC.2004.1312611 27. Wubben, D., Bohnke, R., Rinas, J., Kuhn, V., Kammeyer, K.D.: Efficient algorithm for decoding layered space-time codes. Electronics Letters 37(22), 1348–1350 (2001). DOI 10.1049/el:20010899 28. Zelst, A.V., Hammerschmidt, J.S.: A single coefficient spatial correlation model for multiple-input multiple-output MIMO radio channels. In: in Proc. URSI XXVIIth General Assembly, pp. 1–4 (2002)