Multi-Level SINR Thresholding for ReducedComplexity MIMO Detection

Atul Kumar Sinha, Mohit Agarwal and A. K. ChaturvediDepartment of Electrical Engineering

Indian Institute of Technology Kanpur

Abstract—In previous research works, it was established thatthe post-processing signal-to-interference-and-noise ratio (SINR)distribution of MMSE (Minimum Mean Square Error) detectorshows the potential of high reliability in the high-SINR detectedsymbols. In this paper, a new algorithm with reduced complexityis proposed exploiting the fact that reliability of MMSE estimatedsymbols increases with increase in post-processing SINR. Forhigh SINR region, MMSE estimated symbols are retained asthe final decisions and the remaining symbols are grouped intodifferent regions based on the SINR. Based on the group in whichthe symbols lie, we restrict the search space and feed it into SD(Sphere Decoder). A SD-SE (Schnorr-Euchner) implementationof SD algorithm is used for comparison of results.

Index Terms—Multiple-input multiple-output (MIMO) sys-tems, low-complexity detection, minimum-mean-square-error(MMSE), sphere decoding (SD), signal-to-interference-and-noiseratio (SINR), Maximum Likelihood (ML) detector.

I. INTRODUCTION

Multiple-input multiple-output (MIMO) technique is pre-ferred over single-input single-output (SISO) systems in mod-ern wireless communications, because of its ability to transmitat higher data rate [5]. Moreover, they provide improvedquality of service of wireless systems without the need foradditional frequency bandwidth or transmit power. In MIMOsystem the transmitted symbol is corrupted by noise andinter-symbol-interference. The use of multiple antennas onboth sides of the wireless link requires signicant additionalsignal processing, especially in the receiver, where the MIMOdetection unit separates the spatial streams. Various algorithmshave been proposed for the detection of transmitted symbols.

The optimal maximum-likelihood detection (MLD) thatexhaustively examines all symbol candidates has exponentialcomplexity and may not be applicable in many practicalsystems. The linear Zero-Forcing (ZF) and MMSE equalizersare classic functional blocks which are ubiquitous in digitalcommunications [11]. They are also the building blocks ofmore advanced communication schemes such as the decisionfeedback equalizer (DFE), or equivalently, the vertical BellLabs layered space-time (V-BLAST) architecture [6], [7] andvarious other MIMO transceiver designs.

To achieve near ML performance with lesser complex-ity, Sphere-Decoder (SD) algorithm [4] was proposed whichperforms similar in terms of symbol error rate (SER) withcomparatively less complexity. It requires QR decomposition

of channel matrix to solve the detection problem. With eachsolution point it updates radius of hypersphere. Further, SDcomplexity can be reduced using various other radius estima-tion techniques [13]. Early Termination method proposed in[8] reduces the computational complexity of Sphere Decodingalgorithm using pre-calculated distance. This particular imple-mentation of Sphere Decoder is known as Schnorr-EuchnerDecoding Algorithm and has been used in this paper for faircomparison of results.

In [10], the distribution of SINR at the output of the MMSEdetection is derived for a small number of transmit and receiveantennas using a new approximation for the Gaussian Q-function and the performance of linear MMSE detection underideal fast fading environment was analyzed. In [3], SINRthresholding technique was used to decide the symbols forwhich MMSE estimate with reliable SER can be retained asthe final solution.

In this paper, we propose an algorithm which extends theidea presented in [3]. We use multi-level SINR thresholdingto partition each received symbol into different groups withcorresponding different levels of search space reduction. Thus,the average search space for deciding the MIMO transmittedsymbol is reduced significantly, resulting in lesser complexity.Further, a novel approach for deciding the groups and deter-mining the reduced search space for each received symbol ispresented.

The remainder of the paper is organized as follows. In Sec-tion II, MIMO system model is explained along with integerleast-squares problem, SD and MMSE detection algorithms.Previous research work on this topic is explained in SectionIII. We explain the proposed detecion scheme in Section IV.Section V includes simulation results of proposed detectionscheme and section VI concludes the paper.

II. SYSTEM MODEL

We consider a MIMO system with NT transmit antennas andNR receive antennas (NT ≤ NR). Then the baseband signalmodel is given by

y = Hx+ n (1)

where y ε CNR×1 denotes the received symbol vector and x =[x1...xNT

] ε SNT×1 stands for the transmitted symbol vectorin which each entry xi is independently drawn from a complexconstellation set S with zero mean and variance σ2

x = Es

NT. The

noise n ε CNR×1 is independent identically distributed (i.i.d.)978-1-4673-5952-8/13/$31.00 c© 2013 IEEE

additive white Gaussian noise (AWGN) with zero mean andvariance σ2

n = No and H ε CNR×NT is the Rayleigh fadingchannel with i.i.d. Gaussian entries with zero mean and unitvariance. The channel information is assumed to be perfectlyknown at the receiver and the signal-to-noise ratio (SNR) isdefined as SNR = Es

No.

A. Integer Least-Squares Problem

Integer Least-Squares Problem arises in many problemsincluding its applications in Multiple Input and Multiple Out-put (MIMO) and Orthogonal Frequency Division Multiplexing(OFDM). The integer least-squares problem is the followingminimization problem i.e. searches for the closest lattice pointHx to the received signal y,

arg minxεSNT ×1

‖y −Hx‖2 (2)

where x is unknown. Here ‖.‖ denotes the square-norm of avector. The exhaustive detection scheme searches for all the|S|NT candidate symbol combinations for the optimal solution,where |.| denotes the cardinality of a set. For a general H itis known to be NP hard in the worst-case sense [9], as wellas the average sense [12], [2].

B. SD Algorithm

Sphere decoding is a particular method, which solves integerleast-squares problem by reducing H into upper triangularmatrix using orthogonal transformations to obtain the QRdecomposition:

H = Q

[R0

](3)

where Q ε RNR×NR is orthogonal and R ε RNT×NT is upper-triangular. Partitioning Q = [Q1 Q2], where Q1 is RNR×NT

and Q2 is RNR×(NR−NT ), we get

‖y −Hx‖22 (4)

=

∥∥∥∥y −Q

[R0

]x

∥∥∥∥22

(5)

=

∥∥∥∥Q>y − [R0]x

∥∥∥∥22

(6)

=

∥∥∥∥[Q>1Q>2

]y −

[R0

]x

∥∥∥∥22

(7)

= ‖Q>1 y −Rx‖22 + ‖Q>2 y‖22 (8)

It can be seen that second term is independent of x. Denotingy as Q>1 y, the problem is reduced to the following triangularinteger least square problem

minxεSNT ×1

‖y −Rx‖2 (9)

Sphere Decoder searches for the solution in a pre-determinedhypersphere centered at y.

C. MMSE Algorithm

The MMSE detection performs linear equalization on thereceived signal y to minimize the mean-square error E[||Gy- x||2] between the detected signal and the desired signal. Forthe MIMO system model described in (1), the equalizationmatrix G is given by [1],

G = HH(HHH + σ2n/σ

2xINR

)−1 (10)

where ()1 and ()H denote inverse and hermitian transpose of amatrix, respectively, and INR

is the NR×NR identity matrix.The equalized symbol vector xMMSE is then given by,

xMMSE = Gy = HH(HHH + σ2n/σ

2xINR

)−1y (11)

The MMSE detected symbol vector xMMSE is obtained byelement-wise quantizing (slicing) the equalized symbol vectorto the closest constellation point, i.e., xMMSE = Q(xMMSE),where Q(.) denotes the quantization operation.

III. PREVIOUS RESEARCH WORK

In [3], a low-complexity multiple-input multiple-output(MIMO) detection scheme is proposed by exploiting theMMSE detector concatenated with the sphere decoder (SD),where the MMSE detected symbols with SINR above theSINR threshold are retained as the final decisions and theremaining lower-SINR symbols are to be detected by the SD.The search space was significantly reduced without incurringmuch extra error rate by retaining the high-SINR symbolsdetected in the MMSE.

A. Calculation of SINR of a symbol

Signal to Interference plus Noise Ratio (SINR) is the ratioof signal power with the sum of interference and noise power.SINRi of a particular symbol can be calculated [10] asfollows,

SINRi =βi

1− βi(12)

where βi is (GH)ii. Here G is MMSE equalization matrixand H is the channel matrix.SINRi can be approximated by a Gamma distributed randomvariable Gamma(α,β). Where α and β can be calculated bysolving,

E(SINRi) = αβ = c(1− γ + γµ) (13)

V ar(SINRi) = αβ2 =c2

NR(1− γ + γσ2) (14)

Here,

µ =κ− c(1− γ)− 1

2cγ(15)

σ2 = µ− 1 + c(1 + γ)− κ2cγκ

(16)

B. Symbol estimation

Now symbols with SINR higher than predefined SINRi.e. SINRth are decided by MMSE detector. To keep thenumber of SINR symbols approximately same inverse Gammafunction is used to calculate SINRth i.e.

SINRth = F−1(1− NmNT

) (17)

where F−1(.) is inverse Gamma function with the same αand β parameters defined above and Nm is predefined averagenumber of high-SINR symbols.Now, for the detection of remaining low-SINR symbols,inter-symbol-interference (ISI) of MMSE decided symbols iscancelled from the received symbol vector i.e.

y′ = y − hkx(k)MMSE−SD(Nm) (18)

where k is the index of MMSE decided symbols. After thecancellation, the original signal model is modified as

y′ = H′x+ n, (19)

where H′ is obtained from nulling columns hk of H. Onthe modified signal model, SD is performed to estimate theremaining symbols.

IV. PROPOSED DETECTION SCHEME - MULTI-LEVELSINR THRESHOLDING

The reliability of the MMSE estimate increases with in-crease in the post-processing SINR. This suggests that theprobable distance of the MMSE estimate for a symbol from theactual symbol in the constellation map decreases with increasein post-processing SINR. In what follows, we present a simplescheme that performs MMSE detection and reduces the searchspace of SD algorithm based on multi-level SINR thresholdingwhile still maintaining near ML-performance.

A. Multi-level SINR Thresholding

We perform MMSE detection on the received symbol vectorand partition them into different groups based on pre-definedSINR thresholds, SINRthk

where k ≤ log2M where Mis the Quadrature Amplitude Modulation (QAM) order used.Without loss of generality we assume that SINRth0

>SINRth1 > SINRth2 and so on. A symbol is in groupkif its SINR is between SINRthk

and SINRth(k−1)i.e.

SINRthk≤ SINRsymbol ≤ SINRthk−1

The symbols are in group0 if its SINR is greater thanSINRth0 i.e.

SINRth0≤ SINRsymbol

Thus, the reliability of groupk symbols decreases with in-crease in the value of k. The symbols in group0 have mostreliable MMSE estimates and the MMSE decisions are kept asfinal decisions for these symbols. The search space of symbolsin other groups is reduced as described in section C.

Fig. 1. Determining Restricted Search Space

B. SINR Threshold Calculation

Depending on the desired performance and complexitySINRthk

are precomputed offline. SINRthkare related to

the average number of chosen symbols as,

SINRthk= F−1(1− Nmk

NT) (20)

where Nmkis the average number of symbols whose SINR is

greater than SINRthk. This implies that the average number

of chosen symbols in groupk is Nmk− Nm(k−1)

. The pa-rameters SINRthk

act as a tradeoff factor between error andcomplexity in MIMO detection. Further discussion on choiceof SINRthk

is given in simulation results.

C. Determining restricted search space of particular level

For a symbol which is in groupk, let x and y be the xand y co-ordinates of its MMSE estimate. Let set X be theset which contains (k+1) closest constellation point from x onthe X-projection as shown in Fig.1. The candidate points (x,y)for search space for the symbol are such that xεX and yεY .This can be summarized as follows:

X =

xεxi s.t. | x− xi |≤ (k + 1)d2 if xε(x0, xlog2m−1)i ≤ k if x < x0

i ≥ log2m - k -1 if x > xlog2m−1

where d is the distance between two neighboring constella-tion points on x-axis/y-axis projection. The set Y is definedsimilarly. The search space for group0 points are enclosed bydashed lines and for group1 points are enclosed by solid linesin Fig.1.

Fig. 2. SER of 4x4 MIMO System with 16 QAM

D. Complete Algorithm

The overall proposed algorithm is presented as follows:

Step 1. Obtain from (11) and (12)

xMMSE = [x1MMSE x2MMSE .........xNT

MMSE ]T

and SINR(i)MMSE , i = 1,2,...NT .

Step 2. Calculate the SINR threshold SINRthk(Nmk

)

Step 3. if SINR(i)MMSE ≥ SINRth0(Nm0), for i=1,2,...,NT

the i-th detected symbol is x(i)Proposed(Nm0 )= Q(x

(i)MMSE)

Step 4. Cancel the ISI of MMSE decided symbols from thereceived signal vector as explained earlier and obtain themodified signal model as y′ = H′x+ n.

Step 5. Partition the remaining Symbol vector based onSINR calculated in Step 1 into corresponding groups.

Step 6. Perform SD on the modified signal model based onthe group in which symbols lie with modified search spaceas discussed above in Section C. The final decided symbolvector is then obtained.

V. SIMULATION RESULTS

In this section, we present simulation results to comparethe proposed scheme with MMSE − SD(Nm) [3] andthe conventional SD algorithm through metrics of SymbolError rate (SER) and average number of real multiplications.The simulations are carried out for a MIMO system withTransmitting antennas (NT ) = Receiving antennas (NR) =4 for 16-QAM (Fig.2 and Fig.3) and 32-QAM (Fig.4 andFig.5) configurations. For fair comparison, we apply equivalent

Fig. 3. Complexity of 4x4 MIMO System with 16 QAM

Fig. 4. SER of 4x4 MIMO System with 32 QAM

settings to the Sphere Decoder (SD) operations in all theschemes. The parameters SINRth0

and SINRth1are chosen

such that the error performance of the proposed algorithmconverges to the error performance of MMSE − SD(1)proposed in [3]. SINRth0

is set to 114 dB and SINRth1

is set to 25.34 dB for 4×4, 16-QAM system model. For 4×4,32-QAM MIMO system model the corresponding values areset to 141.57 dB and 31.07 dB respectively. This asserts thatNm0 and Nm1 are 1 and 3 respectively for both configurationswhere Nm0 and Nm1 are the average number of symbolswhose SINR is greater than SINRthk

It is observed from Fig. 2 and Fig. 4 that the proposedalgorithm closely approaches the optimal ML performanceand MMSE − SD(Nm), e.g., within 0.1 dB at SER= 10−2.From Fig. 3 we can see that compared to the MMSE-SD(1)

Fig. 5. Complexity of 4x4 MIMO System with 32 QAM

algorithm, the proposed algorithm achieves 35% and 20%complexity reduction at SNR=10 dB and 15 dB respectively.This algorithm achieves 65% complexity reduction as com-pared to SD algorithm at SNR=10 dB which is better thanthe complexity reduction (40%) achieved by MMSE-SD(1) atsame SNR.

From Fig. 5 (4×4, 32-QAM), at SNR=10 dB complexityreduction of 55% compared to MMSE − SD(1) and 72%compared to SD can be noted. For the same configuration,MMSE − SD(1) achieves only 40% complexity reductionas compared to SD. This shows that the relative performanceof the proposed algorithm improves as the order of QAMincreases.

VI. CONCLUSION

An extension of MMSE-SD(Nm) algorithm presented in [3]is proposed in this paper. The proposed algorithm uses a multi-level SINR thresholding approach to partition the symbols intodifferent groups. The search space of the symbols is restrictedbased on the group in which they lie. This result in thereduced average search space and thus reduces the complexity.Simulation results confirm that this algorithm performs betterin terms of complexity at same Symbol Error Rate.

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