Int. J. Electron. Commun. (AEÜ) 63 (2009) 678–684www.elsevier.de/aeue

Letter

Reliability assessment for MIMO detection and its application

Fan Wanga,∗, Yong Xiongb, Xiumei Yangb

aElectromagnetics Academy, Zhejiang University, Hangzhou 310027, ChinabShanghai Research Center for Wireless Communications, Shanghai 200050, China

Received 14 September 2007; accepted 9 April 2008

Abstract

We propose a reliability assessment (RA) for multiple input and multiple output (MIMO) detection at the receiver, whichcan assess whether the output of MIMO detection is reliable or not. First, an analytical formula of the RA is derived to assessthe reliability of the receiver’s output, a simplified RA is also proposed and a complexity-reduced method is further givenby adjusting the threshold of RA, with some potential performance loss compared to the original method. Next we developa combined detector with simple receiver, advanced receiver and RA module as follows: (1) the initial estimate of transmitsymbols is obtained by the simple receiver, assessed by RA in the RA module; (2) if the estimate satisfies the RA, we con-sider the estimate as the final result; otherwise, the estimate will be through the advanced receiver for further improvement.Finally we give two practical cases of combined detection and run Monte Carlo simulations to confirm the analysis and theefficiency of our proposed combined detection.A key contribution of this paper is a simple analytical formula used to assess whether the detection result is reliable or notat the receiver, which can help the combined detection to sufficiently utilize the output of the simple detection, and estimatepart of transmit symbols only using the simple receiver instead of the advanced receiver with negligible performance loss.� 2008 Elsevier GmbH. All rights reserved.

Keywords: Multiple input–multiple output; Detection; Chi-square distribution; Complexity

1. Introduction

Multiple input and multiple output (MIMO) technique hasattracted many interests due to its high spectra efficiencyand transmission capacity in broadband wireless communi-cations [1,2]. However, the design of low-complexity signalprocessing schemes capable of supporting data rates closeto the MIMO capacity remain a major challenge.

Maximum likelihood (ML) receivers have optimum per-formance with a complexity exponential with the numberof transmit antennas [3], which is always prohibitive inpractical MIMO systems. Ordered successive interference

∗ Corresponding author.E-mail address: beckham@zju.edu.cn (F. Wang).

1434-8411/$ - see front matter � 2008 Elsevier GmbH. All rights reserved.doi:10.1016/j.aeue.2008.04.005

cancellation (OSIC) receiver, proposed in [4], whichdetects the transmit symbols one by one according to thepost-detection SNR and does successive interference can-cellation, can achieve better performance with relativelyhigh complexity, compared to linear receivers. To approachML performance, a lot of works on advanced techniquesare developed, such as (QRD-M) detection [5], particlefiltering (PF) algorithm [6], sequential Gaussian approxi-mation [7], sphere-decoding (SD) [8], which can partiallyreduce the computational complexity with no or negligibleperformance loss compared to ML receivers.

Although the advanced receivers have superior perfor-mance, simple receivers offer a significant computational re-duction. Linear receivers, such as zero-forcing (ZF) receiverand minimum mean square error (MMSE) receiver [9],

F. Wang et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 678–684 679

possess the advantage of having a complexity of O(Nt3),where Nt is the number of transmit antennas.

Unlike previous work, our main concern in this paper isnot the detection technique itself, but to find out a methodto check the reliability of the receiver’s output. As we know,simple receivers offer suboptimal performance. Typically, biterror ratio (BER) performance of MMSE receivers for a 4×4MIMO system with QPSK modulation over Rayleigh flatfading channels in SNR=20 dB is about (10−2) [10], whichmeans that most transmit symbols are correctly recovered byMMSE receivers. It is very attractive that the detection canadopt simple receivers when they can recover the transmitsymbols correctly and adopt advanced receivers when simplereceivers cannot correctly recover the transmit symbols. Theintention of this paper is to present a reliability assessment(RA) to assess whether the estimate of transmit symbols isreliable or not, so we can estimate part of transmit symbolsonly using simple receivers instead of advanced receiverswith no or negligible performance loss.

This paper is organized as follows. The RA, together witha simplified RA and complexity-reduced method by adjust-ing the threshold of RA, is first derived in Section 2. InSection 3 we develop a combined detector with simple re-ceivers, advanced receivers and RA module, give two prac-tical cases of the combined detectors, and run Monte Carlosimulations to validate our analysis. Finally we draw con-clusions in Section 4.

2. Reliability assessment

Consider an MIMO link with Nt transmit antennas andNr receive antennas (Nr�Nt). The transmit vector can bedenoted by S = [s1, . . . , sNt]T, where the superscript Tstands for the transpose, si is the signal transmitted fromantenna i normalized such that E‖si‖2 = 1. Assume a flatfading channel H, where the fading coefficient from the jthtransmit antenna to ith receive antenna is denoted by hi,j ,which is a circularly symmetric zero-mean complex Gaus-sian random variable. The received vector is denoted byY =[y1, . . . , yNr]T. Now the MIMO baseband transmissioncan be classically described as

Y = HS + n (1)

where n is an Nr×1 noise vector with each entry a complexGaussian noise with zero mean and variance �2.

With the assumption of channel fading perfectly knownat the receiver, the detection can be expressed as

S̃ = d(Y, �2, H) (2)

where S̃ is the estimate of S and function d(·) refers tothe detection algorithm. Here we assume that S̃ is the harddecision of S. It can be noticed that �2 is not always neededin MIMO detection, such as ML receivers and ZF receivers.

2.1. Reliability assessment

In this subsection, we derive an RA to assess the initialestimate.

Let us define

� = ‖Y − HS̃‖2/�2 (3)

where ‖ · ‖ represents the Frobenius norm. Following (3),we have

� = ‖H(S − S̃) + n‖2/�2 (4)

If S̃ is the right estimate of transmit symbols, i.e., S = S̃,we can rewrite (4) as

� =Nr∑i=1

(ni)2/�2 (5)

Then � can be seen as the sum of squares of Nr i.i.d. Gaus-sian variables with zero mean and unit variance. So � has achi-square distribution (CSD) with Nr degrees of freedom.The probability density function (PDF) of �, p�2(Nr)(�), isgiven by{ 1

2Nr/2�(Nr/2)e−1/2��Nr/2−1, � > 0

0, � < 0(6)

where �(u) is a gamma function, defined as

�(u) =∫ ∞

0tu−1e(−t) dt

If S̃ is the wrong estimate of transmit symbols, i.e., S �= S̃,we can rewrite (4) as

� =Nr∑i=1

‖Hi (S − S̃) + n2i ‖2/�2 (7)

where Hi is the ith row of H. Then we can find that � has anoncentral CSD with Nr degrees of freedom and noncentralparameter of �, where

� =Nr∑i=1

‖(HiS − S̃))2‖/�2 = ‖(H(S − S̃))2‖/�2 (8)

The PDF of �, q�2(Nr)(�) is very complicated and given by{�Nr/2+k+1e−(�+�)/2

2Nr/2

∑∞k=0

(�/4)k

k!�(Nr/2+k), � > 0

0, � < 0(9)

If the probability density of CSD is higher than that ofall noncentral CSDs, we assess that S̃ is reliable. So we canpresent an ideal RA as

p�2(Nr)(�) > max�∈A

(q�2(Nr,r)(�)) (10)

where A is the set of all possible squared distances as

A = {‖H(S − S̃)2‖/�2|∀�S �= S} (11)

680 F. Wang et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 678–684

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α

prob

abili

ty d

ensi

ty

crossing point

CSDNCSD(1)NCSD(5)NCSD(9)NCSD(20)

Fig. 1. PDF curves of CSD and NCSD with 2 degrees of freedomand noncentral parameters of NCSD are 1, 5, 9, 20.

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prob

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ty d

ensi

ty

crossing point

CSDNCSD(1)NCSD(5)NCSD(9)NCSD(20)

Fig. 2. PDF curves of CSD and NCSD with 4 degrees of freedom,noncentral parameters of NCSD are 1, 5, 9, 20.

where �S is a possible transmitted vector. We should noticethat A is independent of S, and only depends on the channelmatrix H and the modulation format.

Since the analytical expressions of p�2(Nr)(�) andq�2(Nr)(�) are extremely complicated, the closed-form ex-pression of (10) is hard to derive. Here we investigate thecharacteristics of CSD and noncentral chi-square distribu-tions (NCSD) by observing their PDF curves with differentdegrees of freedom and different �.

PDF curves of CSD and NCSD with 2, 4, and 6 degreesof freedom are illustrated in Figs. 1–3, respectively. In eachfigure NCSD curves with �= 1, 5, 9, 20 are given. It is easyto find that there is only one crossing point of p�2(Nr)(�)

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abili

ty d

ensi

ty

crossing point

CSDNCSD(1)NCSD(5)NCSD(9)NCSD(20)

Fig. 3. PDF curves of CSD and NCSD with 6 degrees of freedom,noncentral parameters of NCSD are 1, 5, 9, 20.

and q�2(Nr)(�) for any �. Let �̄(Nr, �) denote the abscissa ofcrossing point; it can be observed that p�2(Nr)(�) is largerthan q�2(Nr)(�) when � < �̄(Nr, �), and p�2(Nr)(�) is smallerthan q�2(Nr)(�) when � > �̄(Nr, �), that is{

p�2(Nr)(�)�q�2(Nr)(�), x� x̄(Nr, �)p�2(Nr)(�) < q�2(Nr)(�), x > x̄(Nr, �)

(12)

We can also find �̄(Nr, �), the abscissa of crossing pointof p�2(Nr)(�) and q�2(Nr)(�), increases with �:

�̄(Nr, �1) > �̄(Nr, �2) if �1 > �2 (13)

From (13), we conclude that �̄(Nr, min(A)) is the mini-mum of �̄(Nr, �). Combing (10)–(13), RA is reduced to

� < �̄(Nr, min(A)) (14)

where

p�2(Nr)(�̄(Nr, min(A))) = q�2(Nr,�)(�̄(Nr, min(A)))

Here we call the value of the right-hand side of (14)threshold, denoted by Th. To compute �̄(Nr, min(A)), nu-merical methods using MATLAB or PDF table can be ap-plied.

2.2. Simplified RA

From (11), it can be found that the complexity of comput-ing min((A) is roughly O(CNt), where C is the number ofdifferent distances dependent on the modulation format, e.g.,C=2 for QPSK modulation. Only over slow fading channels,RA is feasible because min(A) only needs to be computedonce for several consecutive symbol periods. However, RAseems complexity prohibitive in practical systems over fastfading channel.

F. Wang et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 678–684 681

To reduce the complexity of computing min(A) in (14),we make further approximations.

From (11), we have

A = {‖H(−→S − S)‖2/�2|∀−→

S �= S}

={

Nr∑i=1

‖(Hi(−→S − S)2‖2/�2|∀−→

S �= S

}

�

⎧⎨⎩

Nr∑i=1

Nt∑j=1

‖hi,j‖2‖(−→S − S)i‖2/�2|∀−→S �= S

⎫⎬⎭ (15)

where (−→S − S)i denotes the ith element of (

−→S − S). From

(15) we have

min(A)� min(‖hi,j‖2‖(−→S − S)i‖2/�2

= minj (‖Hj‖) • d2min/�

2, ∀−→S �= S (16)

where Hj is the jth column of H and dmin is the minimumdistance of the nearest neighbors in the constellation.

Here we use the right-hand side of (16) to approximatemin(A). It should be noticed that min(A) is set to be a valuelarger than or equal to its exact value. Combining (14) and(16), a simplified RA can be derived as

� < �̄

(Nr, min

j(‖Hj‖) • d2

min/�2)

(17)

Since minj (‖Hj‖) and dmin are easily computed whenH and the modulation format are determined, the computa-tion cost of (17) is greatly reduced compared to (14). Sincemin(A) is approximated by a larger value, the threshold ofthe simplified RA is then larger than that of the original RA,which may result in some performance loss in the accuracyof RA.

2.3. Efficiency analysis and RA with adjustedthresholds

In this subsection we first analyze the efficiency of theproposed RA. As we mentioned earlier, it is the perfectcase that all the right recovered symbols by simple receiversare assessed reliable, while all the wrong recovered symbolsare assessed unreliable. But this is impractical because thereceiver cannot know the exact information of the transmit-ted symbols in the presence of noise and interference.

We have proposed RAs that can assess the reliability oftransmitted symbol estimation in (14) and (17) with differ-ent Th. When �<Th, the estimate is assessed reliable. Theprobability of CSD when x<Th is given by

F�2(Nr)(� < Th) =∫ Th

0p�2(Nr)(�) d� (18)

That is, the probability that the estimate is assessed re-liable when no error detection occurs at the receiver isF�2(Nr)(� < Th). As we have mentioned, it is impractical for

the receivers to recover all the transmitted symbols withouterror. So the probability that the estimate is assessed reliableat a given of RA, which is called efficiency of RA, is thengiven by

Eff(Th) = F�2(Nr)(� < Th)P (S̃ = S)

+∑�S�=S

F�2(Nr,�(�S))(� < Th)P (S̃ = �S) (19)

where

F�2(Nr,�(�S))(� < Th) =

∫ Th

0q�2(Nr,�(�S))

(�) dx

�(�S) = ‖(H−→S − S)‖2/�2

and P(S̃ = S) represents the probability that S̃ equal to thetransmit symbols vector S. Since there are generally no an-alytical expressions of P(S̃ = S) and P(S̃ = �S) for most re-ceivers, we could not get the exact efficiency expressions forRA with threshold Th. However, with high SNR conditions,a few vector error estimations occur, i.e., P(S̃ = S) ≈ 1 and∑

�S�=SP(S̃ = �S) ≈ 0, so (19) can be simplified and approx-imated as

Eff(Th) ≈ F�2(Nr)(� < Th) (20)

From (18) and (19) we can find that the efficiency ofRA increases with Th, which means that the estimate ismore likely to be assessed reliable. Meanwhile, it is notreliable when Th is larger than �̄(Nr, min(A) given in (14).Here we propose an RA with adjusted thresholds, prede-fined and probably larger than �̄(Nr, min(A)). RA withadjusted thresholds has two main advantages, i.e., low com-putation cost without the need to vary with the channel stateinformation (CSI) and higher efficiency, while a disadvan-tage of potential wrong RA. Furthermore, we can achievea performance-complexity tradeoff by adjusting the valueof Th. When Th is set to a value exceeding �̄(Nr, min(A)),performance loss may occur.

3. Application and numerical results

In this section we develop a combined detector with RA.The structure of combined detection is given in Fig. 4. Theproposed detection has two processing steps: (1) the initialestimate of transmit symbols is obtained by the simple re-ceiver, assessed by RA in the RA module; (2) if the estimatesatisfies the RA, we consider the estimate as the final result;otherwise, the estimate will be through the advanced receiverfor further improvement. In the following we investigate theperformance of two practical cases of combined detectionby Monte Carlo Simulations. We assume 4 × 4 MIMO sys-tems, employing QPSK constellation. The MIMO channelis quasi-static Rayleigh flat fading and i.i.d. 10,000 channelrealizations are simulated.

682 F. Wang et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 678–684

Yes

No

ENAdavancedReciver

Y

Y

Received

signals

Estimated

symbolsSimpleReciever

ReliabilityAssessment

S~

S~

S~

Fig. 4. Combined detector with simple receiver, advanced receiver, and reliability assessment module.

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R

MMSECombined Detection (RA)Combined Detection (Simplified RA)Combined Detection (Th=10)Combined Detection (Th=20)OSIC

Fig. 5. BER performance of MMSE receiver, OSIC receiver andcombined detector with RA, simplified RA and RA with adjustedthresholds set to 10 and 20, in 4×4 MIMO BLAST QPSK systems.

We choose MMSE receivers as the simple receiver andOSIC receivers [4] as the advanced receiver. BER perfor-mances of combined detection with RA, simplified RA, andRA with adjusted thresholds are illustrated in Fig. 5. Theadjusted thresholds are set to 10 and 20. It is easy to findthat combined detection with RA can achieve near the per-formance of OSIC detector, and greatly outperforms MMSEdetector, which is consistent with our expectation. There issome performance loss for the combined detection with sim-plified RA compared to combined detection with RA, whichis caused by the approximation in (16) and (17). It is inter-esting to find that combined detection with adjusted thresh-old of Th =10 achieves the approximate OSIC performanceas combined detection with RA, which is very attractive be-cause it means that the combined detection can approachthe performance of OSIC receiver without the necessity tocompute the threshold adaptive to instantaneous CSI, if thethreshold is pre-set to an appropriate value. When we furtherincrease the value of Th, e.g., Th=20, obvious performanceloss occurs.

Fig. 6 shows the experimental and analytical RA effi-ciency of combined detection with different RA thresholds atdifferent SNR levels. The analytical result is approximated

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cy o

f RA

RA (experimental)RA (analytical)Simplified RA (experimental)Simplified RA (anlytical)Th=10 (experimental)Th=10 (analytical)Th=20 (experimental)Th=20 (analytical)

Fig. 6. Comparison of the experimental and analytical RA effi-ciency at different SNR levels corresponding to RA, simplifiedRA, RA with Th = 10 and RA with Th = 20. The analytical RAefficiency is approximated by (20). The dotted are the analyticalresults and the solid are the experimental results.

by (20). First it can be observed that the performance gapbetween the experimental efficiency and the analytical effi-ciency is roughly narrowed with SNR increasing due to theapproximation we made in (20). The figure also shows that,as expected, efficiency of simplified RA is higher than thatof ideal RA, and efficiency of RA (Th = 20) is higher thanthat of RA (Th = 10). We further find that experimental RAefficiencies of all combined detections are higher than 0.6over all SNR ranges, and higher than 0.9 at SNR = 20 dB.Since the RA efficiency represents the probability that weuse simple receivers to obtain the final transmit vector esti-mation, there is great computation complexity reduction forour combined detectors compared to OSIC receivers, espe-cially for high SNR cases. Finally, Fig. 6 validates that theefficiency of RA increases with the threshold.

Another simulation was run on the combined detectorwith linear MMSE receiver as simple receiver and QRD-Mreceiver [5] as advanced receiver. The number of survivorpaths in QRD-M detection is 8, i.e., M =8. The BER perfor-mances of the combined detector with different thresholdsare given in Fig. 7. It shows that combined detection with

F. Wang et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 678–684 683

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R p

erfo

rman

ce

MMSEQRD MCombined Detection (Th=6)Combined Detection (Th=8)Combined Detection (Th=10)Combined Detection (Th=20)

Fig. 7. BER performance of MMSE receiver, ML receiver, andcombined detector with RA with adjusted thresholds set to 6, 8,10, and 20 in 4 × 4 MIMO BLAST QPSK systems.

Table 1. Probability of the QRD-M detector running in combineddetection

SNR (dB) 0 5 10 15 20

Th = 6 0.0540 0.2316 0.3397 0.2439 0.1812Th = 8 0.0116 0.0931 0.2442 0.1550 0.0793Th = 10 0.0022 0.0342 0.1883 0.1300 0.0500Th = 20 0 0.0001 0.0397 0.1015 0.0405

Th = 6 achieves approximately the performance of QRD-M detector. And as is illustrated in Table 1 , the percentageof running QRD-M detector in combined detection is only34% or lower. Again we can find that the performance losswill occur if we increase the value of threshold. Meanwhile,the computational complexity of combined detection goeslower. Therefore, the performance-complexity tradeoff canbe achieved by adjusting the thresholds of RA.

4. Conclusions

In this paper, we present a reliability assessment forMIMO system used to assess whether the detection result isreliable or not at the receiver. Based on this RA, a combineddetector with simple receiver and advanced receiver is fur-ther developed and achieves great computation complexityreduction with no or negligible performance loss comparedto advanced receivers.

The main advantage of RA is to sufficiently utilize theoutput of the simple receiver, and possibly estimate part oftransmit symbols only using the simple receiver instead ofthe advanced receiver with no or little performance loss.

References

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[4] Wolniansky PW, Foschini GJ, Golden GD, Valenzuela RA.V-BLAST: an architecture for realizing very high data ratesover the rich-scattering wireless channel. Int Symp SignalsSyst Electron 1998; 295–300.

[5] Kim KJ, Yue J, Iltis RA, Gibson JD. A QRD-M/Kalman filter-based detection and channel estimation algorithm for MIMO-OFDM systems. IEEE Trans Wireless Commun 2005;4:710–21.

[6] Huang Y, Zhang J, Djuric PM. Detection with particle filteringin BLAST systems. IEEE ICC’03 2003;4:2306–10.

[7] Jia Y, Andrieu C, Piechocki RJ, Sandell M. Gaussianapproximation based mixture reduction for near optimumdetection in MIMO systems. IEEE Commun Lett 2005;9:997–9.

[8] Viterbo E, Boutros J. A universal lattice code decoderfor fading channels. IEEE Trans Inform Theory 1999;45:1639–42.

[9] Foschini GJ. Layered space-time architecture for wirelesscommunication a fading environment when using multipleantennas. Bell Labs Tech J 1996;1:41–59.

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Fan Wang was born in Hubei, China,on March 15, 1980. He received theB.S. degree in Electrical InformationEngineering from Zhejiang University,Hangzhou, China, in 2003. He is cur-rently working towards the Ph.D. de-gree at Zhejiang University, Hangzhou,China. His research interests focus onoptimized MIMO transmitter architec-tures and reduced complexity MIMOreceiver design.

684 F. Wang et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 678–684

Yong Xiong received the B.S. degreein Mathematics from Nanchang Uni-versity, Nanchang, China, in 1996, hisPh.D. degree in operational researchand cybernetics from Shanghai Univer-sity, Shanghai, China, in 2003, and didpostdoctoral work in Shanghai Instituteof Microsystem and Information Tech-nology (formerly Shanghai Institute ofMetallurgy), Chinese Academy of Sci-ence after graduation. He is currently

working as a research engineer in Shanghai Research Center forWireless Communications. His main research interests are proba-bilistic model, space time codes, and mobile computing.

Xiumei Yang received the M.S. degreein communications and information sys-tems from Shandong University, Jinan,China, in 2004. She is currently workingas a research engineer in Shanghai Re-search Center for Wireless Communi-cations, Shanghai, China. Her researchinterest focuses on the design of adap-tive MIMO transceiver.