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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.

A Novel MIMO Detection Scheme with Linear Complexity

Frederik Simoens, Henk Wymeersch? and Marc Moeneclaey Department of Telecommunications and Information Processing, Ghent University, St.-Pietersnieuwstraat 41, B-9000 GENT, Belgium {fsimoens,mm}@telin.UGent.be ? LIDS Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA hwymeers@MIT.edu

Abstract— In this contribution, we present a novel multipleinput multiple-output (MIMO) detection scheme for Quadrature Amplitude Modulation (QAM). The proposed method is based on the slowest descent (SD) method and is suitable for both iterative and non-iterative detectors. Optimal MIMO detection entails maximizing or marginalizing a likelihood function over a very large set of vectors. Similar to other low-complexity detection schemes such as sphere decoding, the SD approach restricts the maximization/marginalization to a small set of candidate vectors. As the size of this set scales linearly with the number of transmit antennas, the SD method bears an exceptionally low complexity. Furthermore, simulation results indicate that the method achieves a close-to-optimal performance, provided that the diversity order is suf?ciently high.

well as long as the system diversity is suf?ciently high. II. P ROBLEM F ORMULATION We consider a discrete-time non-selective block-fading multi-antenna channel with NT transmit- and NR receive antennas. The signals received on the different receive antennas at a given time instant are stacked in the NR × 1 vector r. The relation between the received signal vector r and the NT × 1 transmitted symbol vector a is given by r = Ha + n, (1)

I. I NTRODUCTION Future wireless communication applications call for an increasing spectral ef?ciency and link reliability. Communication systems using multiple transmit and receive antennas at both ends of a wireless link are poised to ful?ll these requirements [1]. Unfortunately, the extensive processing demands of these systems still impedes their widespread deployment. A major challenge for the design of MIMO systems remains the complexity of the detection process. For instance, the optimal maximum-likelihood (ML) detector bears a complexity that scales exponentially with the number of transmit antennas. For this reason, the ML detector cannot be implemented into a practical system. This observation has spurred several research groups to inquire into alternative detector schemes. Among the most promising alternatives we ?nd the sphere decoder [2], [3] and the Markov chain Monte-Carlo approach [4]. While these schemes exhibit a satisfactory performance, their complexity is still prohibitively large from an implementation point of view. In this contribution, we propose an alternative MIMO detector suitable for QAM constellations. The detector is based on the slowest descent (SD) method, which was originally proposed for sequence estimation in a direct sequence spread spectrum context [5]. The technique we will describe can be used for uncoded or coded MIMO systems. Most remarkably, the algorithm bears a complexity that scales only linearly with the number of transmit antennas. Notwithstanding its auspiciously low complexity, the method performs exceptionally

where n denotes the NR × 1 zero-mean Gaussian noise vector and H represents the NR × NT channel matrix. The real and imaginary parts of the noise on each receive antenna are independent, each with a variance equal to σ 2 . The NR × NT channel matrix H is composed of unit variance independent Rayleigh fading complex-valued elements. We further assume that each symbol in a belongs to a complex-valued M -ary constellation ?. A. ML symbol detection for uncoded transmission Assuming all M NT possible symbol vectors a ∈ ?NT are equiprobable, ML symbol detection boils down to ?nding the maximum of the log-likelihood function (LLF): ? aM L = arg max log p(r|a)

a∈?NT a∈?NT

= arg min

r ? Ha

2

(2)

Unfortunately, the complexity of this problem grows exponentially with the number of transmit antennas (i.e., the complexity is of the order M NT ). This observation motivates us to explore alternative detection schemes. B. Iterative detection for coded transmission In a coded MIMO setup, ML detection in general has a complexity that increases exponentially with the block size. Therefore, one usually resorts to a simpler receiver that iterates between MIMO detection and decoding. However, the complexity of the iterative receiver is still of the order M NT

1525-3511/07/$25.00 ?2007 IEEE

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.

? Quantizing the elements in au to the constellation ? leads to the so-called zero-forcing (ZF) solution. It is well-known that Π encoder bits M the ZF solution suffers from noise enhancement and exhibits channel relatively poor performance. P (r|b) PeM (bl ) In a second attempt we could limit our search to symbol demap likelihood ? vectors that are located in the neighborhood of au . This is ?1 decoder Π M?1 computation r the modus operandi in sphere decoding. The basic idea is to limit the search to vectors that lie within a certain radius of Π ? au . As the procedure to ?nd these vectors is quite laborious, PeD (bl ) the sphere decoding alternative is still rather complex. Therefore, we introduce a novel method based on the Fig. 1. A BICM MIMO transceiver. slowest descent method, which was originally proposed in [5]. The idea behind this method is to analyze the slope (as is the case for the ML detector for uncoded transmis- of the log-likelihood function (LLF) in the vicinity of its sion). To illustrate this, we consider the bit-interleaved coded unconstrained maximum and to asses the line(s) of slowest modulation (BICM) set-up depicted in Fig. 1. This scheme descent of this LLF. Symbol vectors that are situated in the has been advocated in [3] as being capacity approaching. In neighborhood of these line(s) of slowest descent are retained BICM, the information bits are encoded, interleaved, grouped for further processing. For the problem at hand this boils down in blocks of length m and mapped onto an M = 2m -ary QAM to restricting the search in (2) to vectors situated close to the constellation ?. Subsequently, these symbols are multiplexed line(s) of slowest descent. The mathematical details of our new method can be sumonto the different transmit antennas, resulting in a vector a. There exists a one-to-one mapping between a symbol vector marized as follows. Since the LLF is quadratic, it can be mN a ∈ ?NT and the string of coded bits b ∈ [0, 1] T , so replaced by its second-order Taylor series expansion about the ? that we can express bl (a) as being the l-th bit after inverse function’s unconstrained maximizer au mapping (demapping) of a. At the receiver side, the MIMO demapper computes extrinlog p (r |?u + e ) = log p (r |? u ) ? eH HH He a a sic information with respect to a bit bl ∈ b and forwards it to . = ψ (e) . (5) the decoder [3], [6]

group mapper a

(M Pe ) (bl = b) = a∈χb l mNT

b

p (r |a )

l =1,l =l

(D) Pe (bl = bl (a)) ,

Let us now consider a straight line {x (γ)} through the ? unconstrained maximum au of the LLF: ? x(γ) = au + γeSD , (6) where γ is real and eSD is a complex-valued unit vector. The aspiration behind the slowest descent method is to ?nd a unit vector eSD such that the decrease of the likelihood along the line de?ned by (6) is as small as possible. It is easily veri?ed that eSD is the normalized eigenvector of HH H corresponding to the minimum eigenvalue1 . However, as we consider complex-valued eigenvectors, the normalized eigenvectors of HH H are determined up to a factor ejθ , θ ∈ [0, 2π[. Let us therefore denote by eSD one of the normalized eigenvectors corresponding to the minimal eigenvalue of HH H. Then, the following set of vectors designate the directions of slowest descent eSD : eSD = ejθ eSD , θ ∈ [0, 2π[ . (7)

(3) (D) for l ∈ {1, . . . , mNT }. Here, Pe (.) stands for the extrinsic information fed back from the decoder to the demapper and χb denotes the set of symbol vectors a for which the l-the bit l bl (a) equals b. Since the summation in (3) goes over 2mNT = M NT terms in total (i.e. for b = 0 and b = 1), (3) and (2) have similar computational complexity. III. T HE S LOWEST D ESCENT M ETHOD In this section, we introduce the Slowest Descent (SD) method as a way to ?nd a smaller set S of symbol vectors over which we perform the maximization/marginalizations (2) and (3). We ?rst develop the algorithm for MIMO ML detection and then extend it to systems with MIMO iterative detection. A. ML symbol detection for uncoded transmission The search required in (2) is cumbersome due to the constraint on a, i.e. a ∈ ?NT . The unconstrained maximization, i.e. the maximization with respect to a ∈ CNT , leads to a very simple closed-form solution: ? au = arg max log p (r |a ) = H H

H a∈RNT ?1

These eigenvectors span the plane of slowest descent in the complex space CNT , which is given by: ? x(γ, θ) = au + γejθ eSD . (8)

The aforementioned results reveal that it is easy to obtain an expression for the plane of slowest descent. All it takes is (4)

1 Note that the eigenvalues of a Hermitian matrix, such as HH H, are necessarily real-valued.

HH r.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.

γ → ?∞ [?u ]i a m1 [eSD ]i m2 m3 γ6+6i

(l)

where √ i denotes the i-th component of v and and mj , j = [v] 1, . . . , M ? 1 denote the real values of the midpoints of two horizontally adjacent symbols in the constellation (see Fig. 2). The values of γ that solve equations (11) and (12) are obtained through elementary deduction: γ2j?1+2i(√M ?1) = (mj ? γ2j+2i(√M ?1) = (mj ?

(l) (l)

([? u ]i )) a ([? u ]i )) a

ejθ eSD ejθ eSD

(l)

(l)

i i

(13) (14)

γ1+6i γ → +∞

Fig. 2. The slowest descent procedure on the i-th antenna for a 16-QAM constellation.

(l)

√ with i = 0, . . . , NT ? 1, j = 1, . . . , M ? 1. Once these γ’s are computed, the vectors in Sθ(l) are obtained √ (l) as a γk , θ(l) , k = 1, . . . , 2NT ( M ? 1). The fact that the i-th component of a γk , θ(l) is ambiguous for k ∈ √ √ [2(i?1)( M ?1)+1, 2i( M ?1)] is a mere implementation issue and is easily resolved. Finally, we obtain S = Sθ(1) ∪ . . . ∪ Sθ(L) . Note that √ S contains at most 2NT ( M ? 1)L + 1 symbol vectors. In general, less symbol vectors will be retained in S due to duplicates that surface when merging the sets Sθ(l) , l = 1, . . . , L. Fig. 2 illustrates the procedure to obtain the elements of Sθ(l) for a 16-QAM constellation. The line of slowest descent is projected on the complex plane related to the i-th transmit antenna. The symbol vectors that end up in Sθ(l) are obtained (l) by evaluating a γk , θ(l) for the γ’s shown as black dots in Fig. 2. For the i-th transmit antenna in particular, the retained symbols are highlighted in black. This procedure is repeated for every transmit antenna. The performance of the scheme is discussed in section IV. B. Iterative detection for coded transmission As we touched upon in section II, the demapper in a coded BICM scheme and the detector in an uncoded set-up exhibit more or less the same computational complexity. The SD technique can be applied to the marginalization (3) by restricting the summation over a as follows: for b ∈ {0, 1}, we introduce Slb = S ∩ χb . For every a ∈ Slb , we l construct a vector a ∈ χ1?b , such that bk (a ) = bk (a), for all l k = l. We then add a to Sl1?b . This is common practice in sphere decoding and ensures that Sl0 and Sl1 are symmetrical and contain exactly the same amount of symbol vectors [4]. (M ) Finally, we compute Pe (bl = b) as follows:

(M Pe ) mNT (l)

? to compute the uncontrained maximum au and to seek for the eigenvector corresponding to the minimal eigenvalue of HH H. Assessing the set of symbol vectors that lie closest to this plane, however, is somewhat more cumbersome. This set of candidate symbol vectors is de?ned by S = {a(γ, θ), γ ∈ R, θ ∈ [0, π[} with a(γ, θ) = arg min

a∈?NT

(9)

a ? x(γ, θ) .

(10)

The minimization in the righthandside of (10) is straightforward. Indeed, a(γ, θ) is obtained by simply quantizing each component of x(γ, θ) to the constellation ?. The impediment arises from the fact that this quantization has to be performed for all possible pairs of variables (γ, θ). To circumvent this problem, we propose to restrain θ to a well-chosen discrete subset of [0, π[, say θ ∈ [θ(1) , . . . , θ(L) ]. Obviously, the best discrete approximation of the continuous interval [0, π[ is π given by [0, L , 2π , . . . , (L?1)π ]. For each of the values θ can L L assume, we determine the set Sθ(l) = a(γ, θ(l) ), γ ∈ R . Obtaining this set is easy as a(γ, θ(l) ) is a piecewise constant function of γ. Hence, the minimization (10) must be carried out only for a ?nite number of values γ. Let us unfold the procedure to determine Sθ(l) for an arbitrary QAM constellation. By de?nition, the zero-forcing solution is part of the set Sθ(l) (corresponding to γ = 0). To determine the other vectors, it suf?ces to ?nd the values of γ for which a(γ, θ(l) ) exhibits a jump. A jump in the i-th component of a(γ, θ(l) ) occurs when ? au + γe or ? au + γejθ eSD

(l)

(bl = b) =

b a∈Sl

p (r |a )

k=1,k=l

(D) Pe (bk = bk (a)) ,

(15)

which contains considerably less terms than (3). C. Computational Complexity We remind that ML detection (2) required a maximization Using SD, the set to be over M NT possible symbol vectors. √ searched now contains at most 2NT ( M ? 1)L + 1 elements. This means that the complexity scales only linearly with the number of transmit antennas. Furthermore, as we will show in

jθ (l)

eSD

i

= mj

(11)

i

= mj ,

(12)

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.

10

0

10 ZF SD L=1 SD L=2 SD L=4 OPT

0

10

?1

10

?1

ZF SD L=1 SD L=2 SD L=4 OPT

10 BER

?2

10

?2

10

?3

10 UNCODED NT=4 NR=7

?3

CODED NT=4 N =5 10

?4

10

?4

R

?20

?15

?10

?5 Eb/N0

0

5

10

?10

?5

0

5

10

15

Fig. 3. BER comparison of different detectors for the uncoded scenario N0 (QPSK, σ 2 = 4E ).

b

Fig. 5. BER comparison of different detectors for the coded scenario (QPSK, N0 σ 2 = 2E ).

b

10

0

10

?1

ZF SD L=1 SD L=2 SD L=4 OPT

16-state rate 1 convolutional code with QPSK signaling. The 2 performance of the coded set-up as shown in the ?gures is after convergence of the iterative detector. Fig. 3 illustrates the performance of the zero-forcing, slowest descent and the optimal maximum likelihood detector for an uncoded set-up. We observe that the novel SD method invokes a signi?cant performance improvement compared to the straightforward ZF detection. As expected, the performance improves when we increase the number of lines of slowest descent (expressed by L). In Fig. 4 and Fig. 5, we compare the performance of the different detectors for a coded set-up. Again, the novel SD method signi?cantly outperforms the straightforward ZF detection. Comparing Fig. 4 with Fig. 5, we observe that the performance of the ZF and SD method deteriorates as the number of receive antennas decreases. This observation could have been anticipated, since, on the one hand, the diversity of the ZF receiver is related to the difference between the number of receive- and transmit antennas [7] and, on the other hand, the ZF-solution is used as a starting point for the SD method. When the difference between the number of receive- and transmit antennas is small (NT = 4, NR = 5), the SD method suffers a signi?cant performance degradation compared to the performance of the optimal detector. On the other hand, when the diversity is suf?ciently high (NT = 4, NR = 7), the gap between the SD performance and the optimal iterative detection narrows to less than 0.5dB for L ≥ 2. Finally, we compare the complexity of the SD based demapper and the optimal demapper. It turns out that in the aformentioned example (with L = 2), the SD method curbs the complexity with a factor 15 and 7.7 for the uncoded and coded scenario, respectively.

10 BER 10

?2

?3

10

?4

CODED NT=4 N =7 R

?15

?10

?5 Eb/N0

0

5

Fig. 4. BER comparison of different detectors for the coded scenario (QPSK, N0 σ 2 = 2E ).

b

section IV, L ≥ 2 renders already good results. A concurrent observation is found for the iterative detector. The optimal iterative detector (3) entails a summation over roughly M NT terms, as opposed to the SD detector (15), which only requires √ a summation over roughly 4NT ( M ? 1)L terms. IV. R ESULTS AND D ISCUSSION In this section, we present simulation results and discuss the restrictions involved with the new method. We consider ML detection for a MIMO system with uncoded QPSK transmission and iterative detection for a BICM MIMO system using a

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.

V. C ONCLUSIONS In this contribution, we proposed a novel MIMO detection algorithm for QAM constellations based on the slowest descent method. The proposed scheme bears an exceptionally low complexity, scaling only linearly with the number of transmit antennas. Nonetheless, the method exhibits a close-tooptimal performance, provided that the diversity is suf?ciently high. ACKNOWLEDGMENTS This work has been supported by the Interuniversity Attraction Poles Program P5/11- Belgian Science Policy and by the Network of Excellence in Wireless Communications (NEWCOM) funded by the European Commission. The ?rst author also gratefully acknowledges the support from the Fund for Scienti?c Research in Flanders (FWO).

R EFERENCES

[1] G.J. Foshini and M.J. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, Mar. 1998. [2] E. Viterbo and J.J. Boutros, “A universal lattice decoder for fading channels,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1639–1642, July 1999. [3] B.M. Hochwald and S. ten Brink, “Achieving near-capacity on a multiple antenna channel,” IEEE Trans. Comm., vol. 51, no. 3, pp. 389–399, Mar. 2003. [4] H.D. Zhu, B. Farhang-Boroujeny and R.R. Chen, “On Performance of Sphere Decoding and Markov Chain Monte Carlo Detection Methods,” IEEE Signal Proc. Letters, vol. 12, no. 10, pp. 669–672, Oct. 2005. [5] P. Spasojevic and C.N. Georghiades, “The Slowest Descent method and its Application to Sequence Estimation,” IEEE Trans. Comm., vol. 49, no. 9, pp. 1592–1604, Sept. 2001. [6] F. Simoens, H. Wymeersch and M. Moeneclaey, “Spatial Mapping for MIMO systems,” in Proc. IEEE Inform. Theory Workshop, San Antonio, Oct. 2004, pp. 187–192. [7] J.H. Winters, J. Salz and R.D. Gitlin, “The Impact of Antenna Diversity on the Capacity of Wireless Communication Systems,” IEEE Trans. Comm., vol. 42, no. 2/3/4, pp. 1740–1751, feb/mar/apr 1994.

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