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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 8, AUGUST 2007 1567

Tree-Search Multiple-Symbol Differential Decodingfor Unitary Space-Time Modulation

Volker Pauli, Student Member, IEEE, and Lutz Lampe, Member, IEEE

AbstractDifferential space-time modulation (DSTM) usingunitary-matrix signal constellations is an attractive solution fortransmission over multiple-input multiple-output (MIMO) fad-ing channels without requiring channel state information (CSI)at the receiver. To avoid a high error floor for DSTM in relativelyfast MIMOfading channels, multiple-symbol differential detection(MSDD) has to be applied at the receiver. MSDD jointly processesblocks of several received matrix-symbols, and power efficiencyimproves as the blocksize increases. But since the search space ofMSDD grows exponentially with the blocksize and also with thenumber of transmit antennas and the data rate, the complexity ofMSDD quickly becomes prohibitive. In this paper, we investigatethe application of tree-search algorithms to overcome the com-plexity limitation of MSDD. We devise a nested MSDD structureconsisting of an outer and a number of inner tree-search decoders,which renders MSDD feasible for wide ranges of system parame-ters. Decoder designs tailored for diagonal and orthogonal DSTMcodes are given, and a more power-efficient variant of MSDD, so-called subset MSDD, is proposed. Furthermore, we derive a tightsymbol-error rate approximation for MSDD, which lends itself toefficient numerical evaluation. Numerical and simulation resultsfor different DSTM constellations and fading channel scenariosshow that the new tree-search MSDD achieves a significantly bet-ter performance-complexity tradeoff than benchmark decoders.

Index TermsDifferential space-time modulation (DSTM),multiple-input multiple-output(MIMO) fading channels, multiple-symbol differential detection (MSDD), sphere decoding, tree-

search decoding.

I. INTRODUCTION

DIFFERENTIAL space-timemodulation(DSTM) with uni-

tary matrix-signal constellations is an attractive solu-

tion for power- and/or bandwidth-efficient transmission over

multiple-input multiple-output (MIMO) fading channels when

channel state information (CSI) is not available at the receiver.

Original work on DSTM, in, e.g., [1][4] considered DSTM

over block-wise constant fading (block-fading) channels and

conventional differential detection (CDD) based on two consec-

utively received symbols. As shown in [2, Section IV-C], CDD

incurs a loss of 3 dB in power efficiency compared to space-time

Paper approved by R. W. Heath, Jr., the Editor for Modulation/Detectionof the IEEE Communications Society. Manuscript received October 12, 2005;revised May 8, 2006 and December 4, 2006. This work was supported in part bythe Deutsche Forschungsgemeinschaft (DFG) under Grant HU 634/6 and in partby theGerman Academic Exchange Service (DAAD). This paper waspresentedin part at the IEEE Global Conference on Telecommunications (GLOBECOM),2005.

V. Pauli is with the Lehrstuhl fur Informationsubertragung, UniversitatErlangenNurnberg, Erlangen 91058, Germany (e-mail: [email protected]).

L. Lampe is with the Department of Electrical and Computer Engineering,University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail:[email protected]).

Digital Object Identifier 10.1109/TSMCC.2007.902500

(ST) transmission with perfect CSI in block-fading channels. In

case of relatively fast fading, CDD further suffers from a high

error floor due to channel variations from one symbol to the

other, and DSTM performs even worse than single-antenna dif-

ferential transmission [5].

To overcome this limitation, multiple-symbol differential de-

tection (MSDD) for DSTM was proposed in [5], [6]. Analo-

gous to MSDD for single-antenna differential phase-shift key-

ing (DPSK) (e.g. [7]), N 1data symbols are jointly detectedbased onNconsecutively received symbols. The performanceof MSDD improves for increasing observation window size

N, and approaches that of coherent detection with perfectCSI. As its complexity grows exponentially with N, a num-ber of suboptimal MSDD algorithms were designed for partic-

ular DSTM constellations and/or values ofN, cf. e.g. [8][10].Furthermore, suboptimal decision-feedback differential detec-

tion (DFDD) and trellis-based noncoherent sequence detection

were devised for DSTM with diagonal constellations [1], [2]

and with orthogonal designs [3], [11] in [5], [12] and [13],

respectively.

Recently, the authors presented an optimal maximum-

likelihood (ML) MSDD algorithm for single-antenna DPSK

[14]. This algorithm is a depth-first tree search or sphere de-

coding (SD) algorithm, and was referred to as multiple-symboldifferential sphere decoding (MSDSD) in [14].1 The complex-

ity of MSDSD was shown to be comparable to that of DFDD

in most cases. In [15], a similar, Fano decoding algorithm for

MSDD of single-antenna DPSK was investigated.

In this paper, we propose the application of tree-search de-

coding for DSTM transmission over MIMO fading channels and

MSDD. In this context, we make the following contributions. Three tree-search decoders are devised for MSDD. These

include MSDSD and Fano decoding similar to the single-

antenna schemes from [14], [15], as well as a new MSDSD

scheme using a Fano-type metric for quicker termination

of the search. The size of the DSTM constellation grows exponentially

with the number of transmit antennas NT and the datarate R. To avoid a corresponding increase in decodingcomplexity, we propose to nest fast CDD algorithms into

the outer tree-search MSDD. Whereas sphere decoding is

typically proposed for conventional CDD, e.g., [4], [12],

we show that stack decoding is better suited when CDD is

integrated into MSDD.

1In the context of this paper, the terms decoding and detection referto the same procedure, and are used interchangeably. When referring to thesignal set of DSTM, the terms constellation and (space-time) code are used

interchangeably.

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1568 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 8, AUGUST 2007

Considering a continuous MIMO fading channel

model, we provide a tight symbol-error rate (SER)

approximation for MSDD, which lends itself to efficient

numerical evaluation. Thereby, we consider the individual

SERs for theN 1consecutive data symbols jointly pro-cessedin oneMSDD block.Numerical results show that the

SERs for the individual symbols in one block vary signifi-cantly. This motivates the design of a more power-efficient

variant of MSDD, which we refer to as subset MSDD. We present a thorough comparison of the proposed tree-

search decoding schemes in terms of SER performance and

computational complexity considering different DSTM

constellations. Numerical and simulation results confirm

that considerable improvements in power-efficiency over

DFDD and CDD are achieved while decoding complexity

remains moderate and, in fact, comparable to that of DFDD.

The paper is organized as follows. Section II introduces the

DSTM system model. The tree-search MSDD algorithms are

developed and optimized for DSTM in Section III. Expressions

for the SER of MSDD are derived and subset MSDD is devisedin Section IV. Numerical and simulated SER-performance and

complexity results are presented and discussed in Section V,

and conclusions are given in Section VI.

In this paper, bold upper case X and lower case x de-

note matrices and vectors, respectively. ()H, ()T, tr{}, ,det{}, , andE{} denote Hermitian transposition, transpo-sition, trace, Frobenius norm, determinant of a matrix, Kro-

necker product, and expectation, respectively. IL is theL Lidentity matrix and diag{X1, . . . ,XL} is an LM LNblock-diagonal matrix with the M NmatricesXlon its main diag-onal. A symmetricL LToeplitz matrix is defined by toeplitz{x1, . . . , xL}.

II. SYSTEMMODEL

We consider a transmission scheme using NT transmit andNRreceive antennas. At the transmitterNTR bits are mapped to(NT NT)-dimensional unitary matricesV[k],whicharetakenfromasetV

={V(l)|l {1, . . . , L}, L= 2NTR}. R isthe data

rate in bit per channel use. In order to facilitate noncoherent

detection, the data symbols V[k]are differentially encoded intotransmit symbols

S[k] = V[k]S[k 1], S[0] = INT . (1)At time = kNT+ i, the transmitter radiates from antenna jthe elementsi,j[k] in theith row and j th column ofS[k], 1i, jNT. The transmit power is independent ofNTatall times:NT

j=1 |si,j[k]|2 =EbR/T, 1iNT, where Eb is the en-ergy per information bit andTis the modulation interval.

We assume a frequency-nonselective MIMO Rayleigh fad-

ing channel. The equivalent complex baseband (ECB) received

signalri,j[k]at time= kNT+ iand antennaj is given by

ri,j [k] =NT=1

si,[k]h,j [] + nj [],

1iNT, 1jNR. (2)

h,j[]andnj []denote, respectively, the complex fading gainbetween transmit antennaand receive antennaj and the zero-mean complex additive spatially and temporally white Gaussian

noise (AWGN) with variance 2n=N0/T effective at the j threceive antenna at time , whileN0is the two-sided noise-powerspectral density in the ECB. The fading channels are assumed

to be spatially uncorrelated with identical temporal correlationaccording to Clarkes fading model [16]. In particular, h,j []iszero-mean complex Gaussian distributed with autocorrelation

hh[]=E{h,j [ + ]h,j[]}= J0(2BhT), where J0()

and BhTdenotethe zeroth-order Besselfunction of the first kindand the maximum normalized fading bandwidth, respectively.

While the fading channel model (2) will be used for analytical

purposes in Section IV and performance evaluation in Section V,

the derivation of low-complexity DSTM detectors requires the

assumption of a quasi-static fading channel (QSFC), i.e., the

channel is assumed constant during NTconsecutive modulationintervals (e.g., [6], [10]). Defining the QSFC matrix H[k]withhi,j[kNT] in row i and columnj, 1

i

NT, 1

j

NR, theQSFC is described by

R[k] = S[k]H[k] + N[k]. (3)

The N NT NR matrix R[k] ofN received matrix symbolscan be expressed as

R[k] =SD[k]H[k] + N[k] (4)

with the N NT N NT block-diagonal matrix SD[k]=diag{S[k N+ 1], . . . ,S[k]} andN NT NR matrices H[k]=HT[k N+ 1], . . . ,H

T[k]T and N[k]= NT[k N+1], . . . ,NT[k]T.We would like to point out that the QSFC model considered in

(3) and (4) is different from the often used block fading model

[1][4], [17], which stipulates that the channel remains constant

during the transmission ofNconsecutive DSTM symbols.

III. TREE-SEARCHMULTIPLE-SYMBOLDIFFERENTIAL

DECODING

In this section, we propose and design tree-search algorithms

for efficient MSDD. To this end, we provide a suitable represen-

tation of the ML MSDD decision rule in Section III-A, based

on which the tree-search decoding algorithms are developed inSection III-B. We then present inner decoders, which are nested

into the outer tree-search decoder and optimized for different

DSTM constellations in Section III-C.

A. ML MSDD

ML MSDD processes blocks R[k] ofN consecutively re-ceived matrix symbols to find estimates for the N 1 data sym-bols V[k]

=VT[k N+ 2], . . . ,VT[k]T corresponding to

the Ntransmit symbolsS[k]=

ST[k N+ 1], . . . ,ST[k]

T

.

In analogy to the single-antenna case (e.g., [7]) consecutively

processed blocks R[k] overlap by at least one matrix symbol,

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PAULI AND LAMPE: TREE-SEARCH MULTIPLE-SYMBOL DIFFERENTIAL DECODING FOR UNITARY SPACE-TIME MODULATION 1569

i.e., the observation window of length Nmoves forward by atmost N 1symbols at a time. For the sake of readability, in thefollowing we omit the reference[k]to time and address subma-trices of the earlier block-matrices via subscripts with Xibeing

the th submatrix ofXand 1i(NorN 1).Using the QSFC model (4) and the fact that SD is a unitary

matrix, the ML MSDD decision rule can be derived as [13,Section V]

V= argmin

VVN1{tr{RHSD

C

1 INTSH

DR}}, (5)

where C=hh+

2nIN

and hh

= toeplitz

{hh[0], hh[NT], . . . , hh[(N 1)NT]}. Using the Choleskyfactorization C1 = UHU with upper triangular U andtr{XXH}=X2, we obtain

V= argmin

VVN1

N1n=1

VnRn,n+ Xn2

(6)

with

Xn= Sn+1

Nj=n+1

SH

j Rn,j (7)

where Rn,j=un,jRj , 1nN 1, njN, and un,j

is the element ofU in rown and column j . It is worth point-

ing out that un,j =p(Nn)jn /

(Nn)e , where p

(n)j and (

(n)e )2

denote thejth coefficient of thenth order linear backward min-imum mean-squared error (MMSE) predictor for the discrete

time random processH[k] + N[k]and the corresponding error

variance, respectively, andp(n)0 =1, n, e.g., [18].

We note that the brute-force search would evaluate (6) for

all 2(N1)NTR V (SN= INT can be chosen without loss ofoptimality), i.e., decoding complexity would be exponential in

N, NT, andR.

B. Efficient Tree-Search Decoding AlgorithmsOuter Decoder

It can be observed that the ML metric in (6) is a sum ofN 1nonnegative scalar terms

2n=VnRn,n+ Xn2, 1nN 1 (8)

which through Xn and (1) depend on symbols Vj , n + 1j

N

1. Thus, ML MSDD constitutes a tree-search prob-

lem, and V from (6) corresponds to the best path in the tree.For an efficient (fast) tree search for MSDD, we concentrate

on the application of depth-first search (DFS) and best-first

search (BFS) algorithms, which achieve the best performance-

complexity tradeoff (see [19] for coherent MIMO detection).

1) Sphere Decoding AlgorithmMSDSD: Sphere decoding

algorithms are DFS-type algorithms. We refer to the application

of SD to accomplish MSDD as MSDSD (see [14] for DPSK).

To formulate the SD algorithm, let us define

d2n=

N1

i=nViRi,i+ Xi2 =d2n+1+ 2n, 1nN 1

(9)

whered2N = 0and d21 is the ML metric in (6). Starting at n =

N 1, the SD algorithm selects candidates for symbols Vnbased on tentative decisions Vj = Vj , n + 1jN 1,and continues to decrement n as long as the current metricdn does not exceed a given maximum metric (radius) , suchthat

dn. (10)If the decoder reaches n= 1, the metric of the currently best

candidate V is used to further reduce the size of the searchspace by updating = d1. Ifdnexceedsfor any value ofn, nis incremented and a new candidate forVn is examined. If the

decoder returns ton= N, i.e., ifdN1> , it means that thereare no further candidates inside the current sphere and that the

ML solution Vhas been found. For the ordering of candidatesfor any Vn we employ the SchnorrEuchner (SE) enumeration

strategy, i.e. we check candidates in order of increasing n, asthis allows for an initialization with and a (usually) fastconvergence to the ML solution [14], [20], [21].

2) Sphere Decoding With Fano-Type MetricMSDSD-FM:

The sphere decoding structure described earlier is very efficient

in finding the ML solution in moderate to high signal-to-noise

ratios (SNRs). However, toward lower SNR the normalized pre-

dictor coefficientspj(i)/(

i)e become very small, and d2nbecomes

less dependent on tentative decisions Vj , n + 1jN 1.In other words, d2n resembles a sum ofN n CDD metrics,and thus, scales almost linearly with N n. It is, therefore,advisable to adjust the metric used in SD to take into account

the lengthN nof the considered tree path, i.e., to introducea bias b. This idea was first proposed by Fano for sequential

decoding of convolutional codes [22], and hence, we refer to itas MSDSD with Fano-type metric (MSDSD-FM). MSDSD-FM

uses the same search strategy as MSDSD but with Fano-type

path metric

d2n=d2n+ (n 1)b (11)

withd2n from (9). We propose to use the expected value of2n,

given Vj = Vj , njN 1, as bias

b=E{2n

Vj = Vj , njN 1}= NTNR, (12)which leads to a simple implementation. The result in (12) is due

to the fact that

Xn is the MMSE estimate forVnRn,n, and

that the filter coefficients are normalized by the corresponding

error variance [see (6)]. It should be noted that different from

MSDSD, MSDSD-FM is suboptimal in that the ML MSDD

solution is not necessarily found. Furthermore, it is worth men-

tioning that an SD with Fano-type metric was presented for

coherent detection of space-time codes in [23].

3) Initial Search Radius: When using the SE enumeration,

no initial radius is required, i.e., init can be assumed.Accordingly, our first strategy is to initialize the MSDSD al-

gorithm with a very large value ofinit. The disadvantage ofthis strategy is a potentially slow convergence or equivalently

high complexity, in cases where the decoder makes false ten-

tative decisions for symbols Vn with n close to N, before

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reaching n= 1for the first time. Therefore, our second strategyis to use a relatively small initfor initialization, and in case nosolution Vis found for this radius, to increase it gradually bysteps ofinit[21]. Since for MSDSD(-FM) the expected metric

of the correct solution equals E{d21| V = V}= (N 1)NTNR(see Section III-B.2), we choose the squared start radius 2init

proportional to that value expressed as

2init= (N 1)NTNRc , c >0 . (13)The value of the constant c influences the overall search time.Simulation results presented in Section V confirm that the use

of a finite initial radius is beneficial especially for DSTM with

large constellations.

4) Fano Decoding AlgorithmMSDD-Fano: The Fano de-

coding algorithm is a BFS type algorithm. It has recently been

applied to MSDD of single-antenna DPSK [15] and to coherent

MIMO detection [19]. In this paper, we propose its application

to MSDD for DSTM using the metric of (11), and refer to the

resulting algorithm as MSDD-Fano. We do not repeat the basicdescription of the basic Fano algorithm, whose details can be

found in [22].

C. Application To Different Signal ConstellationsInner

Decoders

MSDSD, MSDSD-FM, and MSDD-Fano break the exponen-

tial dependence of decoder complexity onN. Also, in order tobreak exponential dependence of decoder complexity on NTandR it is necessary to give some thought on the search for in-dividual candidate symbols Vn. In this regard, it is important tonote that minimizingn is equivalent to the problem of finding

the ML solution for CDD. This observation allows us to imple-ment MSDD for DSTM using a nested structure of an outer and

N 1 identical inner decoders. The outer decoder, which solves(6), initializes an inner decoder at stage n, 1nN 1, withmatrices Rn,n and Xn to find a new candidate Vn that mini-mizesn. It further provides the inner decoder with: 1) a list ofcandidates that have been examined previously given the same

set of tentative decisions Vj , n + 1jN 1, and thus,are to be excluded from the search; and 2) a start radius r toforce n< r such that (10) is still fulfilled. In our investigations,we considered diagonal (cyclic) and dicyclic group codes [24]

and orthogonal [11] and Cayley [4] nongroup codes. Due to

space limitations, however, we present details only for diagonal

and orthogonal constellations.1) Diagonal Constellations: Diagonal constellations are de-

fined by the set [1], [2]

VD=

diag{e 2L c1 , . . . , e 2L cNT}l l {0, . . . , L 1} .

(14)

A straightforward approach, which we will refer to as full search

(FS) inner decoding, is to compute nfor all elements ofVDandto process them in order of increasingn. In this case, MSDDbenefits from the efficiency of the outer tree-search algorithm,

but its complexity is still exponential inNTand R.In order to overcome this exponential dependence of com-

plexity on NT and R we turn to a more sophisticated strat-

egy using a lattice-based CDD algorithm proposed in [25] (see

also [12]). We will refer to this approach as inner lattice decod-

ing (LD). Using the cosine-approximation for small arguments,

the problem of finding the minimizer for ncan be turned into a(degenerated)NT-dimensional lattice-decoding problem of theform

x = argminxNT {Bx t}, (15)

where the NT NT matrix B has non-zero entries bi,j onlyin the first columnj = 1and on the main diagonal i = j , andVn= diag{e 2L c1 , . . . , e 2L cNT}mod(x1,L) (see [12], [25] fordetails). Due to the cosine-approximation,Vnis not necessarilythe optimal solution, but as will be seen in Section V only small

performance degradations result. Since x in (15) and as suchVn can be obtained from a tree search, we propose two tree-

search algorithms, which are particularly suited for efficient

inner decoding.

Sphere Decoding Algorithm: Due to the sparse lower-

triangular structure ofBin (15), a particularly efficient sphere

decoder can be applied. The decoder may use the SE strategy

for finding candidates forx1, and increase i, 2iNTaslong as2i

=|b1,1x1 t1|2 +

ij=2 |bj,1x1+ bj,j xj tj |2 r. Thus, the decoder actuallysearches only one dimension of the alleged NT-dimensionalsearch space. We found that this implementation of the in-

ner sphere decoder is more efficient than performing a QR-

decomposition and subsequent SD as was advocated in [12] for

CDD. The reason for this is that in MSDD previously examinedcandidates are excluded at the very beginning of the search as

they correspond tox1.Stack Decoding Algorithm: One drawback of the SD algo-

rithm is that the search restarts from i = 1and x1=t1/b1,1every time the inner decoder is called to provide the next best

candidate Vn, given the same tentative decisions Vj , n + 1jN 1. The stack algorithm (e.g., [22]) seems to be an in-teresting alternative for the inner decoder, since it maintains a

sorted list of examined paths, and decoding can be continuedif

the inner decoder is called again.

As the regular stack algorithm is not directly applicable for

solving (15), becausexi, 1

i

NT, are not from a finite set,we propose a modified stack decoder. Instead of replacing the

path x(i) = [x1, . . . , xi] currently at the top of the stack withall its extensions x(i+1) = [x(i), xi+1], we only consider itsbest extension x(i+1), and due to the sparse structure of the

matrix in (15), the next candidate forxi is provided only wheni= 1 according to the SE strategy. The search can be terminatedwithout loss of optimality, if the metric of the path at the top

of the stack exceeds the start radius r . As for the regular stackalgorithm, the first path of length NTto reach the top of the stackis the optimal path corresponding to the (next-)best candidate

forVn. It remains to store the stack and to continue with this

stack when the next candidate is required. While, theoretically,

limitation of the stack size may lead to decoding errors, we

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found that limiting the maximum stack size to 2L does notincur a noticeable performance degradation.

2) Orthogonal Designs (ODs): A second important class of

nongroup DSTM codes are ODs. In particular, we consider

DSTM based on Alamoutis code [11], where data symbols Vare taken from the set

VO=

12

a bb a

a, b L PSK

. (16)

For ODs, it is relatively straightforward to show that 2n can beexpressed as

2n= n+ Re{ann} + Re{bnn}, 1nN 1(17)

with n=

2NR

j=1rn,n,1,jxn,1,j + r

n,n,2,jxn,2,j , n

=

2NR

j=1rn,n,1,jxn,2,jrn,n,2,jxn,1,j , andn

=Rn,n2 +

Xn2, wherern,n,i,jand xn,i,jdenote the elements in theithrow andjth column ofRn,nand Xn, respectively. Apparently,

an and bn that minimize 2n in (17) are independent of eachother. Hence, the inner decoding simplifies to two SE enumera-

tions applied to

L-PSK symbolsanandbn, respectively [14].Consequently, outer and inner decoder for MSDD can be

integrated into one decoder searching a(2N 2)-dimensionaltree of

L-PSK symbols. If the MSDSD algorithm from

Section III-B.1 is applied, this decoder is a direct extension of

single-antenna MSDSD from [14] to DSTM in the same way

as for single-antenna MSDSD.

It is worth mentioning that recently, a similar description of

MSDD for DSTM with ODs has been presented in [26]. There,

for the special case ofL = 16(ODs with 4PSK symbols) and

block fading, thedetection problem is formulatedas a 4(N 1)-dimensional search in binary variables. The scheme proposed in[26] can, however, not be extended to arbitrary fading channels

as considered in this paper.

IV. SER ANALYSIS ANDSUBSETMSDD

In this section, we first derive an approximation for the SER

of ML MSDD (Section IV-A). In particular, we consider the

individual SERs for the N 1data symbols in each block V.Evaluation of these individual SERs suggests that decoding of

only a subset of all N 1 symbols can be greatly beneficial.The corresponding subset MSDD algorithm is presented in Sec-

tion IV-B.

A. SER Analysis

It is intuitive that due to different correlations between the

received samples in R, symbol decisions on the N 1 datasymbols in V are not equally reliable. Especially in relativelyfast fading environments it can be expected that symbols located

in the center of the observation window can be detected more

reliably than those at the edges. In order to substantiate this in-

tuitive reasoning, we consider the individual symbol-error rates

SERn forVn, 1nN 1.The pairwise error probability PEP(S S) for detect-

ing S while S= S was transmitted can be expressed as

[27, Section III]

PEP(S S)

=

RHpoles

Res

1

s

NNTi=1

1 + si

rr|SF

NR (18)

where the summation in (18) is taken over all residues corre-sponding to poles located in the right-hand (RH) side of the

complexs-plane, i(X)denotes theith eigenvalue ofX, and

rr|S=SD

C INT

SH

D (19)

F = SD

C

1 INT

SH

DSDC

1 INTSH

D

(20)

C= toeplitz{hh[0], hh[1], . . . , hh[N NT 1]}

+ 2nINNT (21)

SD=diag{S1,:,S2,:, . . . ,SNNT,:} (22)

where Si,: denotes the ith row ofS. Note that C is the au-tocorrelation matrix of the regular fading-plus-noise process

[see (2)]. Depending on the multiplicities of the eigenvalues

j(rr|SF), analytical or numercial evaluation of (18) may bepreferrable [27][30].

With this, theSERn forVncan be upper bounded using theunion bound, averaged over allLN transmit sequences:

SERn 1LN

S

S, Vn =Vn

PEP(S S). (23)

Evaluating (23) is computationally tractable only for small con-

stellations and observation window sizes N. In order to obtaina simpler approximation forSERnwe examine the PEP of (18)

more closely. It can be shown that it depends on Sand Sonlythrough the matrix

P(C INT)PH

SHD SD C1 INT SHDSD C1 INT (24)where we introduced P

= S

H

DSD.

1) Group Constellations: ForLbeing a power of two, diag-

onal (or cyclic) and dicyclic codes [24] fully represent full-rankunitary group constellations. In both cases, P is a sparse ma-

trix with a single one in each row. Whereas Pis constant and

independent ofSDfor diagonal constellations, there are 2N dif-

ferent matricesPin case of dicyclic constellations. This means

that the set ofLN matrices Scan be partitioned intoKequiva-

lence classes Sk

Pwith respect to P, 1kK, whereK= 1and K= 2N for diagonal and dicyclic constellations, respec-

tively. Furthermore, since SH

DSD = diag{SH1S1, . . . ,SHNSN}

with SHnSn V, 1nN, for group constellations, the in-ner sum of (23) is independent of which class representative

S

S

k

Pis chosen. We can, thus, limit the outer summation to

onlyKLN terms (K= 1 orK= 2N).

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To reduce the number of terms of the inner sum in (23), we

propose to take only the dominating instead of all LN1 1relevant error events into account. Following a similar reason-

ing as in [28] for single-antenna DPSK and utilizing the results

of [17] for the design of unitary space-time constellations, these

dominant error events are found to correspond to matrix pairs

{S,S} with the highest correlation =SH

S. A brief exam-ination ofreveals that error events with only one nonidenticaltransmit matrix Sn= Snsuch that

n= Re {tr{SHnSn}}, 1nN (25)

is maximized, which also maximizes. Collecting all matricesS= [ST1 . . . S

T

n1ST

n ST

n+1 . . . ST

N]T with Sn= Sn maxi-

mizingn in sets Sn, 1nN, we can approximateSERn

by

SERn 1K

SSk

Pk=1. . .K

S Sn

PEP(S S) +

S Sn+1

PEP(S S)

(26)

for1nN 1.2) Nongroup Constellations: For nongroup constellations,

e.g., orthogonal [11] and Cayley [4] codes, a simplification

of (23) based on equivalence classes with respect to P is not

possible. Furthermore, it is not guaranteed that the quadruple

Qn={Sn,Sn,Sn+1,Sn+1}with Sn= Sn,Sn+1= Sn+1,

and Sn maximizing (25) is admissible, i.e., that there is aVn Vsuch that Sn+1= VnSn. Hence, the correlation n(25) is not necessarily an appropriate indicator for the domi-

nant error events for every realization ofS. However, we foundfrom numerical evaluations that the SER of nongroup codes

is approximated quite accurately using (26) with a few can-

didates S for which Qn is admissible and Sn according to(25), 1nN 1. For the performance evaluation in Sec-tion V, we usethis approximation with theL admissible matricesS= [INT , (V

(l))T, INT , (V(l))T, . . .]T, 1lL.

3) Evaluation: At this point, it is illustrative to consider

SERn for the example of diagonal DSTM with parametersNT= 3, R= 1, BhT = 0.03, and NR= 1. Fig. 1 shows nu-merical results from (26) and simulation results for the SNR

in terms of10log10(Eb/N0)required to achieve, respectively,SERn = 10

5 (solid lines) and the average error rate SER =(1/N 1)N1n=1 SERn= 105 (dashed lines) as function ofthe positionn, 1nN 1, for different window sizes N.MSDD is implemented using MSDSD with FS inner decoding

(MSDSD-FS). Also included is the SER for coherent detection

assuming perfect CSI.

First, we observe a good agreement between the SER approxi-

mation from (26) and the simulated SER. Second, it can be seen

that the individual error rates SERn are almost identical forsymbols Vn, 2

n

N

2, but significantly deteriorate for

symbols at the edges of the observation window. More specifi-

Fig. 1. Required10log10(Eb/N0) to achieve SER = 105 for positionn in observation window of MSDSD-FS. Parameters: Diagonal constellation,NT = 3,NR = 1, R= 1, andB

hT = 0.03.

cally, there are differences of 58 dB in power efficiency when

comparing edge and nonedge positions.

B. Subset MSDD

The observations from Fig. 1 strongly suggest a variant of

MSDD, which we refer to as subset MSDD. For subset MSDD,

only the decoding decisions for the N 1N 1 symbolsVn located in the middle of the observation window, i.e.,

(N N)/2< n

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TABLE ISUMMARY OFALGORITHMS FORDIFFERENTIALDETECTION FORUNITARY DSTM

Fig. 2. Comparison of various decoders withN= 10 (CDD with N= 2).Required10 log10(Eb/N0) to achieveSER = 105 versus BhT. Diagonalconstellation withNT= 3,NR= 1, R= 1. Lines: numerical results, Mark-

ers: simulation results.

decoder is applicable, which is denoted by an (additional) suffix

-FS (all constellations), -LD (only for diagonal constel-

lations), respectively. As benchmark algorithms, we also con-

sider: 1) CDD(N = 2); 2) DFDD [5], [12]; and 3) the branch-intersect-detect MSD decoder (MSDD-BID) of [10], which is

a suboptimal MSDD algorithm for diagonal constellations. Un-

less specified otherwise, the channel model according to (2) is

applied for simulation.

A. SER Results

We present SER results for three illustrative examples. In

case of DSTM with diagonal constellations we used parameters

[c1, . . . , cNT ]from [2, Table I].1) Diagonal Constellation With R= 1, NT = 3, NR = 1,

and MSDD With N = 10: Fig. 2 compares various decoders(with FS inner decoding) in terms of the SNRrequired to achieve

SER = 105 as function of the normalized fading bandwidthBhT. Both numerical results according to Section IV-A (solidlines) and simulation results (markers) are plotted. As reference

curves, the SNR for MSDSD-FS under block-fading (BhT = 0)and for coherent detection with perfect CSI are also shown.

First, we note the good match of the SER approximation from

Section IV-A and the simulated SER. The slight fluctuations

Fig. 3. Comparison of FS and LD-based inner decoding. SER versus10log10(Eb/N0) for MSDSD with N= 10, MSDSD-S with N= 10 andN= 8, DFDD, and CDD. Diagonal constellation with NT= 3,NR=1, R= 1, andBhT = 0.03. Solid lines: FS; Dashed lines: LD.

of the numerically obtained curves are due to the particular

fading autocorrelation according to the Bessel function. Second,

we observe that CDD-FS suffers from a relatively high error

floor already in moderately fast fading withBhT0.006. Theperformance of MSDD-BID approaches that of ML MSDD for

BhT0.01, but rapidly deteriorates asBhTgrows. MSDSD-FS is considerably more robust to fast fading than MSDD-BID,

and also outperforms DFDD by about 26 dB depending on

the fading bandwidth. Suboptimal MSDD-Fano-FS performs

within 0.30.7 dB of optimal MSDSD-FS. Finally, it can be

seen that subset MSDD (MSDSD-S-FS) significantly improves

performance in the fast fading regime. Almost all the gain in

power efficiency is already accomplished with N = 8, whichcorresponds to a moderate complexity increase by a factor of

1.29compared to MSDD withN= 10.In Fig. 3, we evaluate the performance with inner LD for

an exemplary fading bandwidth of BhT = 0.03. MSDSD,MSDSD-S, and DFDD withN= 10and CDD are considered,and LD inner decoding (dashed lines) is compared with FS

inner decoding (solid lines). It can be seen that the cosine-

approximation involved in LD causes only small performance

degradations in the order of 0.20.3 dB regardless of the partic-

ular detector. We also observed (not shown in Fig. 3) that even

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Fig. 4. Comparison of various decoders with N= 6 (CDD with N= 2)and LD-based inner decoding. SER versus 10log10(Eb/N0) for diagonalconstellation withNT = 3,NR= 1, R= 2, andBhT = 0.03.

smaller performance losses occur at higher ratesR >1, sincethe cosine-approximation becomes more accurate.

2) Diagonal Constellation With R= 2, NT = 3, NR=1, BhT = 0.03, and MSDD With N = 6: The power efficiencyof MSDSD-FM is compared to optimal MSDSD and MSDD-

Fano in Fig. 4. As FS inner decoding is computationally com-

plex for this scenario withL= 64, we consider LD in all cases.The respective curves for CDD-LD, DFDD-LD, and MSDD-

BID are also included for comparison. MSDSD-FM-LD clearly

outperforms CDD-LD and MSDD-BID, which suffer from a

high error floor in this fast fading scenario. We further observethat the use of the suboptimal Fano-type metric results in rela-

tively small performance losses of 0.51.0 dB. As expected,

the performance of MSDSD-FM-LD is very similar to that

of MSDD-Fano-LD. Finally, MSDSD-FM-LD and especially

MSDSD-FM-S-LD achieve significant gains over DFDD-LD,

which has been studied in detail in [12].

3) Orthogonal Design With R= 1, NT = 2, NR= 1, andMSDD WithN= 10: Fig. 5 shows the performance of DSTMwith ODs in terms of the SNR required to achieve SER = 105.MSDSD, using the efficient decoder described in Section III-C,

is compared with DFDD and CDD (N= 2), respectively.Numerical results (see Section IV-A, lines) and simulation

results (markers) are plotted. Since for DSTM with ODs the

NT antennas transmit constantly, the QSFC model is only anapproximation. In order to illustrate the degradation due to the

deviation of the QSFC model from the underlying channel (2),

numerical and simulation results assuming the QSFC model

are also included in Fig. 5 (dashed lines).

First,we observe that SERs fromthe approximation in Section

IV-A and simulated SERs closely match also for this nongroup

constellation. Furthermore, it can be seen that the performance

is limited by channel variations during the transmission of one

ST symbol, which are not accounted for in the QSFC model, and

thus, cause an error floor regardless of the detection algorithm.

MSDSD and DFDD can cope with much faster fading than

Fig. 5. Comparison of various decoders withN= 10 (CDD with N= 2).Required 10log10(Eb/N0) to achieve SER = 105 versus BhT. Orthogonaldesign with NT = 2,NR= 1, and R = 1. Fading channel model according

to (2) (solid lines) and QSFC (dashed lines). Lines: numerical results, Markers:simulation results.

CDD. The performance gains due to MSDSD over DFDD are

between about 1 dB and 4 dB depending onBhT.

B. Computational Complexity

We now compare the computational complexity of the pro-

posed algorithms. In particular, we consider the effects of a finite

initial search radiusinit(see Section III-B) andthe useof sphere

and stack decoding algorithms for inner LD (see Section III-C);and compare the complexity of the proposed MSDD implemen-

tations with those of CDD, DFDD, and MSDD-BID. As a mea-

sure of complexity, we choosethe average numberof real-valued

flops per decoded symbol. Exemplarily, we present results for di-

agonal DSTM with R= 2, NT= 3, NR = 1, BhT = 0.03,andMSDD withN= 6(see Fig. 4 for performance results).

1) Initial Radius: To study the effect of the initial radius

we consider MSDSD-FS (the results are very similar for other

MSDSD algorithms). Fig. 6 depicts the complexity for differ-

ent SNRs as a function of the constant c introduced in (13).While complexity increases if the initial radius is chosen too

small, substantial complexity savings of up to 60% compared to

init can be obtained by choosing an optimal finite startradius. It is interesting to note that for the optimal parameter

c1.7, the transmitted block Vlies inside the sphere with ra-dius init with a probability of about 95% independent of theSNR. It can also be observed that, for this relatively fast fading

example, the complexity of MSDSD with infinite initial radius

may increase for larger SNR. This is due to the fact that for high

SNR the radius after the first tentative decision Vmay be muchlarger than the metric for the ML decision. A finite start radius,

on the other hand, leads to the more intuitive result that the de-

coding complexity decreases monotonically with SNR. All of

the following results assume the optimal finite start radius with

c= 1.7.

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Fig. 6. Complexity of MSDSD-FS withN= 6 versus parameterc [initialsquared radius2

init= (N 1)NTNRc]. Diagonal constellation with NT =

3,NR = 1, R= 2, andBhT = 0.03.

Fig. 7. Complexity versus10 log10(Eb/N0) for different MSDSD algo-rithms withN= 6. Diagonal constellation withNT = 3,NR= 1, R= 2,and BhT = 0.03. Solid lines: without Fano-type metric, Dashed lines: withFano-type metric. LD with sphere and stack decoder, respectively.

2) MSDSD versus MSDSD-FM and Inner Decoding With FS

versus LD: Fig. 7 compares the different possibilities to reduce

the complexity of MSDSD, i.e., MSDSD versus MSDSD-FM

and MSDSD(-FM) with inner FS versus inner LD are consid-

ered. LD is implemented using sphere andstack decoder, respec-

tively. It can be seen that the complexity of MSDSD decreases

rapidly with increasing SNR, since the search quickly terminates

for small enough noise. Inner LD leads to a significant complex-

ity reduction, and the stack decoder is clearly advantageous over

the sphere decoder. Furthermore, applying the Fano-type met-

ric speeds up the search especially in the lower SNR range, as

was anticipated in Section III-B. Combining Fano-type metric

with inner LD using the stack algorithm, i.e., MSDSD-FM-LD,

yields a tremendous complexity reduction by a factor of about

10 to 50 compared to MSDSD-FS in the range of relevant SNR.

Fig. 8. Complexity versus10 log10(Eb/N0)for MSDSD-FM-LD, MSDD-Fano-LD, and benchmark algorithms with N= 6 (CDD-LD with N= 2).

Diagonal constellation with NT = 3,NR= 1, R= 2, and BhT = 0.03.MSDSD-LD-FM with inner stack algorithm.

This gain comes at the expense of a less than 1 dB loss in power

efficiency (see Fig. 4).

3) Comparison With Benchmark Decoders: Fig. 8 compares

the average complexity of MSDSD-FM-LD considered earlier

with those of MSDD-Fano-LD and the benchmark detectors

CDD-LD, DFDD-LD, and MSDD-BID, respectively. While

achieving a similar power efficiency (Fig. 4), MSDSD-FM-LD

requires a lower complexity than MSDD-Fano-LD. The com-

plexity of MSDSD-FM-LD is also well in the range of those

of DFDD-LD and CDD-LD, which are less power efficient and

suffer from a high error floor, respectively. MSDD-BID, whichalso shows a high error floor for this transmission scenario, is

considerably more complex than MSDSD-FM-LD.

We, thus, conclude that MSDSD-FM-LD and its subset

MSDD version with the stack algorithm for inner LD offer

the overall best performance-complexity tradeoff.

VI. CONCLUSION

In this paper, we have investigated tree-search MSDD. We

have proposed a nested decoder structure consisting of an outer

and N 1inner tree-search decoders, where the inner decodersefficiently search the signal space of the potentially large DSTM

constellation. Decoder designs tailored for diagonal and orthog-

onal DSTM codes have been given. Considering the underlying

continuous MIMO fading channel, a tight SER approximation

for MSDD hasbeen obtained, which canbe evaluated efficiently.

The novel subset MSDD scheme further improves power effi-

ciency of MSDD by discarding less reliable decisions at the

edges of the observation window. Numerical and simulation

results presented for different DSTM constellations and fad-

ing channel scenarios have shown: 1) to what extent MSDD

outperforms benchmark decoders increasing the range of fad-

ing bandwidths for which DSTM is applicable; 2) that sphere

decoding with a Fano-type metric (MSDSD-FM) achieves

the best performance-complexity tradeoff with a complexity

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comparable to that of DFDD in most cases; 3) that inner stack

decoding is favorably applied to provide an ordered sequence

of candidate signal points for outer tree-search decoding; and

4) that subset MSDD yields impressive additional performance

gains at a very moderate increase in complexity.

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Volker Pauli (S04) received the Diploma degreein electrical engineering from the University ofErlangen, Erlangen, Germany, in 2003.

He is currently a Research Assistant inthe Telecommunications Laboratory, University ofErlangen. He was a Visiting Scholar at the EcolePolytechnique Federale de Lausanne, Lausanne,Switzerland, in 2002 and the University of BritishColumbia, Vancouver, Canada, in 2003 and 2005.His current research interests include noncoherent

detection, iterative receiver structures, multiple-inputmultiple-output (MIMO)systems, andmulticarriermodulation andequalization.

Lutz Lampe (S98M02) received the Diploma andthe Ph.D. degree from the University of Erlangen,Germany, in 1998 and 2002, respectively, both inelectrical engineering.

He is currently an Associate Professor in the De-partment of Electrical and Computer Engineering,University of British Columbia, Vancouver, Canada.

His current research interests include the areas ofcommunications and information theory applied towireless and powerline transmission.

Dr. Lampe has been a Vice-Chair of the IEEECommunications Society Technical Committee on Power Line Communica-tions since 2004. He was General Chair of the 2005 International Symposiumon Power Line Communications and its Applications (ISPLC 2005) and a Co-Chair of the General Symposium of the 2006 IEEE Global TelecommunicationsConference (Globecom 2006). He is an Editor for the IEEE T RANSACTIONSON WIRELESS COMMUNICATIONS and has been an Associate Editor for theIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY from 2004 to 2007. Hewas the co-recipient of the Eurasip Signal Processing Journal Best Paper Award2005, the Best Paper Award at the IEEE International Conference on Ultra-Wideband (ICUWB) 2006, and the Best Student Paper Awards at the EuropeanWireless Conference 2000 and at the International Zurich Seminar 2002. In2003, he received the Dissertation Award of the German Society of InformationTechniques (ITG).

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