space-time diversity systems based on linear constellation...

294 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003

Space–Time Diversity Systems Based on LinearConstellation Precoding

Yan Xin, Student Member, IEEE, Zhengdao Wang, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE

Abstract—We present a unified approach to designingspace–time (ST) block codes using linear constellation precoding(LCP). Our designs are based either on parameterizations ofunitary matrices, or on algebraic number-theoretic constructions.With an arbitrary number of transmit- and receive-an-tennas, ST-LCP achieves rate 1 symbol/s/Hz and enjoys diversitygain as high as over (possibly correlated) quasi-staticand fast fading channels. As figures of merit, we use diversityand coding gains, as well as mutual information of the under-lying multiple-input–multiple-output system. We show that overquadrature-amplitude modulation and pulse-amplitude modu-lation, our LCP achieves the upper bound on the coding gain ofall linear precoders for certain values of and comes close tothis upper bound for other values of , in both correlated andindependent fading channels. Compared with existing ST blockcodes adhering to an orthogonal design (ST-OD), ST-LCP offersnot only better performance, but also higher mutual informationfor 2. For decoding ST-LCP, we adopt the near-optimumsphere-decoding algorithm, as well as reduced-complexity sub-optimum alternatives. Although ST-OD codes afford simplerdecoding, the tradeoff between performance and rate versuscomplexity favors the ST-LCP codes when , , or the spectralefficiency of the system increase. Simulations corroborate ourtheoretical findings.

Index Terms—Diversity, multiantenna, rotated constellations,space–time (ST) codes, wireless communication.

I. INTRODUCTION

WELL-established by now as a versatile form of diver-sity for wireless applications, spatial diversity is imple-

mented by deploying multiple transmit and/or receive antennasat base stations and/or at mobile units. Because of size andpower limitations at mobile units, multiantenna receive diver-sity is more appropriate for the uplink rather than the downlink.For this reason,transmit diversityschemes have attracted con-

Manuscript received March 7, 2001; revised October 22, 2001 and January25, 2002; accepted February 25, 2002. The editor coordinating the review of thispaper and approving it for publication is A. F. Molisch. This work was supportedin part by the National Science Foundation (NSF) under Grant 9979443 andGrant 012243, and in part by an ARL/CTA Grant DAAD19-01-2-011. This workwas presented in part at Asilomar Conference on Signals, Systems and Com-puters, Pacific Grove, CA, October 2000, at the International Conference onAcoustics, Speech and Signal Processing (ICASSP), Salt Lake, UT, May 2001,and in part at the Global Telecommunications Conference (GLOBECOM), SanAntonio, TX, November 2001.

Y. Xin and G. B. Giannakis are with the Department of Electrical andComputer Engineering, University of Minnesota, Minneapolis, MN 55455USA (email: [email protected]; [email protected]).

Z. Wang was with the Department of Electrical and Computer Engineering,University of Minnesota, Minneapolis, MN 55455 USA. He is now with the De-partment of Electrical and Computer Engineering, Iowa State University, Ames,IA 50011 USA (email: [email protected]).

Digital Object Identifier 10.1109/TWC.2003.808970

siderable research interests recently; see, e.g., [1], [17], [26],[27], [36], and references therein.

It has been widely acknowledged that space–time (ST) codingtechniques can effectively exploit the spatial diversity createdby multiple transmit antennas [27]. Typical examples includeST trellis codes and ST block codes from orthogonal designs(ST-OD). ST trellis codes enjoy maximum diversity and largecoding gains, but their decoding complexity grows exponen-tially in the transmission rate [27], which does not encourageusage of large size constellations. On the other hand, ST-ODcodes [1], [26] offer maximum transmit diversity and can affordlow-complexity linear decoding. Unfortunately, ST-OD codescome with reduced transmission rates, when complex constella-tions are used and the number of transmit antennasis greaterthan two.

An alternative transmit diversity scheme that does not sac-rifice rates, is based on what we term linear constellation pre-coding (LCP). It was originally developed for single-antennatransceivers with an interleaver [4] and later on utilized also formultiantenna systems [7]. Based on the parameterization of realorthogonal matrices, construction of LCP was pursued in [7],[23] based on exhaustive search. Because the search is constel-lation dependent, it becomes infeasible for large size constella-tions. On the other hand, algebraic tools can be used to constructLCP transformations that lead to fading-resilient constellations[4], [5], [12]. These LCP designs are available in closed form,but apply only to particular constellations and-dimensions[5]. Whether algebraically constructed LCP can achieve max-imum diversity and coding gains in ST diversity systems, wasalso left open.

This paper deals with a unified approach to constructing LCPcodes that maximize diversity and coding gains over constel-lations carved from the two-dimensional (2-D) lattice . Weview LCP designs as matrices and prove the existence ofunitaryconstellation precoding (UCP) matrices with maximum diver-sity gain , for any finite constellation. This establishes thetheoretical ground for searching over parameterized UCP ma-trices. For general LCP designs, we provide the upper bound onthe coding gain of all linear precoders to benchmark their per-formance. We generalize the parameterization construction ofUCP codes from real orthogonal matrices [7], [23] to unitarymatrices, which in general can provide larger coding gains. Foralgebraic designs, we construct novel LCP codes that even forcorrelated channels:

1) guarantee maximum diversity gains for any, , re-gardless of the constellation;

2) achieve the upper bound on coding gains over quadra-ture-amplitude modulation (QAM) and pulse-amplitudemodulation (PAM) for certain values of ;

1536-1276/03$17.00 © 2003 IEEE

XIN et al.: SPACE–TIME DIVERSITY SYSTEMS BASED ON LINEAR CONSTELLATION PRECODING 295

Fig. 1. Discrete-time baseband equivalent model.

3) come close to this upper bound on the coding gain forother values of .

We also construct UCP codes adhering to a lower bound onthe coding gain, forany . In addition to diversity and codingcriteria [27], we also employ the maximum average mutual in-formation criterion [14] to evaluate the performance and com-pare ST-LCP with the ST-OD codes of [1], [26], the so-calledquasi-orthogonal ST designs of [16], and the ST linear disper-sive (LD) codes of [14].

This paper is organized as follows. Section II presents thesystem model and the ST-LCP encoding scheme along withpertinent design criteria. Section III provides design methodsbased on the parameterization of unitary matrices and algebraicnumber theoretic tools. Section IV describes the ST-LCP de-coding options, while Section V presents properties of ST-LCPcodes, including a comparison of ST-UCP with ST-OD codesin terms of maximum mutual information. Simulations are pro-vided in Section VI and Section VII concludes the paper.

Notation: Bold lower (upper) case letters are used to denotecolumn vectors (matrices); and represent transpose andconjugate transpose, respectively; Tr denotes trace and

stands for Kronecker product; denotes the ( )thentry of a matrix; denotes an identity matrix;diag denotes a diagonal matrix with diagonalentries ; and denote the real and imaginaryparts, respectively. , , , , and stand for the positiveinteger set, the integer ring, the rational number field, the realnumber field, and the complex number field, respectively;denotes .

II. DESIGN CRITERIA OF ST-LCP

In this section, we introduce ST-LCP and rely on criteria sim-ilar to [27] to deduce its design.

A. ST-LCP Encoding

With reference to Fig. 1, let us consider a wireless link withtransmit and receive antennas over Rayleigh flat fading

channels. The symbol streamfrom a normalized1 constella-tion set is first parsed into signal vectors , and thenit is linearly precoded by an matrix . The precoded

1Average symbol energy ofC is assumed to be one.

block is fed to an ST mapper, which maps to ancode matrix that is sent over the antennas during timeintervals. Specifically, the ( )th entry , istransmitted through the th antenna at theth time interval,where denotes the ( )th entry of a unitary matrix ;i.e., ; and vector denotes theth row of .Defining diag , we can thus write the

transmitted ST-LCP code matrixas

(1)

Square-root Nyquist pulses [23, p. 557] are used as transmitand receive filters in all antennas. After receive filtering andsymbol rate sampling, the signal received by antennaat theth time interval is a noisy superposition of faded transmitted

signals; i.e., , where de-notes the fading coefficient between theth transmitter and the

th receiver antenna with and .We assume that

A1) channel coefficients are uncorrelated with thenoise , zero mean, complex Gaussian, with correla-tion matrix , i.e., is full rank,where ;

A2) channel coefficients are only known to the re-ceiver and remain invariant over intervals (quasi-static flat fading);

A3) noise samples are independent identically dis-tributed (i.i.d.), complex Gaussian with zero mean andvariance per dimension.

Notice that our flat fading channels are allowed to be correlated.Let be the received signal matrix with ( )th entry

; the channel matrix with; and the noise matrix with (

in Fig. 1 denotes the th row of for ). Theinput–output relationship can then be written in matrix form as

(2)

If is chosen to be identity [7], [33], the ST transmissionin (2) reduces to a time-division multiple-access (TDMA)-liketransmission with each antenna pausing for ( ) out of

time intervals. If is a complex Gaussian matrix with


zero-mean i.i.d. entries and is a unitary matrix, the distribu-tion of is the same as the distribution of [19]. Thus, theprobability of error remains invariant to . However, offerssome flexibility that could be used to, e.g., alleviate high-poweramplifier nonlinear effects because it can avoid the unnecessary“on–off” switch for each antenna.

Each transmitted symbol in ST-LCP is a linear combinationof the complex symbol entries in. We will see that by carefullydesigning the precoder , ST-LCP can achieve full diversityand large coding gains at rate 1 symbol/s/Hz. Unlike ST-LCP,ST-OD is linear only in the real and imaginary parts taken sep-arately; ST-OD enables low decoding complexity, by imposingan orthogonality constraint on the code matrix. Unfortunately,this constraint reduces transmission rate when complex constel-lations are used with [26].

B. ST-LCP Design Criteria

At the receiver end, we will rely on to detect in themaximum-likelihood (ML) sense and we will designto opti-mize the ML detection performance. We start with the pairwisematrix error event { } as the event that the ML receiverdecodes diag erroneously, when wasactually sent. Let us define and the ma-trix , where

existence of the correlation matrix square root is ensuredby (A1). Using standard Chernoff bounding techniques [27], wecan upper bound the average pairwise error probability (PEP) as

(3)

where rank , withdenoting a set of indices having cardinality

; and stands for the geometric mean of the product ofthe nonzero eigenvalues of ; i.e.,

.We define the diversity gain, coding gain and kissing number

in terms of as follows.1) Diversity Gain: The overall diversitygain is defined as

, over all distinct pairsof . From the definition of , we infer that the max-imum diversity gain is achieved when the fol-lowing maximum diversity conditionholds true:

(4)

Recalling the fact that is the th coordinate of theprecoded vector , we infer from (4) that in order toachieve the , each vector should be differentfrom all other precoded vectors in all its coordinates. As aresult, from constellation precoded vectorsone can decipher

even if all except one of the coordinates are nullifiedby fading.

2) Coding Gain: For an LCP matrix with a given , thecoding gain is defined as

(5)

When , the coding gain becomes

(6)

where is theminimum productdistance. Note that (4) is equivalent to having .

3) Kissing Number:The product kissing number is de-fined as the total number of pairs of symbol vectorsand withthe same minimum product distance.

For a given diversity gain , the coding gain measuresthe savings in signal-to-noise ratio (SNR) of the LCP system ascompared to an ideal benchmark system of BERat high SNR. Certainly, the diversity gain , the coding gain

, and the kissing number, all depend on the choice of .At high SNR, it is reasonable to maximize the diversity gainfirst, because it determines the slope of the log-log bit-error rate(BER)-SNR curve. Within the class of s that achieve ,the coding gain should be maximized afterwards. If twoshave the same diversity and coding gains, then the one withthe smaller kissing number is preferred. We will not minimizethe kissing number in this paper. However, we will show itsinfluence on the system performance in Section VI. Anotherfactor affecting BER performance is the bit-to-symbol mapping.This should be also optimized in ST-LCP, but here we simplyadopt the Gray mapping [22, p. 170].

III. D ESIGN OFST CONSTELLATION PRECODERS

In our general precoding setup, we do not impose any struc-tural constraints on , except for ensuring that Tr

, which controls the total transmit energy over time in-tervals: . Among all s obeyingthe power constraint, we look for those with maximum diver-sity and high coding gains. We will establish first the existenceof diversity-maximizing precoders (see also [12] and [33]). En-sured by this result, we will next look for an LCP matrixthat maximizes the coding gain of (6) within the class of di-versity-maximizing precoders; the overall optimum LCP matrixwill be selected as [cf. (6)]

(7)

subject to the power constraint .Equation (7) discloses that our precoder design is independentof the channel correlation matrix. For simplicity, we will hence-forth focus on channels with , bearing in mind that ourresults carry over to the correlated case as well.2

2Our subsequent coding gain expressions for i.i.d. channels require just ascalar multiplication by the[det(R )] factor to yield their counter-parts for correlated channels.


To quantify the performance of in (7), we will rely onthe following upper bound on the coding gain that applies to alllinear precoders (see Appendix A for the proof).

Proposition 1: (Upper Bound on the CodingGain): Consider any finite normalized constellationwithminimum (Euclidean) distance .Among all linear precoders obeying the power constraint

, the maximum coding gainis

(8)

In the following two sections, we will provide methodologies fordesigning LCP matrices based either on parameterizations ofunitary matrices, or, on algebraic number theoretic tools.

A. Design Based On Parameterization

Unitary constellation precoding offers a distinct advantageover nonunitary LCP options: a unitarycorresponds to a rota-tion and preserves distances among the-dimensional constel-lation points. On the contrary, a nonunitarydraws some pairsof constellation points closer (and some farther). This distance-preserving property of UCP also guarantees that if such rotatedconstellations are to be used over an additive white Gaussiannoise (AWGN) (or near AWGN) channel, the performance willremain invariant. In practice, the channel condition can also varybetween the two extremes of AWGN and Rayleigh fading, inwhich case a unitary precoder may be preferred [3]. For thesereasons, we first deal with unitary precoders. But prior to de-signing unitary ’s, it is natural to ask whether the unitary classis rich enough to contain ST-LCP precoders with maximum di-versity gains. The following proposition asserts that aunitaryachieving always exists (see Appendix B for the proof).

Proposition 2: (Existence of a Diversity-Maximizing UnitaryPrecoder): As long as the constellation size is finite, there al-ways exists at least one unitarysatisfying(4) and is, thus, ca-pable of achieving the maximum diversity gain for anynumber of transmit ( ) and receive ( ) antennas.

Notice that the fading-resilient constellations in [4], [5], and[12] guarantee maximum diversity gains only for particular con-stellations or -dimensions.

Ensured by Proposition 2, we are now motivated to look fora unitary that maximizes among diversity-maximizingunitary precoders [cf. (7)]. As formulated in (7), findinginvolves multidimensional nonlinear optimization over thecomplex entries of . To facilitate the optimization, we can takeadvantage of the fact that and parameterizeusing real entries taking values from finite intervals. We startwith the simplest case where .

Any real orthogonal precoder for can be expressedas a rotation matrix [7], [23]:

(9)

which is a function of a single parameter . Theprecoder in (9) rotates the constellation points in 2-D so thateach rotated point is different from other rotated points inboth

coordinates. With as in (9), the criterion in (7) needs to beoptimized only over a single parameter. Instead of using thereal orthogonal matrices of [7], [23], we here explore unitaryprecoders in , because they have the potential for larger codinggains than their real counterparts.

It is known that any 2 2 unitary matrix can be parameterizedas [21, p. 7]

(10)

where is a 2 2 diagonal unitary matrix, and.

For , it is possible to construct real orthogonal pre-coders by using Givens matrices [7], [23]. Specifically, any realorthogonal matrix can be factored as a product of

Givens matrices of dimension and anpseudo-identity matrix, which is defined as a diagonal matrixwith diagonal elements 1 [7]. In the following proposition, wegeneralize this result to also include unitary matrices (see Ap-pendix G for the proof).

Proposition 3: (Parameterization of Unitary Matrices):Anyunitary matrix can be written as

where is an diagonal unitary matrix, ,and is a complexGivens

matrix, which is just the identity matrix with the ( )th,( )th, ( )th and ( )th entries replaced by , ,

and , respectively.As multiplication with a diagonal unitary matrix preserves

product distances, in Proposition 3 can be ignored in the op-timization (7). The number of parameters that need to be opti-mized is thus , which are the parametersof the complex Givens rotation matrices. An-alytical solution to this optimization problem appears to be in-tractable. However, for a small number of antennas (say )and small constellation sizes (say ), exhaustive search iscomputationally feasible, as we will illustrate in Section VI.

B. Design Based on Algebraic Tools

The design based on the parameterization of unitary matricesis less practical when either or is large. Fortunately,algebraic number theoretic approaches are possible to yieldclosed-form LCP designs with reasonably large coding gains[5], [12], even when and/or is large. In this section, weintroduce two novel LCP constructions: LCP-A and LCP-B.We prove that LCP-A can achieve the upper bound on thecoding gain over QAM (or PAM) for , where ,or, for , where is an Euler number3 and

mod . We also show that LCP-B, which is unitary forany , has coding gain that is guaranteed to be greater than alower bound.

We start by briefly introducing some necessary definitionsand facts from [12] and [20].

3An Euler number�(P ) is defined as the number of positive integers< P ,which are relatively prime toP .


B.1) Algebraic Number Theory Preliminaries:Notation: denotes the smallest subfield of in-

cluding both and and denotes the smallest subfieldof including both and , where is algebraic over

; i.e., is a root of some nonzero polynomial ;is the ring of Gaussian integers, whose elements are in the

form of with ; denotes the minimalpolynomial of over a field with denotingits degree.

Definitions:

D1) (Cyclotomic Polynomials): If , the th cyclo-tomic polynomial is defined as

, where gcd andand is its degree.

D2) (Extension of an embedding): If is an embedding ofin that fixes in such a way that

, , then is called a -isomorphism of.

D3) (Relative norm of a field): Letdenote the complex roots of the minimal polynomialof over ; and let , , bedistinct -isomorphisms of such that

. Consider and definethe relative norm of from the field as

.D4) (Integral over : An element is said to be integral

over , if is a root of a monic polynomial withcoefficients in . Clearly, every element in isintegral over .

Facts:

F1) The polynomial is the minimal polynomialof and for

any such that gcd .F2) If with , then the of

is with all distinct rootsfor .

F3) If is a finite extension of the field withdegree denoted by , then

forms a basis of over.

F4) The set of elements of , which are integral over, is a subring of containing .

F5) If is integral over , then the relativenorm of from .

F6) If is an odd integer and , then.

F7) If gcd and , then .

Before presenting our constructions that are based on thesefacts, we first prove the following important lemmas (seeAppendices C–E for their proofs).

Lemma 1: If for , thenmod .

Lemma 2: If mod , then the ofis and its degree is .

Lemma 3: All roots of of withsome have unit modulus.

B.2) Algebraic Construction: LCP-A:LCP-A constructs a matrix that applies to any number of

transmit-antennas and subsumes the constructions in[5] and [12] as special cases when is a power of two, wherethe resulting precoder is unitary. When is not a power of two,the construction yields nonunitary LCP matrices.

Let be integral over such that .LCP-A is constructed as follows (see also [5], [12], and [35]):

......

...(11)

where are the roots of andis the normalizing factor ensuring that Tr .

The idea behind LCP-A can be explained as follows. For con-venience, let us ignore the constant for a moment. By F3,the entries in the first row of form a basis of over

. This means that . For all constel-lations carved from the lattice , that is, ,

is integral over based on F4 and the fact thatis integral over . We can, thus, view as the (unique)coordinates of with respect to the basis of the firstrow entries. Defining -isomorphisms

, we have ; entries ofare then the images of these isomorphisms of .

These isomorphisms are the ones required in the definition ofthe relative norm of (cf. D3). The relative norm inthis case also coincides with our definition of product distancein (6). Therefore, for , the minimum product distance,being nonzero and belonging to from F5, is at least one.Of course, we have to also take into account the energy nor-malization and the constant , after which the coding gain is

, where and are constellation dependent pa-rameters (see Proposition 5 next for a complete statement of theresult).

We rely on the following lemma to find values of for whichLCP-A achieves the upper bound on the coding gain (see Ap-pendix F for the proof).

Lemma 4: If in (11) is integral over , then

and the equality holds if and only if all the rootshave unit modulus. For odd , the equality

cannot be achieved.Let us define the set

with for some

Values in this set are special to our goal of maximizing thecoding gain, as we will see soon. Lemma 2 and F2 implythat both for mod andfor , belong to . On the other hand, according toLemmas 3 and 4, odd integers do not belong to . Forinstance, the set and

.


We study next properties of our LCP-A and establish resultson their coding gains. The following proposition provides thelower-bound on coding gains over any constellation carved from

(see Appendix H for the proof):Proposition 4: (Lower Bound on Coding Gains):If one

applies the LCP matrix in (11) to precode the constellationcarved from and normalized by , then thecoding gainsatisfies

(12)

In particular, when QAM (or PAM) constellations are used, weprovide not only the exact coding gain which can be achievedby LCP-A, but also lower and upper bounds on the maximumcoding gain achieved by LCP-A (see Appendix I for the proof).

Proposition 5: (Coding Gains for QAM [orPAM]): Consider a QAM (or PAM) constellation withthe minimum distance of signal points equal to 2, which isnormalized by . For and the linear precoderin (11), the coding gain over the normalized QAM (or PAM)is given by

(13)

Furthermore, the maximum coding gain achieved by LCP-A islower and upper bounded by

(14)

i) For , LCP-A achieves the upper bound in (14)on the coding gain of all linear precoders over QAM (orPAM).

ii) For , LCP-A cannot achieve the upper boundin (14). However, LCP-A at least can achieve the lowerbound in (14), which is a large fraction (70%) of the upperbound.

B.3) Algebraic Construction: LCP-B:As we argued at the beginning of Section III-A, unitary pre-

coders have certain advantages as compared to nonunitary ones.For certain ’s, the precoders designed in LCP-A are not uni-tary.

We here present a construction of unitary precoders for any

diag

where and is the -point inverse fast Fouriertransform (IFFT) matrix whose ( )st entry is given by

. Notice that this LCPmatrix amounts to phase-rotating each entry of the symbolvector and then modulating in a digital multicarrier fashionthat is implemented via . The choice of will be addressedlater in this section.

Next, we state a proposition which provides lower bounds oncoding gains of LCP-B (see Appendix J for the proof).

Proposition 6: (Lower Bounds on Coding Gains of Uni-tary Precoders): For LCP-B, let denote the number of dis-tinct minimal polynomials of ,

TABLE ICODING GAINS OF ST-LCP CODES FORN = 4–10 OVER 4-QAM

, over and let ’s be the degrees ofwith . If , then we have

(15)

where and .To design unitary precoders with large coding gains,

Proposition 6 suggests choosingsuch that the number of distinct minimal polynomials of

, , in Proposition 6 issmall and their degrees are as low as possible, in order to make

small. In particular, when with , we can selectsuch that all , , are

roots of the minimal polynomial over . In thiscase, we have , and (15) coincides with (12) as

. Even though (15) benchmarks the coding gainat high SNR, we have verified through simulations that it israther pessimistic.

Heuristic Rule for Constructing Unitary Precoders:Forany given which is not a power of two, choosefor some such that most of ’s are roots of .This will make small and for .

B.4) Examples of Algebraic Constructions:Example 1: If and , then the upper bound on

for 4-QAM is (14). Obtained via computer simulation,Table I lists the for , 6, 8, 10 and for , 7, 9over 4-QAM constellations with and , whereand denote the coding gains of the precoders from LCP-Aand LCP-B, respectively. We apply the polynomials and

to construct (11) for , 6, 8, 10 and for, 7, 9, respectively. Table I also confirms that the linear

precoders for , 7, 9 provide quite large coding gains evenwhen the construction of (11) cannot achieve the upper boundin (14).

Example 2: If and we choose , then. Notice that with , all

’s except are roots of the minimal polynomial ofwith , while is a root of with .

In this construction, we have . Based on simulations, wefind that for 4-QAM constellations.

IV. DECODING OFST-LCP TRANSMISSIONS

The starting point of our optimal precoder designs was theperformance of ML detection of from (2). Because the com-plexity of ML detection based on exhaustive search is very highwhen and/or is large, we consider in this section three al-ternative decoders for ST-LCP transmissions. The first, spheredecoding (SD), is used to approximate the ML performance at apolynomial (but still relatively high) complexity, while the otheralternatives, Vertical Bell-Labs Layered ST (V-BLAST) [13] orblock minimum mean-square error decision-feedback equaliza-tion (BMMSE-DFE) [2], [25], are used as relatively low-com-


plexity alternatives. Defining , we rewrite (2) as. Using the vec operator to put the columns

of one after the other, we obtain vec as

diag diag vec

(16)

where denotes the th row of corresponding to theth re-ceiver, diag denotes the diagonal matrix generated byand is an block diagonal matrix. The receivedvector in (16) is equivalent to a received block fromuncodedtransmit-antennas to receive-antennas with the channelmatrix being almost always full rank.4 Thanks to the specialstructure of in (16), application of the maximum ratio com-biner yields

diag

where diag andis colored Gaussian noise. The latter can be

prewhitened to obtain

(17)

where and is AWGN. Equation (17) willbe our starting point for ST-LCP decoding.

A. Near-Optimum Decoding

The SD algorithm of [8] and [29] was introduced to reducedecoding complexity provided that the transmitted constella-tion is carved from a lattice. SD takes advantage of the latticestructure of transmitted signals to achieve near ML performancewith a moderate complexity. It has been shown that for a fixedsearching radius and for a given lattice structure, the decodingcomplexity for transmit antennas is approximately[6]. In our simulations, we will consider QAM (or PAM) con-stellations, which are carved from the lattice .

When is real and is complex, we use SD to decode sep-arately the real and imaginary parts ofin (17). When bothand are complex, one should view the complex vector

as a real vector and rewrite the equivalent systemmodel as in [6, eq. (2)]

The computational burden of decoding complex ST-LCP trans-missions will increase accordingly, because we need to applythe SD to a vector. However, the recently proposedcomplex sphere decoder in [15] does not double the size of thesearch lattice vector, thus reducing the complexity.

B. Suboptimum Decoding

The SD algorithm achieves near-ML performance withpolynomial complexity. But when and/or is large, thecomplexity becomes prohibitively high. As reduced-complexityalternatives, we advocate using the V-BLAST [13], or the block

4This holds true because each channel taph in the structured matrix~H isnonzero with probability one.

(B)MMSE-DFE algorithm [2], [25], whose complexity isroughly . Both V-BLAST and (B)MMSE-DFE arethe decoding schemes that are based on decision feedback.The decision feedback here is only used for interferencecancellation, but one could also use it for channel estimation ina decision-directed mode.

Remark: It is known that with linear processing at thereceiver, ST-OD can convert the space–time channel into anumber of parallel AWGN channels. Such a parallel conversionenables the inclusion of an outer channel encoder/decoderbecause soft information can be obtained from these parallelAWGN channels about coded symbols. For ST-LCP, such softinformation output does not seem practically possible unlesssome enumerative search is performed.

V. ST-LCP PROPERTIES

Having described the encoding and decoding options of ourST-LCP system, in this section we present four attractive fea-tures they possess and compare them with competing alterna-tives.

A. Delay Optimality

For the maximum diversity to be achieved, it is knownthat theminimumpossible decoding delay is equal tounder thequasi-static flat fadingassumption; and schemes thatachieve maximum diversity with the minimum delay arecalled delay optimal[10]. ST-LCP is delay-optimal, because

by design. This is not always true for ST-OD, how-ever. For example, when and complex constellationsare used, ST-OD codes require time intervals [26].

B. Mutual Information Optimality

In this section, we will prove that ST-UCP can achieve higheraverage mutual information than ST-OD codes when .

Recalling that for i.i.d. channels the distribution of isidentical to that of , we infer that the maximum average mu-tual information per time interval, , of ST-UCP coincideswith the capacity of spatial cycling [9, eq. (13)]. So we have

(18)

Correspondingly, for ST-OD block transmissions, the max-imum average mutual information per time interval is[24, eq. (4)]

(19)

For real constellations, ST-OD at rate exist for any .But for complex constellations, only Alamouti’s code [1] () is known to have ; for , ST-OD codes offer

; and for , [26].The exponent of in (19), before taking the logarithm, is.

When , this exponent is strictly smaller than the corre-sponding largest exponent ofin (18) which equals one. Basedon this fact, we are able to establish the following propositioncomparing mutual information of ST-UCP with those achiev-able by ST-OD codes (see Appendix K for the proof).


Proposition 7: (Information-Theoretic Comparisons WithST-OD): Under the channel assumptions in (2) and for suffi-ciently large SNR , the maximum average mutual informationof ST-UCP systems is strictly greater than that of ST-ODsystems for .

C. Symbol Detectability

Here, we link with the deterministic notion of zero-forcing equalizability (or symbol detectability). Specifically, weestablish in the appendix that the nonzero minimum product dis-tance [cf. (4)] implies that: if in (16) is nonzero, then the sym-bols are guaranteed to be recoverable (perfectly in the absenceof noise); and this is what we define as symbol detectability.

Proposition 8: (Symbol Detectability):If (4) holds true, thenST-LCP symbols are detectable so long as one entry ofis nonzero.

By exploiting the finite alphabet property of, Propositions2 and 8 assert the surprising result that ST-LCP guarantees de-tectability of the symbols in even from a single receivedsample , provided that among all flatfading channel coefficients,only one .

D. Flexibility in Slow and Fast Fading

By the construction of ST-LCP codes in (1), we have that atevery time interval

where denotes the th column of the unitary matrix in(1). Hence, ST-LCP codes are also suitable for fast fading ac-cording tothe distance criterionof [27]. As ST-LCP only re-quires independence among ’s, it is also applicable to fastfading (as opposed to quasi-static flat fading) channels, whichcan be tracked accurately using the Kalman predictors devel-oped in [18]. However, in fast fading channels, ST-LCP willperform the same as if the channel is slowly fading. To fully ex-ploit the spatio-temporal diversity gain of fast fading channels,one can either use the so-termed “smart-greedy” trellis codes[27], or, capitalize on explicit modeling of the channel varia-tions [11].

VI. SIMULATED PERFORMANCE

In this section, we simulate ST-LCP systems and comparethem with ST-OD, the quasi-orthogonal ST designs of [16] andST LD codes in [14]. For more comparisons, interested readersare referred to [32]. Similar to LCP, quasi-orthogonal designsalso relax the orthogonality imposed by ST-OD codes. We willuse binary phase-shift keying (BPSK) or QAM with constella-tion sizes chosen such that the spectral efficiency of ST-LCPand ST-OD are the same. In all simulations, the real and com-plex part of the AWGN has variance SNR . Thechannel matrix has i.i.d. entries. The average BER is obtainedthrough Monte Carlo simulations, except for ST-OD where arecursive algorithm is used to compute the exact BER [30]. Allsimulations except for Test Case 3 utilize the SD algorithm.

For real precoders, we use the codes in [7] whenand those from [5] when , 4, 6. We construct complex

TABLE IICODING GAINS OF ST-LCP CODES FORN = 2, 3, 4 OVER 4-QAM

precoders for , 4, 6 according to LCP-A in (11),with , being the roots of the poly-nomials (cf. Definition D1 and Fact F2) , and

, respectively; for , for in(11) are chosen to be roots of . When ,we use the parameterization of Proposition 2 and search for thesix Givens matrix parameters: and

. Exhaustive computer search iscarried out over the discrete values obtained by quantizing thefinite continuous intervals of these six parameters. Specifically,we first divide each interval into smaller subintervals. The mid-point of each subinterval is then used as a parameter value andthe coding gain is evaluated. The subinterval whose midpointgives the largest coding gain is further divided into even smallersubintervals for search and the search continues until the codinggains converge. The resulting precoder

is found to have coding gain larger than that of (11) and is thusused.

ST-OD codes for transmit antennas will be denoted aswith the codes taken from [26]. The rates of complex

to are 1, 1/2 or 3/4, 1/2 or 3/4, 1/2, 1/2, respectively. In Testcase 5, the channel capacity is computed using [9, eq. (4)].

Test Case 1 (Complex Versus Real Precoders):Table II liststhe coding gains of real and complex ST-LCP codes (1) over4-QAM constellations for , 3, 4. It also indicates thenumber of distinct pairs with the product distance lessthan 3 ; in the third row denotes the total number of precodedvectors for each . The advantage of complex precoders com-pared to real precoders in coding gains shows up in the last row,whose entries are the ratios (in decibels) between the codinggain of complex precoders and those of real ones. Notice thata complex for has nearly 2 dB larger coding gainthan its real counterpart, while this gain is only about 0.5 dBfor , 3. Fig. 2 compares the BER performance of com-plex and real precoders for , 3 with BPSK. The complexprecoders outperform the real precoders by more than 1 dB atBER 10 . Fig. 3 shows that complex precoders outperformreal ones by about 0.5–1 dB at high SNR.

Our performance analysis is based on PEP, which is known tooffer more accurate approximation of the system performanceat reasonably high SNR [27]. Besides coding gains, the kissingnumber may also play an important role in affecting the systemperformance. The difference in BER between complex and realprecoders is not as significant as the difference in coding gainsand only shows up at high SNR.


Fig. 2. Complex versus real precoders (BPSK,N = 2, 3,N = 1, 1 b/s/Hz).

Fig. 3. ST-OD (256-QAM) versus ST-LCP (16-QAM) (N = 3, 4,N = 2,4 b/s/Hz,R = 1=2).

Test Case 2 (ST-LCP Versus ST-OD):Figs. 4–9 compareST-LCP against ST-OD codes for various combinations of

, spectral efficiencies and rates. Fig. 4 shows thatST-OD codes outperform ST-LCP codes by 1–2 dB when

. Fig. 5 compares complex ST-LCP codes with 4-QAMand rate 1/2 ST-OD codes for , 4 and . Tomaintain the same spectral efficiency, we use 16-QAM forthese ST-OD codes. The simulation shows that ST-LCP nowgains about 2 dB over ST-OD codes. The gain of ST-LCP ismore pronounced when increases as shown in Fig. 6. Fig. 7depicts BER for , 6 and . Again, ST-LCP codeshave an advantage over ST-OD. Fig. 7 also confirms that thecomplex precoder outperforms the real one obtained from [5]by about 1 dB.

For , 4 and , rate 3/4 ST-OD codes with256-QAM are tested and compared with rate 1 ST-LCP codesin Fig. 8. The spectral efficiency in this case is 6 b/s/Hz. Thegain of ST-LCP in SNR is less than 1 dB. Fig. 9 shows that thegain of ST-LCP over ST-OD increases to 3 dB when .

Fig. 4. ST-OD versus ST-LCP (4-QAM,N = 2,N = 1, 2 b/s/Hz,R = 1).


In summary, ST-LCP codes perform better than ST-OD whenat the price of increased decoding complexity.

Test Case 3 (Decoding Options):Fig. 10 depicts the perfor-mance of SD, V-BLAST and BMMSE-DFE in various ST-LCPschemes for , 4 and , at spectral efficiency4 b/s/Hz. It shows that both V-BLAST and BMMSE-DFEcannot achieve the maximum diversity gain; SD outperformsboth alternatives about by 5 dB at BER 10 . However,we observe that even with the suboptimum V-BLAST orBMMSE-DFE decoding, ST-LCP still outperforms ST-OD byabout 7–8 dB at BER 10 . VBLAST uses zero-forcingwith no ordering. The resulting V-BLAST performance is onlyslightly worse than that of BMMSE-DFE.

Test Case 4 (ST-LCP Versus Quasi-Orthogonal ST and LDCodes): Fig. 11 depicts the performance comparison betweenST-LCP and the quasi-orthogonal ST codes of [16] forand with 4-QAM. The decoding of quasi-orthogonalST in [16] was implemented. The diversity gain of ST-LCP is

, while that of the quasi-orthogonal ST codes is only two.


Fig. 6. ST-OD (16-QAM) versus ST-LCP (4-QAM) (N = 3, N = 3,2 b/s/Hz,R = 1=2).


The higher diversity gain of ST-LCP pays off for SNR valuesabove 15 dB. Fig. 12 shows the performance comparison be-tween ST-LCP and LD codes [14, eq. (36)] for and

. Both schemes use the SD algorithm. ST-LCP showsvery similar performance to LD codes, but LD codes have rel-atively higher decoding complexity because they use a largerblock size in this case. For further comparisons between LCPand LD, we refer the readers to [32].

Test Case 5 (Information-Theoretic Comparisons):Fig. 13shows the maximum mutual information comparisons betweenST-UCP and ST-OD for and . ST-ODoutperforms ST-UCP for , but the difference is insignif-icant when . Fig. 14 depicts the maximum averagemutual information for both ST-UCP and ST-OD computedfrom (18)–(19), when and . The ca-pacity loss of ST-OD is significant at high SNR compared withST-UCP. When increases, the capacity loss becomes morepronounced at high SNR. With and spectral efficiency4 b/s/Hz, ST-UCP exhibits about 8 dB gain over ST-OD.



Fig. 10. Decoding options for ST-OD (256-QAM) and ST-LCP (16-QAM)(N = 3, 4,N = 2, 4 b/s/Hz,R = 1=2).


Fig. 11. Quasi-OD versus ST-LCP (4-QAM,N = 4,N = 1, 2 b/s/Hz).

Fig. 12. LD codes versus ST-LCP (4-QAM,N = 3,N = 1, 2 b/s/Hz).

We observe also that the mutual information achieved byST-UCP codes is still far from the channel capacity, especiallywhen is large at high SNR. For example, withand spectral efficiency 4 b/s/Hz, ST-UCP shows about 1 dBloss compared to the channel capacity when , but theloss increases to 2-3 dB when . If we compare the setof curves in Fig. 14 that correspond to the same transmissionscheme (for either ST-UCP or ST-OD), as increases from1 to 6, we notice that the gain of mutual information obtainedby each additional receive antenna becomes smaller when

.

VII. CONCLUSION

A unified approach for exploiting the transmit diversity ina multiantenna environment was developed by utilizing linearconstellation precoding. With any number of transmit- andreceive-antennas, the proposed scheme can achieve a rate of1 symbol/s/Hz, maximum diversity gains, as well as large (in

Fig. 13. Mutual information of ST-UCP and ST-OD (N = 2).

Fig. 14. Mutual information of ST-UCP and ST-OD (N = 3).

some cases, maximum) coding gains over both quasi-static andfast fading, both correlated and i.i.d. channels. Near-optimumand suboptimum decoding options were also provided. Finally,it was shown that ST-LCP codes can achieve better perfor-mance and larger maximum mutual information than ST-ODcodes, when the number of transmit antennas is greater thantwo.

APPENDIX APROOF OFPROPOSITION1

For any which satisfies the power constraint:, we have Tr

Tr . By the definition of the trace and the nonneg-ativity of the diagonal entries of , there exists at least onecolumn of (say the th) with Euclidean norm .Let be a particular pair with ,where is the th column of the identity matrix. Using that

with and the


arithmetic-geometric mean inequality, the square of the productdistance of is as follows:

(20)

Because is chosen arbitrarily and the right-hand side of (20)is independent of , we have

APPENDIX BPROOF OFPROPOSITION2

Let denote the set of all signal vectors over the-dimensional complex space . Every entry of

is a symbol from a finite constellation; hence, is a fi-nite set. We infer from (4) that proving Proposition 2 is equiv-alent to proving that there exists an unitary matrixsuch that has nonzero coordinates for all distinct pairs

. Let denote all possible differences between dis-tinct vectors of , i.e., .It is clear that is finite with cardi-nality . In an -dimensional space, there exists an

vector denoted as , which is neither perpendicularnor parallel to any of the vectors in. Choose as a uni-tary matrix such that the first row vector is taken as. Con-sider as a new set in which the firstcoordinate of any vector is nonzero. Then treat the lastcoordinates of these vectors as a new set of vectors.Since is not parallel to any ( ), all vectors inthis new set are also nonzero vectors. By the same argument, wecan find an vector which is neither perpendicularnor parallel to these vectors in the new set. Choosean unitary matrix such that its first rowvector is . Then, we can construct a new unitary matrixas follows:

Let us define . It is easy tosee that the first two coordinates of vectors in are nonzerowhile the last ( ) coordinates are nonzerovectors. Performing the same construction in ( ) steps,we obtain an unitary matrix suchthat has non zero coordinates, which completes theproof.

APPENDIX CPROOF OFLEMMA 1

If where are distinct primes and ,then can be written as [cf. F8]

(21)

By the definition of , it follows that for. Assume that for some . We have either

with , or, , with .In both cases, we infer that mod by (21); whichcontradicts the assumption that ; hence,

mod .

APPENDIX DPROOF OFLEMMA 2

Case 1: If or , as gcd ,we have from F6, that . Case 2: If

, then by F7.Consider and ; as , wehave ; and thus . By F6, it followsthat .

Therefore, when mod , we have; thus, . Applying the facts that the degree of

over and that for mod , wehave . As

by F1 and since , we have. Hence, when mod ,

is . Since has degree inand is its root, it is the minimal polynomial of

over .

APPENDIX EPROOF OFLEMMA 3

As is a root of and , it followsthat . Hence, every root of mustbe a root of and, thus, all roots of have unitmodulus.

APPENDIX FPROOF OFLEMMA 4

As is integral over , it follows that5 and ; thus, we have

. To force the power constraint:, we set

Moreover, since

we have , where the equality holds if andonly if for . This implies thatand, thus, .

To prove the case for odd , we only need to showthat when is integral over and ,then at least one root of does not have unit modulus.Suppose that all roots of have unit modulus. By

5The proof of this argument relies on the Gaussian Lemma of [20] by applyingthe fact that [j][x] is a unique factorization domain.


using a proof similar to [31, p. 4, Lemma 1.6] for an integralelement over , can be written as for some

. As for , it followsthat and . So

. As is alwaysan even integer for , we know that .But since is always 2, we must have that

. This implies that. By Lemma 1, we deduce that mod . From

Lemma 2, we know that; hence, , which is a contradiction. Thus,

not all roots of have unit modulus.

APPENDIX GPROOF OFPROPOSITION3

We prove this proposition by using induction on. ClearlyProposition 3 holds true for . We know thatunitary matrices can be written recursively as [21, p. 10]

(22)

where is given by

where is an unitary matrix and theparameter .

By the induction hypothesis, we can write

where is an diagonal unitary matrixand ’s are complex Given matrices.Then, we can write as

where diag

and diag . Hence, we findthat

from (22).

APPENDIX HPROOF OFPROPOSITION4

For any distinct and , we define

where and are the th entries of the vectors and , re-spectively. As , arelinearly independent by F3. We infer that ; so

Moreover, it follows from D4 and F4, that is integralover . But using F5, we have that , whichimplies that . As

we can obtain

(23)

Therefore, by the definition of , we infer that

APPENDIX IPROOF OFPROPOSITION5

For the QAM constellation pointswith (the proof for PAM

is similar) and , we have with. From (23) and the facts that is integral

over and is nonzero, we obtain

(24)

Hence, the coding gain is lower bounded by .On the other hand, let us consider a particular pair with

, where is the first column of the identitymatrix. Using the fact that forany LCP-A matrix in (11), we obtain

This, together with (24), establishes that the coding gain ofLCP-A is exactly


A. Proof of (i)

When , we have by Lemmas 3 and 4.So, the coding gain of LCP-A is given by

which equals the upper bound given in (8) of Proposition 1 with.

B. Proof of (ii)

When , not all roots of have unit modulus as isintegral over by [31, Lemma 1.6]. Hence, we have

by Lemma 4. Therefore, the upper bound of (14) cannotbe achieved by LCP-A for . However, we can select

to be the roots of the minimal polynomialover . From (13), we have

(25)

Because the function is decreasing forand , we have from (25) that

APPENDIX JPROOF OFPROPOSITION6

Let and be its cardinality for. Clearly, . Let be the

roots of and without loss of generality, let us assume thatthe first roots of are from and let

. For any distinct and , we define

As is the degree of the minimal polynomial of the roots fromover , using the assumption that , we infer that

are linearly independent. Hence,and

As

we have

(26)

On the other hand, according to the definition ofand sincefor , we have based on Lemma 3, that

We thus obtain

By (26) and , it follows that:

Using the same argument for all and applying thefact that and , we obtain

APPENDIX KPROOF OFPROPOSITION7

To prove the proposition, we first prove the following lemma.Lemma 5: Consider a deterministic variable and al-

most surely positive random variables, , , , ,that are related by and ,where: i) ; ii) ; andiii) and are bounded. It then holds true that

, for sufficiently large values of .Proof of Lemma 5:For sufficiently large values of, we

have

(27)

(28)

(29)

(30)

where (27) is due to ; inequality (28) is due to i) andiii); (29) is due to ii); and (30) is based on Jensen’s inequalityand the concavity of the function.

Proof of Proposition 7: It suffices to show that in (18) and(19), for sufficiently large values of . This canbe established by applying Lemma 5 with: , and


APPENDIX LPROOF OFPROPOSITION8

Arguing by contradiction, suppose that there are two distinctsymbol vectors and that satisfy (16) in the noise-free case;i.e., that . The latter implies that

. Because cannot be identically zero, there exists anonzero entry denoted as . Because , we deducefrom , that [notice the structure of

in (16)]. Because the minimum product distance is nonzero,we infer from (4) that . Therefore, symbol recovery isguaranteed in the noise free case; and hence,is detectable.

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Yan Xin (S’00) received the B.E. degree inelectronics engineering from Beijing PolytechnicUniversity, Beijing, China, in 1992, the M.Sc.degree in mathematics and the M.Sc. degree inelectrical engineering from University of Minnesota,Minneapolis, in 1998 and 2000, respectively. He iscurrently working toward the Ph.D. degree in theDepartment of Electrical and Computer Engineering,University of Minnesota.

His research interests include space–time coding,diversity techniques, and multicarrier transmissions.

Zhengdao Wang(S’00–M’02) was born in Dalian,China, in 1973. He received the B.S. degree inelectrical engineering and information sciencefrom the University of Science and Technology ofChina (USTC), Hefei, in 1996, the M.Sc. degreein electrical engineering from the University ofVirginia, Charlottesville, in 1998, and the Ph.D.degree in electrical engineering from the Universityof Minnesota, Minneapolis, in 2002.

He is now with the Department of Electrical andComputer Engineering, Iowa State University, Ames,

IA. His interests lie in the areas of signal processing, communications, and in-formation theory, including cyclostationary signal processing, blind equaliza-tion algorithms, transceiver optimization, multicarrier, wideband multiple ratesystems, space–time capacity and coding, and error-control coding.


Georgios B. Giannakis(S’84–M’86–SM’91–F’97)received his Diploma in electrical engineering fromthe National Technical University of Athens, Athens,Greece, in 1981 and the M.Sc. degree in electricalengineering, the M.Sc. degree in mathematics, andthe Ph.D. degree in electrical engineering from theUniversity of Southern California (USC), in 1983 and1986, respectively.

From September 1982 to July 1986, he was withUSC. After lecturing for one year at USC, he joinedthe University of Virginia, Charlottesville, in 1987,

where he became a Professor of Electrical Engineering in 1997. Since 1999,he has been a Professor with the Department of Electrical and ComputerEngineering at the University of Minnesota, Minneapolis, where he now holdsan ADC Chair in Wireless Telecommunications. His general interests spanthe areas of communications and signal processing, estimation and detectiontheory, time-series analysis and system identification—subjects on which hehas published more than 150 journal papers, 290 conference papers, and twoedited books. His current research topics focus on transmitter and receiverdiversity techniques for single- and multiuser fading communication channels,precoding and space–time coding for block transmissions, multicarrier, andwideband wireless communication systems.

Dr. Giannakis is the (co-) recipient of four best paper awards from the IEEESignal Processing Society (1992, 1998, 2000, 2001). He also received the So-ciety’s Technical Achievement Award in 2000. He co-organized three IEEE-Signal Processing Workshops, and Guest Edited (co-edited) four special issues.He has served as Editor-in-Chief for the SIGNAL PROCESSINGLETTERS, Asso-ciate Editor for the IEEE TRANSACTIONS ONSIGNAL PROCESSING, and the IEEESIGNAL PROCESSINGLETTERS, as Secretary of the Signal Processing Confer-ence Board, Member of the Signal Processing Publications Board, Member andVice-Chair of the Statistical Signal and Array Processing Technical Committee,and as Chair of the Signal Processing for Communications Technical Com-mittee. He is a Member of the Editorial Board for the PROCEEDINGS OF THE

IEEE and the Steering Committee of the IEEE TRANSACTIONS ONWIRELESS

COMMUNICATIONS. He is a Member of the IEEE Fellows Election Committee,the IEEE Signal Processing Society’s Board of Governors, and a frequent Con-sultant for the telecommunications industry.

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