Space-Time Wireless Systems: From Array Processing to MIMO Communications · 2015-07-29 · Space-Time Wireless Systems: From Array Processing to MIMO Communications Edited by H

Space-Time Wireless Systems:

From Array Processing to MIMO Communications

Edited by

H. Bolcskei, D. Gesbert, C. Papadias, and A. J. van der Veen

Contents

Part I Multiantenna basics page 1

Part II Space-time modulation and coding 3

10 Space-time coding for noncoherent channels J.-C. Belfiore

and A. M. Cipriano 5

Part III Receiver algorithms and parameter estimation 25

Part IV System-level issues of multiantenna systems 27

Part V Implementations, measurements, prototypes, and stan-

dards 29

Author index 31

iii

Part I

Multiantenna basics

Part II

Space-time modulation and coding

10

Space-time coding for noncoherent channels

Jean-Claude Belfiore and Antonio Maria Cipriano

Department of Communication and Electronics

Ecole Nationale Superieure des Telecommunications

This chapter presents some constructions of non coherent space-time codes,

that is, codes for MIMO systems when the channel is known neither at the

transmitter nor at the receiver. Based on the generalized likelihood ratio

test (GLRT) detector, we introduce some optimization criteria and describe

the obtained codes.

10.1 Introduction

In order to achieve high spectral efficiency on wireless channels, we need

multiple antennas at both transmitter and receiver. Information theoretic

results promise considerable capacity gains for wireless communication sys-

tems that use multiple transmit and receive antennas for coherent and non

coherent reception. Coherent reception means that the receiver knows the

channel response but the transmitter not. Non coherent reception means

that neither the transmitter nor the receiver know the channel response. In

this chapter, we propose to show a non exhaustive presentation of the space-

time codes we can use for the noncoherent case when the communication

system uses M transmit antennas and M receive antennas.

General assumptions as well as notations are the following. We assume

a Rayleigh flat fading channel in order to separate space-time processing

and multipath problems. Moreover, this channel is also assumed quasistatic.

That means that channel coefficients do not vary during the transmission of

a codeword with temporal length T . In that case, the received signal can be

expressed as

YT×N = αXT×M .HM×N + WT×N (10.1)

where X is the transmitted codeword, H is the channel response, W is

the i.i.d. Gaussian noise and α is a normalizing factor. Subscripts indicate

5

6 J.-C. Belfiore and A. M. Cipriano

respective dimensions of the complex-valued matrices. In the following, we

will assume a symmetric communication (M = N) and a code length T ≥2M .

10.2 Results from information theory

10.2.1 General results

First investigations for the power-constrained ergodic capacity in the MIMO

case when the channel is not known by the receiver are due to Marzetta and

Hochwald (1999); Hochwald and Marzetta (2000) and Hassibi and Marzetta

(2002). We report the following fundamental results.

(i) Capacity Dependence on T . For any block length T , any number

of receive antennas N and any SNR ρ, the capacity obtained with

M > T and M = T are equal.

(ii) Optimal signal structure. For each value ρ of the SNR, the capa-

city-achieving signal matrix can be written as

X = ΦΛ, Φ ∈ UT,M i.e., Φ†Φ = IM ,

Λ =

λ1 0 · · · 0

0 λ2 · · · 0...

. . ....

0 0 · · · λM

,

where Φ is a T ×M isotropically distributed unitary matrix† and Λ is

an independent M×M real, nonnegative, diagonal matrix. ((·)† is for

the hermitian transpose). In other words, the optimal signal consists

of M orthogonal vectors whose norms are λm ≥ 0, m = 1, . . . ,M .

The general analytical form of the joint density of [λ1, . . . , λM ] is still

unknown.

(iii) Asymptotic capacity for T → ∞. When T → ∞ the capacity

tends to the capacity of the coherent case with perfect CSI at the

receiver. Moreover, all the density distributions p(λm) converge to a

Dirac delta centered in√

T , showing that the information is carried by

the direction of the vectors and not by their norms (see also (Warrier

and Madhow, 2002)).

† An isotropically distributed unitary matrix has a probability density function that is invariantunder left-multiplication by deterministic unitary matrices.

Space-time coding for noncoherent channels 7

10.2.2 Asymptotic results at high SNR

Zheng and Tse (2002) investigated the ergodic capacity in the high SNR

regime. They proved the following results.

(i) Optimal signal structure. For high SNR ρ, the random variables

λm converge to the deterministic constant√

T , as in the case T 1.

So, once again, the optimal signal structure is a matrix whose columns

are isotropically distributed orthogonal vectors

X = [x1 . . . xM ] =√

T Φ , Φ ∈ UT,M . (10.2)

(ii) The information is carried by subspaces. Equation (10.2) states

that a good codebook should set the information in the directions of

vectors xm, m = 1, . . . ,M . However, Zheng and Tse (2002) shows

that the good strategy is to set the information in the whole subspace

ΩX = span(x1, . . . ,xM ) = span(X) . (10.3)

In other words, at high SNR and for H,W with i.i.d. circularly sym-

metric Gaussian variables, they show that the mutual information is

I(X;Y) = I(ΩX;Y). An intuitive explication is that, at high SNR,

we can write

Y = αXH + W ' αXH (10.4)

and we see that the channel stretches and rotates the basis X of

ΩX. Since the channel is unknown, the receiver cannot recover the

particular basis X, but the subspace ΩX is unchanged.

(iii) Capacity and degrees of freedom. The asymptotic capacity (in

bit per channel use) is

C = Knc log2 ρ + cM,N + o(1) (10.5)

where cM,N is a constant depending on M,N and T , while

Knc = M∗(1 − M∗/T ) with M ∗ = min(M,N, bT/2c) (10.6)

is called the number of degrees of freedom or multiplexing gain of the

noncoherent MIMO ergodic channel (Zheng, 2002). Since log2(ρ) is

the high SNR behavior of a classical AWGN SISO channel, Knc can

be interpreted as the number of parallel spatial channels that can

be used at the same time (Zheng and Tse, 2003). M ∗ is the optimal

number of transmit antennas that should be used to communicate.

Using more that M ∗ transmit antennas does not yield any benefit (in

terms of capacity!).


From the previous observations, we can draw some useful conclusions.

Since Knc is the leading coefficient and a 3 dB SNR increase means an

increase of Knc bit/s/Hz in the capacity, the number of degrees of freedom

should be maximized, this means that M ∗ should be T/2 or close to this

value if possible. Moreover, it should be

T ≥ min (2M, 2N) (10.7)

otherwise the additional antennas will not be useful (from a capacity per-

spective).

10.2.3 Asymptotic results for low SNR

In the low SNR regime, results are quite different. In fact, in general, if

ρ → 0, the coherent and noncoherent capacities are asymptotically equal

and their limit is (Zheng and Tse, 2002)

limρ→0

C(ρ)/ρ = N log2 e , bit/s/Hz . (10.8)

So, the relationship with the number of degrees of freedom vanishes. Only

an increase in the number of receive antennas can increase the capacity in

order to collect the small power of the information signal. Moreover, even if

T and M are larger than one, the optimal strategy consists in allocating all

the transmit power to only one antenna during one symbol period.

10.3 Introduction to subspace representations

As it has been seen in Section 10.2.2, for systems with unknown channel,

at high SNR, the information is substantially carried by subspaces. We will

recall here some basic definitions on subspaces.

10.3.1 Basics on subspaces

Let ΩX be an M -dimensional (vector) subspace of CT , with T > M . Given

one of its bases X, we recall (10.3)

ΩX = span(X) with rank(X) = M . (10.9)

The basis X is not unique. In fact, for any nonsingular M × M complex

matrix A, another valid basis of the same subspace is XA. The thin sin-

gular value decomposition† (TSVD) (Golub and Loan, 1996, p. 72) gives

† When T = M , the thin singular value decomposition becomes the common singular valuedecomposition, where all matrices in (10.10) are square.


interesting insights in the structure of a generic basis

X = VΛU† , V ∈ UT,M , U ∈ UM , Λ = diag([λ1 . . . λM ]) (10.10)

where UT,M is the set of the T × M complex matrices with orthonormal

columns, UM is the group of unitary M × M matrices and Λ is a diagonal

matrix whose entries are positive and ordered in decreasing order. A brief

summary is

V†V = IM , U†U = IM , λ1 ≥ λ2 ≥ . . . ≥ λM . (10.11)

10.3.2 Basics on the Grassmann manifold

We give the following basic definition

Definition 10.1 The set of all the M -dimensional complex (real) vector

subspaces ΩX of CT (RT ), with T > M is called the Grassmann manifold

or Grassmannian. It is denoted by GT,M .

The concepts of biorthonormal bases and principal angles is very impor-

tant to characterize a couple of subspaces (Golub and Loan, 1996, p. 603).

Definition 10.2 Given two subspaces ΩX,ΩY ∈ GT,M , two corresponding

bases X and Y are said to be biorthonormal if they are orthonormal and

x†mym′ = 0 for all m 6= m′ , x†

mym = cm with 0 < cm ≤ 1 (10.12)

i.e., for all m = 1, . . . ,M , the mth vector of the first basis is orthogonal to

all the vectors of the other basis except the mth one.

A pair of biorthonormal bases can be obtained by means of the SVD decom-

position of any two orthonormal bases. Let X,Y be two orthonormal but

not necessarily biorthonormal bases of ΩX,ΩY and let

X†Y = UXCU†Y , UX ,UY ∈ UM , C = diag([c1, . . . , cM ]) (10.13)

where 0 < cm ≤ 1 for all m. Then two biorthonormal basis are XUX and

Y UY .

Definition 10.3 Given two biorthonormal bases of the subspaces ΩX,ΩY ∈GT,M , the real positive inner products cm are uniquely written in the follow-

ing form

cm = cos θm , θm ∈ [0, π/2) , m = 1, . . . ,M . (10.14)

θ1, . . . , θM are called the principal angles between subspaces ΩX and ΩY.


Principal angles are unique, while the biorthonormal bases are not (for

example a permutation of two biorthonormal bases gives again two valid

biorthonormal bases).

Definition 10.4 Two subspaces ΩX,ΩY ∈ GT,M are called intersecting

subspaces if the dimension of their intersection is non zero, i.e., dim(ΩX ∩ΩY) > 0. Otherwise they are called nonintersecting.

When two M -dimensional subspaces are intersecting, the dimension of their

intersection is the number of principal angles equal to zero. In this case, some

vectors of the two biorthonormal bases coincide: they span the intersection

(Golub and Loan, 1996, p. 604).

Several distances between subspaces can be defined over the Grassman-

nian (Edelman et al., 1998; Barg and Nogin, 2002). Here we recall the mostly

used ones.

Definition 10.5 Let θ = (θ1, . . . , θM ). Let X and Y two orthonormal bases

of two different subspaces and let X†Y = UXCU†Y the SVD as in (10.13)

with C = cos θ. Then we can define the following distances

(i) The geodesic distance

darc(ΩX,ΩY) = ‖θ‖ =

(M∑

m=1

θ2m

)1/2

(10.15)

(ii) The chordal distance

dc(ΩX,ΩY) = ‖ sin θ‖ =

(M∑

m=1

sin2 θm

)1/2

(10.16)

Another pseudodistance called product distance is often used in the literature

in the case of the MIMO Rayleigh fading channel. It is

dp(ΩX,ΩY) =

(M∏

m=1

sin θm

)1/M

(10.17)

In the following, we will sometimes use the notation d(X,Y), meaning

d(ΩX,ΩY).


10.4 Detection criteria

10.4.1 The noncoherent ML criterion

When the channel statistics of the fading and the noise are known at the

receiver (not their realization), the maximum likelihood (ML) criterion can

be used for noncoherent detection (Proakis, 2000).

With these assumptions, the ML detector is a quadratic receiver and can

be stated as

XML = arg mini=1,...,L

[−Y†FiY + ci] , (10.18)

where

Fi =1

σ2Xi(

σ2

α2INM + X†

iXi)−1X†

i , ci = ln

∣∣∣∣σ2

α2INM + X†

iXi

∣∣∣∣ . (10.19)

L is the code size and α is the normalization factor defined in eq. (10.1).

10.4.2 The GLRT

The GLRT requires neither the knowledge of the fading and noise statis-

tics, nor the knowledge of their realizations (Warrier and Madhow, 2002;

Lapidoth and Narayan, 1998). It is defined as

XGLRT = arg maxi=1,...,L

supH

p(Y|Xi,H). (10.20)

The criterion simplifies in

XGLRT = maxi=1,...,L

y†Xi(X†iXi)

−1X†iy . (10.21)

or equivalently

XGLRT = maxi=1,...,L

tr[Y†Xi(X

†iXi)

−1X†iY]. (10.22)

From (10.21) or (10.22) we see that the GLRT projects the received sig-

nal Y on the different subspaces ΩXi and then calculates the energies of

these projections and chooses the projection that maximizes this energy

(see Fig. 10.1).

From the perspective of the average supersymbol error probability min-

imization, in general, the GLRT gives a suboptimal result with respect to

the ML criterion. However, the GLRT independence on any kind of fading

information makes it an excellent detection rule candidate when the receiver

cannot estimate channel correlations or when the channel has variable statis-

tics.

In the case of i.i.d. fading and unitary codebook the ML and the GLRT


y

a) b)

y

Fig. 10.1. a) The GLRT chooses the subspace with the highest projection energy.b) the coherent ML chooses the closest point.

criteria are equivalent (it trivially comes from (10.19)). In this case the

decision is made according

X = maxi=1,...,L

tr(Y†XiX†iY) = max

i=1,...,L‖Y†Xi‖2

F (10.23)

for which the remarks of (10.23) holds as well.

10.5 Error probability bounds

Let Pij be the Pairwise Error Probability (PEP) between the two codewords

Xi and Xj , i 6= j. The expression of Pij gives useful indications on how to

design the code.

10.5.1 Unitary codebooks

In the case of unitary codebooks and when the ML criterion and GLRT

are equivalent, Hochwald and Marzetta (2000) reports an exact closed-form

analytical expression of the PEP Pij as well as a Chernoff bound that only

depends on the principal angles between ΩXi and ΩXj .

10.5.2 Nonunitary codebooks

In the case of nonunitary codebooks or correlated fading, the noncoherent

ML criterion and the GLRT do not coincide. We give here the asymptotic


expression of the PEP for the GLRT criterion, Let[X†

i

X†j

][Xi Xj

]=

[Rii Rij

Rji Rjj

](10.24)

and assume that the matrix in (10.24) has full rank, (hence T ≥ 2M), the

asymptotic expression is (Brehler and Varanasi, 2001)

P∞ij,GLRT =

[M

Tρ

]NM (2MN − 1

MN

)(1 + |Rii|

|Rjj |

)

|Rii −RijR−1jj Rji|N

(10.25)

Brehler and Varanasi (2001) shows that if the fadings are correlated h ∼CN (0,Kh), the expression (10.25) must be multiplied by a scalar factor

equal to 1/|Kh|.Finally, Brehler and Varanasi (2001) shows that under the assumption of

equal-energy codewords (tr(X†iXi) = P, ∀ i = 1, . . . , L), unitary codebooks

are optimal from an asymptotic PEP minimization perspective. Hence at

high SNR the same signal structure is optimal from a capacity point of view

and from a PEP point of view.

10.6 Diversity for the noncoherent case

In the literature we can find three different definitions of diversity for non-

coherent MIMO block fading systems.

10.6.1 PEP-based diversity

The most widespread and classical definition of diversity, which can be used

for coherent system too (Tarokh et al., 1998), is based on the asymptotic

Pairwise Error Probability or on its Chernoff Bound

Definition 10.6 (PEP-based Definition) Let Pij(ρ) be the Pairwise Er-

ror Probability (P CBij (ρ) be the Chernoff Bound) between the codewords Xi

and Xj, as a function of the SNR ρ. Let Xi and Xj belong to a codebook Cof size L. The codebook C is said to achieve the diversity gain d (or briefly

to have diversity d) iff

mini,j:i6=j

limρ→∞

lnPij(ρ)

ln ρ= −d ,

(min

i,j:i6=jlim

ρ→∞

lnP CBij (ρ)

lnρ= −d

). (10.26)

Since the exponent of the SNR is Nmd,ij we can say that

N ≤ d = N md ≤ MN , md = mini,j:i6=j

md,ij (10.27)


where md,ij is the number of nonzero principal angles between subspaces

ΩXi and ΩXj . When the subspaces are all non intersecting then d reaches

its maximum NM and the codebook is called full-diversity code. A necessary

condition to have a fully diverse code is T ≥ 2M . It can be shown that the

minimization of the PEP is equivalent to the maximization of the minimum

product distance (10.17).

In Brehler and Varanasi (2001) the following proposition is proved, which

holds for every kind of codebook.

Proposition 10.1 If for all couples of codewords Xi and Xj, i 6= j, belong-

ing to C, matrices

[X†

i

X†j

][Xi Xj

]=

[Rii Rij

Rji Rjj

](10.28)

have full rank, then the codebook C achieves full PEP-based diversity. How-

ever, it is necessary that T ≥ 2M .

10.6.2 Error probability-based diversity

A definition, which relates the diversity to the concept of multiplexing gain

(see Section 10.2.2) is given by Zheng and Tse (2003); Zheng (2002). This

definition is based on the average supersymbol error probability and not

on the PEP, moreover its is defined for family of codes whose rate scales

logarithmically with the SNR. This kind of definition is useful to study the

tradeoff between diversity and multiplexing gain.

10.6.3 Algebraic diversity

A definition based on some algebraic properties of the codebooks is intro-

duced by El Gamal et al. (2003). We state it in the case of no coding between

different fading blocks and we give a slightly different but equivalent defi-

nition. Let us suppose one receiving antenna (N = 1) and the absence of

additive noise. Let us define the subspace of channel realizations Hnc(i, j)

that makes the GLRT unable to distinguish between two possible transmit-

ted symbols Xi and Xj as

Hnc(i, j) = h ∈ CM : ∃h1 ∈ C

M , Xih = Xjh1 (10.29)

= h ∈ CM : ∃ h ∈ ΩXi ∩ ΩXj , where h = Xih (10.30)


Definition 10.7 (Algebraic Diversity Gain) The codebook C is said to

achieve the algebraic diversity gain d if

d = N [M − maxi,j:i6=j

dim Hnc(i, j)] = N mini,j:i6=j

[dim(ΩXi + ΩXj ) − dim ΩXj ]

(10.31)

There does not exist a formal proof of the equivalence between the classi-

cal PEP-based diversity and the algebraic diversity, for generic codebooks.

However it has been proved (El Gamal et al., submitted for publication,

2003) the following

Proposition 10.2 If the codebook C is unitary, the algebraic diversity and

the PEP-based diversity are equivalent.

In the general case, it is also clear that when the codebook has full al-

gebraic diversity, then it has also full PEP-based diversity, because with

this assumption, matrices in (10.24) have full rank and Proposition 10.1 can

apply.

10.7 Code design criteria and propositions

Both information theoretical criteria (see (10.2)) and error probability crite-

ria show that, at high SNR, the optimal signals are matrices with orthonor-

mal columns, most research concentrated on codes of this type. However

many propositions, in the literature, present both unitary and nonunitary

codebooks. These propositions differ in code design methods and criteria

and hence in decoding methods. The main ones are

• Codebooks designed by numerical minimization of some cost function re-

lated to the distances of Definition 10.5 (Hochwald and Marzetta, 2000;

Agrawal et al., 2001; Gohary and Davidson, 2004) or by numerical min-

imization of the union bound on the supersymbol/bit error probability

(McCloud et al., 2002; Brehler and Varanasi, 2003) or on the Kullback-

Leibler distance (Borran et al., 2003), an information-theoretic criterion

(Cover and Thomas, 1991).

• Codebooks obtained by some parameterization of unitary matrices (Hoch-

wald et al., 2000; Jing and Hassibi, 2003; Wang et al., submitted for publi-

cation, 2004) or of the Grassmann manifold (Kammoun, 2004; Kammoun

and Belfiore, 2003).

• Codebooks obtained by algebraic construction for some particular cases

(Tarokh and Kim, 2002; Zhao et al., 2004; Oggier et al., submitted for

publication 2003).


• Codebooks that follow the so-called training-based format, i.e., that esti-

mate the channels in the first part of the supersymbols and use the second

part to send information by means of a space-time code designed for co-

herent detection (Brehler and Varanasi, 2003; Dayal et al., 2004; El Gamal

and Damen, 2003; El Gamal et al., 2003, submitted for publication, 2003).

In the following we will report briefly the various propositions with advan-

tages and drawbacks. Some characteristics of these codes are summarized in

Table 10.1.

10.7.1 Numerical optimization designs

These propositions differ in their cost functions and in their (often subopti-

mal) minimization methods. They suffer from common shortcomings:

(i) only low size constellations can be constructed, because of the in-

creasing complexity in the design process;

(ii) the codebook of size L = 2RT has to be stored in the transmitter and

receiver equipments and so, the required memory is exponential in

RT . (R is the transmit rate expressed in bits per symbol period);

(iii) in general no simplified decoding algorithm is available, so that the

GLRT or ML rule must be evaluated for all codewords. Hence, the

decoding complexity is in general exponential in RT .

Small (L ≤ 64) unitary space-time constellations were designed in Hoch-

wald and Marzetta (2000). In Agrawal et al. (2001) the minimum chordal

distance is used as a cost function

d2c,min = min

1≤l<l′≤Ld2

c(Ω(Xl),ΩXl′) = min

1≤l<l′≤L

M∑

m=1

sin2 θm,ll′ (10.32)

where θm,ll′ are the M principal angles between the two subspaces generated

by Xl and Xl′ (see (10.16)). The minimum chordal distance can be related

to the worst case upper bound of the Chernoff Bound on the PEP, a quite

loose approximation of the PEP. Moreover, d2c,min is a good approximation

of the product distance (10.17) only when all subspaces are quasiorthogonal

(Hochwald et al., 2000), an assumption that is not true even for small-size

codebooks when T is comparable to M . The optimization technique used

in Agrawal et al. (2001) is called relaxation method and it generalizes from

Conway et al. (1996) where it was used for real Grassmannians.


Ref. nc. codea designmethod

co. codeb diversity dec.c dec. compl.

Agrawal et al.

(2001)U num. min.

d2

c,min

no nocontrol

no O(2RT )

Gohary andDavidson(2004)

U num. min.d2

cF,min

no nocontrol

S local GLRT

McCloud et al.

(2002); Brehlerand Varanasi(2003)

U num. min.Pe/Pe,bit

no full no O(2RT )

Borran et al.

(2003)N/U num. min. no no

controlnoe O(2RT )

Hochwaldet al. (2000)

U successiverotations

no nocontrol

no O(2RT )

Jing andHassibi (2003);Wang et al.

(submitted forpublication,2004)

U CayleyTransf.

no nocontrol

S sphere dec.

Kammoun andBelfiore (2003)

U exponentialtransf.

yes full con-jectured

S local GLRT

Tarokh andKim (2002)

U algebraic/training

yesd full O O(MN)/O(M2N)

Zhao et al.

(2004)U algebraic/

trainingyes full O O(2TR/2)

Oggier et al.

(submitted forpublication2003)

N/U algebraic no full no O(2TR)

Brehler andVaranasi(2003); Dayalet al. (2004);El Gamal andDamen (2003);El Gamalet al. (2003)

N/U training yes full S sphere dec.

a noncoherent code: U = unitary codebooks, N/U = all kind of codebooks.b coherent code: is the noncoherent code built from a coherent code?

c There exists a simplified decoding? S = yes, but is it suboptimal, O = yes and it is optimal(with respect to the noncoherent ML or GLRT).

d also a proposition not based on coherent codes is presented.e unitary constellations with simplified decoding can be used, however its not the general case.

Table 10.1. Summary of the most quoted propositions in the literature.


Gohary and Davidson (2004) uses another metric, the so-called chordal

Frobenius distance (Edelman et al., 1998)

d2cF (Ω(Xl),ΩXl′

) = 4M∑

m=1

sin2(θm,ll′/2) < d2c(ΩXl

,ΩXl′)

looser than the chordal distance. A simplified decoding method is proposed

that, from some reference point on the Grassmannian, locates a list of candi-

date points over which the GLRT is calculated (local GLRT). This procedure

makes it possible not to calculate the GLRT metric for all the codewords.

However, tables with coordinates of all codewords must be saved in the

memory as well as the codebook, which has no algebraic structure.

In McCloud et al. (2002) and Brehler and Varanasi (2003), constellations

are obtained through numerical search minimizing the asymptotic union

bound on the supersymbol error rate (also called FER—Frame Error Rate)

and the asymptotic union bound on the bit error rate (BER). While the

common drawbacks persist, the advantage of this method is that, being

based on the PEP, it guarantees that the constructed constellation has full

diversity when T ≥ 2M .

Motivated by the fact that unitary codebooks are not optimal at low

SNR or for T comparable to M , Borran et al. (2003) designs nonunitary

codebooks with the Kullback-Leibler distance† criterion. This method has

been proposed because of the intractability of the PEP-based design criterion

when the unitarity assumption for the codewords is no more true.

10.7.2 Parameterization designs

A more structured approach was used in Hochwald et al. (2000), where

an initial matrix generates the whole constellation by successive rotations‡.The parameters of the codebook are chosen by a random search in order

to maximize the minimal chordal distance dc,min as in (10.32). However no

simplified decoding algorithm is reported.

In Jing and Hassibi (2003), a method that constructs unitary codebooks

† The Kullback-Leibler distance between two distributions p1(x) and p2(x) is defined as (Coverand Thomas, 1991)

D(p1‖p2) =

Z

xp1(x)(ln p1(x) − ln p2(x)dx .

‡ In Marzetta et al. (2002) constellations with better statistical properties and spectral efficienciesare constructed. However, the absence of a simplified decoder prevented the authors fromverifying their performances.


is described. The codeword X is obtained via the Cayley transform

X = (IT + jA)−1(IT − jA)

[IM

0

], (10.33)

where the Hermitian matrix A is calculated from a set of fixed Hermitian

matrices Aq as A =∑Q

q=1 αqAq. To ensure the unitarity of X, coefficients

αq are real scalars belonging to a discrete set A whose cardinality is r.

Even if transform (10.33) is invertible (for all Hermitian matrices A with

no eigenvalue equal to −1), the Cayley Transform is nonlinear. The authors

constrain some entries of the set of matrices Aq, the scalar Q (related

to the rate of the system R = Q log2(r)/T ) and make an approximation

on the ML rule so that they get a simplified suboptimal decoding problem

that can be solved via the sphere decoder algorithm (Viterbo and Boutros,

1999). Many variables involved in the design of the code are found by nu-

merical optimization and the set of matrices Aq is chosen to maximize

a criterion that guarantees the diversity for differentially encoded unitary

codebooks (see Hassibi and Hochwald (2002)) but not for noncoherent sys-

tems. Successively, in Wang et al. (submitted for publication, 2004) other

optimization methods are proposed to enhance performances.

Another method based on a parameterization is the one proposed by Kam-

moun and Belfiore (2003). The unitary codewords are obtained via the expo-

nential parameterization or map from a subset of skew-Hermitian matrices

(A = −A†).

X = exp(A)IT,M = exp

[0M −B†

B 0T−M

][IM

0

], (10.34)

where matrices B must satisfy some conditions. In fact matrices B are code-

words from coherent space-time codes (found in El Gamal and Damen (2003)

for example), linearly scaled by a positive real factor αo, which is also called

homothetic factor, and which is the only parameter to optimize for a fixed

codebook. The diversity of the unitary codebook is not simple to link with

the diversity of the coherent code; a conjecture on this topic is proposed in

Kammoun (2004). Codes built in this way can have high spectral efficien-

cies, just by choosing the appropriate coherent code. Only the homothetic

factor must be optimized and not a large number of parameters as in Jing

and Hassibi (2003) or McCloud et al. (2002).


10.7.3 Algebraic designs

These codes are quite different from each other, but they share the same

property: it is possible to control the code parameters thanks to a quite

constraining and powerful algebraic structure.

In Tarokh and Kim (2002) two constructions are presented. The first one,

called generalized PSK constellations, can be used for T = 2M and it can

be written as

Xl =

[cos(φl)IM

sin(φl)IM

], φl = lπ/L, l = 0, 1, . . . , L − 1 . (10.35)

where L is the size of the code. An ML decoder exists whose complexity

is only O(MN), independent of the duration T of the frame and the rate

R = log2(L)/T . However, this is achieved by imposing a strong structure

(10.35) that only exploits M real degrees of freedom among the possible

2M(T − M) real degrees of freedom of the system. Principal angles be-

tween two given codewords Xk and Xl are all the same and are equal to

π(k − l)/2TR. Even if the constellation has full diversity, the minimal prod-

uct distance (10.17) is dp,min = sin(π/L) = sin(π/2TR) and exponentially

decreases with the rate or the duration, so that this method is only efficient

for low spectral efficiencies. A generalization, named complex Givens codes is

proposed (Dayal et al., 2004), which exploits 2M among the 2M(T −M) de-

grees of freedom and only doubles the decoding complexity of the generalized

PSK codes. These codes can also be obtained by the exponential parame-

terization. The second proposition in Tarokh and Kim (2002) is based on

the coherent space-time orthogonal designs (Tarokh et al., 1999) and can be

described in the framework of the training-based codes (Dayal et al., 2004).

Zhao et al. (2004) presents some unitary codes that derive from the or-

thogonal designs, and can be seen as training based codes. They also present

a simplified decoder to perform the ML detection with complexity O(2RT/2),

instead of O(2RT ).

Finally in Oggier et al. (submitted for publication 2003), an investigation

about the maximum number of non intersecting subspaces can be found,

under the condition that the codewords’ entries come from fixed small con-

stellations. The problem is solved when these constellations coincide with

the Galois field GF (q), where q is a power of some prime integer. In this case,

the maximum number of nonintersecting subspaces is (qT −1)/(qM −1). An

upper and lower bound are given in the case of PSK constellations, where the

number of transmit antennas is M = 2. An encoder as well as a simplified

decoder have not yet been proposed.


10.7.4 Training based schemes

The codes

Recently, the research community has carried a growing interest in the so

called training-based schemes (Brehler and Varanasi, 2003; Dayal et al.,

2004; El Gamal and Damen, 2003; El Gamal et al., 2003). In this approach,

each block is divided into two parts, respectively of Tt and Td channel uses

(Tt + Td = T ). In the first Tt channel uses, a pilot signal, known to the

transmitter and the receiver, is sent to get a rough estimation of the channel

(training phase). The remaining Td channel uses are used to send informa-

tion, usually encoded via some coherent space-time code B. So, typically,

codewords are

X =

[ √τT√

1 − τB

], T is the pilot Tt × M matrix, B ∈ B is Td × M

(10.36)

where τ ∈ (0, 1) is a scalar that assigns different transmit power ratios to

the training part. Naturally, the codewords can extend to several blocks of

the fading channel (El Gamal et al., 2003).

The channel estimation performed in the training phase is in a certain

sense quite unusual in estimation practice. In fact, for these noncoherent

systems, channel estimation is performed without having different estimates

of the channel coefficients, but just one. This is due to the statistical inde-

pendence of channel coefficients from one block to the other ones. Moreover,

the block length is so short that repeating the estimation process within the

same block would cause an unacceptable decrease of the spectral efficiency.

In the literature, training for these systems has been ignored for some years

(as remarked by Dayal et al. (2004)) probably for the previous two reasons.

However, since their introduction, training based codes seem to be one of

the best competitors. The advantages of this approach are:

• theory and design knowledge on space-time codes for the coherent channel

(called also coherent space-time codes) can be reused;

• it stems from the previous consideration that simplified decoding tech-

niques of coherent space time codes can be used to decode training based

codes;

• training based codes as in (10.36) achieve the diversity (in the PEP sense)

of the underlying coherent space code B, when B is full-rank and the

fadings are i.i.d. Gaussian random variables (Dayal et al., 2004).


Simplified decoding

Let the received signal be

Y =

[Yt

Yd

]=

[ √τT√

1 − τB

]H +

[Wt

Wd

]. (10.37)

where the channel coefficients are i.i.d. CN (0, 1) and each complex compo-

nent of the additive noise is CN (0, σ2).

The simplified receiver for training based symbols performs two operations

(i) the receiver estimates the channel coefficients via a minimum mean

square error (MMSE) estimator (Hassibi and Hochwald, 2003) from

the signal received during the first M channel uses

H =√

τ(σ2IM + τT†T)−1T†Yt . (10.38)

(ii) the receiver treats the channel estimation as if it was perfect and de-

codes with the coherent ML rule the signal received in the remaining

T − M symbol periods

B = arg minl=1,...,L

‖Yd −BlH‖2F (10.39)

The rule (10.39) corresponds to finding the closest point to a given point

vec(Yd) of CM(T−M). This problem can be efficiently solved via the so-

called sphere decoder algorithm (Viterbo and Boutros, 1999). There exist

different search strategies (see Agrell et al. (2002) and references therein).

We recall the Pohst strategy, which scans the points inside a hypersphere of

fixed radius and when it find a points decrease the radius, and the Schnorr-

Euchner strategy, which searches, as well, in a hypersphere but scans the

points in a different order (i.e., it first searches for points in the nearest

hyperplanes inside the sphere).

References

Agrawal, D., Richardson, T. J., and Urbanke, R. L. (2001). Multiple-antenna signalconstellations for fading channels. IEEE Trans. Inform. Theory, 47 (6), 2618–2626.

Agrell, E., Eriksson, T., Vardy, A., and Zeger, K. (2002). Closest Point Search inLattices. IEEE Trans. Inform. Theory, 48 (8), 2201–2214.

Barg, A. and Nogin, D. Y. (2002). Bounds on packings in the Grassmann manifold.IEEE Trans. Inform. Theory, 48 (9), 2450–2454.

Borran, M. J., Sabharwal, A., and Aazhang, B. (2003). On design criteria andconstruction of noncoherent space-time constellations. IEEE Trans. Inform.Theory, 49 (10), 2332–2351.


Brehler, M. and Varanasi, M. K. (2001). Asymptotic error probability analysisof quadratic receivers in Rayleigh–fading channels with applications to a uni-fied analysis of coherent and noncoherent space–time receivers. IEEE Trans.Inform. Theory, 47 (9), 2383–2399.

—— (2003). Training-codes for the noncoherent multi-antenna block-Rayleigh-fading channel. In Proc. of the Conf. Inform. Sciences and Systems (JohnsHopkins University).

Conway, J. H., Hardin, R. H., and Sloane, N. J. A. (1996). Packing Lines, Planes,etc.: Packings in Grassmannian Spaces. Experimental Mathematics, 5 (2), 139–159.

Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory (Wiley,New York).

Dayal, P., Brehler, M., and Varanasi, M. K. (2004). Leveraging coherent space-time codes for noncoherent communication via training. IEEE Trans. Inform.Theory, 50 (9), 2058–2080.

Edelman, A., Arias, T. A., and Smith, S. T. (1998). The Geometry of Algorithmswith Orthogonality Constraints. Siam J. Matrix Anal. Appl., 20 (2), 303–353.

El Gamal, H., Aktas, D., and Damen, M. O. (2003). Coherent Space–Time Codes forNoncoherent Channels. In Proc. IEEE Global Telecommunications Conference(San Francisco, CA).

—— (submitted for publication, 2003). Noncoherent Space–Time Coding: an Al-gebraic Perspective. IEEE Trans. Inform. Theory.

El Gamal, H. and Damen, M. O. (2003). Universal Space–Time Coding. IEEETrans. Inform. Theory, 49 (5), 1097–1119.

Gohary, R. H. and Davidson, T. N. (2004). Non-coherent MIMO communication:Grassmannian constellation and efficient detection. In Proc. of the Interna-tional Symposium of Information Theory 2004 (ISIT2004), 65 (Chicago, USA).

Golub, G. H. and Loan, C. F. V. (1996). Matrix Computations (Johns HopkinsUniversity Press), third edn.

Hassibi, B. and Hochwald, B. M. (2002). Cayley differential unitary space–timecodes. IEEE Trans. Inform. Theory, 48 (6), 1485–1503.

—— (2003). How much training is needed in multiple-antenna wireless links? IEEETrans. Inform. Theory, 49 (4), 951–963.

Hassibi, B. and Marzetta, T. L. (2002). Multiple–antennas and isotropically randomunitary inputs: the received signal density in closed form. IEEE Trans. Inform.Theory, 48 (6), 1473–1484.

Hochwald, B. M. and Marzetta, T. L. (2000). Unitary space–time modulation formultiple-antenna communications in Rayleigh flat fading. IEEE Trans. Inform.Theory, 46 (2), 543–564.

Hochwald, B. M., Marzetta, T. L., Richardson, T. J., Sweldens, W., and Urbanke,R. (2000). Systematic design of unitary space–time constellations. IEEE Trans.Inform. Theory, 46 (6), 1962–1973.

Jing, Y. and Hassibi, B. (2003). Unitary Space–Time Modulation via Cayley Trans-form. IEEE Trans. Signal Processing, 51 (11), 2891–2904.

Kammoun, I. (2004). Codage Spatio–temporel sans connaissance a priori du canal.Ph.D. thesis, Ecole Nationale Superieure des Telecommunications, Paris.

Kammoun, I. and Belfiore, J.-C. (2003). A new family of Grassmannian space-timecodes for non-coherent MIMO systems. IEEE Commun. Lett., 7 (11), 528–530.

Lapidoth, A. and Narayan, P. (1998). Reliable Communications Under ChannelUncertainty. IEEE Trans. Inform. Theory, 44 (6), 2148–2177.


Marzetta, T. L., Hassibi, B., and Hochwald, B. M. (2002). Structured unitaryspace–time autocoding constellations. IEEE Trans. Inform. Theory, 48 (4),942–950.

Marzetta, T. L. and Hochwald, B. M. (1999). Capacity of a mobile multiple–antenna communication link in Rayleigh flat fading. IEEE Trans. Inform.Theory, 45 (1), 139–157.

McCloud, M. L., Brehler, M., and Varanasi, M. K. (2002). Signal design andconvolutional coding for noncoherent space-time communication on the block-Rayleigh-fading channel. IEEE Trans. Inform. Theory, 48 (5), 1186–1194.

Oggier, F., Sloane, N., Calderbank, A., and Diggavi, S. (submitted for publication2003). Nonintersecting Subspaces Based on Finite Alphabets. IEEE Trans.Inform. Theory. URL http://www.research.att.com/~njas/.

Proakis, J. G. (2000). Digital Communications (McGraw-Hill), 4th edn.Tarokh, V., Jafarkhani, H., and Calderbank, A. R. (1999). Space–time block codes

from orthogonal–designs. IEEE Trans. Inform. Theory, 45 (5), 744–765.Tarokh, V. and Kim, I.-M. (2002). Existence and construction of noncoherent

unitary space-time codes. IEEE Trans. Inform. Theory, 48 (12), 3112–3117.Tarokh, V., Seshadri, N., and Calderbank, A. R. (1998). Space–time codes for high

data rate wireless communications: Performance criterion and code construc-tion. IEEE Trans. Inform. Theory, 44 (2), 744–765.

Viterbo, E. and Boutros, J. (1999). A universal lattice decoder for fading channels.IEEE Trans. Inform. Theory, 45 (5), 1639–1642.

Wang, J., Wang, X., and Madihian, M. (submitted for publication, 2004). Optimumdesign of noncoherent Cayley unitary space–time codes. IEEE Trans. WirelessCommun.

Warrier, D. and Madhow, U. (2002). Spectrally efficient noncoherent communica-tion. IEEE Trans. Inform. Theory, 48 (3), 651–668.

Zhao, W., Leus, G., and Giannakis, G. (2004). Orthogonal design of unitary constel-lations for uncoded and trellis coded non–coherent space–time systems. IEEETrans. Inform. Theory, 50 (6), 1319–1327.

Zheng, L. (2002). Diversity-Multiplexing Tradeoff: A Comprehensive View of Mul-tiple Antenna Systems. Ph.D. thesis, University of California, Berkeley. URLhttp://web.mit.edu/lizhong/www/.

Zheng, L. and Tse, D. N. C. (2002). Communication on the Grassmann manifold:A geometric approach to the noncoherent multiple–antenna channel. IEEETrans. Inform. Theory, 48 (2), 359–383.

—— (2003). Diversity and multiplexing: a fundamental tradeoff in multiple–antenna channels. IEEE Trans. Inform. Theory, 49 (5), 1073–1096.

Part III

Receiver algorithms and parameter estimation

Part IV

System-level issues of multiantenna systems

Part V

Implementations, measurements, prototypes, andstandards

Author index

31

Index

algebraic diversity, 15

biorthonormal bases, 9

Cayley transform, 19chordal distance, 10codebook, 7

degrees of freedom, 7diversity gain, 13

exponential parameterization, 19

full diversity, 14

geodesic distance, 10Givens codes, 20GLRT, 11Grassmann manifold, 9Grassmannian, 9

multiplexing gain, 7

pairwise error probability, 12principal angles, 9product distance, 10

Rayleigh, 5

sphere decoder, 19subspace, 7

training-based, 16

unitary codebook, 11

32

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