Download Constellations Matched to the Rayleigh Fading Channel

106
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 42, NO 1, JANUARY 1996
oristellations atehed to
e Rayleig Fading C
Xavier Giraud and Jean Claude Belfiore, Member, IEEE
Abstract- We introduce a new technique for designing signal
sets matched to the Rayleigh fading channel. In particular, we
look for n-dimensional ( n 2 2) lattices whose structure provides nth-order diversity. Our approach is based on a geometric
formulation of the design problem which in turn can be solved
by using a number-geometric approach. Specifically, a suitable
upper bound on the pairwise error probability makes the design
problem tantamount to the determination of what is called a
critical lattice of the body
The lattices among which we search for an optimal solution are
the standard embeddingsin R" of the numlber ring of some totally
real number field of degree n over &. Simiulation results confirm
that this approach yields lattices with considerable coding gains.
Index Terms-Rayleigh fading channel, diversity, lattices, number fields.
I. INTRODUCTION
N SIGNAL selection for digital transimission, considerable
attention has recently been devoted to multidimensional
constellations [ I]-[3]. In particular, lattice signaling is deemed
to provide an excellent tradeoff between performance and
implementation complexity. The idea is to carve a finite set of
signals C out of a lattice A.
Virtually all the literature on applications of lattices has
focused on the Gaussian channel, the relevant mathematical
tool being sphere-packing theory [4], [5]. Here we introduce
new mathematical tools for the design of a lattice matched to
a given channel. The lattice design problem is formulated as
a number-geometric problem. For the Glaussian channel, the
sphere packing problem stems as an example whereas for the
Rayleigh fading channel, we are led to determine what number
geometry recognizes as dense admissible lattices of
Manuscript received November 22, 1994; revised July 12, 1995. The material in this paper was presented at the 1993 IEEE International Symposium
on Information Theory, San Antonio, TX.
The authors are with Ecole Nationale SupCrieure des TClCcommunications
(ENST), Paris, France.
Publisher Item Identifier S 0018-9448(96)00028-4.
This approach quantitatively establishes properties that a lattice should have on the Rayleigh fading channel and why the
packing formulation does not apply.
Making use of diversity is of prime importance in a fading
environment. Explicit antenna diversity and implicit time
diversity from interleaved trellis-coded modulation are methods of transmission quality enhancement that can be jointly
employed without loss of power and bandwidth. In this work,
we focus on time diversity. For standard signal constellations
like PSK or QAM, there is at least one signal pair having
all coordinates but one identical. This is known as orderone diversity. The effort to go beyond this limitation includes
two-dimensional signal sets construction (QAM or PSK) and
trellis-coded modulation schemes designed to maximize the
number of positions where the transmitted and erroneous
sequences differ. The diversity order, i.e., the number of
different signal coordinates, becomes the slope of the error
probability curve, leading to substantial performance improvements. Most of the proposed systems combined PSK and
TCM codes; they are restricted to low spectral efficiency
( 51.5 bits/dimension) because the optimization criterion for
the codes search is too difficult to handle. However, as we
emphasized above, multidimensional M-QAM modulations
schemes suffer significant degradation over the fading channels
as a consequence of their poor built-in time diversity. We show
in Section 111 that the S-admissibility of a lattice is desirable
because it guarantees that any constellation carved out of A
offers an nth-order diversity on the Rayleigh fading channel.
Use of number-field theory allows us to build a family
of S-admissible lattices. In fact, it is well known that the
standard embeddings in Rn of the number ring of totally
real number fields of degree n over Q yields a family of
S-admissible lattices. We provide a list for the totally real
number fields which produce the densest possible lattices that
this method can give, up to dimension 8. We also determine a
generating matrix for each of these lattices, and identify those
whose generating matrix is circulant, since they are easier to
decode. Computer simulation results show that the resulting ndimensional signal constellations exhibit nth-order diversity.
In comparison to conventional 16-QAM, corresponding to a
rate of 2 bits per dimension, our signaling schemes yield
a "coding" gain ranging from 9.5 to 15 dB for the same
efficiency at a symbol error rate of
This approach
provides a simple quantitative method to design n-dimensional
lattice constellations with pre-fixed spectral efficiency and
built-in nth-order diversity. Further, gain can be obtained
0018-9448/96$05.00 0 1996 IEEE
GIRAUD AND BELFIORE: CONSTELLATIONS MATCHED TO THE RAYLEIGH FADING CHANNEL
107
by combining these constellations with appropriate coding
schemes [6], [7], [21].
11. A FRAMEWORK
FOR THE LATTICEDESIGNPROBLEM
A. Admissibility
Let C = {x} denote an n-dimensional finite signal set with
IC] signals. At the output of the transmission channel, upon
observation of the n-dimensional vector y, the decision on the
transmitted signal is based on the maximization of the metric
m(y, z). Under this decision rule we may define the painvise
error probability p(z i k), z # 2,as the probability that the
metric m(y, 2)be larger than m(y, 2) when 2 was transmitted.
Now, if
p ( z -2) 2 E
e)
p(
1
< -xJ'p(z-+2)
- IC1
<&.IC(.
x X#x
This straightforward inequality is the basis for our discussion.
DeJnition 2.1: For any given E > 0, a signal set C is said
to be ''&-admissible" if for any pair z,x of distinct signals
in C the painvise error probability p ( z --t 2) is smaller than
or equal to E .
This definition can be extended to encompass points oT R"
that do not belong to C. To do this, let S; denote the set of
n-dimensional vectors t whose metric is larger than or equal to
that of x with probability greater than E when z is transmitted,
i.e., such that
P b ( Y , t >2 m(y,z) 14 > E .
Suppose that we are building a constellation C. We start with
a signal zo and let Sj, denote the set of points t such that,
if they were chosen as channel signals, then the condition
p(zo -+ t ) 5 E would not be met. Hence, we ought to pick
the next channel symbol, 2 1 , outside S& . Next we choose 2 2
outside S& U S& , etc. By means of the mapping fi on R" x R"
-
(z,t>
p: {(z,t)
p [ m ( y , t ) 2 m(y,z) 1.1,
+I,
ifx#t
ifz=t
the set SE can be rewritten as
SE = {t E R", $ ( x , t ) > E } .
(1)
Fig. 1. The set of nonadmissible points with respect to
5.
fc"-ktted as the problem of finding IC1 points in R" that
be surrounded by their regions
in
a way that each
region contains only one signal. We hasten to observe that the
present formulation is not a packing problem: in fact, these
regions may overlap and may not be convex, or finite. They
s$
In that case, the decoding metric to minimize changes at each
signaling interval and the decision region are n-dimensional
ellipsoid because of the fading affecting each coordinate (see
Section VIII). Loosely, the shape of the body S can be viewed
as the region obtained when these ellipsoids are averaged on
the fading.
B. Translation Invariance
We choose to specialize our analysis to the case of a
translation-invariant mapping fi
--f
+
so that S;, = 5': + zz' or Sz = 3: S i for all z ( 0 is the
origin).
1) The Packing Formulation: If S is an open subset of R",
a system consisting of the translates ( S X ) , ~ C is called an
( S ,C)-packing of S if for any two distinct points z,t E C,
the sets z S and t S are disjoint. We assume now that S i
is open, bounded, convex, and o-symmetric, i.e., 5': contains,
with each of its points z the entire open interval ( - - = , E ) .
In that case, the next result establishes a link between our
formulation and the packing formulation of the constellation
design problem.
Proposition 2.3: A signal set C finite or not is &-admissible
if and only if (fSo",C) is a packing of ;SE.
Proofi
+
+
+
3( f1
V z , t E R n x+-S,
n t+-s: # Q W ~ - X E S ;
It is the set of the nonadmissible points with respect to z as
illustrated by Fig. 1. We assume here that 5
': is an open set.
*t€Z+SE
If this is not the case, then we take its topological interior.
*t€SSj.
Definition 2.1 can be replaced by
Dejinition 2.2: The signal set C is said to be &-admissible Suppose that C is S-admissible and let z and t be two distinct
if each region SE contains only one signal from C, that is, z points in C then t @ S;, hence
S; n C = {z}.
(
It should be clear at this point that, for a given P ( e ) and a
given cardinality IC\,the problem of finding a signal set whose
error probability does not exceed a preassigned value can be
(,+ is;) n
(t+
is:)
= 0.
Therefore, (;So",C) is a packing of %SE.The converse is
similar.
0
EEE TRANSACTIONS ON INFORMATION THEORY, VOL. 42, NO. 1, JANUARY 1996
108
independent sets (a1, , a,) and (el,. . . ,e,) determine the
same lattice if and only if they are connected by a unimodular
transformation, that is, (el, . . . , e,) = U ( a l ,. .. , a,) where
U is an integral matrix with determinant rfl. Consequently,
( e l , . - . , e n ) i s a b a s i s o f A = A Z nw h e n e l , , . . , e , E h a n d
ldet (el,.. . , e n ) ]= ldet Al. The number ldet AI is an invariant
of A denoted by det (A) and called the determinant of A.From
a geometrical viewpoint, the determinant of the lattice A is the
volume of the fundamental parallelotope .
Constellarion C
(I>
P = (Ala1
+ . ‘ . + Anan, 0 5 A,
5 l}
or of the volume of the Voronoi cell of A. Loosely, det (A) is
the volume of n-space per lattice point.
The choice of translation invariance and Proposition 2.2
make it natural to restrict our attention to lattices. We have
Proposition 2.4: A lattice A, or any of its translate a A,
is €-admissible if and only if S: n A = (0).
Proofi We denote by t, the translation U H U x. Now
let x E Q + A. Since L,(a + A) = A, we have
+
+
5’: n (U + A) = {x}
U { 0 } = t-,(s; n ( U + A))
e { o } = t-,(S;) n t-,(a + A))
U {o} = 5
’: n A.
xts,
n
0
For practical applications we want our signal set to be as
dense as possible, i.e., we want to maximize the number of
lattice points per unit volume. Due to Proposition 2.4, this
(c)
translates into the search for a lattice A such that S: n A = { o }
Fig. 2. The constellation design problem. (a) Two sets of nonadmissible whose determinant is a minimum.
points with respect to 0. (b) Admissibility. (c) The packing formulation.
We are now ready to formulate and solve the signal-desi n
5
problem mentioned before. The mathematical framework for
For example, the constellation design problem for the this is provided by the geometry of numbers, a branch of
Gaussian channel is tantamount to the sphere packing problem mathematics aimed at describing the behavior of a body with
since this channel is translation-invariant and the Euclidean respect to a lattice [8]. In the language of geometry of
numbers, any nonempty o-symmetric open set S c R” is
ball 5’: is convex and o-symmetric.
Fig. 2 illustrates our approach: we denote by SI and 5’2 the called an 0-symmetric star body. For simplicity’s sake, in
set of inadmissible points with respect of the origin of two the following w e refer to it as a “star body.” If S is a star
fictitious translation-invariant channels (Fig. 2(a)). The five- body, a lattice A is called S-admissible when S n A = (0).
point constellation C is shown admissible for both channels The Malher’s selection theorem 18, ch. 3, Th. 91 asserts
by centering SI and Sz, respectively, at each point of the that if there exists an S-admissible lattice, then there exists
constellation C and verifying that the other channel signals an S-admissible lattice with minimal determinant, called a
are outside the region (Fig. 2(b)). Since SIis convex, the critical lattice of S.Such a lattice is not necessarily unique.
constellation design problem is tantamount to finding dense This result is of prime theoretical interest since it states the
packing of S1 (Fig. 2(c)); however, S2 is not convex hence existence of an S-admissible lattice which maximizes the
number of lattice points per unit volume. In the language
the packing formulation does not apply.
2) The Lattice Design Problem: We briefly recall that a lat- of the geometry of numbers, the lattice design problem is
tice A is a discrete additive subgroup of R” and that the tantamount to determining a critical lattice of Sz provided that
. . . mnan where SE is o-symmetric.
points of A can be written x = mlal
If in addition, 5’: is convex, a critical lattice of S: achieves
ml, . . . , m, are integers and ( a l ,. . . ,a,) is a system of
independent points of A called a basis of A. We anticipate that the maximum density of a lattice packing of $5’: as a
we shall write a lattice in R” in the form a12 @ ‘ . . @ a,Z.
consequence of Proposition 2.3. In the Gaussian case for
If A denotes the matrix whose columns are the canonical example, our formulation reduces to the lattice packing of
coordinates of a l , . . . ,a,, we write A = AZ”. Two linearly spheres [4], [5].
Equivalenrpackmg problem
for convex body
+ +
GIRAUD AND BELFIORE: CONSTELLATIONS MATCHED TO THE RAYLEIOH FADING CHANNEL
109
Deccder
t
z
Fig. 3. A general baseband transmission model.
111. PAIRWISE ERRORPROBABILITY AND THE
RAYLEIGH
FADINGCHANNEL
We now specialize the formulation of the previous section
to the case of a Rayleigh fading channel with ideal channel
state information. The fading channels are characterized by
time-varying distributions which limit the effectiveness of
the modulation schemes designed for the Gaussian channel. Acquisition and exploitation of some kind of channel
state information has proven advantageous in a fading environment. Here, we shall assume that such information is
available through a channel estimator [9], [lo]. Besides, we Fig. 4. Behavior of Se
assume that the channel is memoryless by means of perfect
interleaving\de-interleaving performed at the coordinate level.
Hence, from (1) and (2) we have
Digital information from the source is mapped to signals of
C by taking a block of IC bits and selecting one of 111 = 2k
signals z in C. The transmission rate is p = log,M/n
bitstdimension.
A general baseband transmission model is shown in Fig. 3. For a given E the two bodies Sz (I?) and S; (I?) are homothetic,
Corresponding to the transmission of Z, the channel outputs i.e., S:(r') is a scaled version of Sz(r); therefore, they
Y = (yl,..*>yn)?where
have homothetic critical lattices. Consequently, the design of
- Yk = ( a k l x k w k ,
a lattice matched to the Rayleigh fading channel reduces to
- ak N ( 0 ,y2), so that [ a k l is Rayleigh-distributed,
finding a critical lattice of
- wk is a sample of a white Gaussian noise process -N
(0>0".
+
N
Y2
The signal-to-noise ratio (SNR) is defined as 7 = --E
202
where E is the average signal energy per dimension.
The metric used here is
zE
n
m ( y , x )=
(Yz - lQi1G)2
i=l
so that from [l 11 we have
because SE= XSz(r) with X =
m,i.e.
S;(r) iff AX
E SE
Fig. 4 shows SEin two dimensions for different values of
Now
f
n
E.
\
xERn, n x % < 1
i=l
and hence if A is S-admissible, it is also SE-admissible.Moreover, all pairs of distinct points of A have their coordinates
distinct, and therefore the pairwise error probability varies
asymptotically as the inverse of P.Thus a constellation
carved out of A yields nth-order diversity. Consequently, at
low error rate the design of a lattice matched to the Rayleigh
fading channel reduces to finding a critical lattice of the body
where r = 7/4E. As the SNR grows to infinity, this upper
bound is asymptotically tight. The painvise error probability
varies asymptotically as the inverse of the signal-to-noise ratio.
We now determine the set S:(r) of inadmissible points with
respect to the origin, a critical lattice of which we should find
in order to solve the signal design problem for the Rayleigh S .
channel.
Although (2) is not the exact expression of the pairwise Iv. LATTICES
MATCHED
TO THE RAYLEIGH FADING
CHANNEL
error probability, the theoretical framework of Section I1 is
In
the
previous
section
we
have
formulated
our
signalapplicable but the performance may not be optimal. The
design
problem
in
terms
of
finding
an
S-admissible
lattice
expression (2) may be viewed as the pairwise error probability
in
R".
The
questions
are
now
the
following:
of some fictitious channel worse than the fading channel.
- How to prove that a lattice is S-admissible?
In this respect, an admissible constellation for the fictitious
- How to reduce the set of lattices to investigate?
channel is admissible for the fading channel.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 42, NO. 1, JANUARY 1996
110
The first question does not seem tlo admit an easy answer. In principle, to prove that a lattice A is S-admissible
we should examine an infinity of points. We need suitable
mathematical tools provided by number-field theory to do this.
The fundamentals are summarized in the following, where the
basic definitions and results are listed. For further details, the
interested reader is referred to [121, [13].
V. ALGEBRAIC
NUMBER FIELDS
We describe here how a lattice can be deFcribed in numbertheoretical terms. A hint to this formulation is provided for
instance by observing that the familiar square lattice Z2 can
be represented in the form
z2= { a + bi,
a E 2,b E 2)
m.
where i =
Thus 2' is, in a sense to be specified, connected to i , a root of the polynomial with integer coefficients
x2 + 1. Similarly, for the hexagonal lattice A2 we can write,
with w = @ 3 / 3
A2 = { U
+ bw,
U
E 2, b
E Z}
which shows that A2 is connected to w,a root of the polynomial with integer coefficients x3 - 1.
Any (real or complex) root E of a polynomial with rational
coefficients is called an algebraic number. The monic polynomial of lowest degree whose root is is called the minimum
polynomial of <,and is denoted M t . The roots of ME are said
to be conjugate of E ; when and its conjugates are real, is
said to be totally real. If E is a root of a monic polynomial
with integer coefficients, then 5 is called an algebraic integer
(for example, E = & is an algebraic integer with minimal
polynomial z2- 5). The set of algebraic integers forms a ring
A.
Let E be an algebraic number, if the degree of ME is n,then
(I,E , . . . , gn-') is a basis of the Q-vector space
Q(E)
={YO+Ylt+
..
*
+qn-lEn-l, (Yo,
(71,
Fig. 5. The lattice a ( M ) where M
= 2 @ &Z.
is totally real then the embeddings 01, . . . , O, range in R. For
example, the two embeddings of K = Q(&) in C are
01
=Id~:a+b&t+a+b&
02: a + b& H a - b&
and they range in R.In the following, we shall always assume
that K is totally real.
By means of these embeddings two mathematical objects
relevant for our problem can be built.
i) The n o m of an element a E K ; it is defined as
N ( a ) = cr1(a).. . a,(a).
ii) The assignment
cr: a
. . . , a,(a))
H (cr1(a),
which defines an additive homomorphism from K into
R" with trivial kernel. The embedding cr unveils the
geometrical nature of a structured subset of K . For
example, consider
K = Q([) is actually a field sandwiched between Q and
the field C of complex numbers; its dimension over Q. n,is
called the degree of K over Q. For example, (1,d)
is a basis
M = Z @ 6 2 c Q (&) .
of Q(&), i.e., Q(&) is the set of all linear combinations
a b& with (a, b ) E Q2. Conversely, every field sandwiched
M is dense in R, hence the way the elements of M are
between Q and C finite-dimensionalover Q is primitive, i.e.,
organized is hidden by the line representation of R. By
has the form &(E) for some algebraic number 5 E C.
means of U , the set M is mapped onto the lattice of R2
A subfield K of C obtained as a finite-dimensional vector
space over the rationals Q is called a nun%ber$eld. We denote
0 the ring of the algebraic integers of IC, i.e., 0 = K n A,
it is called the number ring of K .
as shown in Fig. 5.
With K a number field of degree n over Q, an embedding
of K into C is a map cr: K H C which is both a field and
VI. BUILDINGS-ADMISSIBLE
LATTICES
a Q-linear homomorphism. There are exactly n embeddings
of K into C: we denote them 01, . . . , 0., They are described
by writing K = &(E) and observing that [ can be mapped A. The Construction
to any of the n roots of ME.Each root determines a unique
Now let (p1, . . . , U
, ),
be an n-tuple of K , a totally real
embedding of K into C , and every embedding must arise in number field of degree n.Let R = [a&)] denote the n x n
this way since E must be mapped to one of the roots of M t . matrix whose jth column is the vector of the conjugates of p3.
The first such embedding is cr1 = IdK, the identity of K . If [ Then ldet 01' is called the discriminant of (p1,
+
GIRAUD AND BELFIORE: CONSTELLATIONS MATCHED TO THE RAYLEIGH FADING CHANNEL
111
xxy=4
xxy=3
xxy= 2
xxy=1
Fig. 6. The body S and the S-admissible lattice A o where 0 is the number ring of
is denoted Disc(pz).This discriminant is nonzero if and only
if (PI,.
. ,pun)is a basis of K . (For example, (1,(1 & ) / 2 )
is a basis of Q(&) with discriminant 5.)
We are now ready to show how a lattice can be described
in number-theoretical terms. Let (pI,.. . ,p,) be a basis of
K . Then the set M of all linear combinations of PI,.. . , p,
with integer coefficients is a free abelian group of rank n, with
every z E M uniquely represented in the form
+
II: = z l p l +
where the
5,
. . . + 2,pn
&(A)
Proposition 6.2: o ( 0 ) is an S-admissible lattice, denoted
by Ao.
Pro03 First, o ( 0 ) is a lattice of R" because 0 is a
free abelian group of rank n in K . Next, if z E O\{O},
then N ( x ) E 2\{0}. Consequently, II: E O\{O} implies
IN(z)I L 1.
Finally, if a E Ao\{o} then
a = (a1,. . . ,a,) = o(x) = (o1(z), . . . , a,(z))
for some z E O\{O} and
are integers. We write
I n
M = p12 c3 . * . e3 p n 2 .
Any two bases of M correspond through a unimodular matrix:
consequently, their discriminants are equal, i.e., all bases of M
have the same values Disc ( M ) ,called the discriminant of M .
The map o transforms M into a lattice of R" as follows.
Define arc = (ol(pk), . . . ,o,(pk)). Then
AM = a ( M )= a i 2 CB . .
a,Z
is a lattice of R", with
det2 (AM) = Disc ( M ) .
The next proposition is central to our construction, its proof
can be found in [12].
Proposition 6.1: Let 0 denote the number ring of K
i) The norm of a E 0\{0} is an element of z\{O} [12, p.
22, Corrolary 21.
ii) 0 is a free abelian group of rank n [12, p. 30, Corrolary].
Now we can prove our key result by combining Section V-i),
ii) and the proposition above.
e
I
I
therefore, a
S hence A 0 n S = { o } , proving that A 0 is
S-admissible.
0
A basis of K which is a 2-basis of 0 is called an integral
basis of K .
In order to illustrate this result, we take
It is an algebraic
number whose mimimal polynomial is M,,pj(z)= x2 - 2. The
vector space
a.
Q ( h ) = {a+ b h ,
(a,b)E
Q2}
&a,
has dimension 2. Since the roots of M , , ~ ~ ( Iviz.,
I:),
are both real, fi is totally real. The two embeddings are
01: f i H fi (the identity map) and 0 2 : fi H -&'
(conjugation): they range in R. An integral basis of Q(fi)
is (1,fi),
it generates the S-admissible lattice
shown in Fig. 6.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 42, NO. 1, JANUARY 1996
112
Pro03 Since a: is an algebraic integer of Q(a)then 8 =
is an algebraic integer of &(a)too. Consequently, ak
and 2are algebraic integers of Q ( a ) ,and hence p k = ak+a""
is a real algebraic integer of &(a).Therefore, it is an integer
of K .
An element E E & ( a )written
TABLE I
TOTALLY
REAL NUMBER
FIELDWITH SMALLEST ABSOLVE DISCRIMINANT
aq-l
E = Xlcy + X z a 2 + .
with A, E Q is real iff
E
' '
+ X,-laQ-'
= <, i.e.
&(PI, P a root of
X 7 + X 6 - 6 X 5 - 5X4 4-8X3 + 5 X 2 - 2X
-1
E = XlPl
+ X2P2 + . .. + X(q-1)/2p(q-1)/2
$(a),where
2a2 = 7 t 2 J z + ( 1
and PI,. . . , 4 q - l ) / 2 i s a basis of K .
In a similar way,
+ ./2,(dGE)
From now on, the embedding in E" of a number ring will
be called an integer lattice. The discriminant of 0 is called the
absolute discriminant of K denoted by d K and (det ( A O ) ) ~=
d K . Since d K is an integer, there exists a number field K,
whose absolute discriminant d, is the smallest absolute value
of d K as K varies over all totally real fields of degree n, and
by Hermite's theorem there are only finitely many number
fields with discriminant d,. Since we are interested in signal
constellations with the smallest energy for a given size, we
shall call the corresponding integer lattice optimal.
< = Ala: +
A2a2
+ . . . + Xq.-lQP--l
with A; E 2 is a real algebraic integer iff X I
X q - 2 , . . .. Thus
c=
ALP1
1
Xq-l,
A2
=
+ X2P2 + . . + X(q-1)/2P(q-1)/2
'
and consequently PI,. . . ,p(q--l)/zis an integral basis of K .
Finally
q--3
B. Optimal Integer Lattices for 2 5 n 5 8
Disc (1,Pl,P4,. * . , P 1
Optimal integer lattices for dimension n 5 8 are already
known; we denote them A,". For n = 2 and 3 they are also
critical lattices of S [8]. For dimension n 2 4, it is not known
whether they are critical or not. Table I lists for each dimension
the smallest absolute discriminant d, together with a field K,
with absolute discriminant d,; its ring of integers is denoted
0,.
) = Disc (Pl,' . , P ( q - l ) / 2 )
'
9-3
and therefore (1,P I ,,@ , . . . , PIz ) is equally an integral basis
of K .
0
The absolute discriminant of K is
q-3
d~ = Disc (pl,.. . ,p(q-l)/2)= q ( 7 )
[18]. This result provides also an upper bound for the critical
determinant of S for dimension n = ( q - 1)/2:
C. Integer Lattices Derived from Cyclotomic Extensions
If q > 2 is a prime then the Eisenstein criterion [19] shows
that @ ( X ) = 1 + X
X 2 ' . . X4-I is irreducible in
A(S) 5
=q ( 9 ) .
Q [ X ] .The number field & ( a ) ,where cy = ei(2T/q),is called
the qth cyclotomic extension of Q and a 2 , .. . ,a9-l are the
conjugates of a. Therefore, Q ( a )is the splitting field of @ ( X ) . In the following, the integer lattice of K will be referred to
Its ring of algebraic integers is 0 = Z [ a ]Since
.
the field & ( a ) as a cyclotomic lattice.
Examples:
is not real, in order to build an S-admissible lattice we examine
e If n = 2 ( q = 5), then d K
= 5 and K = K z ; the
K = R n Q ( a ) .As it will be shown in Section VII-Al), K
embedding Ay of 0 2 in R2 is a critical lattice of S [8].
is totally real.
If n = 3 ( q = 7), then d K = 49 and K = K3; the
Proposition 6.3 (Integral Basis of K): The degree of K is
cyclotomic lattice is A?, it is again a critical lattice of
( q - 1)/2, and an integral basis of K is
+
p1=
2cos
(f),.",
Moreover, K =
+
+
p(q-1)/2
Q[pl] and 0
= 2cos
=
(
Z@1].
( q - 1)/2 x
'"1
-
9
.
S PI.
0
When n = 5 ( q = ll), the corresponding cyclotomic
lattice yields A;", the den-sest integer lattice matched to
S.However, this may not be a critical lattice of S.
~
GIRAUD AND BELFIORE: CONSTELLATIONS MATCHED TO THE RAYLEIGH FADING CHANNEL
VII. GENERATING
MATRIXOF INTEGER LATTICES
Here we list a generating matrix for each one of the
optimal integer lattices of Table I. Since simplified detection
algorithms can be used for lattices with circulant generating
matrix referred to as circulant lattices [7], we have identified
those lattices that are circulant.
In addition to the results already introduced in Section V,
we shall use the two following facts:
Given a E K , the elements a1 ( a ) ,. . . ,a,(a) are the
conjugates of a.
If the range of the n embeddings of K into C is K
itself, then { g1, . . . ,a,} is a group under the composition,
denoted Aut ( K ) .In that case, K is the splitting field of
some polynomial.
113
The embeddings of K6 in R are such that
a(&) = & and a
(
COS
or a ( c o s
or .(cos
a(&) = -& and a ( cos
(7))
= cos(?)
($))
= cos($)
(T)) (7)
(7)) (F)
= cos
= cos
or a ( cos
(F))
or a ( cos
(5)) (T).
= cos
($1
= cos
Every embedding of K6 into R ranges in K6 and
Aut (&) N 2 / 6 2 . The embedding a such that a(&) =
-A and a( cos
= cos ($) is a generator of Aut (&).
Let
A. Examples of Circullant Lattices
(9))
I ) Cyclotomic Lattices: A remarkable example is provided
by cyclotomic fields. Let y > 2 be a prime, and a = e 2 ( 2 T / q ) .
The following properties are relevant here. Let Aut ( & ( a ) )
denote the set of automorphisms of Q ( a ) :
p=2cos($)
x 2 c 0 s ( 7 ) = ( & - l ) x cos($).
i) Aut ( Q ( a ) )is isomorphic to the multiplicative group of
the field GF ( q ) [ 191.
Since gcd (d2, d 3 ) = 1, the number ring 0 6 arises as
ii) If T E Aut ( & ( a ) )is such that .(a) = a2,then it is a
,
generator of Aut ( & ( a ) )and
2COS
0 3 cl3 2 cos
0 3
(X)
is an integral basis of &(a).
The conjugates of
are PZ,...,P(~-~)/~
and K =
Q ( P 1 , . . . ,,L?(q-1)/2).
Therefore, the embeddings of K in
C range in K itself; they are the restrictions to K of the
automorphisms of &(a).In particular, K is totally real.
Let a be the automorphism of K induced by 7. From i) and
ii) above, Aut ( K ) is cyclic with a as generator. Since
(%)
hence the family (ak(,u)kE{o,
. . , 5 ) ) is an integral basis of
And a generating matrix of AF is
(
+ &) cos (%);(-I - &)cos
(-1 + &) cos
; (-1 - h)
cos
Circ (-1
(7)
K6.
(7);
(5)
;
( - l + h ) c o s ($);(-l-h)cm(?)).
B. Examples of Noncirculant Lattices
the family
(PI,a(pl), . . . ,a? (PI)) is
an integral basis of
K . and
Circ (aO(P1),a(P1),- , a % ? ) )
is a generating matrix of Ro.
e If n = 2, ( q = 5 ) then ao((pl)= P1,al(P1) = P2; a
generating matrix of A? is Circ (pl, p2).
If n = 3, (y = 7) then ao(P1) = PI, al(P1) =
Pz, a'(/%) = P 3 ; a generating matrix of AT is
circ(P1 PZ P3).
I
7
If n = 5, ( q = 11) then ao(pl) = P1, al(P1)=
Pz, az@i) = P 4 , a3(P1) = P 3 , and a4(P1)= P 5 ; a
generating matrix of A? is Circ (PI, P2,P4, P3,P5).
2) The Six-Dimensional Case: Dimension six provides an
example of noncyclotomic lattices with circulant matrix. The
field Ks arises as a quadratic extension of K 3 ,it is the smallest
and cos
=
field containing K3 = &( cos
(F))
KG= K3 (cos
(F))
= K3 @ cos
(y)
(F)
K3
Proposition 7.1 (A Necessary Condition): Let K = Q ( p )
be a totally real number field of degree n and 0 its number
ring; if the integer lattice ho is circulant then
i) K is the splitting field of the minimal polynomial of p
over Q .
ii) Aut ( K ) is a cyclic group.
Pro08 We assume that the embeddings of K in R have
been numbered so that a1 is the identity embedding. Let
(PI, . . . ,p,) be an integral basis of K such that A = ( a z ( p 3 ) )
be a circulant generating matrix of ho.Then 0%
(,u3)E K , and
hence the embeddings azrange in K . In addition, the minimal
polynomial of p is
&(I)= ( X - O l ( P ) ) . . . ( X - ~ n ( P > )
and therefore K is the splitting field of Mp. Finally, since
the matrix A is circulant, a 2 ( p z ) = ,uL,+l (here we use the
convention that n + 1 -+ l), and consequently, a2 generates
) ~
~ + 1 ( ,0 5 k 5 n - 1). 0
Aut ( K ) ;more precisely, ( ~ 2 =
In the following, V (bl , . . . , b,) denotes the Vandermonde
matrix.
114
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 42, NO. 1, JANUARY 1996
I
Fig. 7.
16-ary signal set carved out of A?.
1) The Four-Dimensional Case: The minimal polynomial
of p = I,/is M p ( X ) = X4 - 14X2 29;if F is
a splitting field of MO,then its degree is IF : Q1 = 8 and
Aut (F)is the dihedral group 0 4 [is].ConsequeGly, AT is
not circulant.
The four embeddings of K4 in R are such that
+
m(p) =p
01(p) = p
Q ( P ) = -P
04(P)
= -P
+
d-.
where p =
Besides, E = (1 p ) / 2 is an algebraic
integer and Disc (1,E , E’, E3) = 725. Hence a generating
matrix of AT is
.(-
l+p
2
’
1
1+p 1-p 1-p
--2
’
2
’
2
.
lo-’
15
10
20
25
.@
/ N o [W i,
Fig. 8. Symbol error events.
pomts closest to the origin of A F . This corresponds to a
transmission rate of 2 bits per dimension. 16-QAM (i.e., the
two-dimensional signal set { -3, -1,1,3}2) was taken as a the
baseline. The 16-ary constellation carved out of AY is shown
by Fig. 7.
The decoding metric to minimize was
n
2) The Seven- and Eight-Dimensional Case: The situation
is very similar to the four-dimensional case. The optimal
seven-dimensional integer lattice is obtained with K7 = Q ( p )
where p is a root of
If F denotes a splitting field of P , then its degree is
[F : Q] = 7! and Aut(F) is the symmetric group 5’7 [16].
Hence AT is not circulant. If p1 = p, p ~. ., . ,p 7 are the seven
roots of P , then the seven embeddings of K7 in R are such
that a , ( p l ) = pz. Besides (1,p, p 2 , . . . , p6) is an integral basis
of AT. The situation is similar in the eight-dimensional case
~71.
VIII. SIMULATION
m(z,?/)=
E(!/.-
IQz/G)2,
3: E
Comparing the received signal with all the points of C is
prohibitively complex, so we used a simplified detection algorithm whose performance was found to be indistinguishable
to that of exhaustive search [7].
Fig. 8 shows the symbol error rate P e ( n ) versus bit energyto-noise ratio for the Rayleigh fading channel. Observe that the
n-dimensional modulation scheme offers nth-order diversity
on the Rayleigh fading channel. We emphasize that any
constellation carved out the lattice A,” has built-in nth-order
rlivprsitv whntever
it< c i 7 ~
We recall that the bit error rate Pe(b) depends on the
mapping, i.e., on the assignment of the information bits to
the signal points. The rate Pe(b) is related to the symbol error
rate, Pe(n), by
RESULTS
Simulation of the communication performance of these
lattices was performed for 2 5 n 5 6 by choosing spherically
shaped n-dimensional signal sets built with the M = 22n
c.
z=1
--Pe(n)
1
np
5 Pe(b) 5 P e ( n )
where p is the number of bits per dimension.
GIRAUD AND BELFIORE CONSTELLATIONS MATCHED TO THE RAYLEIGH FADING CHANNEL
115
IX. CONCLUSION
In this paper we have developed a new tool for the design
of signal sets based on lattices for a variety of non-Gaussian
n
2
3
4
5
6
7
8
channels. These signal sets are obtained as a subset of the
standard embedding in R” of the number ring 0 of some
yz(AF) 0.89 0.82 0.77 0.73 0.73 0.63 0.70
totally real number field K of degree n.
Application of this tool to the Rayleigh fading channel proACKNOWLEDGMENT
vides n-dimensional signals endowed with nth-order diversity
for any transmission rate. The densest lattices obtained in this
The authors wish to thank E. Biglieri for his detailed
way up to dimension 8 were listed, and performance checked comments and suggestions which have significantly improved
by simulation.
present work.
We hasten to stress that our optimization in the set of S admissible lattices in R” was actually carried out in the subset
REFERENCES
of integer lattices. So the question naturally arises of how large
G. Ungerboeck, “Trellis-coded modulation with redundant signal sets,
this family is among the set of S-admissible lattices. Because
Part 11,” IEEE Commun. Mag., vol. 25, no. 2, Feb. 1987.
L. F. Wei, “Trellis-coded modulations with multidimensional constelS-critical lattices in dimension 2 and 3 are integer lattices,
lations,” IEEE Trans. Inform. Theory, vol. 1T-33, pp. 483-501, July
it is legitimate to conjecture that the family of integer lattices
1987.
comprises dense S-admissible lattice: although we were unable
G. D. Forney Jr., “Coset codes-Part I and Part 11,”IEEE Trans. Inform.
Theory, vol. 34, pp. 1123-1187, Jan. 1988.
to prove this conjecture, we have accumulated a number of
J. Leech and N. 5. A. Sloane, “Sphere packings and error-correcting
partial results [7], not reported here for sake of brevity, that
codes,” Can. J. Math., vol. 23, pp. 718-745, 1971.
seem to substantiate it.
G. D. Forney Jr., R. G . Gallager, G. R. Lang, .F. M. Longstaff, and
S . U. Qureshi, “Efficient modulation for band-limited channels,” ZEEE
Finally, the lattice constellations proposed in this paper have
J. Select. Areas. Commun., vol. SAC-2, pp. 632-647, Sept. 1984.
a low sphere packing density as pointed out in [7]. Specifically,
X. Giraud and J. C. Belfiore, “Coset codes on constellations matched to
the fundamental gain of a lattice A is defined as
the fading channel,” presented at the 1994 Intemational Symposium on
e;
numerical results are
In the case of interest, ~ 2 ( A r=)
shown in Table 11. This consideration has motivated Boutros
et al. to relax the requirement on diversity and investigate
lattices, the diversity of which is half their dimension. They
build rotated version of the some of the densest lattices with
respect to the Euclidean distance out of which they carved
constellations with good properties on both the Gaussian
channel (a high sphere packing density) and the Rayleigh
fading channel (an 2th-order diversity) [20]. Recently, we
have proposed a unified framework in order to build lattice
constellations matched to both the Rayleigh fading channel
and the Gaussian channel [21]. The method encompasses
the situations where the interleaving is done on the real
components or on the complex components. In the latter case,
a simpler construction than that of [20] of lattices congruent
to the densest lattices with respect to the Euclidean distance
is proposed. Besides, we find a way to increase the sphere
packing density of S-admissible lattices with built-in nth-order
diversity in order to build constellations with good properties
on the Gaussian channel and an nth-order diversity on the
Rayleigh fading channel.
Information Theory, Trondheim, Norway.
X. Giraud, “Constellations matched to the fading channels,” Ph.D.
dissertation, ENST, Paris, 1994.
P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers. Amsterdam, The Netherlands: North Holland Math. Library Elsevier, 1987.
S. A. Fechtel and H. Meyer, “Near-optimal tracking of time-varying
digital radio channels using a-priori statistical information,” in Coded
Modulation and Bandwidth Eficient Transmission (Tirrenia, Italy, Sept.
1991), pp. 367-377.
N. W. K. Lo, D. D. Falconer, and A. U. H. Sheikh, “Channel interpolation for digital mobile radio communications,” in Proc. ICC, June
1991, pp. 773-777.
D. Divsalar and M. K. Simon, “The design of treillis coded MPSK for
fading channels: Performance criteria,” IEEE Trans. Commun., vol. 36,
pp. 1004-1012, Sept. 1988.
D. A. Marcus, Number Fields (University Texts). New York SpringerVerlag, 1977.
Z. I. Borevich and I. R. Shafarevich, Number Theory. New York,
Academic Press, 1966.
H. Hasse, Number Theory. New York, Springer-Verlag, 1980.
M. Pohst, “Berechnung kleiner Diskriminanten total reeller algebraisher
Zahlkorper,” J. Reine Angew. Math., vol. 278/279, pp. 278-300, 1975.
__, “The minimum discriminant of seventh degree totally real algebraic number fields,” in Number Theory and Algebra, H. Zassenhaus,
Ed. New York Academic Press, 1977, pp. 235-240.
M. Pohst, J. Martinet, and F. Diaz, “The minimum discriminant of totally
real octic fields,” J. Number Theory, vol. 36, pp. 145-159, 1990.
H. Cohn, “Note on fields of small discriminant,” Proc. Amer. Math.
Soc., vol. 3, pp. 713-14, 1952.
Hungerford, Algebra (Graduate Texts in Mathematics). New York,
Springer-Verlag, 1987.
J. Boutros, E. Viterbo, C. Rastello, and J. C . Belfiore, “Good lattice
constellations for both Rayleigh and Gaussian channels,” submitted to
IEEE Trans. Inform. Theory, 1994.
X. Giraud, E. Boutillon, and J. C. Belfiore, “Modulation schemes for
fading channels,” submitted to IEEE Trans. Inform. Theory, 1995.