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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003
Space–Time Diversity Systems Based on Linear
Constellation Precoding
Yan Xin, Student Member, IEEE, Zhengdao Wang, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE
Abstract—We present a unified approach to designing
space–time (ST) block codes using linear constellation precoding
(LCP). Our designs are based either on parameterizations of
unitary matrices, or on algebraic number-theoretic constructions.
transmit- and
receive-anWith an arbitrary number of
tennas, ST-LCP achieves rate 1 symbol/s/Hz and enjoys diversity
gain as high as
over (possibly correlated) quasi-static
and fast fading channels. As figures of merit, we use diversity
and coding gains, as well as mutual information of the underlying multiple-input–multiple-output system. We show that over
quadrature-amplitude modulation and pulse-amplitude modulation, our LCP achieves the upper bound on the coding gain of
all linear precoders for certain values of
and comes close to
this upper bound for other values of
, in both correlated and
independent fading channels. Compared with existing ST block
codes adhering to an orthogonal design (ST-OD), ST-LCP offers
not only better performance, but also higher mutual information
for
2. For decoding ST-LCP, we adopt the near-optimum
sphere-decoding algorithm, as well as reduced-complexity suboptimum alternatives. Although ST-OD codes afford simpler
decoding, the tradeoff between performance and rate versus
complexity favors the ST-LCP codes when
,
, or the spectral
efficiency of the system increase. Simulations corroborate our
theoretical findings.
Index Terms—Diversity, multiantenna, rotated constellations,
space–time (ST) codes, wireless communication.
I. INTRODUCTION
W
ELL-established by now as a versatile form of diversity for wireless applications, spatial diversity is implemented by deploying multiple transmit and/or receive antennas
at base stations and/or at mobile units. Because of size and
power limitations at mobile units, multiantenna receive diversity is more appropriate for the uplink rather than the downlink.
For this reason, transmit diversity schemes have attracted con-
Manuscript received March 7, 2001; revised October 22, 2001 and January
25, 2002; accepted February 25, 2002. The editor coordinating the review of this
paper and approving it for publication is A. F. Molisch. This work was supported
in part by the National Science Foundation (NSF) under Grant 9979443 and
Grant 012243, and in part by an ARL/CTA Grant DAAD19-01-2-011. This work
was presented in part at Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, October 2000, at the International Conference on
Acoustics, Speech and Signal Processing (ICASSP), Salt Lake, UT, May 2001,
and in part at the Global Telecommunications Conference (GLOBECOM), San
Antonio, TX, November 2001.
Y. Xin and G. B. Giannakis are with the Department of Electrical and
Computer Engineering, University of Minnesota, Minneapolis, MN 55455
USA (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 Department 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) coding
techniques can effectively exploit the spatial diversity created
by multiple transmit antennas [27]. Typical examples include
ST trellis codes and ST block codes from orthogonal designs
(ST-OD). ST trellis codes enjoy maximum diversity and large
coding gains, but their decoding complexity grows exponentially in the transmission rate [27], which does not encourage
usage of large size constellations. On the other hand, ST-OD
codes [1], [26] offer maximum transmit diversity and can afford
low-complexity linear decoding. Unfortunately, ST-OD codes
come with reduced transmission rates, when complex constellais greater
tions are used and the number of transmit antennas
than two.
An alternative transmit diversity scheme that does not sacrifice rates, is based on what we term linear constellation precoding (LCP). It was originally developed for single-antenna
transceivers with an interleaver [4] and later on utilized also for
multiantenna systems [7]. Based on the parameterization of real
orthogonal matrices, construction of LCP was pursued in [7],
[23] based on exhaustive search. Because the search is constellation dependent, it becomes infeasible for large size constellations. On the other hand, algebraic tools can be used to construct
LCP 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 maximum diversity and coding gains in ST diversity systems, was
also left open.
This paper deals with a unified approach to constructing LCP
codes that maximize diversity and coding gains over constel. We
lations carved from the two-dimensional (2-D) lattice
view LCP designs as matrices and prove the existence of unitary
constellation precoding (UCP) matrices with maximum diver, for any finite constellation. This establishes the
sity gain
theoretical ground for searching over parameterized UCP matrices. For general LCP designs, we provide the upper bound on
the coding gain of all linear precoders to benchmark their performance. We generalize the parameterization construction of
UCP codes from real orthogonal matrices [7], [23] to unitary
matrices, which in general can provide larger coding gains. For
algebraic designs, we construct novel LCP codes that even for
correlated channels:
, re1) guarantee maximum diversity gains for any ,
gardless of the constellation;
2) achieve the upper bound on coding gains over quadrature-amplitude modulation (QAM) and pulse-amplitude
modulation (PAM) for certain values of ;
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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 for
other values of .
We also construct UCP codes adhering to a lower bound on
the coding gain, for any . In addition to diversity and coding
criteria [27], we also employ the maximum average mutual information criterion [14] to evaluate the performance and compare ST-LCP with the ST-OD codes of [1], [26], the so-called
quasi-orthogonal ST designs of [16], and the ST linear dispersive (LD) codes of [14].
This paper is organized as follows. Section II presents the
system model and the ST-LCP encoding scheme along with
pertinent design criteria. Section III provides design methods
based on the parameterization of unitary matrices and algebraic
number theoretic tools. Section IV describes the ST-LCP decoding options, while Section V presents properties of ST-LCP
codes, including a comparison of ST-UCP with ST-OD codes
in terms of maximum mutual information. Simulations are provided in Section VI and Section VII concludes the paper.
Notation: Bold lower (upper) case letters are used to denote
represent transpose and
column vectors (matrices); and
denotes trace and
conjugate transpose, respectively; Tr
stands for Kronecker product;
denotes the (
)th
denotes an
identity matrix;
entry of a matrix;
denotes a diagonal matrix with diagonal
diag
;
and
denote the real and imaginary
entries
stand for the positive
parts, respectively. , , , , and
integer set, the integer ring, the rational number field, the real
number 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 similar to [27] to deduce its design.
A. ST-LCP Encoding
With reference to Fig. 1, let us consider a wireless link with
transmit and
receive antennas over Rayleigh flat fading
channels. The symbol stream from a normalized1 constellasignal vectors , and then
tion set is first parsed into
matrix . The precoded
it is linearly precoded by an
1Average
symbol energy of C is assumed to be one.
block
is fed to an ST mapper, which maps
to an
code matrix that is sent over the
antennas during
time
)th entry
, is
intervals. Specifically, the (
transmitted through the th antenna at the th time interval,
denotes the (
)th entry of a unitary matrix ;
where
; and vector
denotes the th row of .
i.e.,
diag
, we can thus write the
Defining
transmitted ST-LCP code matrix as
(1)
Square-root Nyquist pulses [23, p. 557] are used as transmit
and receive filters in all antennas. After receive filtering and
received by antenna at the
symbol rate sampling, the signal
th time interval is a noisy superposition of
faded transmitted
, where
designals; i.e.,
notes the fading coefficient between the th transmitter and the
th receiver antenna with
and
.
We assume that
channel coefficients are uncorrelated with the
A1)
, zero mean, complex Gaussian, with correlanoise
, i.e.,
is full rank,
tion matrix
;
where
channel coefficients are only known to the reA2)
intervals (quasiceiver and remain invariant over
static flat fading);
noise samples are independent identically disA3)
tributed (i.i.d.), complex Gaussian with zero mean and
per dimension.
variance
Notice that our flat fading channels are allowed to be correlated.
received signal matrix with ( )th entry
Let be the
;
the
channel matrix with
; and
the
noise matrix with
(
in Fig. 1 denotes the th row of
for
). The
input–output relationship can then be written in matrix form as
(2)
is chosen to be identity [7], [33], the ST transmission
If
in (2) reduces to a time-division multiple-access (TDMA)-like
) out of
transmission with each antenna pausing for (
time intervals. If
is a complex Gaussian matrix with
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003
zero-mean i.i.d. entries and is a unitary matrix, the distribuis the same as the distribution of [19]. Thus, the
tion of
probability of error remains invariant to . However, offers
some flexibility that could be used to, e.g., alleviate high-power
amplifier nonlinear effects because it can avoid the unnecessary
“on–off” switch for each antenna.
Each transmitted symbol in ST-LCP is a linear combination
of the complex symbol entries in . We will see that by carefully
designing the precoder , ST-LCP can achieve full diversity
and large coding gains at rate 1 symbol/s/Hz. Unlike ST-LCP,
ST-OD is linear only in the real and imaginary parts taken separately; ST-OD enables low decoding complexity, by imposing
an orthogonality constraint on the code matrix . Unfortunately,
this constraint reduces transmission rate when complex constel[26].
lations are used with
B. ST-LCP Design Criteria
to detect in the
At the receiver end, we will rely on
maximum-likelihood (ML) sense and we will design to optimize the ML detection performance. We start with the pairwise
} as the event that the ML receiver
matrix error event {
diag
erroneously, when was
decodes
and the
maactually sent. Let us define
, where
trix
is ensured
existence of the correlation matrix square root
by (A1). Using standard Chernoff bounding techniques [27], we
can upper bound the average pairwise error probability (PEP) as
(3)
rank
, with
denoting a set of indices having cardinality
; and
stands for the geometric mean of the product of
nonzero eigenvalues
of
; i.e.,
the
.
We define the diversity gain, coding gain and kissing number
in terms of as follows.
1) Diversity Gain: The overall diversity gain is defined as
, over all distinct pairs
. From the definition of
, we infer that the maxof
is achieved when the folimum diversity gain
lowing maximum diversity condition holds true:
where
When
, the coding gain becomes
(6)
is the minimum product
where
.
distance. Note that (4) is equivalent to having
3) Kissing Number: The product kissing number is defined as the total number of pairs of symbol vectors and with
the same minimum product distance .
, the coding gain
measures
For a given diversity gain
the savings in signal-to-noise ratio (SNR) of the LCP system as
compared to an ideal benchmark system of BER
at 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 gain
first, because it determines the slope of the log-log bit-error rate
,
(BER)-SNR curve. Within the class of s that achieve
should be maximized afterwards. If two s
the coding gain
have the same diversity and coding gains, then the one with
the smaller kissing number is preferred. We will not minimize
the kissing number in this paper. However, we will show its
influence on the system performance in Section VI. Another
factor affecting BER performance is the bit-to-symbol mapping.
This should be also optimized in ST-LCP, but here we simply
adopt the Gray mapping [22, p. 170].
III. DESIGN OF ST CONSTELLATION PRECODERS
In our general precoding setup, we do not impose any structural constraints on , except for ensuring that Tr
, which controls the total transmit energy over
time in. Among all s obeying
tervals:
the power constraint, we look for those with maximum diversity and high coding gains. We will establish first the existence
of diversity-maximizing precoders (see also [12] and [33]). Ensured by this result, we will next look for an LCP matrix
that maximizes the coding gain
of (6) within the class of diversity-maximizing precoders; the overall optimum LCP matrix
will be selected as [cf. (6)]
(4)
is the th coordinate of the
Recalling the fact that
, we infer from (4) that in order to
precoded vector
, each
vector
should be different
achieve the
in all its coordinates. As a
from all other precoded vectors
one can decipher
result, from constellation precoded vectors
even if all except one of the
coordinates
are nullified
by fading.
2) Coding Gain: For an LCP matrix with a given , the
is defined as
coding gain
(5)
(7)
.
subject to the power constraint
Equation (7) discloses that our precoder design is independent
of the channel correlation matrix. For simplicity, we will hence, bearing in mind that our
forth focus on channels with
results carry over to the correlated case as well.2
2Our subsequent coding gain expressions for i.i.d. channels require just a
scalar multiplication by the [det( )]
factor to yield their counterparts for correlated channels.
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XIN et al.: SPACE–TIME DIVERSITY SYSTEMS BASED ON LINEAR CONSTELLATION PRECODING
To quantify the performance of
in (7), we will rely on
the following upper bound on the coding gain that applies to all
linear precoders (see Appendix A for the proof).
Proposition 1: (Upper Bound on the Coding
with
Gain): Consider any finite normalized constellation
.
minimum (Euclidean) distance
Among all linear precoders obeying the power constraint
, the maximum coding gain
is
(8)
In the following two sections, we will provide methodologies for
designing LCP matrices based either on parameterizations of
unitary matrices, or, on algebraic number theoretic tools.
A. Design Based On Parameterization
Unitary constellation precoding offers a distinct advantage
over nonunitary LCP options: a unitary corresponds to a rotation and preserves distances among the -dimensional constellation points. On the contrary, a nonunitary draws some pairs
of constellation points closer (and some farther). This distancepreserving property of UCP also guarantees that if such rotated
constellations are to be used over an additive white Gaussian
noise (AWGN) (or near AWGN) channel, the performance will
remain invariant. In practice, the channel condition can also vary
between the two extremes of AWGN and Rayleigh fading, in
which case a unitary precoder may be preferred [3]. For these
reasons, we first deal with unitary precoders. But prior to designing unitary ’s, it is natural to ask whether the unitary class
is rich enough to contain ST-LCP precoders with maximum diversity gains. The following proposition asserts that a unitary
achieving
always exists (see Appendix B for the proof).
Proposition 2: (Existence of a Diversity-Maximizing Unitary
Precoder): As long as the constellation size is finite, there always exists at least one unitary satisfying(4) and is, thus, cafor any
pable of achieving the maximum diversity gain
number of transmit ( ) and receive ( ) antennas.
Notice that the fading-resilient constellations in [4], [5], and
[12] guarantee maximum diversity gains only for particular constellations or -dimensions.
Ensured by Proposition 2, we are now motivated to look for
that maximizes
among diversity-maximizing
a unitary
unitary precoders [cf. (7)]. As formulated in (7), finding
involves multidimensional nonlinear optimization over the
complex entries of . To facilitate the optimization, we can take
and parameterize
advantage of the fact that
using
real entries taking values from finite intervals. We start
.
with the simplest case where
can be expressed
Any real orthogonal precoder for
as a rotation matrix [7], [23]:
(9)
. The
which is a function of a single parameter
precoder in (9) rotates the constellation points in 2-D so that
each rotated point is different from other rotated points in both
297
coordinates. With
as in (9), the criterion in (7) needs to be
optimized only over a single parameter . Instead of using the
real orthogonal matrices of [7], [23], we here explore unitary
precoders in , because they have the potential for larger coding
gains than their real counterparts.
It is known that any 2 2 unitary matrix can be parameterized
as [21, p. 7]
(10)
where
is a 2
2 diagonal unitary matrix,
and
.
, it is possible to construct real orthogonal preFor
coders by using Givens matrices [7], [23]. Specifically, any real
orthogonal matrix can be factored as a product of
Givens matrices of dimension
and an
pseudo-identity matrix, which is defined as a diagonal matrix
with diagonal elements 1 [7]. In the following proposition, we
generalize this result to also include unitary matrices (see Appendix G for the proof).
Proposition 3: (Parameterization of Unitary Matrices): Any
unitary matrix can be written as
where
is an
diagonal unitary matrix,
,
and
is a complex Givens
with the (
)th,
matrix, which is just the identity matrix
,
,
( )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 optimization (7). The number of parameters that need to be opti, which are the parameters
mized is thus
of the
complex Givens rotation matrices. Analytical solution to this optimization problem appears to be in)
tractable. However, for a small number of antennas (say
), exhaustive search is
and small constellation sizes (say
computationally feasible, as we will illustrate in Section VI.
B. Design Based on Algebraic Tools
The design based on the parameterization of unitary matrices
or
is large. Fortunately,
is less practical when either
algebraic number theoretic approaches are possible to yield
closed-form LCP designs with reasonably large coding gains
and/or
is large. In this section, we
[5], [12], even when
introduce two novel LCP constructions: LCP-A and LCP-B.
We prove that LCP-A can achieve the upper bound on the
, where
,
coding gain over QAM (or PAM) for
, where
is an Euler number3 and
or, for
mod . We also show that LCP-B, which is unitary for
any , has coding gain that is guaranteed to be greater than a
lower bound.
We start by briefly introducing some necessary definitions
and facts from [12] and [20].
3An Euler number (P ) is defined as the number of positive integers <
which are relatively prime to P .
P,
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003
B.1) Algebraic Number Theory Preliminaries:
denotes the smallest subfield of
inNotation:
denotes the smallest subfield
cluding both and and
including both
and , where is algebraic over
of
; i.e., is a root of some nonzero polynomial
;
is the ring of Gaussian integers, whose elements are in the
with
;
denotes the minimal
form of
denoting
polynomial of over a field with
its degree.
Definitions:
D1)
D2)
D3)
D4)
, the th cyclo(Cyclotomic Polynomials): If
tomic polynomial is defined as
, where
gcd
and
and
is its degree.
(Extension of an embedding): If is an embedding of
in that fixes
in such a way that
,
, then is called a
-isomorphism of
.
(Relative norm of a field): Let
denote the complex roots of the minimal polynomial
; and let
,
, be
of over
distinct
-isomorphisms of
such that
. Consider
and define
from the field
as
the relative norm of
.
: An element is said to be integral
(Integral over
, if is a root of a monic polynomial with
over
. Clearly, every element in
is
coefficients in
.
integral over
Facts:
F1)
F2)
F3)
F4)
F5)
F6)
F7)
is the minimal polynomial
The polynomial
of
and
for
such that gcd
.
any
with
, then the
of
If
is
with all distinct roots
for
.
is a finite extension of the field
with
If
, then
degree denoted by
forms a basis of
over
.
, which are integral over
The set of elements of
, is a subring of
containing
.
is integral over
, then the relative
If
.
norm of from
, then
If is an odd integer and
.
and
, then
.
If gcd
Before presenting our constructions that are based on these
facts, we first prove the following important lemmas (see
Appendices C–E for their proofs).
for
, then
Lemma 1: If
mod .
mod , then the
of
Lemma 2: If
is
and its degree is
.
of
with
Lemma 3: All roots of
have unit modulus.
some
B.2) Algebraic Construction: LCP-A:
LCP-A constructs a matrix that applies to any number of
and subsumes the constructions in
transmit-antennas
is a power of two, where
[5] and [12] as special cases when
is not a power of two,
the resulting precoder is unitary. When
the construction yields nonunitary LCP matrices.
such that
.
Let be integral over
LCP-A is constructed as follows (see also [5], [12], and [35]):
..
.
..
.
..
.
(11)
are the roots of
and
where
is the normalizing factor ensuring that Tr
.
The idea behind LCP-A can be explained as follows. For confor a moment. By F3,
venience, let us ignore the constant
form a basis of
over
the entries in the first row of
. This means that
. For all constel, that is,
,
lations carved from the lattice
is integral over
based on F4 and the fact that
is integral over
. We can, thus, view
as the (unique)
with respect to the basis of the first
coordinates of
-isomorphisms
row entries. Defining
, we have
; entries of
are then the images of these isomorphisms of
.
These isomorphisms are the ones required in the definition of
the relative norm of
(cf. D3). The relative norm in
this case also coincides with our definition of product distance
, the minimum product distance,
in (6). Therefore, for
from F5, is at least one.
being nonzero and belonging to
Of course, we have to also take into account the energy nor, after which the coding gain is
malization and the constant
, where and
are constellation dependent parameters (see Proposition 5 next for a complete statement of the
result).
for which
We rely on the following lemma to find values of
LCP-A achieves the upper bound on the coding gain (see Appendix F for the proof).
, then
Lemma 4: If in (11) is integral over
and the equality holds if and only if all the roots
have 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 the
coding gain, as we will see soon. Lemma 2 and F2 imply
for
mod
and
that both
for
, belong to . On the other hand, according to
do not belong to . For
Lemmas 3 and 4, odd integers
instance, the set
and
.
XIN et al.: SPACE–TIME DIVERSITY SYSTEMS BASED ON LINEAR CONSTELLATION PRECODING
We study next properties of our LCP-A and establish results
on their coding gains. The following proposition provides the
lower-bound on coding gains over any constellation carved from
(see Appendix H for the proof):
Proposition 4: (Lower Bound on Coding Gains): If one
in (11) to precode the constellation
applies the LCP matrix
and normalized by
, then the coding gain
carved from
satisfies
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TABLE I
CODING GAINS OF ST-LCP CODES FOR
with
, over
. If
N
= 4–10 OVER 4-QAM
and let ’s be the degrees of
, then we have
(12)
(15)
In particular, when QAM (or PAM) constellations are used, we
provide not only the exact coding gain which can be achieved
by LCP-A, but also lower and upper bounds on the maximum
coding gain achieved by LCP-A (see Appendix I for the proof).
Proposition 5: (Coding Gains for QAM [or
with
PAM]): Consider a QAM (or PAM) constellation
the minimum distance of signal points equal to 2 , which is
. For
and the linear precoder
normalized by
in (11), the coding gain over the normalized QAM (or PAM)
is given by
(13)
Furthermore, the maximum coding gain achieved by LCP-A is
lower and upper bounded by
(14)
, LCP-A achieves the upper bound in (14)
i) For
on the coding gain of all linear precoders over QAM (or
PAM).
, LCP-A cannot achieve the upper bound
ii) For
in (14). However, LCP-A at least can achieve the lower
bound in (14), which is a large fraction (70%) of the upper
bound.
B.3) Algebraic Construction: LCP-B:
As we argued at the beginning of Section III-A, unitary precoders have certain advantages as compared to nonunitary ones.
For certain ’s, the precoders designed in LCP-A are not unitary.
We here present a construction of unitary precoders for any
diag
where
and
is the -point inverse fast Fourier
transform (IFFT) matrix whose (
)st entry is given by
. Notice that this LCP
matrix amounts to phase-rotating each entry of the symbol
vector and then modulating in a digital multicarrier fashion
that is implemented via
. The choice of will be addressed
later in this section.
Next, we state a proposition which provides lower bounds on
coding gains of LCP-B (see Appendix J for the proof).
Proposition 6: (Lower Bounds on Coding Gains of Unitary Precoders): For LCP-B, let denote the number of disof
,
tinct minimal polynomials
and
.
where
To design unitary precoders with large coding gains,
Proposition 6 suggests choosing
such that the number of distinct minimal polynomials of
,
, in Proposition 6 is
small and their degrees are as low as possible, in order to make
small. In particular, when
with
, we can select
such that all
,
, are
over
. In this
roots of the minimal polynomial
,
and (15) coincides with (12) as
case, we have
. Even though (15) benchmarks the coding gain
at high SNR, we have verified through simulations that it is
rather pessimistic.
Heuristic Rule for Constructing Unitary Precoders: For
which is not a power of two, choose
any given
for some
such that most of
’s are roots of
.
for
.
This will make small and
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, 9
and
, where
over 4-QAM constellations with
and
denote the coding gains of the precoders from LCP-A
and 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
, 7, 9 provide quite large coding gains even
precoders for
when the construction of (11) cannot achieve the upper bound
in (14).
and we choose
, then
Example 2: If
. Notice that with
, all
’s except
are roots of the minimal polynomial
of
with
, while
is a root of
with
.
In this construction, we have
. Based on simulations, we
for 4-QAM constellations.
find that
IV. DECODING OF ST-LCP TRANSMISSIONS
The starting point of our optimal precoder designs was the
performance of ML detection of from (2). Because the complexity of ML detection based on exhaustive search is very high
and/or
is large, we consider in this section three alwhen
ternative decoders for ST-LCP transmissions. The first, sphere
decoding (SD), is used to approximate the ML performance at a
polynomial (but still relatively high) complexity, while the other
alternatives, Vertical Bell-Labs Layered ST (V-BLAST) [13] or
block minimum mean-square error decision-feedback equalization (BMMSE-DFE) [2], [25], are used as relatively low-com-
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003
plexity alternatives. Defining
, we rewrite (2) as
. Using the vec operator to put the columns
one after the other, we obtain
vec
as
of
diag
diag
vec
(16)
denotes the th row of corresponding to the th rewhere
denotes the diagonal matrix generated by
ceiver, diag
and
is an
block diagonal matrix. The received
uncoded
vector in (16) is equivalent to a received block from
receive-antennas with the channel
transmit-antennas to
matrix being almost always full rank.4 Thanks to the special
structure of in (16), application of the maximum ratio combiner yields
V. ST-LCP PROPERTIES
diag
where
diag
and
is colored Gaussian noise. The latter can be
prewhitened to obtain
(17)
and is AWGN. Equation (17) will
where
be our starting point for ST-LCP decoding.
A. Near-Optimum Decoding
The SD algorithm of [8] and [29] was introduced to reduce
decoding complexity provided that the transmitted constellation is carved from a lattice. SD takes advantage of the lattice
structure of transmitted signals to achieve near ML performance
with a moderate complexity. It has been shown that for a fixed
searching radius and for a given lattice structure, the decoding
transmit antennas is approximately
complexity for
[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 separately the real and imaginary parts of in (17). When both
and are complex, one should view the
complex vector
as a
real vector and rewrite the equivalent system
model as in [6, eq. (2)]
The computational burden of decoding complex ST-LCP transmissions will increase accordingly, because we need to apply
vector. However, the recently proposed
the SD to a
complex sphere decoder in [15] does not double the size of the
search lattice vector, thus reducing the complexity.
B. Suboptimum Decoding
The SD algorithm achieves near-ML performance with
and/or
is large, the
polynomial complexity. But when
complexity becomes prohibitively high. As reduced-complexity
alternatives, we advocate using the V-BLAST [13], or the block
4This holds true because each channel tap h
nonzero with probability one.
(B)MMSE-DFE algorithm [2], [25], whose complexity is
. Both V-BLAST and (B)MMSE-DFE are
roughly
the decoding schemes that are based on decision feedback.
The decision feedback here is only used for interference
cancellation, but one could also use it for channel estimation in
a decision-directed mode.
Remark: It is known that with linear processing at the
receiver, ST-OD can convert the space–time channel into a
number of parallel AWGN channels. Such a parallel conversion
enables the inclusion of an outer channel encoder/decoder
because soft information can be obtained from these parallel
AWGN channels about coded symbols. For ST-LCP, such soft
information output does not seem practically possible unless
some enumerative search is performed.
H
in the structured matrix ~ is
Having described the encoding and decoding options of our
ST-LCP system, in this section we present four attractive features they possess and compare them with competing alternatives.
A. Delay Optimality
to be achieved, it is known
For the maximum diversity
is equal to
that the minimum possible decoding delay
under the quasi-static flat fading assumption; and schemes that
are
achieve maximum diversity with the minimum delay
called delay optimal [10]. ST-LCP is delay-optimal, because
by design. This is not always true for ST-OD, howand complex constellations
ever. For example, when
time intervals [26].
are used, ST-OD codes require
B. Mutual Information Optimality
In this section, we will prove that ST-UCP can achieve higher
.
average mutual information than ST-OD codes when
is
Recalling that for i.i.d. channels the distribution of
identical to that of , we infer that the maximum average mu, of ST-UCP coincides
tual information per time interval,
with the capacity of spatial cycling [9, eq. (13)]. So we have
(18)
Correspondingly, for ST-OD block transmissions, the maximum average mutual information per time interval is
[24, eq. (4)]
(19)
exist for any .
For real constellations, ST-OD at rate
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 .
, this exponent is strictly smaller than the correWhen
sponding largest exponent of in (18) which equals one. Based
on this fact, we are able to establish the following proposition
comparing mutual information of ST-UCP with those achievable by ST-OD codes (see Appendix K for the proof).
XIN et al.: SPACE–TIME DIVERSITY SYSTEMS BASED ON LINEAR CONSTELLATION PRECODING
Proposition 7: (Information-Theoretic Comparisons With
ST-OD): Under the channel assumptions in (2) and for sufficiently large SNR , the maximum average mutual information
of ST-UCP systems is strictly greater than that of ST-OD
.
systems for
TABLE II
CODING GAINS OF ST-LCP CODES FOR
301
N
= 2, 3, 4 OVER 4-QAM
C. Symbol Detectability
with the deterministic notion of zeroHere, we link
forcing equalizability (or symbol detectability). Specifically, we
establish in the appendix that the nonzero minimum product distance [cf. (4)] implies that: if in (16) is nonzero, then the symbols are guaranteed to be recoverable (perfectly in the absence
of noise); and this is what we define as symbol detectability.
Proposition 8: (Symbol Detectability): If (4) holds true, then
are detectable so long as one entry of
ST-LCP symbols
is nonzero.
By exploiting the finite alphabet property of , Propositions
2 and 8 assert the surprising result that ST-LCP guarantees desymbols in even from a single received
tectability of the
, provided that among all
flat
sample
.
fading channel coefficients, only one
D. Flexibility in Slow and Fast Fading
By the construction of ST-LCP codes in (1), we have that at
every time interval
where
denotes the th column of the unitary matrix in
(1). Hence, ST-LCP codes are also suitable for fast fading according to the distance criterion of [27]. As ST-LCP only re’s, it is also applicable to fast
quires independence among
fading (as opposed to quasi-static flat fading) channels, which
can be tracked accurately using the Kalman predictors developed in [18]. However, in fast fading channels, ST-LCP will
perform the same as if the channel is slowly fading. To fully exploit 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 variations [11].
VI. SIMULATED PERFORMANCE
In this section, we simulate ST-LCP systems and compare
them with ST-OD, the quasi-orthogonal ST designs of [16] and
ST LD codes in [14]. For more comparisons, interested readers
are referred to [32]. Similar to LCP, quasi-orthogonal designs
also relax the orthogonality imposed by ST-OD codes. We will
use binary phase-shift keying (BPSK) or QAM with constellation sizes chosen such that the spectral efficiency of ST-LCP
and ST-OD are the same. In all simulations, the real and comSNR . The
plex part of the AWGN has variance
channel matrix has i.i.d. entries. The average BER is obtained
through Monte Carlo simulations, except for ST-OD where a
recursive algorithm is used to compute the exact BER [30]. All
simulations except for Test Case 3 utilize the SD algorithm.
For real precoders, we use the codes in [7] when
and those from [5] when
, 4, 6. We construct complex
precoders
for
, 4, 6 according to LCP-A in (11),
, being the
roots of the polywith
,
and
nomials (cf. Definition D1 and Fact F2)
, respectively; for
,
for
in
. When
,
(11) are chosen to be roots of
we use the parameterization of Proposition 2 and search for the
and
six Givens matrix parameters:
. Exhaustive computer search is
carried out over the discrete values obtained by quantizing the
finite continuous intervals of these six parameters. Specifically,
we first divide each interval into smaller subintervals. The midpoint of each subinterval is then used as a parameter value and
the coding gain is evaluated. The subinterval whose midpoint
gives the largest coding gain is further divided into even smaller
subintervals for search and the search continues until the coding
gains converge. The resulting precoder
is found to have coding gain larger than that of (11) and is thus
used.
transmit antennas will be denoted as
ST-OD codes for
with 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 Test
case 5, the channel capacity is computed using [9, eq. (4)].
Test Case 1 (Complex Versus Real Precoders): Table II lists
the coding gains of real and complex ST-LCP codes (1) over
, 3, 4. It also indicates the
4-QAM constellations for
of distinct
pairs with the product distance less
number
than 3 ; in the third row denotes the total number of precoded
vectors for each . The advantage of complex precoders compared to real precoders in coding gains shows up in the last row,
whose entries are the ratios (in decibels) between the coding
gain of complex precoders and those of real ones. Notice that
for
has nearly 2 dB larger coding gain
a complex
than its real counterpart, while this gain is only about 0.5 dB
, 3. Fig. 2 compares the BER performance of comfor
, 3 with BPSK. The complex
plex and real precoders for
precoders outperform the real precoders by more than 1 dB at
BER 10 . Fig. 3 shows that complex precoders outperform
real ones by about 0.5–1 dB at high SNR.
Our performance analysis is based on PEP, which is known to
offer more accurate approximation of the system performance
at reasonably high SNR [27]. Besides coding gains, the kissing
number may also play an important role in affecting the system
performance. The difference in BER between complex and real
precoders is not as significant as the difference in coding gains
and only shows up at high SNR.
302
Fig. 2. Complex versus real precoders (BPSK,
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003
N
= 2, 3, N = 1, 1 b/s/Hz).
Fig. 3. ST-OD (256-QAM) versus ST-LCP (16-QAM) (
4 b/s/Hz, R = 1=2).
N
= 3, 4, N = 2,
Test Case 2 (ST-LCP Versus ST-OD): Figs. 4–9 compare
ST-LCP against ST-OD codes for various combinations of
, spectral efficiencies and rates . Fig. 4 shows that
ST-OD codes outperform ST-LCP codes by 1–2 dB when
. Fig. 5 compares complex ST-LCP codes with 4-QAM
, 4 and
. To
and rate 1/2 ST-OD codes for
maintain the same spectral efficiency, we use 16-QAM for
these ST-OD codes. The simulation shows that ST-LCP now
gains about 2 dB over ST-OD codes. The gain of ST-LCP is
increases as shown in Fig. 6. Fig. 7
more pronounced when
, 6 and
. Again, ST-LCP codes
depicts BER for
have an advantage over ST-OD. Fig. 7 also confirms that the
complex precoder outperforms the real one obtained from [5]
by about 1 dB.
, 4 and
, rate 3/4 ST-OD codes with
For
256-QAM are tested and compared with rate 1 ST-LCP codes
in Fig. 8. The spectral efficiency in this case is 6 b/s/Hz. The
gain of ST-LCP in SNR is less than 1 dB. Fig. 9 shows that the
.
gain 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).
Fig. 5. ST-OD (16-QAM) versus ST-LCP (4-QAM) (N = 3, 4, N = 1,
2 b/s/Hz, R = 1=2).
In summary, ST-LCP codes perform better than ST-OD when
at the price of increased decoding complexity.
Test Case 3 (Decoding Options): Fig. 10 depicts the performance of SD, V-BLAST and BMMSE-DFE in various ST-LCP
, 4 and
, at spectral efficiency
schemes for
4 b/s/Hz. It shows that both V-BLAST and BMMSE-DFE
cannot achieve the maximum diversity gain; SD outperforms
10 . However,
both alternatives about by 5 dB at BER
we observe that even with the suboptimum V-BLAST or
BMMSE-DFE decoding, ST-LCP still outperforms ST-OD by
10 . VBLAST uses zero-forcing
about 7–8 dB at BER
with no ordering. The resulting V-BLAST performance is only
slightly worse than that of BMMSE-DFE.
Test Case 4 (ST-LCP Versus Quasi-Orthogonal ST and LD
Codes): Fig. 11 depicts the performance comparison between
ST-LCP and the quasi-orthogonal ST codes of [16] for
and
with 4-QAM. The decoding of quasi-orthogonal
ST in [16] was implemented. The diversity gain of ST-LCP is
, while that of the quasi-orthogonal ST codes is only two.
XIN et al.: SPACE–TIME DIVERSITY SYSTEMS BASED ON LINEAR CONSTELLATION PRECODING
N
N
= 3,
Fig. 8. ST-OD (256-QAM) versus ST-LCP (64-QAM) (N = 3, 4, N = 1,
6 b/s/Hz, R = 3=4).
Fig. 7. ST-OD (16-QAM) versus ST-LCP (4-QAM) (N = 5, 6, N = 1,
2 b/s/Hz, R = 1=2).
Fig. 9. ST-OD (256-QAM) versus ST-LCP (64-QAM) (N = 3, 4, N = 2,
6 b/s/Hz, R = 3=4).
The higher diversity gain of ST-LCP pays off for SNR values
above 15 dB. Fig. 12 shows the performance comparison beand
tween ST-LCP and LD codes [14, eq. (36)] for
. Both schemes use the SD algorithm. ST-LCP shows
very similar performance to LD codes, but LD codes have relatively higher decoding complexity because they use a larger
block size in this case. For further comparisons between LCP
and LD, we refer the readers to [32].
Test Case 5 (Information-Theoretic Comparisons): Fig. 13
shows the maximum mutual information comparisons between
and
. ST-OD
ST-UCP and ST-OD for
, but the difference is insignifoutperforms ST-UCP for
. Fig. 14 depicts the maximum average
icant when
mutual information for both ST-UCP and ST-OD computed
and
. The cafrom (18)–(19), when
pacity loss of ST-OD is significant at high SNR compared with
increases, the capacity loss becomes more
ST-UCP. When
and spectral efficiency
pronounced at high SNR. With
4 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. 6. ST-OD (16-QAM) versus ST-LCP (4-QAM) (
2 b/s/Hz, R = 1=2).
= 3,
303
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003
Fig. 11.
Quasi-OD versus ST-LCP (4-QAM,
N
= 4, N = 1, 2 b/s/Hz).
Fig. 13.
Mutual information of ST-UCP and ST-OD (
N
= 2).
Fig. 12.
LD codes versus ST-LCP (4-QAM,
N
= 3, N = 1, 2 b/s/Hz).
Fig. 14.
Mutual information of ST-UCP and ST-OD (
N
= 3).
We observe also that the mutual information achieved by
ST-UCP codes is still far from the channel capacity, especially
is large at high SNR. For example, with
when
and spectral efficiency 4 b/s/Hz, ST-UCP shows about 1 dB
, but the
loss compared to the channel capacity when
. If we compare the set
loss increases to 2-3 dB when
of curves in Fig. 14 that correspond to the same transmission
increases from
scheme (for either ST-UCP or ST-OD), as
1 to 6, we notice that the gain of mutual information obtained
by each additional receive antenna becomes smaller when
.
some cases, maximum) coding gains over both quasi-static and
fast fading, both correlated and i.i.d. channels. Near-optimum
and suboptimum decoding options were also provided. Finally,
it was shown that ST-LCP codes can achieve better performance and larger maximum mutual information than ST-OD
codes, when the number of transmit antennas is greater than
two.
APPENDIX A
PROOF OF PROPOSITION 1
which satisfies the power constraint:
, we have Tr
Tr
. By the definition of the trace and the nonneg, there exists at least one
ativity of the diagonal entries of
.
column of (say the th) with Euclidean norm
be a particular pair with
,
Let
is the th column of the identity matrix. Using that
where
with
and the
For any
VII. CONCLUSION
A unified approach for exploiting the transmit diversity in
a multiantenna environment was developed by utilizing linear
constellation precoding. With any number of transmit- and
receive-antennas, the proposed scheme can achieve a rate of
1 symbol/s/Hz, maximum diversity gains, as well as large (in
XIN et al.: SPACE–TIME DIVERSITY SYSTEMS BASED ON LINEAR CONSTELLATION PRECODING
arithmetic-geometric mean inequality, the square of the product
is as follows:
distance of
305
By the definition of
, it follows that
for
. Assume that
for some
. We have either
with
, or,
, with
.
mod by (21); which
In both cases, we infer that
; hence,
contradicts the assumption that
mod .
(20)
Because is chosen arbitrarily and the right-hand side of (20)
is independent of , we have
APPENDIX B
PROOF OF PROPOSITION 2
denote the set of all
signal vectors over the
Let
-dimensional complex space
. Every entry of
is a symbol from a finite constellation ; hence,
is a finite set. We infer from (4) that proving Proposition 2 is equivunitary matrix
alent to proving that there exists an
has nonzero coordinates for all distinct pairs
such that
. Let denote all possible differences between dis, i.e.,
.
tinct vectors of
is finite with cardiIt is clear that
. In an
-dimensional space, there exists an
nality
vector denoted as , which is neither perpendicular
as a uninor parallel to any of the vectors in . Choose
tary matrix such that the first row vector is taken as . Conas a new set in which the first
sider
coordinate of any vector is nonzero. Then treat the last
coordinates of these vectors as a new set of
vectors.
(
), all vectors in
Since is not parallel to any
this new set are also nonzero vectors. By the same argument, we
vector which is neither perpendicular
can find an
vectors in the new set. Choose
nor parallel to these
unitary matrix such that its first row
an
vector is . Then, we can construct a new unitary matrix
as follows:
Let us define
. It is easy to
are nonzero
see that the first two coordinates of vectors in
) coordinates are
nonzero
while the last (
) steps,
vectors. Performing the same construction in (
unitary matrix
such
we obtain an
has non zero coordinates, which completes the
that
proof.
APPENDIX C
PROOF OF LEMMA 1
If
then
where are distinct primes and
can be written as [cf. F8]
,
(21)
APPENDIX D
PROOF OF LEMMA 2
or
, as gcd
,
Case 1: If
. Case 2: If
we have from F6, that
, then
by F7.
and
; as
, we
Consider
; and thus
. By F6, it follows
have
.
that
mod , we have
Therefore, when
; thus,
. Applying the facts that the degree of
over
and that
for
mod , we
. As
have
by F1 and since
, we have
. Hence, when
mod ,
is
. Since
has degree
in
and is its root, it is the minimal polynomial of
over
.
APPENDIX E
PROOF OF LEMMA 3
and
As is a root of
. Hence, every root of
that
and, thus, all roots of
be a root of
modulus.
, it follows
must
have unit
APPENDIX F
PROOF OF LEMMA 4
As
5
is integral over
, it follows that
and
; thus, we have
. To force the power constraint:
, we set
Moreover, since
we have
, where the equality holds if and
for
. This implies that
only if
and, thus,
.
, we only need to show
To prove the case for odd
and
,
that when is integral over
does not have unit modulus.
then at least one root of
of
have unit modulus. By
Suppose that all roots
5The proof of this argument relies on the Gaussian Lemma of [20] by applying
the fact that [j ][x] is a unique factorization domain.
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003
using a proof similar to [31, p. 4, Lemma 1.6] for an integral
, can be written as
for some
element over
. As
for
, it follows
and
. So
that
. As
is always
, we know that
.
an even integer for
is always 2, we must have that
But since
. This implies that
. By Lemma 1, we deduce that
mod . From
Lemma 2, we know that
; hence,
, which is a contradiction. Thus,
have unit modulus.
not all roots of
APPENDIX G
PROOF OF PROPOSITION 3
We prove this proposition by using induction on . Clearly
. We know that
Proposition 3 holds true for
unitary matrices can be written recursively as [21, p. 10]
APPENDIX H
PROOF OF PROPOSITION 4
For any distinct
and , we define
where
and
are the th entries of the vectors and , re,
are
spectively. As
; so
linearly independent by F3. We infer that
Moreover, it follows from D4 and F4, that
. But using F5, we have that
over
. As
implies that
is integral
, which
(22)
we can obtain
where
is given by
(23)
where
is an
unitary matrix and the
.
parameter
By the induction hypothesis, we can write
where
is an
’s are
and
Then, we can write
diagonal unitary matrix
complex Given matrices.
as
Therefore, by the definition of
, we infer that
APPENDIX I
PROOF OF PROPOSITION 5
For the QAM constellation points
with
, we have
is similar) and
. From (23) and the facts that
and is nonzero, we obtain
over
(the proof for PAM
with
is integral
(24)
where
and
that
is lower bounded by
.
Hence, the coding gain
with
On the other hand, let us consider a particular pair
, where is the first column of the identity
for
matrix. Using the fact that
any LCP-A matrix in (11), we obtain
diag
diag
. Hence, we find
This, together with (24), establishes that the coding gain of
LCP-A is exactly
from (22).
XIN et al.: SPACE–TIME DIVERSITY SYSTEMS BASED ON LINEAR CONSTELLATION PRECODING
A. Proof of (i)
, we have
by Lemmas 3 and 4.
When
So, the coding gain of LCP-A is given by
which equals the upper bound given in (8) of Proposition 1 with
.
307
On the other hand, according to the definition of and since
for
, we have based on Lemma 3, that
We thus obtain
By (26) and
, it follows that:
B. Proof of (ii)
, not all roots of have unit modulus as is
When
by [31, Lemma 1.6]. Hence, we have
integral over
by Lemma 4. Therefore, the upper bound of (14) cannot
. However, we can select
be achieved by LCP-A for
to be the roots of the minimal polynomial
over
. From (13), we have
Using the same argument for all
and
fact that
and applying the
, we obtain
(25)
Because the function
and
is decreasing for
, we have from (25) that
APPENDIX J
PROOF OF PROPOSITION 6
and
be its cardinality for
. Clearly,
. Let
be the
and without loss of generality, let us assume that
roots of
roots of
are from
and let
the first
. For any distinct and , we define
Let
APPENDIX K
PROOF OF PROPOSITION 7
To prove the proposition, we first prove the following lemma.
and alLemma 5: Consider a deterministic variable
,
most surely positive random variables , , , ,
that are related by
and
,
; ii)
; and
where: i)
and
are bounded. It then holds true that
iii)
, for sufficiently large values of .
Proof of Lemma 5: For sufficiently large values of , we
have
(27)
(28)
(29)
(30)
As
is the degree of the minimal polynomial of the roots from
over
, using the assumption that
, we infer that
are linearly independent. Hence,
and
As
we have
(26)
; inequality (28) is due to i) and
where (27) is due to
iii); (29) is due to ii); and (30) is based on Jensen’s inequality
function.
and the concavity of the
Proof of Proposition 7: It suffices to show that in (18) and
for sufficiently large values of . This can
(19),
,
and
be established by applying Lemma 5 with:
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003
APPENDIX L
PROOF OF PROPOSITION 8
Arguing by contradiction, suppose that there are two distinct
symbol vectors and that satisfy (16) in the noise-free case;
. The latter implies that
i.e., that
. Because
cannot be identically zero, there exists a
. Because
, we deduce
nonzero entry denoted as
, that
[notice the structure of
from
in (16)]. Because the minimum product distance is nonzero,
. Therefore, symbol recovery is
we infer from (4) that
guaranteed in the noise free case; and hence, is detectable.
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Yan Xin (S’00) received the B.E. degree in
electronics engineering from Beijing Polytechnic
University, Beijing, China, in 1992, the M.Sc.
degree in mathematics and the M.Sc. degree in
electrical engineering from University of Minnesota,
Minneapolis, in 1998 and 2000, respectively. He is
currently working toward the Ph.D. degree in the
Department 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 in
electrical engineering and information science
from the University of Science and Technology of
China (USTC), Hefei, in 1996, the M.Sc. degree
in electrical engineering from the University of
Virginia, Charlottesville, in 1998, and the Ph.D.
degree in electrical engineering from the University
of Minnesota, Minneapolis, in 2002.
He is now with the Department of Electrical and
Computer Engineering, Iowa State University, Ames,
IA. His interests lie in the areas of signal processing, communications, and information theory, including cyclostationary signal processing, blind equalization algorithms, transceiver optimization, multicarrier, wideband multiple rate
systems, space–time capacity and coding, and error-control coding.
XIN et al.: SPACE–TIME DIVERSITY SYSTEMS BASED ON LINEAR CONSTELLATION PRECODING
Georgios B. Giannakis (S’84–M’86–SM’91–F’97)
received his Diploma in electrical engineering from
the National Technical University of Athens, Athens,
Greece, in 1981 and the M.Sc. degree in electrical
engineering, the M.Sc. degree in mathematics, and
the Ph.D. degree in electrical engineering from the
University of Southern California (USC), in 1983 and
1986, respectively.
From September 1982 to July 1986, he was with
USC. After lecturing for one year at USC, he joined
the 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 Computer
Engineering at the University of Minnesota, Minneapolis, where he now holds
an ADC Chair in Wireless Telecommunications. His general interests span
the areas of communications and signal processing, estimation and detection
theory, time-series analysis and system identification—subjects on which he
has published more than 150 journal papers, 290 conference papers, and two
edited books. His current research topics focus on transmitter and receiver
diversity techniques for single- and multiuser fading communication channels,
precoding and space–time coding for block transmissions, multicarrier, and
wideband wireless communication systems.
Dr. Giannakis is the (co-) recipient of four best paper awards from the IEEE
Signal Processing Society (1992, 1998, 2000, 2001). He also received the Society’s Technical Achievement Award in 2000. He co-organized three IEEESignal Processing Workshops, and Guest Edited (co-edited) four special issues.
He has served as Editor-in-Chief for the SIGNAL PROCESSING LETTERS, Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and the IEEE
SIGNAL PROCESSING LETTERS, as Secretary of the Signal Processing Conference Board, Member of the Signal Processing Publications Board, Member and
Vice-Chair of the Statistical Signal and Array Processing Technical Committee,
and as Chair of the Signal Processing for Communications Technical Committee. He is a Member of the Editorial Board for the PROCEEDINGS OF THE
IEEE and the Steering Committee of the IEEE TRANSACTIONS ON WIRELESS
COMMUNICATIONS. He is a Member of the IEEE Fellows Election Committee,
the IEEE Signal Processing Society’s Board of Governors, and a frequent Consultant for the telecommunications industry.
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