![Fig. 4. (a) Finding the optimum decision statistic vectord[i] for each ofm levels: i = 1; 2; . . . ;m. (b) Search overm (i + 1) candidates for the spatial matched filter vector for the best decision statistic at leveli.](/figures/fig-4-a-finding-the-optimum-decision-statistic-vectord-i-for-2h4c9jjw.webp)
![Fig. 1. High-level view of a vertical BLAST communication link. Assumptions: narrowband operation, allm constellations are the same size, and the transmitter does not know the channel instantiation, but the receiver learns it. Specifications:n; (when m = 1; is the average SNR between a transmit–receive pair), probability [error-free burst], andPtot, the total power transmitted out of allm antennas. The number of transmit antennasm and the quadrature amplitude modulation (QAM) constellation size are optimized for maximum burst throughput.](/figures/fig-1-high-level-view-of-a-vertical-blast-communication-link-syx7g1nt.webp)
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 11, NOVEMBER 1999 1841
Simplified Processing for High Spectral
Efficiency Wireless Communication
Employing Multi-Element Arrays
Gerard J. Foschini, Glen D. Golden, Reinaldo A. Valenzuela, Fellow, IEEE, and Peter W. Wolniansky
Abstract— We investigate robust wireless communication in
high-scattering propagation environments using multi-element
antenna arrays (MEA’s) at both transmit and receive sites. A
simplified, but highly spectrally efficient space–time communi-
cation processing method is presented. The user’s bit stream is
mapped to a vector of independently modulated equal bit-rate
signal components that are simultaneously transmitted in the
same band. A detection algorithm similar to multiuser detection
is employed to detect the signal components in white Gaussian
noise (WGN). For a large number of antennas, a more efficient
architecture can offer no more than about 40% more capacity
than the simple architecture presented. A testbed that is now
being completed operates at 1.9 GHz with up to 16 quadrature
amplitude modulation (QAM) transmitters and 16 receive anten-
nas. Under ideal operation at 18 dB signal-to-noise ratio (SNR),
using 12 transmit antennas and 16 receive antennas (even with
uncoded communication), the theoretical spectral efficiency is 36
bit/s/Hz, whereas the Shannon capacity is 71.1 bit/s/Hz. The 36
bits per vector symbol, which corresponds to over 200 billion
constellation points, assumes a 5% block error rate (BLER) for
100 vector symbol bursts.
Index Terms— Antenna diversity, multi-element arrays
(MEA’s), space–time processing, wireless communications.
I. INTRODUCTION
W
E investigate wireless communication using multi-
element antenna arrays (MEA’s) at both the transmit
and receive sites to achieve very high spectral efficiencies in
a high-scattering environment. It has been reported [1]–[4]
that MEA’s along with space–time processing can aggressively
exploit multipath propagation effects for communication. We
present a single link study of communication of a user’s signal
when the bit stream is demultiplexed and the transmitted signal
vector components convey distinct bit substreams, one sub-
stream per transmit antenna. The method presented, designed
to be of limited complexity regarding the spatial processing
required, can demonstrate robust high-capacity operation.
The three-step detection processing of the user’s vector sig-
nal that is received in additive white Gaussian noise (AWGN)
brings together some well-established procedures. There is
zero forcing combining of the received signal components
(the value of this type of combining is already established
for space division multiple access systems). Substreams are
Manuscript received December 1, 1998; revised May 1, 1999.
The authors are with Lucent Technologies, Bell Labs Innovations, Holmdel,
NJ 07733-0400 USA.
Publisher Item Identifier S 0733-8716(99)08267-0.
sorted at the receiver based on how “good” their channels
are. The substream with the best conditions is detected first;
its contribution is subtracted from the total received vector
signal. The same process is repeated until all substreams are
detected. Then the original bit stream is reconstituted. Later,
we quantify the communication efficiency possible, observing
that gains offered by increased complexity can sometimes be
modest.
The context of our study is a propagation environment
resulting in significant decorrelation of the electromagnetic
field sampled by the receive array elements. This decorrelation
is exploited to create many parallel subchannels. The number
of effective subchannels is related to both the degree of
decorrelation and the number of antennas. The propagation
environment is represented by a matrix
where the th
element is the transfer function from the
th transmitter to
the
th receiver. As in [1]–[4], we will assume ideal Rayleigh
propagation, meaning that the entries of the
matrix are
independently distributed complex Gaussian variables. The
channel, assumed unknown to the transmitter, is learned at
the receiver by measuring the response to a training sequence
[5]–[6]. Only long-term statistical knowledge of the ensemble
of possible channels is assumed to have been fed back to the
transmitter.
We assume burst mode digital communication in which the
channel is static during the burst. We allow that the channel
characteristic may change from burst to burst. Consequently,
channel capacity is treated as a random variable. Some key
applications are fixed wireless and wireless LAN’s.
To facilitate first implementation, various aspects of the dual
site MEA system are kept simple. Narrowband operation is
assumed so that the delay spread can be kept to a small fraction
of the symbol period, and thus the channel characteristic is
nominally flat across the frequency band. We also concentrate
initially on an uncoded system. However, for a sufficiently
long burst, the infinite time horizon idealization common in
information theory provides us with meaningful initial insights
as to what coding would offer for implementations beyond our
current concern.
A feasibility/research testbed implementing the algorithm
will operate at 1.9 GHz with up to 16 transmit and 16 receive
antennas. We specify the burst length at some target block
error rate (BLER). A block error occurs when a burst contains
at least one bit in error. Along with the number of transmit
antennas
and receive antennas , a key system parameter is
0733–8716/99$10.00 1999 IEEE
1842 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 11, NOVEMBER 1999
Fig. 1. High-level view of a vertical BLAST communication link. Assumptions: narrowband operation, all
m
constellations are the same size, and the
transmitter does not know the channel instantiation, but the receiver learns it. Specifications:
n;
(when
m
=1
;
is the average SNR between a
transmit–receive pair), probability [error-free burst], and
P
tot
, the total power transmitted out of all
m
antennas. The number of transmit antennas
m
and
the quadrature amplitude modulation (QAM) constellation size are optimized for maximum burst throughput.
the average signal-to-noise ratio (SNR), . This is the average
of the SNR’s measured by a probe receive antenna element, as
a test transmit antenna and the probe antenna are independently
moved over their respective volumes. (Alternately, assuming
that the propagation environment changes substantially from
burst to burst,
could be defined as the SNR seen by a
single receive antenna averaged over a large number of bursts
from a single transmitter.) We defined
with ,
but for the ideal Rayleigh environment,
can be arbitrary
in the definition of
so long as the total radiated power
is constrained. Consequently, if
is increased, there is
proportionately less power per transmit antenna. Then
is
independent of the number of transmit antennas.
There is exploding literature on related communication
subjects involving spatial processing and/or the related topic of
multiuser detection. References [7]–[30] are a relatively small
but wide-ranging sample.
II. M
ATHEMATICAL MODEL FOR
WIRELESS CHANNELS EMPLOYING MEA’s
We take a complex baseband view involving a fixed linear
matrix channel with AWGN. As indicated, although fixed, the
channel will often be taken to be random. Time is taken to be
discrete. A high-level system view is given in Fig. 1. We need
to list more notations and some basic assumptions.
• Noise at receiver
: complex -D AWGN with statis-
tically independent components of identical power
at
each of the
receiver branches.
• Transmitted signal
: the total power is constrained to
regardless of the number of transmit antennas [the
dimension of
]. The bandwidth is narrow enough that
we can treat the channel frequency characteristic as flat
over frequency.
• Received signal
: -D received signal so that at
each time, there is one complex vector component per
receive antenna. When there is only one transmit antenna,
the transmitter radiates power
, and we denote the
resulting average power at the output of any of the
receiving antennas by
.
• Average SNR at each receiver branch:
.
• Burst size:
: the number of vector symbols in one burst.
• Matrix channel impulse impulse response: the discrete
time response is denoted by the matrix delta function
with columns and rows. So, except for ,
is the zero matrix. Consistent with the narrowband
assumption, we use
for the (flat matrix) Fourier
transform of
and write suppressing the frequency
dependence. It will be convenient to represent the matrix
channel response in normalized form
. Specifically,
related to
, we have the matrix , where the equation
defines its relationship to .
Therefore,
.
• The ideal Rayleigh propagation environment: for this
environment, the n
m entries of the matrix are
outcomes of independently identically, distributed (i.i.d.)
complex Gaussian variables of unit variance. The sum-
mary information mentioned earlier that is fed back to the
transmitter, which is assumed not to know
realizations,
is the number of receive antennas
and the average SNR
.
Using
for convolution, the basic vector equation describ-
ing the channel is
(1)
The two vectors added on the right-hand side are complex
-D vectors (2 real dimensions). Using the narrowband
assumption, we simplify, replacing convolution by product
and write
(2)
FOSCHINI et al.: HIGH SPECTRAL EFFICIENCY WIRELESS COMMUNICATION EMPLOYING MEA’S 1843
III. VERTICAL INSTEAD OF DIAGONAL PROCESSING
Reference [1] explains the diagonally layered architecture of
an advanced system [diagonal-Bell Labs layered space–time
(D-BLAST)] as opposed to the less complex vertical BLAST
(V-BLAST) system which is our focus here. Space is the
point discrete space defined by the transmit antenna
elements. V stands for vertical, referring to our layering
space–time with successive transmitted vector signals—a se-
quence of consecutive vertical columns in space–time. The
diagonally layered architecture of [1] requires encoding the
transmitted symbol information along space–time diagonals.
There are communication efficiency advantages to such a
diagonally layered architecture. However, advanced coding
techniques, now a topic of research, are needed in this ap-
proach, and such complications were judged to be best avoided
in a first implementation. Moreover, with diagonal layering,
some space–time is wasted at the start and end of a burst.
This boundary waste becomes negligible as the burst length
increases. However, for lengths of initial interest to us, namely,
, the waste would be significant. For the initial
prototype, our limiting of burst length permits us to avoid,
for now, difficult channel tracking issues. Finally, with a
diagonal system, the complication of the careful avoidance
of catastrophic error propagation is a concern. Therefore, we
focus here on the vertical algorithm with no coding.
1
We will
see that the uncoded vertical architecture often attains a hefty
fraction of the bit rates of the diagonal approach.
IV. T
HE VERTICAL DETECTION PROCESS
Fig. 1 illustrates V-BLAST. We will take the different
QAM signals to be statistically independent (but otherwise)
identical modulations. Each of the QAM modulated compo-
nents of the vector transmit process conveys a distinct bit
substream. For expositional convenience, we express
as
(3)
and rewrite (2) in the form
. Detection
amounts to estimating the
QAM components of the vector
from the received vector .
From [1], the three key aspects of spatial processing of
a received vector signal in detection of any substream: i)
interference nulling: interference from yet to be detected
substreams is projected out; ii) interference canceling: inter-
ference from already detected substreams is subtracted out;
and iii) compensation: stronger elements of the received signal
compensate the weaker elements. (See [4] for a highly compact
formulation of the detection process analyzed in detail here.)
We suppress
, writing , , and for the -D vectors
, , and at any fixed time . We write for the
matrix
which we assume is essentially perfectly known
to the receiver: in practice, it is accurately learned in a training
1
An eight-transmit and 12-receiver antenna V-BLAST system is operating
at the Crawford Hill, Bell Labs location in Holmdel, NJ. In initial indoor
experiments at 18 dB SNR and 95% required BLER, 21 bits/symbol in the
form of seven eight-QAM streams has already been achieved (as compared to
24 bits/symbol theoretically possible in ideal Rayleigh propagation). At about
20% rolloff, 21 bits/symbol amounts to 17 bit/s/Hz. At higher SNR’s (22–35
dB) experimental efficiencies of 20–40 bit/s/Hz have been attained.
phase using, say, a burst preamble or midamble. The receiver’s
knowledge of the
-D vectors comprising the matrix
will be used in the processes of interference cancellation,
nulling, and compensation. Based on the realization of
,
the list of
components are reordered with
a parenthesized subscript conveying the order in which the
components are to be detected. So
is an
dependent permutation of the components of the vector .
The compensation step provides the optimum permutation for
minimizing the probability of error in the large SNR limit.
Assuming a reordering, we iteratively form
-D vectors
called spatial matched filters. These are
used in
scalar products to project to a scalar sequence
comprising the decision statistics for
.
The
need only be constructed once per
burst and the same matched filters reused for each vector
symbol. Fig. 2 shows the processing in the decision process
for
in an (8, 12) example. We next explain the iterative
decision process for the general (
) case.
A. The Interference Cancellation Step
Assume that the receiver has detected the first
and
that the
decisions were error free. Then we can cancel the
interference from these decided components of
. To express
this, it is useful to write
in terms of its -D columns so
then
. We note that the received signal is
(4)
Defining each of the
as that multiplying in
(4), we rewrite
using this reordering
(5)
The first square-bracketed sum involves only correctly de-
tected signal components and is subtracted from
in a manner
similar to decision feedback equalization (DFE). We denote
the resulting
-D vector
(6)
B. The Interference Nulling Step and the Use
of Spatial Matched Filters
The interference nulling
2
step frees the process of detecting
from interference stemming from the simultaneous
transmission of
. To avoid this
interference from the as yet undecided components, we
project
orthogonal to the dimensional subspace
spanned by
. To express this projec-
tion let
be the orthonormal set of
vectors obtained from the
using the
2
In the high SNR asymptote, the advantage of not nulling, but instead
maximizing SNR, plus self-interference, is negligible.
1844 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 11, NOVEMBER 1999
Fig. 2. Vector symbol by symbol detection. From the
n
-dimensional received signal
r
, an optimum stack of interference-free decision statistics for the
m
components of the vector symbol
q
is formed. An
m
=12
,
n
=16
example is shown.
Gram–Schmidt process. Denoting the result of the projection
by
, we write
(7)
Each of the
components of is the sum of a known
multiple of
and noise. In so far as processing ,we
have the setup of standard maximum ratio combining. So
the decision statistic for
is given by a scalar product
with optimized for detection of an -
fold diversity interference-free signal in vector AWGN. To
recall how such a
is obtained, note that the noise power
of
is proportional to the squared norm .We
can also say that the optimized signal-to-noise ratio SNR
for this decision statistic has the signal power proportional to
. To see why, define to denote what the vector is
in the absence of noise. SNR
is optimized when is any
multiple of
. This follows by applying the Cauchy–Schwarz
inequality to the signal power term in the numerator of SNR
.
C. The Compensation Step: Optimizing the Order of Detection
Next, we discuss the compensation feature. The desired
detector minimizes the probability of making a decision error
in a burst. To minimize this probability, it turns out that we
need to stack the
decision statistics for the components
to accord with the following criterion:
maximize minimum
SNR (8)
Next, we show that this criterion corresponds to minimizing
the probability of burst error.
1) Establishing the Criterion—Maximize Minimum
SNR : With points in each planar
constellation, the number of constellation points in each
vector is
no. vector constellation points
no. 2-D constellation points
[no. substreams]
(9)
Let
be the probability that a vector symbol has at least
one error. We sum probabilities over the
disjoint events that
register where the first occurrence of a transmitted vector in
error occurs. We get
Erroneous Block
(10)
is obtained by summing probabilities over the disjoint
events as to where the first stack level in error occurs. If the
errors made at various levels were statistically independent,
one could write
SNR SNR (11)
where
is the well-known function (see e.g., [31]) for
the probability of bit error of a two-dimensional (2-D) con-
stellation as a function of SNR for large SNR. Namely, for
-point QAM constellations
SNR
SNR (12)
SNR decays exponentially with SNR , implying that
in the small noise asymptote
SNR where SNR is
the minimum of the
SNR s in the stack. However, (11) is
not strictly correct since decisions made at lower stack levels
bias decisions at higher levels so independence is not justified.
FOSCHINI et al.: HIGH SPECTRAL EFFICIENCY WIRELESS COMMUNICATION EMPLOYING MEA’S 1845
Fig. 3. Myopic optimization from the bottom level up gives the global max–min. The example depicts how a stack of 12 interference-free decision
statistics can be improved.
Yet SNR is asymptotically correct. The biases are
asymptotically inconsequential for (11) as we show.
Scale the constellations, and hence the (optimized) decision
regions, to be the same at each stack level. Then, at each level
there is a different AWGN variance
, .
For a small noise analysis, take the
to share a positive
factor
and analyze . We use to denote the
probability that the noise at the
th level exceeds threshold
(more precisely—translates the decision statistic outside the
optimum planar decision region of a correct decision).
First, we examine some simpler hypothetical cases. Take
with independent Bernoulli trials at times
until the event that the noise exceeds threshold
occurs. The time until the first threshold is exceeded is
(asymptotically)
. For blocks of length , the expected
time to first erroneous block is
. Equivalently, the
rate of erroneous blocks is
. Generalize to ,
where at the
th level, successive trials take place with noise
variance
. However, unlike the case that ultimately interests
us, the trials at the various levels are statistically independent
of each of other. There are just
cases like the first one
running in parallel with different variances for the trials at the
levels. Let denote the largest of the probabilities .
The time until the first threshold is exceeded is asymptotically
, and again for blocks of length the rate of erroneous
blocks is
, which is also the probability of an erroneous
block.
Next, we consider the interesting case when the lower levels
feed decisions to the higher levels for cancellation purposes,
and erroneous decisions are passed up the stack levels. It
is convenient to introduce the artifice of a “genie” that acts
whenever an error is made at a lower level than the level
of the greatest noise variance. The genie, while, say, leaving
such an error in place at the level at which it occurs, corrects
the error only in that the cancellation process is made to
proceed at higher levels without the error. This is nothing
other than the previous case. It is clear that the asymptotic
rate at which erroneous blocks occur is left invariant if we
put the genie back in the bottle. This is simply because the
genie acts in a comparatively asymptotically trivial fraction
of the cases where errors are made. So, genie or no genie,
errors of consequence in the small noise asymptote occur at
the level of greatest noise variance. These errors occur at the
same asymptotic rate
whether the genie is present or not.
The rate of erroneous blocks is again
, where SNR ,
and so the max–min criterion is established.
2) Myopic Optimization Equals Global Optimization: By
myopic optimization, we mean: starting at the bottom stack
level and continuing iteratively up to the
th level, always
choose that decision statistic among all the options that max-
imizes the SNR for that level. With myopic optimization, we
need only consider
options in filling the totality of
all
stack levels, as opposed to a thorough evaluation of all
stacking options. We next prove that it is in fact globally
optimum to form the stack in a myopic fashion.
Start it at the bottom level (1) and iterate up to level
.
Hold the level (1) competition for the best (highest SNR)
decision statistic. Say that some substream, call it
, wins.
We will prove that it is optimal to decide
first. Suppose that
we do not decide
first, instead deciding another substream
first. Let s denote the SNR’s associated with
a stacking that has
at the bottom. Now consider the following
alteration of that stack to produce a new stack. Move
to the
bottom level and displace those components up one level from
up to those at the level occupied originally by . So a simple
cycling of substreams at the bottom was used to make the new
stack. Fig. 3 illustrates the process that we are describing here;
in general, for the special case
.
Let S
be the SNR’s of the new stack. We now
show that each of the
upper case SNR’s is bounded below
by at least one of the lower-case SNR’s. Since substreams
above
in the original stack have not changed their level, we
can say that s
S for those. For each substream moving
up one level in the stacking, we can say that the new S
can
only exceed the original s
of that substream. This is because
the imposed constraint of projecting orthogonal to
has been
removed. Certainly s
S since won the competition