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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008 5191
Approaching MIMO-OFDM Capacity With
Zero-Forcing V-BLAST Decoding and Optimized
Power, Rate, and Antenna-Mapping Feedback
Rui Zhang, Member, IEEE, and John M. Cioffi, Fellow, IEEE
Abstract—This paper studies capacity-approaching transmis-
sion schemes for the multiple-input multiple-output orthogonal
frequency-division multiplexing (MIMO-OFDM) channel, under
the assumption that the channel state information (CSI) is com-
pletely known at the receiver but only partially available at the
transmitter via a limited-rate feedback channel. A vertical Bell
Labs layered space-time (V-BLAST)-based transmission structure
is considered, where multiple data streams are independently en-
coded at the transmitter (i.e., horizontal encoding) and successively
decoded at the receiver by the zero-forcing-based generalized de-
cision feedback equalizer. A closed-loop V-BLAST extension is
presented whereby transmit powers, rates, and antenna mappings
for multiple data streams at different OFDM tones are jointly
optimized at the receiver and then returned to the transmitter
via the feedback channel. Two low-complexity algorithms for
optimization of feedback parameters are proposed: one is based
on the Lagrange dual-decomposition method and the other is a
greedy algorithm. Antenna and tone grouping techniques by ex-
ploiting the MIMO-OFDM channel space-frequency correlations
are also proposed to reduce the feedback complexity. Simulation
results show that by only a moderate amount of feedback, the
proposed closed-loop V-BLAST scheme improves substantially
the throughput of the conventional open-loop V-BLAST scheme
without feedback and, furthermore, approaches closely the
MIMO-OFDM channel capacity achievable by the eigenmode
transmission that requires the complete CSI at the transmitter.
Index Terms—Adaptive coding and modulation (ACM), convex
optimization, multiple-input multiple-output (MIMO), orthog-
onal frequency-division multiplexing (OFDM), partial channel
feedback, spatial multiplexing, V-BLAST.
I. INTRODUCTION
M
ULTIPLE-input multiple-output orthogonal frequency-
division multiplexing (MIMO-OFDM) is a promising
technology for support of high-rate and broadband transmis-
sions over frequency-selective fading channels with multiple
transmit and multiple receive antennas. When the channel state
information (CSI) is perfectly known at both the transmitter and
receiver, the MIMO-OFDM channel capacity can be achieved
Manuscript received August 10, 2007, revised June 26, 2008. First published
July 25, 2008; current version published September 19, 2008. The associate ed-
itor coordinating the review of this manuscript and approving it for publication
was Dr. Geert Leus. This paper was presented in part at the IEEE Global Com-
munications Conference, San Francisco, CA, Nov. 27–Dec. 1, 2006.
R. Zhang is with the Institute for Infocomm Research, 119613 Singapore
(e-mail: rzhang@i2r.a-star.edu.sg).
J. M. Cioffi is with the Department of Electrical Engineering, Stanford Uni-
versity, Stanford, CA 94305 USA (e-mail: cioffi@stanford.edu).
Digital Object Identifier 10.1109/TSP.2008.928965
by signaling through the channel’s eigenmodes at each OFDM
tone along with “water-filling”-based power and rate adapta-
tions [1]. However, the eigenmode transmission relies on the
complete CSI at the transmitter. For systems where the trans-
mitter can acquire the CSI only via a feedback channel from the
receiver, the amount of feedback overhead can be substantial for
MIMO-OFDM channels with a large number of antennas and/or
a large number of multipath delays with significant path gains.
Consequently, a great deal of research has studied partial CSI
feedback schemes for MIMO and MIMO-OFDM channels.
Generally speaking, the optimal strategy (e.g., [2]–[4]) for
transmit adaptation over block-fading (BF) MIMO channels
under a limited-rate feedback, say,
feedback bits for each
block transmission, is to partition the multidimensional space of
random MIMO channels into 2
disjoint regions and associate
each region with one unique set of transmit parameters that may
include the number of transmitted data streams as well as the
transmit power, rate, and beamforming vector assigned to each
data stream. Thereby, for each block transmission, the receiver
can first determine the region where the instantaneous MIMO
channel is located and then feed back the
-bit representation
of this region to the transmitter for adapting the transmission
accordingly. Finding optimal MIMO channel partitions and
their corresponding transmit parameters for arbitrary number
of transmit and receive antennas is in general still an open
problem in literature. Furthermore, such optimization not
only relies on the complete knowledge on the fading channel
distribution but also usually requires a costly computational
complexity. Therefore, study on robust and low-complexity
MIMO feedback schemes is still an important area for research.
One commonly adopted feedback strategy for MIMO chan-
nels is to send back efficient representation of precoding
(beamforming) vectors, one for each transmitted data stream
(e.g., [5]–[10]). The precoding vector is usually drawn from
a finite codebook that is known to both the transmitter and
receiver and predetermined based on the feedback rate and
some assumed statistical properties of the channel (e.g.,
isotropic fading distribution, transmit antenna correlations,
and so on). Then, at each fading state, the receiver selects
from the codebook one precoding vector (or precoding matrix
if spatial multiplexing with more than one data stream used)
corresponding to the best achievable performance [e.g., bit
error rate (BER) minimization or capacity maximization] to
return to the transmitter via the feedback channel. The simplest
design for precoding vectors is probably the transmit-antenna
selection (e.g., [11]–[13]) for which the codebook consists
1053-587X/$25.00 © 2008 IEEE
5192 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008
of merely indexes of transmit antennas and the feedback in-
formation becomes the binary index of the selected antenna
for transmission. Recently, reduced-feedback precoding for
MIMO-OFDM channels by exploiting channel correlations
across adjacent OFDM tones has also been considered in, e.g.,
[14]–[18]. Although the MIMO precoding technique has been
shown in the literature to be able to improve substantially both
the MIMO channel capacity and transmission reliability, there
is usually a common drawback associated with this approach:
when the distribution of the fading channel deviates drastically
from the presumed one, the channel-mismatched codebook
may cause a notable performance degradation.
This paper considers a
special form of MIMO feedback that
does not contain any precoder information (i.e., the precoding
matrix at the transmitter is constantly the identity matrix) but in-
stead contains transmit power and rate adaptations based solely
on the instantaneous MIMO channel realization. The proposed
feedback scheme does not require any knowledge on the MIMO
channel distribution and, hence, may be found useful when the
fading environment exhibits heterogeneous statistical properties
such that a constant precoding codebook having a limited size
is unlikely to be effective for all possible channel realizations.
There are two commonly deployed encoding methods for
MIMO spatial multiplexing, known as “horizontal” encoding
and “vertical” encoding, respectively [19]. For the former,
each data stream for spatial multiplexing is first independently
encoded and then transmitted by a different antenna, while for
the latter, a single encoder is used to spread coded information
bits across all transmit antennas. The decoding methods for the
horizontal and vertical encoding are also different. The former
allows for parallel or successive decoding (by, e.g., the vertical
Bell Labs layered space-time (V-BLAST) receiver [20], [21]
or the equivalent generalized decision feedback equalizer
(GDFE) [22]), while the latter requires maximum likelihood
(ML)-based or approximate ML-based joint decoding (e.g.,
[23]). Successive decoding usually has a lower complexity
to implement than joint decoding; however, it has a critical
issue to tackle in practice on decoding error propagation. For
instance, in the originally proposed open-loop V-BLAST [20],
[21], equal power and rate are assigned to all data streams
because of the lack of channel knowledge at the transmitter.
As a result, the achievable rate of the open-loop V-BLAST
is limited by the transmit antenna with the smallest channel
capacity because incorrectly decoding one data stream may
cause failures in decoding the remaining data streams due to
error propagation. Although changing the decoding order of
transmit antennas can prevent error propagation to some extent,
the achievable rate of the open-loop V-BLAST is still far from
the MIMO channel capacity [21].
This paper incorporates per-antenna-based power and rate
feedback into the V-BLAST, termed the closed-loop V-BLAST,
as an effective solution to improve the achievable rate of the
open-loop V-BLAST. First, because of power control ap-
plied to each transmit antenna, more reliable transmission for
each data stream is achievable and hence error propagation
is effectively avoided. Secondly, adaptive rate allocation to
different transmit antennas makes the achievable throughput of
the closed-loop V-BLAST no longer limited by one transmit
antenna with the smallest channel capacity. Note that if there
is only one transmit antenna assigned positive transmit rate
and power, the closed-loop V-BLAST becomes identical to
the transmit-antenna selection. The closed-loop V-BLAST
has been considered for MIMO channels [24]–[27] as well
as MIMO-OFDM channels [28]. This paper differs from the
above prior work in the following aspects. First, this paper
considers the V-BLAST receiver that utilizes the zero-forcing
(ZF)-based successive decoding (also known as the ZF-GDFE
[22]), while the receiver in [27] and [28] employs the min-
imum mean-squared error (MMSE)-based successive decoding
(also known as MMSE-GDFE [22]). Consequently, the corre-
sponding feedback power and rate optimization for these two
receiver structures are also different. Secondly, the criterion
adopted in [24]–[26] for feedback optimization is the mini-
mized BER with fixed-rate transmission, while in this paper a
generalized optimization framework is provided for evaluating
the performance limit of the closed-loop V-BLAST.
The main contributions of this paper are summarized as
follows. This paper formulates joint optimization of feed-
back parameters in the closed-loop V-BLAST as a convex
optimization problem. For the ZF-based V-BLAST receiver,
selection of active transmit antennas as well as their decoding
order can produce different tradeoffs in the resultant spatial
subchannel gains at each OFDM tone. Therefore, these pa-
rameters need to be jointly optimized along with feedback
transmit powers and rates at the receiver. An exhaustive search
for the optimal antenna selection and decoding order (probably
different from one OFDM tone to another) at all OFDM tones
is shown to be practically infeasible for MIMO-OFDM due
to the prohibitive computational complexity. This paper thus
presents two low-complexity algorithms for this problem.
The first algorithm is based on the Lagrange dual-decomposi-
tion method, which breaks the original problem into parallel
subproblems, each independently solving the corresponding
optimization at one OFDM tone. Consequently, the overall
complexity becomes linear in the number of OFDM tones. The
second approach is a simple greedy algorithm that provides
a suboptimal solution but has a close-to-optimal performance
with even lower complexity than the dual decomposition. This
paper also considers modifications of the developed algorithms
for further reduction of the feedback complexity by exploiting
space–frequency correlations in the MIMO-OFDM channel.
The remainder of this paper is organized as follows. Section II
provides the MIMO-OFDM channel model. Section III illus-
trates the closed-loop V-BLAST architecture. Section IV
formulates the optimization problem for determining feedback
parameters and presents various solutions to this problem.
Section V provides the simulation results. Section VI concludes
this paper.
Notation: Scalars are denoted by lower case letters, e.g.,
.
Boldface lower case letters are used for vectors, e.g.,
, and
boldface upper case letters for matrices, e.g.,
. denotes
the trace of a square matrix
. For any general matrix ,
and denote its transpose and conjugate transpose, respec-
tively, and
denotes its th element. denotes the iden-
tity matrix.
denotes a diagonal matrix with all the di-
agonal elements represented by the vector
. denotes the
ZHANG AND CIOFFI: APPROACHING MIMO-OFDM CAPACITY 5193
Euclidean norm of a complex vector . denotes statistical
expectation.
denotes the space of matrices with com-
plex entries. The distribution of a circular symmetric complex
Gaussian vector with the mean vector
and the covariance ma-
trix
is denoted by , and means “distributed as.”
The notations
and denote, respectively, the
maximum and the minimum between two real numbers
and
, and .
II. C
HANNEL
MODEL
This paper considers an MIMO-OFDM channel with
transmit antennas, receiver antennas, and orthogonal
OFDM tones. It is assumed that the transmission is on a block
basis and the channel is slow-fading. For simplicity, the BF
model is considered in this paper, i.e., the channel remains
constant during each transmission block but can probably
vary from block to block. Let
denote the fading state of the
channel during each block transmission and assume that
is
ergodic and stationary. The cyclic prefix in each OFDM symbol
is assumed to be sufficiently long to suppress any possible
intersymbol interference caused by the multipath propagation.
By applying standard OFDM modulation and demodulation at
the transmitter and the receiver, respectively, the demodulated
signal for one transmission block of interest can be expressed as
(1)
In the above,
and are
the received and transmitted signal vector at the
th OFDM
tone, respectively,
; denotes
the OFDM symbol index and
is the number of OFDM
symbols in each transmission block;
is
the frequency-domain channel matrix at the
th tone and is
expressed as
, where
its
th column represents the channel associated with
the
th transmit antenna, ;
is the additive noise at the receiver, and it is assumed that
. Assuming that the time-domain channel
responses have
delayed multipath taps,
then denotes the time-domain channel matrix for the th tap,
. The frequency-domain channel matrix
can then be expressed as
(2)
for
. Assuming that the transmitted signals at dif-
ferent tones,
, , are independent vectors with
zero means and covariance matrices,
(the
expectation is taken over
), the average transmit power over
all OFDM tones for each transmission block of interest is then
given by
(3)
In general,
s and can be changed according to the fading
state
of each transmission block, though for brevity they
are not written here explicitly as functions of
. The average
Fig. 1. The closed-loop V-BLAST structure for MIMO-OFDM.
signal-to-noise ratio (SNR) over different fading states at the
receiver is then defined as
SNR
(4)
It is further assumed that
, .
It then follows from (4) that
.
III. C
LOSED-LOOP V-BLAST FOR MIMO-OFDM
This section presents an extension of the open-loop
V-BLAST transmission structure in [20] and [21] for
MIMO-OFDM, referred to as the closed-loop V-BLAST. The
closed-loop V-BLAST differs from the open-loop V-BLAST
in that there exists a feedback channel that enables the receiver
to send to the transmitter the optimized transmit adaptation
based on the instantaneous channel realization, which is as-
sumed to be perfectly known at the receiver. Fig. 1 provides an
illustration of the closed-loop V-BLAST architecture, which is
described in more detail as follows.
• Demultiplexer (Demux): All information bits of each trans-
mission block corresponding to a target rate
are first di-
vided into
, ,
1
data streams, each
carrying a portion of the total bits equal to
,
, where denotes the rate for the th data
stream.
• Encoder: Each data stream is then independently pro-
cessed for channel coding and modulation. Modulated
symbols of each data stream are then divided into
con-
secutive
-tuples, each tuple assigned to one of OFDM
symbols. For the
th data stream, the signals assigned to
the
th tone of the th OFDM symbol are denoted by
, , .
• Power Amplifier:
is then amplified according to
its assigned data stream and tone position at each OFDM
symbol. For the
th data stream, , ,
at every
th tone are amplified according to , and the
outputs are denoted by
.
1
As will be shown later in this paper, is required by the
considered ZF-based V-BLAST receiver.
5194 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008
• Antenna Mapping: Let .
Next, each element of
is mapped to a corresponding
transmit antenna. This antenna mapping is the same for
all OFDM symbols but probably different from tone to
tone.
2
Let the antenna mapping at tone
be specified by
the vector
,
e.g.,
denotes the transmit antenna at tone assigned
to carry the first data stream. Define the
matrix,
, parameterized by , as follows:
if
otherwise.
(5)
The input and output after the antenna mapping can then
be expressed as
(6)
where
is the transmitted signal vector given in (1).
Clearly,
not only selects (out of ) active transmit
antennas at tone
to carry data streams but also speci-
fies the mapping between these selected antennas and their
assigned data streams.
• OFDM Modulator: Let
. is then
modulated as the
th OFDM symbol, ,
radiated by the
th transmit antenna, .
To summarize, a total number of
data streams with
transmit rate
, respectively, are independently
encoded and simultaneously transmitted (i.e., horizontal en-
coding). If
is zero for any , then , , and no
data are assigned to the
th data stream. Hence, the closed-loop
V-BLAST dynamically adapts the number of data streams or
the degree of spatial multiplexing. Notice that although one
transmit antenna may be assigned to carry a data stream at
one tone but not assigned any data stream at another tone, in
general all transmit antennas may be active. This is in contrast
to the transmit antenna selection in the case of a flat-fading
MIMO channel for which some transmit antennas
are completely switched off if they are not assigned any data
stream. In general, there may be three sets of parameters for the
receiver to feed back in the closed-loop V-BLAST:
i) rate assignments for different data streams:
,
;
ii) power assignments for each data stream at different
OFDM tones:
, , ;
iii) transmit antennas assigned to each data stream at different
tones:
, .
For the V-BLAST, each independently encoded data stream
is analogous to a virtual user in an equivalent Gaussian mul-
tiple-access channel (MAC). Therefore, each data stream of
the V-BLAST can be decoded by employing the well-known
multiuser detection techniques [29], which in general can be
either linear or nonlinear, and either ZF-based or MMSE-based.
2
This is because active transmit antennas and their corresponding decoding
orders optimized at the receiver, as will be shown later in this paper, may also
be variable at different OFDM tones in order to exploit the channel frequency
diversity. As a result, transmit antenna mapping needs to ensure that the signals
at different OFDM tones from the
th data stream are all decoded th in the
order by the successive decoding at the receiver.
Depending on the receiver structure, the optimization of feed-
back parameters in the closed-loop V-BLAST can also be
different. In [27] and [28], the closed-loop V-BLAST with
MMSE-based successive decoding has been studied, while in
this paper the ZF-based successive decoding is considered.
The MMSE-based V-BLAST receiver is the optimal multiuser
detector that achieves the capacity region of the Gaussian MAC
[30], and is thus also optimal for the closed-loop V-BLAST.
However, as shown in [28], the optimization of feedback
parameters for the MMSE-based successive decoding is asso-
ciated with a prohibitive computational complexity, especially
when the number of transmit and/or receive antennas is large.
Furthermore, it is hard to incorporate practical (non-Gaussian)
modulation and coding constraints into the feedback opti-
mization for the MMSE-based receiver [28]. In this paper, the
ZF-based V-BLAST receiver is considered to overcome the
above difficulties.
The ZF-based V-BLAST receiver with successive decoding
is parameterized by a set of projection vectors for each data
stream at different OFDM tones, denoted by
,
, . Notice that is dependent
on
, which specifies the assigned transmit antennas for data
streams at tone
. For some given , the QR decomposition
of the channel matrix
at tone (for brevity, the fading state
is dropped) after the column-wise selection and permutation
based on
can be expressed as
(7)
where
satisfies , is a pos-
itive
diagonal matrix, and is a monic
lower triangular matrix. The projection vector for the
th data
stream at tone
, , can then be obtained as the th column
of
. The ZF-based V-BLAST receiver first applies the pro-
jection vectors to the received signals to obtain
(8)
(9)
where
and . From
(6), (7), and (9), and by using the fact that
,
can be simplified as
(10)
Assuming that the decoding order for data streams is from
one to
, from (10) it is observed that the th data stream
is interfered merely by the decoded data streams one to
1,
which correspond to the lower and off-diagonal elements
of
and thus can be subtracted from . It is also
observed that the interferences from not-yet-decoded data
streams
1to have been completely removed by the ZF
operation of the projection vector. Notice that because
has the size of , the maximum value of needs to
be
such that the first decoded data stream can
successfully remove the interference from the other
data streams using the ZF-based projection. Assuming that
the interference subtraction is perfect, the equivalent channel
for the
th data stream can be characterized by channel
ZHANG AND CIOFFI: APPROACHING MIMO-OFDM CAPACITY 5195
gains, denoted by , at each tone , where is obtained
from
. The
receiver then jointly decodes the signals for the
th data stream
, . Note that the overall
decoding delay due to successive decoding is approximately
OFDM symbol periods, which is usually tolerable in practice if
is a small number.
Given the set of antenna mappings
and power assign-
ments
, the achievable rate of each data stream ,
, averaged over all OFDM tones, can be obtained
as
(11)
Note that the factor 1/2 in front of the achievable rate expression
is because
is measured in bits/real dimension (or b/s/Hz).
In the above,
denotes the “gap”[31], which is assumed to be
equal for all data streams and accounts for the rate loss from
the actual channel capacity owing to nonideal (non-Gaussian)
signaling. The gap is usually determined by the required BER
and the employed modulation and coding scheme (MCS). For
example, for the uncoded M-ary quadrature amplitude modula-
tion with the BER of 10
, has the value of 8.8 dB. With a ca-
pacity-achieving MCS,
can become close to one (0 dB). It was
shown in [32] that it is possible to employ a universal Gaussian
codebook with different powers assigned to codeword symbols
based on their experienced fading gains to achieve the capacity
of a scalar fading channel. Practical capacity-approaching MCS
for a scalar fading channel may be, e.g., turbo or low-density
parity-check code combined with bit-interleaved coded modu-
lation. This result implies that in the closed-loop V-BLAST, by
employing a scalar capacity-achieving code for each data stream
along with optimal power assignments at different OFDM tones,
each data stream can transmit reliably at a maximum rate given
in (11) with
(0 dB). Furthermore, it is sufficient to feed
back the overall rate per data stream
, instead
of its exact rate components at different tones.
IV. O
PTIMAL FEEDBACK PARAMETERS
This section first provides an optimization framework
for determining the optimal feedback parameters for the
closed-loop V-BLAST. Then, two low-complexity algorithms
for this problem are presented. Lastly, this section presents
modifications of the proposed algorithms for further feedback
complexity reduction.
A. Optimization Problem
In this paper, transmit optimization is assumed to achieve the
goal of minimizing the average transmit power for each trans-
mission block to support a target average transmit rate
. Sim-
ilar optimization techniques can be developed for maximizing
the transmit rate of each block under some given transmit power
constraint. The following optimization problem is thus consid-
ered for each transmission block:
TABLE I
C
OMPARISON OF
COMPUTATIONAL
COMPLEXITY
Minimize
Subject to
(12)
The optimization variables in the above problem are the antenna
mapping vectors
and the power assignments . After
they are determined, the associated rate assignments for each
data stream
can be obtained accordingly from (11). For
a given set of
, are uniquely determined from the
QR decomposition and, hence, the optimal values for
in
the above problem can be easily found by using the standard
water-filling algorithm [33]. The main difficulty for solving the
problem at hand lies in the search for the optimal values of
.
Note that the problem in (12) might not be convex because the
left-hand-side (LHS) function of its constraint is a maximum of
a set of concave functions, and is thus not necessarily concave
[34]. Therefore, it is unclear whether this problem can be solved
by using standard convex optimization techniques. Next, three
candidate algorithms are considered for this problem, and their
computational complexities are summarized in Table I.
1) Exhaustive Search: A direct approach to solve the
problem at hand is to search over all possible antenna mappings
at all OFDM tones and find the optimal one with the smallest
power consumption
. For a given set of , the complexity
is mainly due to two parts: i) QR decomposition to obtain
channel gains
and ii) water-filling algorithm to compute
optimal values of
. The first part requires the complexity
of
for each tone
3
and hence in total.
The second part has the complexity of
. At each tone,
there are
possible ways of assigning each of data
streams a different transmit antenna. Hence, the number of
distinct antenna mappings over all tones is
, which
is
. The total computational complexity for the ex-
haustive search can thus be shown to be
.
This complexity is in the exponential of both
and , and
hence becomes infeasible even for a small number of tones and
transmit antennas.
4
2) Lagrange Dual-Decomposition Method: A more efficient
algorithm to find the optimal antenna mapping than the ex-
haustive search is based on the Lagrange dual-decomposition
method [36]. The first step of this method is to introduce the
3
QR decomposition can be implemented by either the Gram–Schmidt algo-
rithm or the Householder transform [35].
4
When computing the QR decomposition, a further reduction of complexity
is possible by utilizing the fact that some different antenna mappings may have
similar QR decompositions. Nevertheless, even if the complexity for computing
QR decompositions is ignored, it can be easily verified that a total complexity
of
is still necessary for the exhaustive search.
5196 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008
positive dual variable associated with the rate constraint in
(12) and to write the Lagrangian [34] of the original (primal)
problem as
(13)
Next, the Lagrange dual function [34] is defined as
(14)
The dual problem is then defined as
, and the op-
timal value of the dual problem is denoted by
. From
(13) and (14),
is a concave function because it is a point-
wise minimum of a family of affine functions of
[34]. Hence,
the optimal value of
, denoted by , which maximizes the dual
function
, can be found by using convex optimization tech-
niques. The corresponding optimal value of the dual problem
then serves as a lower bound for the optimal value of the orig-
inal problem, denoted by
[34]
(15)
In general, the duality gap between
and is nonzero for
a nonconvex optimization problem. However, for the problem
at hand, it is shown in Appendix I that the duality gap is in-
deed zero. Hence,
can be obtained by first minimizing the
Lagrangian
to obtain the dual function and then maxi-
mizing
over all positive values of .
Consider first the problem for obtaining
with a given .
From (13) and (14), it is interesting to observe that
can
also be written as
(16)
where
(17)
Hence,
can be obtained through solving independent
subproblems, each for
, . The above practice
is usually referred to as the dual decomposition.
5
For tone with
a given antenna mapping
, the minimization in (17) over ,
, is a convex optimization problem. Hence, by
the Karush–Kuhn–Tucker (KKT) conditions [34], the optimal
values of
at tone need to satisfy the following equations
simultaneously:
(18)
5
Other applications of the Lagrange dual-decomposition method for resource
allocation in communication systems can be found in, e.g., [28], [37]–[39].
(19)
where
is the positive dual variable associated with the con-
straint
, for , respectively. From (18) and
(19), the optimal values for
can be obtained as the following
“water-filling” solutions:
(20)
By substituting them into (17), it follows that
(21)
From (21),
can be obtained by the minimization over all
possible
. The remaining task is to find that maximizes
over all , which can be done by a bisection search
[34] for
. In summary, the following algorithm can be used to
solve the problem at hand.
Algorithm 4.1:
• Initialize , .
• Repeat
1. Set
;
2. From (21), find the optimal
, for ;
3. Set
, ,
;
4. If
, set
; otherwise set .
• Until
where is a small positive constant
that controls the algorithm accuracy.
In the above algorithm, is an upper bound for . One
possible method to obtain
is provided in Appendix II.
The computational complexity of the dual-decomposition
method can be obtained as follows. The number of itera-
tions of the bisection search for
requires a complexity of
. In each iteration, like the exhaustive search, the
complexity can be shown to be
. Hence, the
total complexity is
, which is a sig-
nificant (a factor of
) reduction compared to the exhaustive
search. Nevertheless, the complexity is still in the exponential
of
.
3) Greedy Algorithm: The greedy algorithm iteratively finds
a suboptimal transmit antenna mapping at each OFDM tone. In
the first iteration, the algorithm applied to tone
computes the
norm of the channel gain vector
for each transmit an-
tenna,
, and then selects the one with the largest
norm to carry the
th data stream that is decoded th in the
order at the receiver. In the second iteration, each remaining
ZHANG AND CIOFFI: APPROACHING MIMO-OFDM CAPACITY 5197
projects itself into the null space of the selected channel gain
vector obtained in the first iteration, and the transmit antenna
with the largest vector norm after the projection is assigned to
carry the
th data stream. This process iterates until all
data streams at tone are assigned with different transmit
antennas. The following algorithm summarizes the greedy al-
gorithm for finding the antenna mapping at tone
.
Algorithm 4.2:
• Initialize
and .
• Repeat
1.
;
2.
;
3.
;
4. For
, ,
;
5.
.
• Until
.
The computational complexity of the greedy algorithm is
mainly due to the channel QR decompositions and can be shown
to be
, which becomes linear in both and
and is thus a significant reduction compared to
for the dual decomposition. With the antenna mappings
found by the greedy algorithm, the corresponding power and
rate assignments can be easily found by using the standard
water-filling algorithm.
The greedy algorithm is different from the rule proposed in
[21], [40], and [41] for finding decoding orders of transmit an-
tennas in the open-loop V-BLAST. In the above prior work, it
is assumed that there is no feedback power and rate for each
transmit antenna, and, hence, equal power and rate are assigned
to each data stream radiated by a corresponding transmit an-
tenna. In this case, the rule for assigning decoding orders for
transmit antennas is quite intuitive, i.e., always selecting the
transmit antenna that has the largest channel gain (after pro-
jecting into the null space of channel vectors of those not-yet-de-
coded) to be decoded first. This rule is to minimize the error
prorogations into subsequent decoding stages. In contrast, the
closed-loop V-BLAST has feedback power control to protect
each data stream, and hence the criterion for selecting transmit
antennas and their corresponding decoding orders changes to
maximize the sum-rate of all data streams (or equivalently, min-
imize the total transmit power given the sum-rate). As a result,
the decoding orders given by the greedy algorithm are simply
reverted, i.e., always selecting the transmit antenna that has the
largest channel gain among the remaining unassigned antennas
to be decoded last.
B. Feedback Reduction
The feedback parameters and their associated complexities in
the closed-loop V-BLAST are summarized as follows:
i) rate assignments
: ;
TABLE II
F
EEDBACK
COMPLEXITY OF
CLOSED-LOOP
V-BLAST
ii) antenna mappings : ;
iii) power assignments
: .
This section presents two techniques to further reduce the
amount of feedback for antenna mappings and power as-
signments, respectively. Table II summarizes the feedback
complexity for the closed-loop V-BLAST before and after
applying these techniques.
1) Tone Grouping: The technique of tone-grouping or tone-
clustering reduces the amount of feedback for antenna map-
pings
by dividing OFDM tones into disjoint groups,
(each group consists of adjacent tones, and
for convenience
is assumed to be divisible by ) and then as-
signing all tones in each group the same set of antenna mappings
, . The feedback complexity for antenna map-
pings becomes
after applying this technique.
The tone grouping utilizes the fact that the MIMO channels of
adjacent OFDM tones are usually highly correlated and, hence,
their optimal antenna mappings are also very likely to be sim-
ilar. As a result, these adjacent tones can be grouped together
and assigned with an identical antenna mapping without a no-
table performance degradation. Some cases of the tone grouping
with different values of
are listed as follows.
•
: The receiver feeds back the complete antenna
mappings for all tones.
•
: The receiver feeds back the antenna mappings
for
disjoint bands, where is the number of delayed
multipaths in (2) and
is approximately equal to the
channel coherence bandwidth.
•
: The receiver does not feed back any antenna map-
pings, and a fixed set of antenna mappings is used for all
tones. Without loss of generality, this paper assumes that
, , in this case.
With the tone grouping, the optimal feedback parameters can
also be obtained by solving the problem in (12) with an addi-
tional constraint
if (22)
The above constraint can be easily incorporated into the algo-
rithms previously developed. Algorithm 4.1 remains mostly un-
changed except that the dual-decomposition method needs to be
applied to
disjoint grouped frequency bands. Algorithm 4.2
needs to modify the criterion for determining
for the th
group,
. In this paper, it is simply assumed that
the sum of channel gains of each transmit antenna from all tones
in each group (e.g.,
for transmit antenna in the
th group) is the criterion for assigning antenna mapping in the
greedy algorithm.
2) Antenna Grouping: The goal of antenna grouping is
to reduce the amount of power feedback in the closed-loop
5198 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008
Fig. 2. Space-frequency channel gains by the V-BLAST receiver with
ZF-based successive decoding for a Rayleigh-fading MIMO-OFDM channel
with
, , and .
V-BLAST. Suppose that the set of antenna mappings at
all tones is already resolved by using either Algorithm 4.1 or
Algorithm 4.2 or their modifications after applying the tone
grouping. The antenna grouping collects the transmit antennas
at all OFDM tones corresponding to the same assigned data
stream (or equivalently, the same decoding order) into a group
and assigns them an identical power. More specifically, the
following new constraints can be added to the problem in (12):
(23)
As a result, the total feedback complexity for powers is reduced
from
to after the antenna grouping is applied,
which can be a significant reduction for MIMO-OFDM with a
large number of tones.
In addition to reducing the feedback complexity, the antenna
grouping is also able to achieve closely the performance with a
complete power feedback. This can be explained by observing
the channel gains
from all tones of each
group
. Fig. 2 plots these channel gains for a Rayleigh-fading
MIMO-OFDM channel with
, ,
and
. It is assumed that the antenna mappings at
different tones are obtained by the greedy algorithm (Algorithm
4.2). It is observed that the channel gains of each group can
have very different fading statistics compared to those of the
other groups. The channel gains corresponding to a higher group
index
(i.e., decoded later) have a larger mean and smaller vari-
ations (i.e., less fading) compared to those from lower group
indexes. Based on these observations, it follows that it might
not be necessary to allocate different powers to transmit an-
tennas at all tones because most of the performance gain over
the equal-power allocation can be obtained by only varying the
powers assigned to different groups. The optimal power allo-
cation to each group can be obtained by solving the following
modified problem of (12):
Minimize
Subject to (24)
The solutions to the above problem can be obtained by mod-
ifying the standard water-filling algorithm. The derivation is
omitted here for brevity, and the optimal power values can be
expressed as
, , where is the unique
root of the following equation:
(25)
for
, respectively, and is the “water level” with
which the rate constraint becomes equality in (24).
V. S
IMULATION
RESULTS
This section presents simulation results to evaluate the per-
formance of the closed-loop V-BLAST and compares it to that
of the conventional open-loop V-BLAST and other existing
MIMO feedback schemes in the literature. The MIMO-OFDM
channel is assumed to have
tones and
equal-energy multipath delays. The Rayleigh-fading channel
model is assumed for simplicity, and thus the time-domain
channel matrices
, , are independent random
matrices, each having entries independently distributed as
. Two antenna configurations are considered: i)
and ii) .
Monte Carlo simulations are used to average the fading effects
over 5000 independently generated MIMO-OFDM channels. If
not stated otherwise, it is assumed that the capacity-achieving
code is used and thus the gap
is one (0 dB). A constant-rate
transmission is assumed for each fading block, and the re-
quired average transmit power over all randomly generated
MIMO-OFDM channels, which can be equivalently repre-
sented by the average SNR defined in (4), is used as the
performance measure. Note that the maximum constant-rate
that is achievable over all the fading states given an average
power constraint is known as the channel delay-limited capacity
[42], which is achievable by the eigenmode transmission along
with water-filling-based power and rate adaptations [43].
Figs. 3 and 4 compare the achievable rates of the closed-loop
V-BLAST with the full transmit power, rate, and antenna-map-
ping feedback and the open-loop V-BLAST without feedback,
for the 4
4 and 2 4 antenna configuration, respectively.
Two algorithms for finding antenna mapping in the closed-loop
V-BLAST, namely, the Lagrange dual-decomposition method
(Algorithm 4.1) and the greedy algorithm (Algorithm 4.2), are
also compared. For both antenna configurations, it is observed
that the greedy algorithm sustains a rate almost equal to that ob-
tained by the Lagrange dual-decomposition method for all SNR
ZHANG AND CIOFFI: APPROACHING MIMO-OFDM CAPACITY 5199
Fig. 3. Comparison of the achievable rates of the closed-loop and open-loop
V-BLAST for MIMO-OFDM channels with
.
Fig. 4. Comparison of the achievable rates of the closed-loop and open-loop
V-BLAST for MIMO-OFDM channels with
and .
values. This result demonstrates the usefulness of the greedy al-
gorithm in achieving the close-to-optimal performance given its
very low computational complexity. Two receiver structures for
the open-loop V-BLAST are considered: the MMSE-based and
ZF-based successive decoding, respectively.
6
It is observed that
the throughput improvement by the closed-loop V-BLAST over
the open-loop V-BLAST is quite substantial for both antenna
configurations. This is because the closed-loop V-BLAST has
better decoding reliability by power control, as well as more
flexible transmit rate assignments and antenna mappings.
6
For this open-loop V-BLAST, equal power and rate are assigned to all
OFDM tones of four transmit antennas in the 4
4 case and of two randomly
selected transmit antennas in the 2
4 case. For both antenna configurations,
decoding order of transmit antennas at the receiver are chosen to minimize the
decoding error propagation like in [40] and [41]. For each randomly generated
MIMO-OFDM channel, the sum-power for all transmit antennas is chosen to be
the minimum for the assigned equal rate to be achievable for all data streams.
Fig. 5. Comparison of the achievable rates of the closed-loop V-BLAST under
different amounts of feedback for transmit antenna mappings for MIMO-OFDM
channels with
and . denotes the number of grouped bands
for the tone grouping.
Fig. 5 compares the achievable rates of the closed-loop
V-BLAST with different amounts of feedback for transmit-an-
tenna mappings by applying the tone grouping. Four cases
with different numbers of grouped bands are compared with
decreasing amount of feedback:
, , ,
and
. The 2 4 antenna configuration is considered.
For all cases except
(fixed antenna mapping), the
greedy algorithm is used to find antenna mappings for different
grouped bands. In this simulation, the full power and rate
feedback is assumed. It is observed that fixed antenna mapping
at all tones suffers from a severe rate loss compared to the other
three cases with antenna-mapping feedback. This is because in
the case of
, antenna selection diversity at different
tones is crucial to the achievable rate and, hence, the feedback
of antenna mappings (this feedback also selects two active
transmit antennas from four available ones) at different tones
boosts up the achievable rate significantly. It is also observed
that
(i.e., the number of grouped bands is equal to that
of independent multipaths of the MIMO-OFDM channel) is
a good choice for balancing the feedback complexity and the
achievable rate.
Fig. 6 shows the achievable rates of the closed-loop
V-BLAST under different amounts of power feedback for
the 4
4 antenna configuration. Three schemes with their
corresponding power feedback complexities are considered:
i) full power feedback:
;
ii) reduced power feedback after the antenna grouping is ap-
plied:
;
iii) equal-power feedback for which the same feedback
power is assigned to all transmit antennas at all tones:
.
The full rate and antenna-mapping feedback are assumed, and
the greedy algorithm is used to find the antenna mapping at each
tone. It is observed that for all SNR values, the antenna grouping
achieves almost the identical rate by the full power feedback. At
asymptotically high SNR, it is observed that the achievable rates
5200 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008
Fig. 6. Comparison of the achievable rates of the closed-loop V-BLAST under
different amounts of power feedback for MIMO-OFDM channels with
.
Fig. 7. Comparison of the achievable rates of the closed-loop V-BLAST and
other closed-loop feedback schemes for MIMO-OFDM channels with
.
of all three power feedback schemes converge to be identical,
and the gain by adapting power assignments becomes minimal.
In contrast to rate and antenna-mapping feedback, it is observed
that the throughput gain by power feedback is much less sub-
stantial. Nevertheless, power feedback (at least the equal-power
feedback) is still necessary in the closed-loop V-BLAST be-
cause it ensures the reliability of successive decoding.
Figs. 7 and 8 compare the achievable rates by the proposed
closed-loop V-BLAST and three other feedback schemes for
the 4
4 and 2 4 antenna configuration, respectively. These
feedback schemes for comparison include the eigenmode trans-
mission with the complete CSI feedback, a vertical-encoding
scheme with ZF-based linear receivers (e.g., [44]) and opti-
mized power feedback,
7
and the per-tone-based transmit-an-
7
For this scheme, the ZF-based linear receiver decomposes the MIMO
channel at each tone into
subchannels. Then, water-filling
based power allocations for sub-channels at all OFDM tones are computed.
Note that for this scheme with the 2
4 antenna configuration, two randomly
selected transmit antennas out of four available ones are used to carry two
transmitted data streams.
Fig. 8. Comparison of the achievable rates of the closed-loop V-BLAST and
other closed-loop feedback schemes for MIMO-OFDM channels with
and
.
tenna selection with feedback of selected antennas and
optimized power assignments.
8
For the closed-loop V-BLAST,
the greedy algorithm is used for determining transmit antenna
mappings with
, and the equal-power feedback
and the full rate feedback are assumed. Note that the eigen-
mode transmission achieves the channel delay-limited capacity,
which is also the upper bound for the achievable rate by any
MIMO-precoding feedback scheme with a finite codebook
size. It is observed that the rate gap between the closed-loop
V-BLAST and the channel delay-limited capacity is very
small for
in Fig. 7 but increases when
in Fig. 8. This is so because the closed-loop V-BLAST with
per-antenna-based feedback is incapable of capturing the full
transmit beamforming gain by the eigenmode transmission
or MIMO-precoding feedback, which becomes more domi-
nant at high SNR. On the other hand, it is observed that the
throughput gain achievable by the closed-loop V-BLAST
(with reduced feedback) over the closed-loop ZF-based linear
receiver (with full power feedback) is still substantial for both
antenna configurations. This is mainly because of the gain by
successive decoding over linear decoding. Similarly, substan-
tial throughput gains are also observed for the closed-loop
V-BLAST over the per-tone-based transmit-antenna selection
feedback, especially at high SNR when spatial multiplexing
gain achievable by the closed-loop V-BLAST becomes more
dominant over diversity gain by transmit-antenna selection.
For all previous simulation results, it is assumed that the feed-
back delay is negligible compared to the channel coherence time
and, hence, the channel is unchanged during each feedback pe-
riod. Next, the performance of the closed-loop V-BLAST is
evaluated under the channel estimation error due to the feedback
delay. For simplicity, it is assumed that the actual time-domain
channel matrices
, at each fading state are
8
For this scheme, a greedy algorithm similar like Algorithm 4.2 is used to
select one transmit antenna at each tone (probably different from tone to tone).
Then, the power allocations for selected antennas over different tones are opti-
mized by the water-filling algorithm.
ZHANG AND CIOFFI: APPROACHING MIMO-OFDM CAPACITY 5201
Fig. 9. Additional transmit power margin to compensate for the channel vari-
ation in the closed-loop V-BLAST for MIMO-OFDM channels with
.
equal to , where is the channel matrix
used at the receiver for assigning feedback parameters like in
previous simulations,
is the error matrix that is identically
but independently distributed compared to
, and the param-
eter
, , controls the amount of channel error due to
the feedback delay. Notice that
is equivalent to no channel
estimation error and
for the complete channel mismatch
between the transmitter and the receiver sides. For simplicity,
the greedy algorithm with
, the equal-power feed-
back, and the full rate feedback are assumed for the closed-loop
V-BLAST. The 4
4 case is considered.
Fig. 9 plots the additional transmit power, termed power
margin, assigned equally to all data streams at the transmitter to
ensure a block error probability
9
of 10 with a fixed transmit
rate of 6 b/s/Hz versus the channel variation parameter
. Note
that the transmit power margin is used for compensating the
channel estimation error at the receiver. It is observed that the
closed-loop V-BLAST is very robust to channel estimation
errors, e.g., only 1.7 dB power margin is required for
.
At last, Fig. 10 shows the achievable rate of the closed-loop
V-BLAST when practical non-Gaussian MCS is applied. It is
assumed that each data stream uses the same adaptive MCS
that has an SNR gap
and a discrete bit-loading granularity
. The greedy algorithm with , the per-antenna
power feedback, and the full rate feedback are assumed for the
closed-loop V-BLAST, and the 4
4 case is considered. Be-
cause the antenna mapping at different tones has been resolved
by the greedy algorithm, the V-BLAST receiver decomposes
the MIMO channel at each OFDM tone into parallel scalar
channels, and therefore the optimal discrete bit-loading algo-
rithm [45] can be applied. It is observed that the closed-loop
V-BLAST with nonzero
only supports a discrete set of
transmit rates, and increasing
from 0.5 to 1 bit results in a
further transmit power loss of 0–0.5 dB under the same gap
dB.
9
Since successive decoding is used at the receiver, a decoding error of the
whole block is declared if any of the data streams is not decoded correctly.
Fig. 10. The achievable rate of the closed-loop V-BLAST under practical adap-
tive MCS for MIMO-OFDM channels with
.
VI. C
ONCLUDING REMARKS
This paper presents a practical partial-channel-feedback
scheme to support capacity-approaching spatial multiplexing
for the frequency-selective fading MIMO-OFDM channel. The
proposed scheme is a closed-loop extension of the well-known
V-BLAST transmission scheme. Though the conventional
open-loop V-BLAST is severely compromised in practice
owing to its poor diversity performance and error propagation,
the proposed closed-loop V-BLAST overcomes these difficul-
ties by adaptively assigning transmit powers, rates, and antenna
mappings at all OFDM tones. It is shown by simulation results
that even with a moderate amount of feedback by applying
antenna and tone grouping, the closed-loop V-BLAST is still
able to approach closely the MIMO-OFDM channel capacity
achievable by the eigenmode transmission. The main chal-
lenge for determining feedback parameters for the closed-loop
V-BLAST is optimization of transmit antenna mappings to-
gether with transmit powers and rates. This paper presents
low-complexity algorithms for this problem and reveals some
new insights on finding good transmit antenna mappings for
the closed-loop V-BLAST.
This paper studies the feedback power and rate optimization
for each independent fading block. With consecutive block
transmission and the sufficiently large channel coherence time,
there is likely strong correlation between the MIMO-OFDM
channels during consecutive block transmission. As a result, if
a constant-rate transmission per block is required, the optimal
power and rate assignments for consecutive blocks cannot
change abruptly. This observation has an important conse-
quence, i.e., only feedback of the increments or decrements
of powers and rates (also known as “energy swapping” and
“bit swapping,” respectively, in [31]) between the current
transmission block and its preceding one is sufficient. This fact
can be used to reduce further the amount of feedback for each
transmission block. By doing this, the time-correlated CSI at
the receiver is conveyed to the transmitter by incrementally
encoding the feedback transmit powers, rates, and antenna
mappings.
5202 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008
APPENDIX I
P
ROOF OF
ZERO
DUALITY
GAP
This appendix proves that the duality gap between the primal
and dual problem in Section IV-A-2) is indeed zero. First, be-
cause
is concave, from (16) and (21), it follows that the
optimal dual variable
satisfies , i.e.,
(26)
Let
denote the optimal values for in (26), .
By substituting the optimal powers corresponding to
,
, into the LHS of (12) and
using (26), it can be verified that the achievable sum-rate
is indeed equal to
. As a result, from (13), it follows that
. Because can be no less than
, the obtained must be optimal for the primal problem
as well. Hence,
is established.
APPENDIX
II
U
PPER BOUND OF
An upper bound for the optimal dual variable in Al-
gorithm 4.1 can be obtained as follows. Let
and
be any set of antenna mappings and power assignments, respec-
tively, satisfying
(27)
The following equalities/inequalities can be shown:
(28)
(29)
(30)
(31)
where (29) is from the zero duality gap, (30) is due to
the fact that
are not the Lagrangian minimizers in
(14) for
, and (31) is by substituting (27) into (13). Let
. From (31), it follows that
.
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Rui Zhang (S’00–M’07) received the B.S. and M.S.
degrees in electrical and computer engineering from
National University of Singapore, Singapore, in
2000 and 2001, respectively, and the Ph.D. degree
in electrical engineering from Stanford University,
Stanford, CA, in 2007.
Since 2007, he has been a Research Fellow with
the Institute for Infocomm Research, Singapore.
His recent research interests include cognitive radio
networks, cooperative communication systems, and
multiuser MIMO transmission systems.
John M. Cioffi (S’77–M’78–SM’90–F’96) received
the B.S. degree from the University of Illinois, Ur-
bana-Champaign, in 1978 and the Ph.D. degree from
Stanford University, Stanford, CA, in 1984, both in
electrical engineering.
He was with Bell Laboratories from 1978 to 1984
and IBM Research from 1984 to 1986. He has been
a Professor of electrical engineering with Stanford
University since 1986. He founded Amati Commu-
nications Corporation in 1991 (purchased by Texas
Instruments in 1997) and was Officer/Director from
1991 to 1997. He currently is on the Board of Directors of ASSIA (Chair),
Afond, Teranetics, and ClariPhy. He is on the Advisory Board of Portview Ven-
tures, Wavion, MySource, and Amicus. His specific interests are in the area of
high-performance digital transmission. He has published more than 250 papers
and has received more than 80 patents.
Dr. Cioffi is a member of the National Academy of Engineering. He received
the Hitachi America Professorship in Electrical Engineering from Stanford
(2002), the IEEE Kobayashi Medal (2001), the IEEE Millennium Medal
(2000), the IEE J. J. Tomson Medal (2000), the 1999 University of Illinois
Outstanding Alumnus Award, the 1991 IEEE Communications Magazine
Best Paper Award, the 1995 ANSI T1 Outstanding Achievement Award, the
National Science Foundation Presidential Investigator Award (1987–1992), the
ISSLS 2004 Outstanding Paper Award, and the Marconi Fellow Award (2006).
