Grassmannian Beamforming on Correlated MIMO
Channels
David J. Love
Robert W. Heath, Jr.
School of Electrical and Computer Engineering
Purdue University
West Lafayette, IN 47907
[email protected]
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
Austin, TX 78712
[email protected]
Abstract— The diversity gains available from multiple-input
multiple-output (MIMO) wireless systems are well documented.
These gains are realizable through the use of transmit beamforming and receive combining. Transmit beamforming relies on
the assumption of channel knowledge at the transmitter, an assumption that is often unrealistic. Limited feedback beamformers
have been designed over the past few years, but they concentrate
primarily on the assumption of a spatially uncorrelated Rayleigh
fading channel matrix. This paper addresses the design of limited feedback beamformers for transmit and receive correlated
MIMO channels. In particular, we use a technique where the
receiver chooses the beamforming vector from a codebook of
possible vectors and conveys this vector over a limited feedback
channel. We show how this method obtains full diversity order.
Monte Carlo simulations show performance close to optimal
beamforming.
I. I NTRODUCTION
Multiple-input multiple-output (MIMO) wireless systems
have received much interest over the last few years because
of their large capacity and diversity gains over single antenna
systems [1]. Despite the theoretical capacity benefits, there is
still much interest in the diversity-only usage of MIMO systems. There are two main approaches to diversity-only MIMO:
space-time coding and beamforming. Of these two schemes,
beamforming is the clear winner in terms of probability of
error performance because it actually adapts to the channel
using closed-loop techniques [2].
Closed-loop methods work by using forwardlink channel
knowledge at the transmitter. While this knowledge can
sometimes be obtained using forward and reverse channel
reciprocity, many systems such as those using frequency
division duplexing (FDD) will not have a priori channel
knowledge. The usage of limited feedback has been proposed
to overcome this lack of channel knowledge in beamforming
systems [3]–[7]. These algorithms work by using a codebook
of beamforming vectors that is designed off-line, fixed for
all channel realizations, and known at both the transmitter
and receiver. Under the assumption of spatially uncorrelated
Robert Heath was supported in part by the Texas Advanced Research
(Technology) Program under Grant No. 003658-0614-2001.
Rayleigh fading matrix channels, [4], [5] derive a beamforming design criterion based on thinking of the codebook
vectors as points in the Grassmann manifold. Unfortunately,
this analysis does not extend to correlated Rayleigh channel
models.
This paper analyzes the design of limited feedback beamforming in correlated Rayleigh models. We consider the European Union Information Society Technologies (IST) MultiElement Transmit and Receive Antennas (METRA) model [8],
[9]. This model is a realistic channel model that has been
proposed for usage in the IEEE 802.11N study group [10]
and studied for usage in the IEEE 802.20 working group
on mobile broadband wireless access [11]. This model has
also been experimentally validated in [9], [12]. We find that
a beamforming codebook design criterion can be obtained by
rotating the Grassmannian line packing beamformers studied
in [4], [5] by the square root of the transmit correlation matrix.
In proving this result, we show that receive correlation does
not affect the distribution of the optimal beamforming vector.
This means that in the presence of only receive correlation
the design criterion in [4], [5] are still applicable. We also
show that the optimal beamforming direction is independent of
the beamformer channel gain even in the presence of receive
correlation. This motivates the feedback of only direction
information.
This paper is organized as follows. Section II reviews the
beamforming framework for MIMO systems. The effect of
receive correlation is analyzed in Section III. A codebook design criterion is developed in Section IV. Section V shows that
the proposed criterion can yield full diversity performance.
Monte Carlo beamforming simulations results are presented
in Section VI. We conclude in Section VII.
II. S YSTEM OVERVIEW
Transmitters employing sufficiently spaced multiple antennas allow signals to be designed over the dimension of space.
A beamforming Mt antenna transmitter works by transmitting
an Mt -dimensional complex vector
xk = wsk
at the k th channel usage where sk is a single-dimensional
real (sk ∈ R) or complex (sk ∈ C) symbol and w is
an Mt -dimensional beamforming vector with each entry wi
representing a complex weight and phase shift on the ith
transmit antenna. A memoryless, Rayleigh fading MIMO
system with Mt transmit antennas and Mr receive antennas
using beamforming and optimal coherent maximum likelihood
detection can be represented by the input/output relationship
y = kHwk2 s + n
(1)
where H is an Mr ×Mt channel matrix, k·k2 is the two-norm,
and n is a noise term distributed according to CN (0, N0 ). We
removed the channel use index in (1) because the receiver only
performs symbol-by-symbol detection. Note that (1) is dependent on the optimal detection assumption, which is equivalent
to the assumption of receive maximum ratio combining. We
will assume that the transmit constellation is normalized such
that it has average power given by Esk [|sk |2 ] = E where | · |
denotes absolute value.
We will assume that H follows a transmit-receive correlation Rayleigh fading channel model. This assumption will be
expressed by modeling the MIMO fading as
H = RR GRT
RR R∗R
(2)
R∗T RT
where
is the receive correlation matrix,
is the
transmit correlation matrix, and G is a random matrix with
independent and identically distributed CN (0, 1) entries. In
this paper, we will assume that both RR and RT are full rank.
We will model the channel as being constant over multiple
transmissions before changing independently. We will assume
that the receiver has perfect knowledge of H and that the
transmitter has no knowledge of the current realization of H.
The vector w will be restricted to lie in a codebook
F = {f1 , f2 , . . . , fN }. The receiver, which has full channel
knowledge, will choose w as a function of the current channel
realization in order to minimize the instantaneous probability
of error and maximize the instantaneous capacity. This corresponds to maximizing the receive signal-to-noise ratio (SNR)
given by
kHwk22 E
Γr E
γr =
=
.
(3)
N0
N0
Because E and N0 are fixed, this corresponds to choosing
w = argmax kHf k22 .
(4)
f ∈F
The optimization in (4) can be implemented by simply evaluating the cost function over each of the N codebook vectors.
Once the vector is chosen, the vector can be conveyed from
the receiver to the transmitter using dlog2 N e bits of feedback.
As in [13], we will call the term Γr in (3) the effective
channel gain of the effective single-input single-output channel created through beamforming and receive detection. Note
that the average transmitted symbol power is given by kwk22 E.
Thus we will restrict kwk2 = 1 in order to enforce a transmit
power constraint. This means that the codebook F is a finite
set of unit vectors.
The design of the codebook F will be dependent on
the distribution of the optimal, unquantized beamforming
vector wopt . This vector is actually the right singular vector
of H corresponding to the largest singular value and sets
kHwopt k22 = kHk22 (where kHk2 is the largest singular
value of H). Note that this vector is not unique because
kHejθ wopt k22 = kHwopt k22 for any θ ∈ [0, 2π).
III. E FFECT OF R ECEIVE C ORRELATION
We will first analyze the effect of receive only correlation.
This analysis will be instrumental in understanding the fully
correlated case. This analysis also serves as an extension to
the work in [4], [5].
Consider the matrix given by
e = RR G
H
with RR and G defined as in (2). Let the singular value
decomposition be denoted as
e = UL ΛU∗
H
R
where UL is an Mr × Mr unitary matrix, UR is an Mt × Mt
unitary matrix, Λ is a real diagonal matrix with decreasing
diagonal entries λ1 ≥ · · · ≥ λmin(Mt ,Mr ) , and ∗ denotes
conjugation and transportation.
The beamforming structure of the MIMO system is completely specified with knowledge of Λ and UR at the transmitter. The optimal beamforming vector is given by wopt =
UR [1 0 · · · 0]T with channel gain λ21 . The following Lemma
characterizes both the distribution of UR and its relationship
to Λ.
Lemma 1: UR is a uniformly distributed unitary matrix
that is independent of Λ if G has i.i.d. CN (0, 1) entries.
Proof: If G is a spatially uncorrelated complex normal
matrix, its distribution does not change if pre- or poste is
multiplied by a fixed unitary matrix [14]. Thus the matrix H
distribution invariant to post-multiplication by any fixed unie is a right-rotationally invariant random
tary matrix. Thus H
matrix. The singular value matrices Λ and UR are actually
defined by the eigenvalue decomposition
e ∗H
e = UR Λ2 U∗ .
H
R
(5)
The matrix in (5) is isotropically random [15]. Thus the matrix
UR is uniformly distributed on the group of unitary matrices
and is independent of Λ.
The ramifications of Lemma 1 are that the codebooks and
analysis in [4], [5] are still valid even when the receive
antenna array is correlated. This greatly expands both the
contribution and applicability of the Grassmannian beamforming techniques design therein to a wide range of more realistic
system models. Unfortunately, we have still not answered the
question of dealing with transmit correlation.
IV. C ODEBOOK D ESIGN C RITERION
We will now consider the fully correlated channel model
in (2). We will make use of the results in Section III during
this analysis.
Assuming the full correlation model of (2), the optimal
beamforming vector will yield kHwopt k22 = kHk22 . We will
take the “communications vector quantization” approach of
[4] to define a system parameter and try to minimize the
loss in that system parameter due to quantization. Because
improving the probability of error and capacity both relate
to maximizing Γr = kHwk22 , we will attempt to minimize
the loss in our channel gain due to quantization. Thus, we
consider a distortion function
¤
£
(6)
D(F) = EH kHk22 − kHwk22
with w chosen according to (4).
We can bound the distortion in (6) as
h
i
£
¤
e T wk2
EH kHk22 − kHwk22 = EH kHk22 − kHR
2
h
¯ ∗
¯2 i
≤ EH kHk22 − λ21 ¯w̃opt
RT w¯
(7)
£
¤
£ ¤
= EH kHk22 − EH λ21 ·
h¯
¯2 i
∗
EH ¯w̃opt
RT w¯
(8)
where (7) is a result of zeroing out all but the largest singular
e w̃opt is the dominant right singular vector of H,
e
values of H,
and (8) follows from Lemma 1. This distortion only has one
term that depends on the form of the codebook F. Thus, we
∗
would like to quantize the vector
.
¯ ∗ RT w̃¯opt
2
The modified correlation ¯w̃opt RT w¯ is a subspace correlation because it only depends on the column space of w̃opt
and w. The column space of a vector is a line. Thus just
as in [4], [5] we should think of F as a set of lines rather
than a set of vectors. The set of lines in the Mt -dimensional
vector space is called the Grassmann manifold G(Mt , 1) [4].
A distance between points on G(Mt , 1) can be defined by
the sine of the angle between the corresponding lines in CMt
[16]. For example, if P1 , P2 ∈ G(Mt , 1) with corresponded
unit vectors v1 , v2 ∈ CMt then
q
2
d(P1 , P2 ) = 1 − |v1∗ v2 | .
(9)
We will use a technique called companding to compress
R∗T w̃opt . The basic idea of companding is to transform a
source for quantization and then transform it back before
usage [17]. We will design our codebook using ideas from
the companding literature.
Our compander will work by designing a quantizer for
wopt obtained by inverting the square root of the correlation matrix. Codebooks for uncorrelated MIMO beamformers
were designed in [4] and found to relate to the problem
of Grassmannian line packing. Grassmannian line packings
would design a codebook C = {c1 , c2 , . . . , cN } such that
mink6=l d(Pck , Pck ) where d is defined as in (9) and Pck is
the column space of ck .
A companding approach would then use the codebook
obtained from multiplying each element of C by R∗T . This
approach, however, does not maintain our unit vector power
constraint requirement. Thus we will normalize our codebook
to enforce the unit vector requirement.
Therefore to perform the codebook design, we will take
a two-step approach of designing a codebook for a spatially
uncorrelated source and then rotating our codebook. This leads
to the following criterion.
Correlated Grassmannian Beamforming: Design F by
picking {c1 , c2 , . . . , cN } that maximize
q
1 − |c∗k cl |2
δ = min
1≤k<l≤N
and setting
fi =
R∗T ci
.
kR∗T ci k2
This criterion actually allows the system to adjust the
limited feedback beamforming to current correlation conditions. The transmitter spatial correlation matrix can usually be
estimated at the transmitter without any additional feedback.
This would allow only one beamforming codebook, designed
using the Grassmannian beamforming criterion in [4], to be
stored in a system because the same codebook could be
adapted to any transmit and receive correlation structure.
V. D IVERSITY P ERFORMANCE
The error rate reduction benefits of MIMO systems are
usually characterized in terms of diversity gain. The diversity
gain of a MIMO system is the asymptotic log-log slope of the
probability of error curve. More precisely, a system is said to
have diversity gain d if [1]
d=
log Ps (SNR)
.
SNR→∞
log SNR
lim
Specifically, we should be able to determine if the designed
beamforming vectors provide full diversity order for any form
of transmit and receive correlation. The following theory
answers this important question.
Theorem 1: A Grassmannian beamforming system employing beamforming and combining over correlated Rayleigh
fading channels provides full diversity order if the vectors
in the beamforming codebook span CMt .
Proof: Suppose that the beamforming vectors span CMt .
Because there are only Mt Mr independently fading parameters, the diversity order is bounded by Mt Mr . We can construct an invertible matrix B = RT F = RT [fk1 fk2 · · · fkMt ]
where 1 ≤ ki ≤ N for all i. Since the matrix is invertible,
we can define a singular value decomposition
∗
B = VL ΦVR
(10)
where VL and VR are Mt × Mt unitary matrices and Φ is
a diagonal matrix with diagonal entries φ1 ≥ φ2 ≥ . . . ≥
φMt > 0.
For this system,
2
Γr = max kHwk2
w∈W
°
°2
≥ max °Hfkq °2
1≤q≤Mt
°
°2
°
°
= max °RR GVL Φ (VR )q °
1≤q≤Mt
measurements in [12]. The limited feedback codebooks were
restricted to six bits of feedback. Unrotated beamforming
outperforms antenna selection by 4.5dB. Note that simply
rotating and normalizing the codebooks from [4] yields performance approximately the same as full channel knowledge
beamforming. Both unquantized and rotated Grassmannian
beamforming perform around 0.6dB better than unrotated
beamforming.
0
2
10
2
= max kRR GΦ(VR )q k2
1≤q≤Mt
1
2
kRR GΦVR k1
Mr
1
2
kRR GΦk2
≥
Mt Mr
1
2
≥
° −1 °2 kGΦk2
°
°
Mt Mr R
≥
R
Antenna Selection
Unrotated
Rotated
Full Channel MRT
(11)
(12)
−1
(13)
2
1
2
2
≥
° −1 °2 φMt max max |gp,q |
1≤p≤M
1≤q≤M
°
°
r
t
Mt Mr RR 2
(14)
10
Prob. Symbol Error
d
−2
10
d
where (A)p represents the pth column of a matrix A and = denotes equivalence in distribution. Eq. (11) used the invariance
of complex normal matrices to unitary transformation [14],
and (14) follows from the matrix norm bounds described in
[18].
Let Ps (Error | E/N0 ) denote the probability of symbol
error conditioned on the SNR. Noting that the maximum over
2
the entries |gp,q | is the effective channel gain for selection
diversity (SD) systems, we thus have that
−3
10
−5
0
5
10
Es/N0
Fig. 1. Probability of symbol error performance comparison on a 4 × 5
transmit correlated channel MIMO system.
This is the probability of symbol error for an uncorrelated Rayleigh fading SD system with array gain shift of
1
2
2 φMt which provides a diversity of order Mt Mr .
Mt Mr kR−1
R k2
We have thus proven that the system achieves a diversity order
of Mt Mr .
Experiment 2: The second experiment was for a transmit
and receive correlation 3 × 2 MIMO system using four bits of
feedback. The correlation matrices were generating using the
IEEE 802.11N techniques outlined in [21] with a 5o angle
spread and three clusters. The results are shown in Fig. 2.
Interestingly, the rotated and normalized approach provides
performance halfway between the unrotated codebook design
and the full channel knowledge case.
Experiment 3: The final experiment focused on a receiveonly correlation setting. We considered a 3×4 system with the
receive correlation matrix taken from the “Pico Decorrelated”
measurements in [12]. Five bits of feedback were considered.
The simulation results are shown in Fig. 3. Grassmannian
beamforming performs within 0.6dB of the full channel
knowledge case.
VI. S IMULATIONS
VII. S UMMARY AND C ONCLUSIONS
Correlated Grassmannian beamforming was simulated to
against antenna selection [19], codebooks designed in [4], and
unquantized maximum ratio transmission [20].
Experiment 1: This experiment addresses the probability
of symbol error for a 4 × 5 beamforming system with only
transmit correlation. The results are shown in Fig. 1. The
correlation matrices were taken from the “Micro Correlated”
We proposed a limited feedback beamforming technique
for spatially correlated Rayleigh fading channels. This new
method is actually a simple modification to the existing limited feedback beamforming (or Grassmannian beamforming)
method discussed in [4]. The new codebooks are only required
to be rotated and normalized versions of the traditional
codebooks. Monte Carlo simulation results show that this
Ps (Error | E/N0 ) ≤
" Ã
EG Ps
¯
¯
Error ¯¯
φ2Mt
°
°2 γ rSD
°
Mt Mr °R−1
R 2
with
γ rSD = max
max |gp,q |
1≤p≤Mr 1≤q≤Mt
2
!#
(15)
E
.
N0
0
10
R EFERENCES
Prob. Symbol Error
Antenna Selection
Unrotated
Rotated
Full Channel MRT
−1
10
−2
10
−8
−6
−4
−2
E /N
s
0
2
4
0
Fig. 2. Probability of symbol error performance comparison on a 3 × 2
transmit and receive correlated channel MIMO system.
0
10
Antenna Selection
Grassmannian beamforming
Full Channel MRT
−1
Prob. Symbol Error
10
−2
10
−3
10
−8
−6
−4
−2
0
Es/N0
2
4
6
8
10
Fig. 3. Probability of symbol error performance comparison on a 3 × 4
MIMO system with receive correlation.
new approach provides large performance gains over limited
feedback beamformers designed for uncorrelated channels.
Note that in this paper we have only considered a rotation
based approach to the correlated design. It would be of
interest to derive the optimal design strategy to maximize
performance. It is likely that an optimal design would have
a difficult codebook design that would make the technique
impractical. It is also of interest to find efficient ways to
search over these subspace codebooks. Currently, only a brute
force search is used. It might be possible to use other coding
techniques to localize the beamforming vector required for
feedback to a small search sphere in the Grassmann manifold.
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