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Blind MIMO communication based on subspace estimation

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Blind MIMO communication based onSubspace Estimation
T. Dahl, S. Silva, N. Christophersen, D. Gesbert
T. Dahl, S. Silva, and N. Christophersen are at the Department of Informatics, University of Oslo, P.O. Box 1080, N-0316Blindern, Norway, while D. Gesbert is at the Mobile Communication Department, Eurecom, BP 193, F-06904, France.Their e-mails are: tobias@iﬁ.uio.no, silvanam@iﬁ.uio.no, nilsch@iﬁ.uio.no, gesbert@eurecom.fr, respectively.
DRAFT
2
Abstract
A new method is proposed for blindly estimating the top singular modes in a reciprocal MIMO (Multiple Input MultipleOutput) channel, while at the same time using these modes for multi-stream communication without the need for trainingdata. The uplink and downlink parties obtain the relevant singular modes from the received data blocks as the eigen-vectors/eigenvalues of the spatial empirical correlation matrices. The only requirement is that the separate data streams arestatistically uncorrelated. The approach relies on a key and simple “need to know” observation about MIMO transmission :In order both to transmit and receive, one party needs only the singular values and
one set
of singular vectors, say the leftset, while the other party needs only the corresponding right set of singular vectors in addition to the singular values; noparty needs both sets of vectors, and other aspects of
H
are irrelevant to both parties. Advantages of this blind approachinclude no need for higher order statistic based estimation and convergence in one iteration.
EDICS: 1-MAPP MULTICHANNEL SIGNAL PROCESSING APPLICATIONS1-ACOM Antenna arrays and multichannel processing for communications
Keywords
MIMO systems, channel identiﬁcation, singular value decomposition (SVD), singular modes, eigen-modes.
I. Introduction
Wireless MIMO (Multiple Input Multiple Output) systems are capable of delivering large increases incapacity through utilization of parallel communication channels [5], [6], [13]. Most wireless communicationsystems assume knowledge of the channel at the receiver, in the MIMO case a channel
matrix
. Channelestimation techniques can be divided into training-based techniques (e.g. V-BLAST [5], [7]) and blindor semi-blind methods (e.g. [4],[10],[11],[12],[14], [15]). Training-based methods have the advantage of relative computational simplicity at the cost of a reduced data rate. Blind methods are typically morecomplex and may not be accurate enough, but avoid the use of training data and thus have the potentialfor increased payload rates.If the channel matrix is known at both the transmitter and the receiver, the singular modes of the matrixchannel can be used to transport independent data streams (to increase data rate), while maximizing theSNR on each stream. For example in one extreme, one may choose to exploit the top mode only (associatedwith the largest singular value) in order to maximize the spatial diversity advantage. Also, when combinedwith an optimization of the number of streams, one can realize the optimum trade-oﬀ between rate anddiversity maximization. Traditionally, the problems of (blind) estimation and that of transmission areaddressed in a decoupled manner, but here the two problems are solved simultaneously.In [2] and [3], we proposed a technique for direct blind identiﬁcation of the main singular modes, withoutestimating the channel matrix itself. The technique is related to the iterative numerical Power methodfor ﬁnding eigenvectors of a matrix and basically required only a QR decomposition ([8]). In estimatingthe eigenstructure directly, it overcomes many of the problems associated with other classes of algorithms
DRAFT
3
([4],[10],[11],[14],[15]), such as slow convergence, use of higher order statistics and requirements aboutstatistical independence. J.B. Andersen ([1]) had previously noted the that vectors transmitted iterativelyforwards and back converge towards the singular vectors of the channel matrix.In this paper, we present a new and conceptually simpler blind method achieving the same goal. Thenew approach is based on standard ideas from signal subspace estimation in array processing which, toour knowledge, have so far not been utilized in the context of blind MIMO transmission of the spatialmultiplexing type. The present method relies on a Singular Value Decomposition (SVD) of the receiveddata block iterated between the transmitter and the receiver.The paper is laid out as follows: In Sec. 2.A we present the channel model, and in Sec. 2.B werecapitulate MIMO communication using singular modes of
H
and
H
∗
. In Sec. 2.C we show how arraysignal subspace estimation in a simple way is brought to bear on the problem of singular mode transmissionwithout prior knowledge of
H
. In Sec. 3 some initial results are given and discussed.
II. Mathematical Background
A. Channel Model
We assume two-way communication through a
N
(
receive
)
×
M
(
transmit
)
ﬂat-fading MIMO channelmatrix
H
∈
C
N
×
M
:
Y
Rec
=
HX
Send
(uplink)
(1)
X
Rec
=
H
T
Y
Send
(downlink)
(2)where
X
Send
∈
C
M
×
n
and
Y
Send
∈
C
N
×
n
are the transmitted data blocks of length
n
uplink (UL)and downlink (DL), respectively. Such a model describes a TDD (Time Division Duplex) system providedthe ping-pong time - the time between the beginning of a DL frame and the beginning of the next ULframe - is small compared to the channel coherence period. For algorithm derivation purposes, we assumeno channel noise, but return to in Sec. 2.C. It is also convenient to work with a channel where
H
T
in (2)is eﬀectively replaced by the complex conjugate transpose
H
∗
. This is achieved by letting the transmitdata
Y
Send
be complex conjugated prior to transmission, and the received data block
X
Rec
be complexconjugated prior to any further processing. In this way, (2) may be replaced by
X
Rec
=
H
∗
Y
Send
(downlink)
(3)
B. Communication using singular vectors
The rank of
H
is denoted by
K
0
≤
min
(
N,M
)
and its SVD is
H
=
USV
∗
.
S
is the diagonal matrix of singular values
σ
1
≥
σ
2
≥ ··· ≥
σ
K
0
>
0
, and
U
= [
u
1
,...,
u
K
0
]
∈
C
N
×
K
0
(4)
V
= [
v
1
,...,
v
K
0
]
∈
C
M
×
K
0
(5)
DRAFT
4
are unitary matrices whose columns can be used as receive and transmit vectors
{
u
i
}
and
{
v
i
}
, respectively.Clearly, one can select a number
K
(
K
≤
K
0
) of vectors for communication through orthogonal singularmodes.Assume, initially, that the singular vectorsare known: One party (e.g. the base station) knows the top
K
left singular vectors
{
u
i
}
,
i
= 1
,
···
,K
of
H
and the other side (subscriber unit) knows the correspondingsubset of right singular vectors
{
v
i
}
,
i
= 1
,
···
,K
. The top singular values
{
σ
1
,σ
2
,...,σ
K
}
are known toboth parties. Let
U
K
= [
u
1
,
u
2
,...,
u
K
]
,
V
K
= [
v
1
,
v
2
,...,
v
K
]
and
S
K
=
diag
{
σ
1
,σ
2
,...,σ
K
}
, denotethese subsets arranged into matrices, and let
C
x
,
C
y
∈
C
K
×
n
(
n
≥
K
) be the
symbol matrices
comprisingthe UL and DL symbol blocks, respectively. Each row of a symbol matrix represents an individual datastream. The elements of these matrices are symbols from a modulation constellation (e.g. BPSK, QPSK,16PSK,16QAM). Using these (known) singular vectors and values and neglecting noise, the UL transmitdata block
X
Send
=
V
K
C
x
would be received as
Y
Rec
=
HX
Send
=
USV
∗
V
K
C
x
=
U
K
S
K
C
x
. Thelast equality is obtained by noting that
V
∗
V
K
=
I
K
0
×
K
which has
1
s in the
K
main diagonal elementsand zeros elsewhere. From this it follows that
USI
K
=
U
K
S
K
.Decoding is then simply performed through
ˆ
C
x
=
S
−
1K
U
∗
K
Y
Rec
. The DL transmission block is set upas
Y
Send
=
U
K
C
y
and decoding is carried out at the subscriber unit correspondingly. Note that bothparties require
S
K
, but the key remark is that no party requires knowledge of
H
. In fact, the base stationboth receives and transmits using only
U
K
, while the subscriber unit only needs
V
K
. In addition, if onlyphase modulation is used (e.g. QPSK), the singular values are not even necessary for decoding. Next weshow how each party, based on what they receive during normal operation in a noisy environment, canestimate the singular values and the required singular vectors in only one iteration.
C. Obtaining the singular modes through subspace estimation using noisy data
Starting from, say, the subscriber unit, one transmits
X
Send
=
V
Ini
C
x
where
V
Ini
∈
C
M
×
M
is anarbitrary unitary matrix. This is received as
Y
Rec
=
HV
Ini
C
x
+
N
, where each element of the complexAWGN term
N
has zero mean and variance
σ
2
N
. The spatial correlation matrix at the base station is then
Y
Rec
Y
∗
Rec
=
HV
Ini
C
x
C
∗
x
V
∗
Ini
H
∗
+
NN
∗
+
HV
Ini
C
x
N
∗
+
NC
∗
x
V
∗
Ini
H
∗
Assuming a large enough block length
n
this simpliﬁes to
1
n
Y
Rec
Y
∗
Rec
≈
HH
∗
+
σ
2N
I
=
USV
∗
VSU
∗
+
σ
2N
I
=
US
2
U
∗
+
σ
2N
I
(6)where we have assumed independent data streams leading to
C
x
C
∗
x
≈
nI
(for an average squaredsymbol modulo of 1), used the fact that
V
Ini
V
∗
Ini
=
I
since
V
Ini
is unitary, and ﬁnally assuming that thenoise is statistically independent of the data. The ﬁrst data block
C
x
goes to waste.Estimates of the
K
top left singular vectors,
ˆ
U
K
, and singular values,
ˆ
σ
i
of
H
are obtained as thecorresponding eigenvectors and square root of the eigenvalues of the normalized spatial correlation matrix
1
n
Y
Rec
Y
∗
Rec
. Note that the estimates of the singular vectors will be unbiased whereas the singular values
DRAFT
5
will be incremented by the noise variance. If only phase modulation is used, the singular values are notof concern. Once this is completed, DL information is transmitted as
Y
Send
=
ˆU
K
C
y
. Forming thenormalized spatial correlation matrix at the subscriber unit, we have:
1
n
X
Rec
X
∗
Rec
≈
H
∗
U
K
U
∗
K
H
+
σ
2N
I
=
VSU
∗
U
K
U
∗
K
USV
∗
+
σ
2N
I
(7)Here
U
K
is in general not a unitary matrix and
U
K
U
∗
K
=
I
. However,
U
∗
U
K
=
I
K
0
×
K
implying
1
n
X
Rec
X
∗
Rec
≈
V
K
S
2K
V
∗
K
+
σ
2N
I
From this correlation matrix, the subscriber unit estimates the
K
required singular values and vectors,decodes the received information, and transmits uplink
X
Send
=
V
K
C
x
. Note that the method givesconsistent estimates of the eigenvectors, even in the presence of noise.The stability of this approach deserves a separate comment. Once started, the procedure transmitsapproximate singular vectors of the channel matrix through the channel. As noted in [3], this is essentiallyidentical to the method of orthogonal iteration ([8]) used to ﬁnd a set of dominant eigenvectors of a matrix.The procedure is therefore numerically stable and also robust in the presence of noise as demonstrated inthe simulations below.Note that the singular vectors can only be determined up to multiplication by a complex number of unitnorm. This is a standard ambiguity in blind estimation methods, and is overcome by using diﬀerentialcoding.
III. Results and discussion
Figure 1 shows the eﬀect of block length on BER vs. SNR for a 4TX x 4Rx channel using QPSKmodulation. The results are for the average of the two top singular modes and also includes the case forknown singular vectors. For each block length, a plateau in BER is reached for increasing SNR and theplateau decreases with block length. This is because the approximations in equations 6 and 7 improve withincreasing block length and for any ﬁnite length there remains an eﬀective noise component. Diﬀerentialcoding leads to a 3 dB loss in SNR and this explains the diﬀerence between the case for true singularvectors and the two largest block cases at low SNR.Figure 2 shows the BER against SNR for each of the two top singular modes in 4TX x 4Rx system forboth true and estimated singular vectors, using a block length of 1000. The diﬀerence in BER betweenthe two modes is small in the plateau phase.
A. Discussion
We have demonstrated blind communication via subspace estimation for a reciprocal MIMO channel.We exploit the fact that the data streams are uncorrelated in a spatial multiplexing scenario. Singularvectors and values are estimated on a "need-to-know" basis. No party knows the full matrix
H
, onlyone set of singular vectors and the singular values. The algorithm is also suitable for tracking, since it
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