JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN ELECTRONICS
AND COMMUNICATION ENGINEERING
V-BLAST DETECTION USING DUAL-LATTICE
1
1
SUMITRA SHAH , 2 JAYMIN BHALANI
, 2 Department of Electronics & Communication,
Charotar Institute of Technology and Science
Changa, Ta: Patlad Gujarat,India
[email protected] , [email protected]
ABSTRACT : A new view of the vertical bell labs layered Space-Time (V-BLAST) system by interpreting the detection
process on the dual lattice, and propose two O(N3) complexity ordering algorithms. One algorithm updates the dual
basis by using the backward Greville formula, while the other is to apply the sorted Gram-Schmidt orthogonization to the
dual basis. The demonstrate the connection between the two algorithms as well as their relations with existing
algorithms. The dual-lattice view does not only result in computational saving, it also seamlessly integrates the ordering
and nulling process.
Keywords: MIMO, SIC, V-BLAST, Dual Lattice, Gram Schmidt, Greville formula
1.
INTRODUCTION
where
Spatial multiplexing main objective is to transmit data
N
in parallel through a multi-input multi-output (MIMO)
y, n ∈ C denote the channel output and noise
channel. There are many detection schemes for MIMO
vectors, respectively
& it is many classified in linear & non linear. V-Blast is
M×N
one of the non-linear technique have gained much
H∈C
is the M × N full-rank matrix of channel
attention due to their capabilities to improve the
coefficients with N ≤ M.
transmission reliability and/or increase the channel
capacity. In order to reduce detection complexity due to
In zero-forcing (ZF) detection, y is multiplied on the
the enormous data rate, in this technique of successive
left by the pseudo inverse of H, giving the detection
interference cancelation (SIC) & the performance of
rule
SIC depends on the order in which the data sub streams
†
are detected. The purpose of V-BLAST is that it
xˆ= Q (H y)
(2.2)
maximizes the minimum signal-to-noise ratio (SNR).
This emerging technique is used in MIMO
Where, Q (.) denotes the quantization.
communication systems, i.e. decision feedback
detection for code-division multiple access (CDMA)
One way to SIC detection is to perform the QR
systems. The standard implementation of the ordering
decomposition H = QR, where Q is an orthonormal
4
matrix and R is an upper triangular matrix. Multiplying
algorithm requires O (N ) computational complexity for
†
3
(1) on the left with Q we have
an N × N MIMO system. A number of O (N )
algorithms have been developed in recent years. In this
†
paper, we interpret V-BLAST ordering from the
Y = Q y = Rx + n
(2.3)
viewpoint of lattices. When Quadrature Amplitude
Modulation (QAM) symbols are sent through a linear
In SIC, the estimate is substituted to remove the
MIMO channel, the received signal vector is a point in
interference term,
a lattice. The dual of the lattice has nice properties for
MIMO detection, which leads to two new ordering
(2.4)
algorithms for V-BLAST.
for n= N, N-1… 1
V-BLAST DETECTION
Let x =(x1... xN) T be the N × 1 data vector, where each
symbol xn is chosen from a finite subset of the complex
integer lattice Z + iZ. With scaling and shifting, one has
the generic (N × M) MIMO system model,
y = Hx + n
(2.1)
The order of detection is crucial to the error
performance of SIC. Ordering amounts to permuting the
columns of H, i.e., multiplying H on the right with a
(square) permutation matrix P so that some preset
criterion is met. In V-BLAST, this is done successively
from bottom up; it always chooses the column with the
ISSN: 0975 – 6779| NOV 09 TO OCT 10 | Volume 1, Issue 1
Page 10
JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN ELECTRONICS AND
COMMUNICATION ENGINEERING
maximum SNR at each stage of detection. It is proven
that this greedy search strategy actually maximizes the
minimum SNR among all N! possible orders.
Lattice is useful tool to study the v-blast detection. An
N-dimension complex lattice L L(H) with basis H is
generated as the complex-integer linear combination of
the set of linearly independent vectors {h1, h2... hN }. As
the received vector y is a noisy version of a point in the
n
with n column, is the
remaining matrix after deleting of
is a ZF
nulling vector for sublattice Ln, by we have with n
columns, is the remaining matrix after deleting columns
,,…,
from H, for which detection has already
been done.
n
lattice {Hx|x ∈ Z + iZ } the detector aims to search for
a point in the lattice that is as close to y as possible.
The decision region of ZF is a parallelogram centered at
x = 0, known as the fundamental parallelogram, which
is define as the region {y|y = Ha, −1/2 ≤ an < 1/2}.
Accordingly, the distance is given by,
(2.5)
Where
denotes the angle between hn and the linear
space spanned by the other N − 1 basis vectors.
QR decomposition can be implemented by the GramSchmidt orthogonization. Let
be the
matrix of the Gram-Schmidt vectors of H. One has the
relation. The decision region of SIC is the rectangle
{y|y = Ha, −1/2 ≤ an < 1/2}. Correspondingly, the
distance is given by,
n = 1, 2… N
(2.6)
Where
is the angle between hn and the linear space
spanned by the first n − 1 basis vectors h1... hn−1. Since
the SNR at the n-th detection stage, assuming correct
2
ˆ
At the first stage n=N, the V-blast algorithm chooses
the row of
with minimum length, and the
corresponding column is detected. This procedure is
then repeated on remaining matrix for n=N-1,…,1. This
formulation is better interpreted from the viewpoint of
dual lattice.
A. DUAL BASIS
Dual lattices are usually defined for real lattices. Here
we extend the definition to complex lattices. The dual
lattice L* of a complex lattice L is defined as those
vectors u, such that the inner product <u, v>= Z + iZ,
for all v  L. Correspondingly, the dual basis is given
by
=
, which satisfies,
(hn, hn* )= δn,m
Where,
(3.3)
is the Kronecker delta
The nulling vectors for ZF detection are defined in as
those vectors vn, n =1...N, satisfying,
(vn, hk) = δn,k
(3.4)
By (3.3) we have the following property:
2
feedback, is proportional to | | = |h n| , V-BLAST
ordering amounts to successively choosing the n-th
Gram-Schmidt vector with the maximum length for n =
N, N-1…1.
2.
, where
PROPERTY 1: The n-th column vector of the dual
basis H* is the n-th nulling vector for ZF.
Furthermore, since the angle between hn & hn* is π/2θn, we have,
DUAL-LATTICE VIEWPOINT
Successive Interference Cancellation (SIC) can be
performed by dealing with the pseudo inverse of H. In
this way of SIC detection, the n-th nulling vector is
define as the unique minimum-norm vector satisfying
(3.1)
The detection process as
(3.2)
for n = N,…,1.
The nulling vector wn is the n-th row of the matrix
(3.5)
Thus, the following relation holds for the dual basis:
(3.6)
It indicates that we can have large distances in ZF if the
vectors of the dual basis are short.
Let Ln l(
) be the n-dimensional sub lattice of
L, with Ln = L. the dual lattice Ln* of Ln has basis
=
.since the n-th column
of
is a ZF nulling vector for sub lattice Ln, by
we have,
ISSN: 0975 – 6779| NOV 09 TO OCT 10 | Volume 1, Issue 1
Page 11
JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN ELECTRONICS AND
COMMUNICATION ENGINEERING
(3.7)
Here,
the angle is because
should be understood
as the angle in Ln. Now it is very clear that minimizing
is equivalent to
maximizing. Therefore,
V-BLAST ordering can be interpreted as successively
finding the shortest column vector in the dual basis of
the sub-lattice.
B. FLIPPED DUAL BASIS
in detection only. The standard Greville formula gives
the pseudo inverse of a matrix to which a new column
is appended. Our purpose here is converse: we have to
update the pseudo inverse when a column is removed
from a matrix. To this end, we derive the backward
Greville formula, omitting the details Backward
Greville Formula: Suppose that the matrix.
Backward Greville Formula: An = [An-1, an] = [a1...
an-1, an] 
has Pseudo inverse . Partition
into the form
Equation (3.3) is the standard definition of dual basis. It
is further flipped in the left-right do that becomes
(3.8)
(4.1)
then the pseudo inverse of An¡1 is given by otherwise,
Flipping bring an elegant relation between the GramSchmidt vectors of a basis & it’s dual
(3.9)
where ,…
are the Gram-Schmidt vectors of the
flipped dual basis H*
(4.2)
Where
If
in the range of
(3.10)
It shows that the detector can have large distances if the
flipped dual basis has short Gram-Schmidt vectors.
The inversely proportional relation (3.10) between the
lengths of the basis vectors leads to another new
interpretation of VBLAST ordering. That is, it permutes
the flipped dual basis in such an order that the lengths
of its Gram-Schimidt vectors are successively
minimized for n = 1, 2,…..,N, instead of maximizing
that of the primal basis for n = N,N – 1,…,1. The former
is much more tractable. Moreover, (3.9) implies the
following property that is especially appealing to the
implementation of the detector.
PROPERTY 2: The n-th Gram-Schmidt vector of the
flipped dual basis H* is the (N ¡ n + 1)-th nulling vector
for SIC, namely
, and
(4.3)
otherwise.
By an in the range of An-1, we mean that there exists a
vector t  Cm such that an = An-1t. In other words, and
is in the subspace spanned by the columns a1... an-1.
Since the channel matrix is assumed to be of full rank,
this case never occurs; otherwise the columns will be
linearly dependent. Accordingly, we only have to use
the case (4.3) for our purposes. To use the backward
Greville formula in dual-basis updating, we slightly
modify (4.2) and (4.3). Put the dual basis for an in the
form
(4.4)
then the dual basis for An¡1 can be expressed by
(3.11)
3.
BACKWARD GREVILLE RECURSION
The O(N4) complexity of the V-BLAST ordering is due
to re-computing the pseudo inverse at each stage. The
idea of our algorithm is to avoid the re-computation of
the pseudo inverse after a column is deleted from the
matrix. Specifically, it employs the backward Greville
formula to recursively compute the pseudo inverses.
Note that, although the standard forward Greville
formula was applied in, it was not used in ordering, but
(4.5)
Where, we rearrange the second term so that the
computation complexity is O(nN) for column length
N of the dual basis; otherwise it will be increased by
an order due to matrix multiplication.
ISSN: 0975 – 6779| NOV 09 TO OCT 10 | Volume 1, Issue 1
Page 12
JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN ELECTRONICS AND
COMMUNICATION ENGINEERING
Algorithm 1 gives the whole detection process. The
matrix
represents the current dual basis, which is
being updated and is getting narrower as the algorithm
progresses. The vector k stores the indexes of the
detection order.
Implementation: (Dual ordering and detection based
on backward Greville recursion)
Step 1: Initial sorting and nulling
Set
= H*, k = (
. Find the index j for the
column of
with the minimum length. Exchange the j
and N-th columns of
and also kj and kN. Then,
project the received signal y onto the nulling vector
(the last column of
and perform the detection
(4.6)
Step 2: Interference cancelation
Subtract the detected signal from the received signal
and average.
dual basis for n = 1, 2… N. This ordering can be
realized by slightly modifying the standard GramSchmidt orthogonization. The remaining columns of H*
are according to their length orthogonal to the linear
space spanned by the Gram-Schmidt vectors already
obtained. This results in significant computational
savings because only a single Gram-Schmidt
orthogonization process is needed. Algorithm 2
describes the ordering and detection process, where the
Gram-Schmidt matrix
is continuously being
updated, and p is the vector of indexes.
Implementation: (Dual ordering based on sorted
Gram- Schmidt orthogonization)
Step 1: Initial sorting and nulling
Set
=H* p =
. Find the index j for the
column of the dual basis with the minimum length,
exchange the first and j- the columns of , and also p1
and pj . Then, project the received signal y onto the
nulling vector and perform the detection
(5.1)
(4.7)
Step 2: Interference cancelation
Subtract the detected signal from the received signal
Step 3: Dual-basis updating
From H¤ get the updated dual basis
(5.2)
(4.8)
Step 4: Recursion
Repeat the above three steps until all symbols are
detected. That is, at the n-th (n = N -1… 1) stage, find
the j-th column vector of
with the minimum length,
exchange the j and n-th columns of
, and also kj and
kn. Then perform the detection, subtraction, and basis
updating
(4.9)
Step 3: Projection
Project the other columns of * to the orthogonal
complement of
, are detected.
(5.3)
Step 4: Recursion
Repeat the above three steps until all symbols are
detected. That is, at the n-th (n = 2… N) stage, let,
(5.4)
(4.10)
(4.11)
Where,
matrix
denotes the n-th column of the updated
4.
SORTED GRAM-SCHMIDT
ORTHOGONIZATION
In this Section, we present another O(N3) algorithm of
dual ordering. As demonstrated earlier, V-BLAST
ordering is equivalent to successively minimize the
lengths of the Gram- Schmidt vectors for the flipped
exchange the n and j-th columns of , and also pn and
pj . Then perform the detection, subtraction, and
projection
In Algorithm 2, , for n = 1…N are used as the nulling
vectors. Therefore, the algorithm does not only have O
(N3) complexity, it also seamlessly integrates the
ordering and nulling processes. Moreover, the
projection procedures (4.7), (4.10) are in fact the socalled the modified Gram-Schmidt orthogonization, as
recognized in. It has fairly sound numerical stability.
Note that Wubben et al.’s sorted QR decomposition
results in suboptimal ordering. Proposition 1 show that,
interestingly, it will result in optimal ordering even if
ISSN: 0975 – 6779| NOV 09 TO OCT 10 | Volume 1, Issue 1
Page 13
JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN ELECTRONICS AND
COMMUNICATION ENGINEERING
applied to the dual basis without flipping. This is
because flipping (as any column permutation) makes no
difference to sorted Gram- Schmidt orthogonization,
i.e., the sorted basis will be the same as that without
flipping.
Given H*
4N3
6N3
6N3
2N3
Dual
ascending
Sorted QR
Dual
ordering
Noisepredictive
De
correlating
Given H
Squareroot
Scheme 1: Applying the sorted Gram-Schmidt
orthogonization to either the standard or the flipped
dual basis achieves the optimal V-BLAST ordering.
2N3
-
-
2N3
Table-1 Complexity Comparison for VBLAST
5.
COMPLEXITY COMPARISONS
A close look at the two proposed algorithms tells us that
they are not only equivalent, but are also the same. In
particular, we have the correspondence
and
In order to see this, notice that both
and
are the shortest column of the dual basis H*,
and (4.10) can be rewritten as
; m=n+1,..,N
(6.1)
Which, represents the projection to the orthogonal
complement of .
This is precisely done in the Gram-Schmidt process in
Equation (6.1). Therefore, there is a line-by-line
correspondence between the two algorithms, although
they are developed by using different mathematical
tools. A suboptimal order was given in that, in essence
sorts the dual basis in ascending-length order
(6.2)
Where, we assume the detection starts from n = N.
Surprisingly, this simple ordering is significantly better
than sorting the primal basis in the same order, and its
performance is also close to that of optimal V-BLAST
ordering. The reason can be explained by comparing it
with the proposed algorithms. Clearly, the first column
vector on the left of (6.2) has the minimum length in the
dual basis, i.e., it in fact corresponds to the initial
sorting stage. In MIMO fading channels, the first stage
of detection dominates the error performance.
Therefore, the error performance of this suboptimal
ordering is expected to be close to that of the optimum
ordering, and specifically, achieves the same order of
diversity. This has been observed in computer
simulations
Further comparison with existing algorithms shows that
the ordering part of the proposed algorithms is the same
as that of the noise-predictive algorithm of Waters and
Barry. Their algorithm exploits the correlation within
the noise vector H†n after the nulling operation of the
ZF detector (1.2). The noise at a later stage can be
linearly predicted from the previous samples. The
standard method of obtaining the prediction coefficients
is to solve the Wiener-Hoff equation, it is efficiently
implemented by recursively project the rows of Hy onto
the orthogonal complement of the subspace spanned by
the rows already chosen. This corresponds to the
orthogonality principle interpretation of the WienerHoff equation, i.e., the prediction error has to be
orthogonal to the previous samples in order to achieve
the minimum mean square error.
Nonetheless, the proposed dual-ordering algorithms do
lead to some computational savings. After obtaining the
detection index, the noise-predictive algorithm has to
calculate the prediction coefficients by inverting a
triangular matrix, whose complexity is about N3=3. In
contrast, the nulling vectors are readily available once
the ordering is done in the dual ordering algorithms
Table-1 compares the complexity of V-BLAST
detection algorithms. We assume M = N for
convenience, which is the case of most interest in
practice. We also assume that the modified GramSchmidt orthogonization is employed by all algorithms
as in The lengths of column vectors in dual-ordering
algorithms can be recursively computed , whose
complexity is O(N2) (direct computation would require
2N3=3 complex operations.) Such lower-order terms are
neglected in Table-1.
It is remarkable that, given H*, the proposed dualordering algorithms have almost the same complexity
as sorting the dual-basis vectors in ascending order.
Another merit of the dual-ordering algorithms is that
ordering
and
detection
can
be
performed
“simultaneously”. All other three optimal algorithms
have to “wait” until the ordering is finished (The
square-root algorithm allows simultaneous ordering and
ISSN: 0975 – 6779| NOV 09 TO OCT 10 | Volume 1, Issue 1
Page 14
JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN ELECTRONICS AND
COMMUNICATION ENGINEERING
detection as well, but the complexity will be higher).
Thus, the dual-ordering algorithms reduce the
processing delay when ordering and detection are
implemented in a pipeline structure.
6.
CONCLUSION
We have shown that the dual lattice is a useful tool for
the study of V-BLAST detection. The dual basis
consists of the zero forcing nulling vectors, while the
Gram-Schmidt vectors of the flipped dual basis
correspond to the SIC nulling vectors. Hence, VBLAST ordering can be interpreted as successively
finding the shortest column vectors from the dual basis
of the sub lattice, or successively finding the shortest
Gram- Schmidt vectors of the dual basis. This
explanation led to two new ordering algorithms: one is
based the backward Greville formula which updates the
dual basis itself, the other applies the sorted GramSchmidt orthogonization to the dual basis. The dualordering algorithms are not only simpler, they also
allows a pipeline structure to implement ordering and
nulling.
7.
REFERENCES
[1] P. W. Wolniansky, G. J. Foschini, G. D. Golden,
and R. A. Valenzuela, “V-BLAST: architecture for
realizing very high data rates over richscattering
wireless channel,” in Proc. Int. Symp. Signals,
Syst., Electron. (ISSSE), Pisa, Italy, Sept. 1998, pp.
295–300.
[2] M. K. Varanasi, “Decision feedback multiuser
detection: A systematic approach,” IEEE Trans.
Inform. Theory, vol. 45, pp. 219–240, Jan. 1999.
[3] W. Zha and S. Blostein, “Modified decorrelating
decision-feedback detection of BLAST space-time
system,” in Proc. IEEE Int. Conf. Commun., May
2002, pp. 335–339.
[4] W. Wai, C. Tsui, and R. Cheng, “A low complexity
architecture of the V-BLAST system,” in Proc.
IEEE WCNC, Sept. 2000, pp. 310–314.
[5] Z. Luo, M. Zhao, S. Liu, and Y. Liu, “Greville-toinverse-Greville algorithm for V-BLAST systems,”
in Proc. Int. Conf. Commun. (ICC), Istanbul,
Turkey, June 2006.
ISSN: 0975 – 6779| NOV 09 TO OCT 10 | Volume 1, Issue 1
Page 15