SPENCER et al.: ZERO-FORCING METHODS FOR DOWNLINK SPATIAL MULTIPLEXING 463
the receivers, so channel diagonalization (if desired) must been
done entirely by the transmitter. Perfect diagonalization is only
possible for
and can be achieved using channel inver-
sion, e.g., by choosing
, where is the pseudo-in-
verse of
[15], [27]. On the other hand, when each of the
users has multiple antennas, complete diagonalization of the
channel at the transmitter is suboptimal since each user is able
to coordinate the processing of its own receiver outputs. If we
define the network channel and modulation matrices
and
as
the optimal solution under the constraint that all interuser
interference be zero is one, where
is block diagonal.
Like channel inversion, a block-diagonal solution imposes two
conditions—one on the dimensions and one on the independence
of the component
matrices—although it will be shown in
the next section that the conditions are somewhat less strict for
a block-diagonal solution. However, there is still a limitation
on how many users can be accomodated simultaneously. These
conditions are not as restrictive as they appear when viewed
in the context of a system that uses SDMA in conjunction
with other multiple access methods (TDMA, FDMA, etc.).
Consider a base station with a small number of antennas
and a large group of users, where an SDMA-only solution
is impractical. A more realistic implementation would divide
the users into subgroups (organized so that the dimension
requirements are satisfied within each group) whose members
are multiplexed spatially, while the subgroups themselves
are assigned different time or frequency slots. The linear
independence condition can be met by intelligently grouping
the users to avoid placing two users with highly correlated
channels in the same subgroup.
An algorithm for achieving a block-diagonal solution is pre-
sented in the following section.
III. B
LOCK DIAGONALIZATION
ALGORITHM
This section outlines a procedure for finding the optimal
transmit vectors
such that all multiuser interference is zero.
Since the resulting product
will be block diagonal, the
algorithm is referred to here as BD. Note that when
for
all users, this simplifies to a complete diagonalization, which
can be achieved using a pseudo-inverse of the channel. While
complete diagonalization could also be applied when
and would have the advantage of simplifying the receiver (each
antenna would receive only one signal), it comes at the cost
of reduced throughput or requiring higher power at the trans-
mitter, particularly when there is significant spatial correlation
between the antennas at the receiver. The two approaches are
compared in the simulation results of Section VI.
A. BD for Throughput Maximization
To eliminate all multi-user interference, we impose the con-
straint that
for . With a sum power con-
straint, the achievable throughput for the resulting block-diag-
onal system is
(5)
(6)
where
represents the sum capacity of the system, and in-
dicates the Hermitian transpose. If we define
as
(7)
the zero-interference constraint forces
to lie in the
null space of
. This definition allows us to define the
dimension condition necessary to guarantee that all users
can be accomodated under the zero-interference constraint.
Data can be transmitted to user
if the null space of
has a dimension greater than 0. This is satisfied when
rank
. So for any , block diagonalization is
possible if
rank rank . Thus, it is
theoretically possible to support some situations where both
and rank (for example, the {3,3} 4
channel). Assuming the dimension condition is satisfied for all
users, let
rank , and define the singular
value decomposition (SVD)
(8)
where
holds the first right singular vectors, and
holds the last ( ) right singular vectors. Thus, forms
an orthogonal basis for the null space of , and its columns are,
thus, candidates for the modulation matrix
of user .
Let
represent the rank of the product . In order for
transmission to user
to take place under the zero-interference
constraint,
is necessary. In general, is bounded by
[28]. A sufficient condition
for
is that at least one row of is linearly independent
of the rows of
. To satisfy this condition, one should take
care to avoid spatially multiplexing users with highly correlated
channel matrices. Note that both the dimension and indepen-
dence conditions allow certain cases that cannot be handled by
channel inversion. The channel inversion approach would re-
quire that all rows of
be linearly independent of . While
this is not necessary for block diagonalization, it would still be
beneficial, resulting in a higher value of
and, thus,
greater degrees of freedom for the final solution. Assuming that
the independence condition is satisfied for all users, we now de-
fine the matrix
.
.
.
(9)
The system capacity under the zero-interference constraint can
now be written as
(10)