<s>
Precoding	B-Protocol
is	O
a	O
generalization	O
of	O
beamforming	B-Algorithm
to	O
support	O
multi-stream	O
(	O
or	O
multi-layer	O
)	O
transmission	O
in	O
multi-antenna	O
wireless	O
communications	O
.	O
</s>
<s>
In	O
conventional	O
single-stream	O
beamforming	B-Algorithm
,	O
the	O
same	O
signal	O
is	O
emitted	O
from	O
each	O
of	O
the	O
transmit	O
antennas	O
with	O
appropriate	O
weighting	O
(	O
phase	O
and	O
gain	O
)	O
such	O
that	O
the	O
signal	O
power	O
is	O
maximized	O
at	O
the	O
receiver	O
output	O
.	O
</s>
<s>
When	O
the	O
receiver	O
has	O
multiple	O
antennas	O
,	O
single-stream	O
beamforming	B-Algorithm
cannot	O
simultaneously	O
maximize	O
the	O
signal	O
level	O
at	O
all	O
of	O
the	O
receive	O
antennas	O
.	O
</s>
<s>
In	O
point-to-point	O
systems	O
,	O
precoding	B-Protocol
means	O
that	O
multiple	O
data	O
streams	O
are	O
emitted	O
from	O
the	O
transmit	O
antennas	O
with	O
independent	O
and	O
appropriate	O
weightings	O
such	O
that	O
the	O
link	O
throughput	O
is	O
maximized	O
at	O
the	O
receiver	O
output	O
.	O
</s>
<s>
In	O
multi-user	B-Architecture
MIMO	I-Architecture
,	O
the	O
data	O
streams	O
are	O
intended	O
for	O
different	O
users	O
(	O
known	O
as	O
SDMA	O
)	O
and	O
some	O
measure	O
of	O
the	O
total	O
throughput	O
(	O
e.g.	O
,	O
the	O
sum	O
performance	O
or	O
max-min	O
fairness	O
)	O
is	O
maximized	O
.	O
</s>
<s>
In	O
point-to-point	O
systems	O
,	O
some	O
of	O
the	O
benefits	O
of	O
precoding	B-Protocol
can	O
be	O
realized	O
without	O
requiring	O
channel	O
state	O
information	O
at	O
the	O
transmitter	O
,	O
while	O
such	O
information	O
is	O
essential	O
to	O
handle	O
the	O
inter-user	O
interference	O
in	O
multi-user	O
systems	O
.	O
</s>
<s>
Precoding	B-Protocol
in	O
the	O
downlink	O
of	O
cellular	O
networks	O
,	O
known	O
as	O
network	B-Architecture
MIMO	I-Architecture
or	O
coordinated	O
multipoint	O
(	O
CoMP	O
)	O
,	O
is	O
a	O
generalized	O
form	O
of	O
multi-user	B-Architecture
MIMO	I-Architecture
that	O
can	O
be	O
analyzed	O
by	O
the	O
same	O
mathematical	O
techniques	O
.	O
</s>
<s>
Precoding	B-Protocol
is	O
a	O
technique	O
that	O
exploits	O
transmit	O
diversity	O
by	O
weighting	O
the	O
information	O
stream	O
,	O
i.e.	O
</s>
<s>
To	O
prevent	O
a	O
potential	O
misunderstanding	O
here	O
,	O
precoding	B-Protocol
does	O
not	O
cancel	O
out	O
the	O
impact	O
of	O
the	O
channel	O
,	O
but	O
it	O
aligns	O
the	O
vector	O
containing	O
the	O
transmit	O
symbols	O
(	O
i.e.	O
</s>
<s>
Most	O
classic	O
precoding	B-Protocol
results	O
assume	O
narrowband	O
,	O
slowly	O
fading	O
channels	O
,	O
meaning	O
that	O
the	O
channel	O
for	O
a	O
certain	O
period	O
of	O
time	O
can	O
be	O
described	O
by	O
a	O
single	O
channel	O
matrix	O
which	O
does	O
not	O
change	O
faster	O
.	O
</s>
<s>
In	O
practice	O
,	O
such	O
channels	O
can	O
be	O
achieved	O
,	O
for	O
example	O
,	O
through	O
OFDM	B-Protocol
.	O
</s>
<s>
The	O
precoding	B-Protocol
strategy	O
that	O
maximizes	O
the	O
throughput	O
,	O
called	O
channel	O
capacity	O
,	O
depends	O
on	O
the	O
channel	O
state	O
information	O
available	O
in	O
the	O
system	O
.	O
</s>
<s>
If	O
the	O
channel	O
matrix	O
is	O
completely	O
known	O
,	O
singular	O
value	O
decomposition	O
(	O
SVD	O
)	O
precoding	B-Protocol
is	O
known	O
to	O
achieve	O
the	O
MIMO	O
channel	O
capacity	O
.	O
</s>
<s>
In	O
multi-user	B-Architecture
MIMO	I-Architecture
,	O
a	O
multi-antenna	O
transmitter	O
communicates	O
simultaneously	O
with	O
multiple	O
receivers	O
(	O
each	O
having	O
one	O
or	O
multiple	O
antennas	O
)	O
.	O
</s>
<s>
From	O
an	O
implementation	O
perspective	O
,	O
precoding	B-Protocol
algorithms	O
for	O
SDMA	O
systems	O
can	O
be	O
sub-divided	O
into	O
linear	O
and	O
nonlinear	O
precoding	B-Protocol
types	O
.	O
</s>
<s>
The	O
capacity	O
achieving	O
algorithms	O
are	O
nonlinear	O
,	O
but	O
linear	O
precoding	B-Protocol
approaches	O
usually	O
achieve	O
reasonable	O
performance	O
with	O
much	O
lower	O
complexity	O
.	O
</s>
<s>
Linear	O
precoding	B-Protocol
strategies	O
include	O
maximum	O
ratio	O
transmission	O
(	O
MRT	O
)	O
,	O
zero-forcing	B-Protocol
(	O
ZF	O
)	O
precoding	B-Protocol
,	O
and	O
transmit	O
Wiener	O
precoding	B-Protocol
.	O
</s>
<s>
There	O
are	O
also	O
precoding	B-Protocol
strategies	O
tailored	O
for	O
low-rate	O
feedback	O
of	O
channel	O
state	O
information	O
,	O
for	O
example	O
random	O
beamforming	B-Algorithm
.	O
</s>
<s>
Nonlinear	O
precoding	B-Protocol
is	O
designed	O
based	O
on	O
the	O
concept	O
of	O
dirty	O
paper	O
coding	O
(	O
DPC	O
)	O
,	O
which	O
shows	O
that	O
any	O
known	O
interference	O
at	O
the	O
transmitter	O
can	O
be	O
subtracted	O
without	O
the	O
penalty	O
of	O
radio	O
resources	O
if	O
the	O
optimal	O
precoding	B-Protocol
scheme	O
can	O
be	O
applied	O
on	O
the	O
transmit	O
signal	O
.	O
</s>
<s>
This	O
suboptimal	O
approach	O
cannot	O
achieve	O
the	O
weighted	O
sum	O
rate	O
,	O
but	O
it	O
can	O
still	O
maximize	O
the	O
weighted	O
sum	O
performance	O
(	O
or	O
some	O
other	O
metric	O
of	O
achievable	O
rates	O
under	O
linear	O
precoding	B-Protocol
)	O
.	O
</s>
<s>
The	O
optimal	O
linear	O
precoding	B-Protocol
does	O
not	O
have	O
any	O
closed-form	O
expression	O
,	O
but	O
it	O
takes	O
the	O
form	O
of	O
a	O
weighted	O
MMSE	O
precoding	B-Protocol
for	O
single-antenna	O
receivers	O
.	O
</s>
<s>
The	O
precoding	B-Protocol
weights	O
for	O
a	O
given	O
user	O
are	O
selected	O
to	O
maximize	O
a	O
ratio	O
between	O
the	O
signal	O
gain	O
at	O
this	O
user	O
and	O
the	O
interference	O
generated	O
at	O
other	O
users	O
(	O
with	O
some	O
weights	O
)	O
plus	O
noise	O
.	O
</s>
<s>
Thus	O
,	O
precoding	B-Protocol
can	O
be	O
interpreted	O
as	O
finding	O
the	O
optimal	O
balance	O
between	O
achieving	O
strong	O
signal	O
gain	O
and	O
limiting	O
inter-user	O
interference	O
.	O
</s>
<s>
Finding	O
the	O
optimal	O
weighted	O
MMSE	O
precoding	B-Protocol
is	O
difficult	O
,	O
leading	O
to	O
approximate	O
approaches	O
where	O
the	O
weights	O
are	O
selected	O
heuristically	O
.	O
</s>
<s>
A	O
common	O
approach	O
is	O
to	O
concentrate	O
on	O
either	O
the	O
numerator	O
or	O
the	O
denominator	O
of	O
the	O
mentioned	O
ratio	O
;	O
that	O
is	O
,	O
maximum	O
ratio	O
transmission	O
(	O
MRT	O
)	O
and	O
zero-forcing	B-Protocol
(	O
ZF	O
)	O
precoding	B-Protocol
.	O
</s>
<s>
ZF	B-Protocol
precoding	I-Protocol
aims	O
at	O
nulling	O
the	O
inter-user	O
interference	O
,	O
at	O
the	O
expense	O
of	O
losing	O
some	O
signal	O
gain	O
.	O
</s>
<s>
ZF	B-Protocol
precoding	I-Protocol
can	O
achieve	O
a	O
performance	O
close	O
to	O
the	O
sum	O
capacity	O
when	O
the	O
number	O
of	O
users	O
is	O
large	O
or	O
the	O
system	O
is	O
interference-limited	O
(	O
i.e.	O
,	O
the	O
noise	O
is	O
weak	O
compared	O
to	O
the	O
interference	O
)	O
.	O
</s>
<s>
A	O
balance	O
between	O
MRT	O
and	O
ZF	O
is	O
obtained	O
by	O
the	O
so-called	O
regularized	O
zero-forcing	B-Protocol
(	O
also	O
known	O
as	O
signal-to-leakage-and-interference	O
ratio	O
(	O
SLNR	O
)	O
beamforming	B-Algorithm
and	O
transmit	O
Wiener	O
filtering	O
)	O
All	O
of	O
these	O
heuristic	O
approaches	O
can	O
also	O
be	O
applied	O
to	O
receivers	O
that	O
have	O
multiple	O
antennas	O
.	O
</s>
<s>
Also	O
for	O
multiuser	B-Architecture
MIMO	I-Architecture
system	O
setup	O
,	O
another	O
approach	O
has	O
been	O
used	O
to	O
reformulate	O
the	O
weighted	O
sum	O
rate	O
optimization	O
problem	O
to	O
a	O
weighted	O
sum	O
MSE	O
problem	O
with	O
additional	O
optimization	O
MSE	O
weights	O
for	O
each	O
symbol	O
in	O
.	O
</s>
<s>
Note	O
that	O
the	O
optimal	O
linear	O
precoding	B-Protocol
can	O
be	O
computed	O
using	O
monotonic	O
optimization	O
algorithms	O
,	O
but	O
the	O
computational	O
complexity	O
scales	O
exponentially	O
fast	O
with	O
the	O
number	O
of	O
users	O
.	O
</s>
<s>
In	O
closed-loop	O
systems	O
,	O
the	O
feedback	O
capabilities	O
decide	O
which	O
precoding	B-Protocol
strategies	O
that	O
are	O
feasible	O
.	O
</s>
<s>
Zero-forcing	B-Protocol
precoding	I-Protocol
may	O
even	O
achieve	O
the	O
full	O
multiplexing	O
gain	O
,	O
but	O
only	O
provided	O
that	O
the	O
accuracy	O
of	O
the	O
channel	O
feedback	O
increases	O
linearly	O
with	O
signal-to-noise	O
ratio	O
(	O
in	O
dB	O
)	O
.	O
</s>
<s>
Quantization	O
and	O
feedback	O
of	O
channel	O
state	O
information	O
is	O
based	O
on	O
vector	B-Algorithm
quantization	I-Algorithm
,	O
and	O
codebooks	O
based	O
on	O
Grassmannian	O
line	O
packing	O
have	O
shown	O
good	O
performance	O
.	O
</s>
<s>
Other	O
precoding	B-Protocol
strategies	O
have	O
been	O
developed	O
for	O
the	O
case	O
with	O
very	O
low	O
channel	O
feedback	O
rates	O
.	O
</s>
<s>
Random	O
beamforming	B-Algorithm
(	O
or	O
opportunistic	O
beamforming	B-Algorithm
)	O
was	O
proposed	O
as	O
a	O
simple	O
way	O
of	O
achieving	O
good	O
performance	O
that	O
scales	O
like	O
the	O
sum	O
capacity	O
when	O
the	O
number	O
of	O
receivers	O
is	O
large	O
.	O
</s>
<s>
In	O
this	O
suboptimal	O
strategy	O
,	O
a	O
set	O
of	O
beamforming	B-Algorithm
directions	O
are	O
selected	O
randomly	O
and	O
users	O
feed	O
back	O
a	O
few	O
bits	O
to	O
tell	O
the	O
transmitter	O
which	O
beam	O
gives	O
the	O
best	O
performance	O
and	O
what	O
rate	O
they	O
can	O
support	O
using	O
it	O
.	O
</s>
<s>
When	O
the	O
number	O
of	O
users	O
is	O
large	O
,	O
it	O
is	O
likely	O
that	O
each	O
random	O
beamforming	B-Algorithm
weight	O
will	O
provide	O
good	O
performance	O
for	O
some	O
user	O
.	O
</s>
<s>
In	O
spatially	O
correlated	O
environments	O
,	O
the	O
long-term	O
channel	O
statistics	O
can	O
be	O
combined	O
with	O
low-rate	O
feedback	O
to	O
perform	O
multi-user	O
precoding	B-Protocol
.	O
</s>
<s>
As	O
the	O
beamforming	B-Algorithm
weights	O
are	O
selected	O
from	O
the	O
statistics	O
,	O
and	O
not	O
randomly	O
,	O
this	O
approach	O
outperforms	O
random	O
beamforming	B-Algorithm
under	O
strong	O
spatial	O
correlation	O
.	O
</s>
<s>
In	O
multiuser	B-Architecture
MIMO	I-Architecture
systems	O
where	O
the	O
number	O
of	O
users	O
are	O
higher	O
than	O
the	O
number	O
of	O
transmit	O
antennas	O
,	O
a	O
multiuser	O
diversity	O
can	O
be	O
achieved	O
by	O
performing	O
user	O
scheduling	O
before	O
applying	O
zero-forcing	B-Protocol
beamforming	B-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
achieve	O
multiuser	O
diversity	O
and	O
apply	O
zero-forcing	B-Protocol
precoding	I-Protocol
,	O
the	O
CSI	O
of	O
all	O
users	O
are	O
required	O
at	O
the	O
base	O
station	O
.	O
</s>
<s>
This	O
category	O
includes	O
Costa	O
precoding	B-Protocol
,	O
Tomlinson-Harashima	O
precoding	B-Protocol
and	O
the	O
vector	O
perturbation	O
technique	O
.	O
</s>
<s>
Consider	O
a	O
downlink	O
multi-user	B-Architecture
MIMO	I-Architecture
system	O
where	O
a	O
base	O
station	O
with	O
transmit	O
antennas	O
and	O
single-antenna	O
users	O
.	O
</s>
<s>
where	O
is	O
the	O
(	O
normalized	O
)	O
data	O
symbol	O
and	O
is	O
the	O
linear	O
precoding	B-Protocol
vector	O
.	O
</s>
<s>
while	O
the	O
suboptimal	O
ZF	B-Protocol
precoding	I-Protocol
makes	O
sure	O
that	O
for	O
all	O
i	O
≠	O
k	O
and	O
thus	O
the	O
interference	O
can	O
be	O
removed	O
in	O
the	O
SINR	O
expression	O
:	O
</s>
<s>
Remarkably	O
,	O
the	O
optimal	O
receive	O
filters	O
are	O
the	O
same	O
as	O
the	O
weighted	O
MMSE	O
precoding	B-Protocol
vectors	O
,	O
up	O
to	O
a	O
scaling	O
factor	O
:	O
</s>
<s>
Observe	O
that	O
the	O
coefficients	O
that	O
was	O
used	O
in	O
the	O
weighted	O
MMSE	O
precoding	B-Protocol
are	O
not	O
exactly	O
the	O
optimal	O
power	O
coefficients	O
in	O
the	O
uplink	O
(	O
that	O
maximize	O
the	O
weighted	O
sum	O
rate	O
)	O
except	O
under	O
certain	O
conditions	O
.	O
</s>
<s>
This	O
important	O
relationship	O
between	O
downlink	O
precoding	B-Protocol
and	O
uplink	O
receive	O
filtering	O
is	O
known	O
as	O
the	O
uplink-downlink	O
duality	O
.	O
</s>
<s>
As	O
the	O
downlink	O
precoding	B-Protocol
problem	O
usually	O
is	O
more	O
difficult	O
to	O
solve	O
,	O
it	O
often	O
useful	O
to	O
first	O
solve	O
the	O
corresponding	O
uplink	O
problem	O
.	O
</s>
<s>
The	O
precoding	B-Protocol
strategies	O
described	O
above	O
was	O
based	O
on	O
having	O
perfect	O
channel	O
state	O
information	O
at	O
the	O
transmitter	O
.	O
</s>
<s>
If	O
the	O
same	O
precoding	B-Protocol
strategies	O
are	O
applied	O
,	O
but	O
now	O
based	O
on	O
inaccurate	O
channel	O
information	O
,	O
additional	O
interference	O
appears	O
.	O
</s>
<s>
This	O
is	O
an	O
example	O
on	O
limited	O
feedback	O
precoding	B-Protocol
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
beamforming	B-Algorithm
vectors	O
are	O
distorted	O
as	O
,	O
where	O
is	O
the	O
optimal	O
vector	O
and	O
is	O
the	O
error	O
vector	O
caused	O
by	O
inaccurate	O
channel	O
state	O
information	O
.	O
</s>
<s>
where	O
is	O
the	O
additional	O
interference	O
at	O
user	O
according	O
to	O
the	O
limited	O
feedback	O
precoding	B-Protocol
.	O
</s>
