An Overview of Algorithms for Downlink Transmit Beamforming A. Lee Swindlehurstand Chris Peel Brigham Young University phone: 801-422-4343 email:
[email protected] email:
[email protected] Quentin Spencer Distribution Control Systems, Inc. email:
[email protected] Abstract Downlink beamforming refers to the problem of using an array of antennas at a particular node (e.g., a basestation) in a wireless network to communicate simultaneously with multiple co-channel users. The users in the network may have a single antenna, and hence no ability for spatial discrimination, or they may have multiple antennas and the ability to perform some type of interference suppression. The primary issue is how to balance the need for high, received signal power for each user against the interference produced by the signal at other points in the network. In this presentation, we describe several approaches to this problem: channel inversion, regularized channel inversion, channel block diagonalization, coordinated transmit/receive beamforming, and dirty-paper coding. While the basic idea behind these algorithms is the same, namely the use of channel information at the transmitter to predict and then counteract the interference produced at each node in the network, each of the algorithms is based on achieving a different performance objective. Typical performance criteria include zero-interference transmission, minimum transmit power subject to a minimum signal-to-interference plus noise ratio at each receiver, or maximum throughput subject to a given transmit power constraint. We compare the various goals of the above algorithms, and detail their respective advantages and disadvantages in terms of computational complexity, required transmit power, network throughput, and assumed receiver capabilities. The results of several simulation studies are presented to quantify these comparisons.
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An Overview of Algorithms for Downlink Transmit Beamforming A. Lee Swindlehurst, Chris Peel, Quentin Spencer Brigham Young University Provo, UT USA
Presentation Outline Background
"*
- Single-User vs. Multi-User MIMO Scenarios - Mathematical Notation
Algorithms for Single-Antenna Users
"*
-
Channel Inversion Regularized Channel Inversion Vector Modulo Pre-Coding Interference-Balancing Methods for Power Control
Algorithms for Multiple-Antenna Users
"*
- Joint Transmit/Receive Beamforming - One vs. Multiple Sub-Channels per User
* Experimental Results * Summary A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
Single-User, Point-to-Point MIMO
T s I•
Space-
Wireless
Time Encoder
Channel
N Transmit Antennas "*Receive "* Under
Space-
Time Decoder
S
M Receive Antennas
processing is centralized, assume CSI at receiver (rCSI)
ideal conditions (e.g., independent Rayleigh fading) - capacity grows linearly with min(N,M) - capacity growth is independent of CSI at the transmitter (xCSI)
"*Ifchannel
A. Lee Swindlehurst
T-
is rank deficient, xCSI is much more important Electrical & Computer Engineering
Brigham Young University
Multi-User MIMO Downlink STime
s, M
T
SpaceTime
Decoder
S
Wireless -
T
(Channel
Encoder
Space-• Time
S311••/•
T
Space-
$2
Decoder
NTransmitTie3 ._ Decoder
Antennas
M Total Receive Antennas
* Receive processing is distributed, only local rCSI is available * Without xCSI, no capacity growth at high SNR due to interference, even in the ideal case A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
MU-MIMO Capacity Example 8x8 Single User
8x{4,4} Multi-User
8 Total Receive Antennas
8 Transmit Antennas
8 Receive Antennas
8 Transmit Antennas
One User 50
Uninformed Transmit --
Full CSI: Water-Filling
0
10
15
20
Two Users
50
Transmit Full CSI: Block Diagonal
-Uninformed
•25
-
rd
0
A. Lee Swindlehurst
5
10 SNR (dB)
Electrical & Computer Engineering
-
15
-
20
Brigham Young University
Mathematical Notation Assume P users, each with m antennas, d data streams transmitted to each user (m _ d). Signal received by user 1: P
x -HTsl +LHkTks + n, k=2 mxNsub-channel matrix
N x d transmit beamformer
dataother intended for users
Assume each user's data stream is scaled so that
:
X=
s, = 1.
H1 [T
_
System equation:
Noise and unmodeled interference
•
1
r
... T_]
s1 +
-
+
xI
-XPTotal # of receive antennas: M= mP
A. Lee Swindlehurst
Hp
Sp-
np-
HTs+n
Electrical & Computer Engineering
Brigham Young University
Presentation Outline
Algorithms for Single-Antenna Users -
Channel Inversion Regularized Channel Inversion Vector Modulo Pre-Coding Interference-Balancing Methods for Power Control
A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
Special Case: Single Antenna Users Assume each user has m
=
1 antenna, and N _ M = P.
Channel Inversion Transmitter "pre-inverts" the channel. Transmit beamformers are columns of the pseudo-inverse:
Tc, =yH*(HH*)I To maintain fixed transmit power p, must scale signal:
Y=
H
s*(HH*) 1 S Ideally, channel inversion eliminates all inter-user interference:
x = HTcjs+n =yHH*(HH*)1S+n
ys+n
"*problems
obviously arise when channel is (nearly) rank deficient "*problem isn't noise amplification, instead it is signal attenuation due to power constraints "* what about in the "ideal" case, e.g., with independent Rayleigh fading? A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
Capacity of Channel Inversion Assume elements of H are independent, Rayleigh w/ unit variance, N = P = M and s - Q (0, ).
A bad sigqn:XA = s*(HH*-lS
is distributed as* p(X)= N and
E(X)=-o
!!
Capacity** for large N:
lim Cc=-
N-->oo
:
(Tp2
Iog 2 (e)
No capacity growth with # of users/antennas
*
Hochwald & Vishwanath, Proc. 4 0th Allerton Conf., October, 2002
**
Peel, Hochwald & Swindlehurst, Proc. 4lstAIlerton Conf., October 2003
A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
Regularized Channel Inversion A simple fix is to regularize the inverse: (To maximize SINR @ the receivers, choose*
N2
Trci
y H*(HH*+
N1yp *Peel, Hochwald, Swindlehurst: "Avector perturbation technique for near-capacity multi-antenna, multi-user submitted to IEEE Trans. Comm.) communication, Part I",
Linear growth with N is recovered:
N=1 0
p=lOdB 30
Sum Capacity
Sum CapacityI
70
Inversion ...................... , Channel inversion
Regularized Inversion Channel inversion
-Regularized
25
-.-
-
..
• 50
i
00 . •"i
.------.... ...... .....:. .......... ........ 1..................
Ni
ci, 40"
.........
30 ..
0
C). 20-
.....u .
.. . ....... .. .. .... .. .. .. .. .. .. .. .. S..
2
6
8
-
N A. Lee Swindlehurst
.,.* ........ ...
.. ........... .....................
-10
Electrical & Computer Engineering
-5
..
I-
10 . ...... ... .. .......... , • . .
10
.........................
.........
.............
0
.. ............ ....... ........ '- .............
. .... :. ...... ....• ... .......... . . . . . . . . . . . .
5
p (dB)
10
15
20
25
Brigham Young University
What's the real issue? "* The
regularization "quick fix" helps, but there is still a significant performance gap
"* Ultimately,
problems arise when s happens to lie in the direction of the "largest" singular vector of (HH)- 1
"* IDEA:
could we perturb s to eliminate this possibility, i.e., min (s +s)*(HH*)l(s + s) Sp
and still decode s at the receivers, without knowledge of Sp ? "* Yes,
using a little "trick": modulo pre-coding* *M. Tomlinson, "New automatic equaliser employing modulo arithmetic," Elec. Letters, March, 1971 H. Harashima & H. Miyakawa, "Matched transmission technique for channels with intersymbol interference," IEEE Trans. Comm., August, 1972
A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
Vector Modulo Pre-coding Use a perturbation of the form
SP =,Cc =t(a+ jb) where -cis real and a,b are vectors of integers. For channel inversion: x = HTc,(S + s±)+ n = YHH*(HH*)'(s + s±)+n =
ys + y-c + n
Assuming receivers know y, perfect decoding is possible w/out noise using the mod-function: "O"=
At receiver k,
(mod-function applied to
real & imaginary parts separately)
fj- xk = f,(sk +u ak +ibk) = s, -cmust be chosen large enough to avoid mod-function ambiguities: A. Lee Swindlehurst
dmax =constellation "size"
u= 2(dmax + A/2),
Electrical & Computer Engineering
A=constellation "spacing"
Brigham Young University
Vector Modulo Pre-coding (cont.) * Choose c to solve integer-lattice least-squares problem (sphere encoding): min (s+Ta+Tjb)*(HH*)-'(s+Ta+Tjb) aeL/bGL
* : --> ,oo = standard ch. inversion
10
Tx antennas, 10 users (1Rx each), 4QPSK.
............. 100
"* can
regularize this approach too
......
- - - - - -. .---. . ---. . . .-
--. ----
--. -- -. .. ----. --. ---. -. -------__--
* provides linear capacity growth w/N
... ....
.. . . . . . . . .. . . .
........
-i
i
i-
-
-
-
-
- ,,
10 -----------------------------------..........\ , N
4--
get closer, one must resort to
o
"dirty paper" techniques, which lead
A. Lee Swindlehurst
. .I,.
.... - , .... .... ... :: : ...--:: --.. ... ---------------......... .. 0~~~..
'-
to much more complex receivers
' '
°--
with outer turbo code, this approach
"* to
. .. . . '
101 n
gets within 3-4 dB of capacity
- --- --- -....
. . . . . . . . . . . . . . . ------:.--. .--. -. .-.:. .----
•
--.-------.---.---.---.---.---.-------.---...--..... -
-
10
-3
-2
Channel Inversion ------------Successive Inversion-------a R
_._
vector Perturbation V, Regularized Pert. 0 2 4
Electrical & Computer Engineering
I
6 p (dB)
8
------10 12 14
Brigham Young University
-
An Alternative Based on Power Control (Interference Balancing) With a single antenna at each receiver, the columns of T= [ t, ... tp ] represent the transmit beamformers for each user, and each channel is a row vector Hi = hi*
xi = htisi +I hktkSk + n k~i
~
interference
from signals
sent to other users
In the power controlformulation, minimize total transmitted power subject to a certain QoS constraint, usually measured by SINR:
P
*
minLtkk,= tk k=1
s.t.
t*h.h*ht Z tkkkt hht+ l~
2
>- P ,
i
P
~
k#i
Can be posed as a convex, semi-definite optimization & efficiently solved using (for example) interior-point methods. See Rashid-Farrokhi, Liu, Tassiulas, "Transmit beamforming and power control for cellular wireless systems," IEEE J. Sel. Areas in Comm., October, 1998 Bengtsson & Ottersten "Optimal and sub-optimal transmit beamforming," Handbook of Antennas in Wireless Comm., CRC Press, August, 2001
A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
Presentation Outline
Algorithms for Multiple-Antenna Users - Joint Transmit/Receive Beamforming - One vs. Multiple Sub-Channels per User
A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
Multiple-Antenna Users Recall system equation for P users with m antennas each:
H-
X=
:
1
... T -]s,
-
:
+
= HTs+n "* If
N > M = mP, we could diagonalize HT, but such an approach would be sub-optimal, since it would ignore spatial discrimination at the receivers
"* From
the standpoint of capacity, it is better to b/ock-diagonalize HT, and then use water-filling to allocate power to all available spatial channels* *Q. Spencer and A. Swindlehurst, "Zero-forcing methods for downlink spatial multiplexing in multi-user MIMO channels," IEEE Trans. Sig. Proc., February, 2004
* Disadvantage is that capacity may be achieved at the expense of weak users; e.g., 1-2 strong users may take a dominant share of available power A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
Multiple-Antenna User Example 4 xmit antennas, 4 total rcv antennas, Rayleigh fading channel Capacity CCDFS form7 - 4 at 10 dB SNR ... Inversion
Block Diag. Blind Tx
S0.9" Ica 0.8
L
,0. A
x
060.6 {1,1,1,1} x'4'
-
0.5 0.4-
L
{2,2} x 4
2 0.2 00
1 User
-
2
4
6
8
10
12
14
16
Capacity (bits/use) A. Lee Swindlehurst
Electrical & Computer Engineering
Brigham Young University
Joint Tx-Rx Design Problem: 1 Sub-Channel per User It is unlikely that N > M = mP. Can multiplex up to N data streams with N antennas. Assume 1 stream is sent to each of P _
