Interference mitigation via scheduling for the MIMO broadcast channel with limited feedback.

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Interference Mitigation via Scheduling for the
MIMO Broadcast Channel with Limited Feedback
Tae Hyun Kim
The Department of Electrical and Computer Engineering
The University of Illinois at Urbana-Champaign, IL 61801-2302
tkim56@illinois.edu
Sunghyun Choi
School of Electrical Engineering and INMC
Seoul National University, Seoul 151-744, Korea
schoi@snu.ac.kr
Abstract—The multiple-input multiple-output (MIMO) broad-
cast channel uses multiple antennas at the transmitter to con-
currently deliver information to multiple receivers. Despite the
theoretical capacity gains over single user MIMO, the actual
throughput of the system with limited feedback on channel state
information suffers due to interspatial stream interference. On
top of the orthogonal beamforming with limited feedback [1],
this paper introduces two novel stream schedulers to mitigate the
interference. By the proposed adaptive spatial stream schedulers,
an adaptive number of spatial streams are scheduled and thus,
reduces the interference, which also achieves power gains by
concentrating power into fewer streams. The two proposed
algorithms are evaluated and shown to effectively mitigate the
interference.
I. INTRODUCTION
The multiple-input multiple-output (MIMO) broadcast
channel uses multiple antennas at the transmitter to support
concurrent transmission to different receivers. The receivers
in a cellular network with a MIMO broadcast-capable base
station are spatially separated and served at the same time. This
advantage of concurrent services has attracted much research
attention with several approaches that have been proposed thus
far (see [2] and the references therein).
In the MIMO broadcast channel transmit signals are typi-
cally precoded with an aid of channel state information (CSI)
at the transmitter so as to minimize the interspatial stream
interference.1For this purpose, CSI is quantized and fed back
from the receivers to the transmitter by using a finite size
codebook, which eventually introduces the interference in the
system due to the imperfection of the quantized CSI. This pa-
per focuses on the mitigation of such interference in one type
of MIMO broadcast systems, namely, orthogonal beamforming
with limited feedback (OBLF) [1], which is also known as per
user unitary rate control in standard associations [3].
The countermeasures to mitigate this interference are sum-
marized as follows: enlarge the codebook, perform joint pro-
cessing at the transmitter and receivers using multiple receive
antennas [4], schedule users with small mutual interference
This material is based in part upon work supported by National Sci-
ence Foundation award 0831670, the DARPA IT-MANET program, Grant
W911NF-07-1-0028 and the IT R&D program of MKE/IITA [2009-F-044-01,
Development of cooperative operation profiles in multicell wireless systems].
1In this paper, we refer to interspatial stream interference as the interference.
It is also called by other names such as multiuser interference and co-channel
interference.
and schedule a smaller number of concurrent streams and
concentrate transmit power into them. Our interests in this
work is the last one; interference mitigation via less streams
can be achieved by a scheduler at the transmitter that can find a
subset of streams that yield the largest sum throughput. Many
schedulers have been proposed for systems with perfect CSI
at the transmitter (CSIT) and recently for quantized CSIT [1],
[5], [6], but scheduling for mitigating the interference has not
been considered so far.
Based on OBLF, this paper proposes two spatial stream
schedulers that choose a subset of streams to mitigate the inter-
ference in the MIMO broadcast channel with limited feedback.
The proposed spatial stream schedulers select an adaptive
number of spatial streams to maximize sum throughput by
controlling the amount of the interference; they are designed
to find a subset of spatial streams with small interference
and concentrate transmit power to the chosen, thus achieving
power gain. Their computational performance is also analyzed.
Our contributions are summarized as follows. First, adaptive
schedulers for practical OBLF systems are proposed and
their computational performance is analyzed. Second, we
have shown that adaptive scheduling is essential to have at
least logarithmic throughput increase as signal-to-noise-ratio
(SNR) grows, thanks to the power gain as well as reduced
the interference. Lastly, one of our adaptive schedulers is
designed to rely on a high feedback rate to make instantaneous
and adaptive decisions at the transmitter, showing that such
decisions are highly expensive in terms of system resources.
In what follows, Section II gives the considered system
model. Adaptive schedulers and their computational perfor-
mance are introduced in Section III. Section IV gives analysis
on the simulations results. Finally, Section V concludes this
paper.
II. SYSTEM MODEL
A. Signal Model
We consider a MIMO broadcast system with one transmitter
and multiple receivers that have a single antenna. The number
of antennas at the base station (transmitter) is denoted by M.
Assuming narrow band signaling and flat fading, the multi-
path fading channel from the base station to a certain receiver
is described by h ∈ CM×1. The discrete time received signal

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delivered by beamformer k is2
?
yk=
Es
|K|hH?
wkxk
? ?? ?
information
+
?
interference
k?∈K\{k}
?
wk?xk?
???
?
+ n,
(1)
where n ∈ C is a zero-mean additive complex Gaussian
noise with variance N0. Esis the mean symbol energy of the
received signal. After scheduling M or less spatial streams by
the transmitter, wk∈ CM×1are chosen as transmit beamform-
ers. K is a set of beamformer indices that create spatial streams
to the receivers. The term xkrepresents the transmitted symbol
associated with beamformer wk. We assume that mean energy
of the symbol xkis statistically normalized (E{|xk|2} = 1).
From (1), the SINR for a spatial stream conveyed on
beamformer wkis
Es|hHwk|2
Es
φk=
?
k?∈K\{k}|hHwk?|2+ N0· |K|,
(2)
which depends upon our choices of the set of beamformers
(K) and transmit beamformers themselves (wk).
B. Orthogonal Beamforming with Limited Feedback (OBLF)
Our proposals against the interference are on top of OBLF.
The needed details of OBLF for exposition are given in this
section.
Let us decompose CSI into channel direction information
(CDI) and magnitude. Limited feedback is an index feedback
of the codeword which is desirable for the transmitter to know
as CDI. In OBLF, M orthogonal codewords form a subset of
the codebook, which will be referred to as a (orthogonal) code
set. Multiple code sets constitute the entire codebook, which
is defined as
F := {wk: k ∈ ∪V
where wkis the k-th codeword and Vi= {(i − 1)M + 1,(i −
1)M + 2,··· ,i · M} for i = 1,··· ,V denotes the set of
codeword indices in i-th orthogonal code set. The codewords
wk,k ∈ Viare thus orthogonal to each other in the same set.
A code set index for a certain codeword is easily obtained by
i = ?(k − 1)/M? + 1.
In OBLF, the codeword for feedback is found by the
criterion
k∗= argmax
i=1Vi} = {wk: k = 1,··· ,M · V }, (3)
k
hHwk
?h?
,
(4)
which is to select the codeword whose CDI has the least inner
angle to the channel direction.
As implicitly shown in (3), in OBLF the codewords in a
code set are used for transmit beamforming as beam weights
for M spatial streams. If a receiver reports that its CDI is
the k-th codeword, the other codewords in the code set of k-
th’s are close to orthogonal to the channel. Therefore, if the
reported channel direction is indeed identical to the k-th’s, the
2We use superscript T for transpose, superscript H for conjugate transpose,
and superscript † for the matrix pseudo inverse. The expression ? · ? denotes
the usual Euclidean vector norm.
spatial stream destined to that direction will not interfere with
other streams.
Provided that the transmitter chooses code set Vifor beam-
forming, a receiver can decompose an incoming signal to
individual SNRs corresponding to the beamformers in Viby
applying different transmit beamformers. Define individual
SNR by γn:=Es
of the considered signal. The transmitter can collect this set of
information from the receivers and reconstruct SINRs of any
combinations of spatial streams by
N0|hHwn|2where n is the beamformer index
φTX
k
=
γk
?
n∈K\{k}γn+ |K|,
(5)
which may be slightly different from φk at the receiver due
to quantization for feedback. If such a set of individual SNR
information is not necessary, the receivers may feedback SINR
itself only. For later discussion, the numerator of (5), γk,
is named as an individual information SNR while individual
SNRs refer to all γn,n ∈ K.
III. SPATIAL STREAM SCHEDULERS
In this section, previous schedulers for OBLF are explained,
and then, schedulers that choose an adaptive number of spatial
streams are proposed to improve sum throughput.
A. Previous Schedulers for OBLF
The purpose of spatial stream scheduling is to maximize
sum throughput by selecting a subset of available spatial
streams, which is formulated as
K?= argmax
K⊂Vi:1≤i≤V
max
1≤jk≤Jk:k∈K
?
k∈K
log
?
1 + φTX
k,jk(K)
?
, (6)
where φTX
number of spatial streams using codeword k, which have been
reported by receivers to the transmitter. An index jk is used
to indicate a certain spatial stream out of Jk’s.
The term jkis also used to refer to the individual informa-
tion SNR of a certain spatial stream; the term γk,jkdenotes
γkfor the jk-th spatial stream among all streams that use the
beamformer k, i.e., the numerator of φTX
of generality, it is assumed for the following discussion that
γk,j1≥ γk,j2for all 0 < j1< j2≤ Jk,∀k, which is done by
sorting spatial streams in a descending order with respect to
their individual information SNRs γk,jk.
Let us denote the number of receivers by Z. The brute-force
search for an optimal K?takes O(M2Z) time, assuming M <
Z. Therefore, spatial streams that maximize sum throughput
are found only when Z is small or moderate. Let us call this
brute-force search as OptSch for discussion.
For OBLF, it is introduced in [1] a user scheduler that
always chooses M streams, which is described in Algorithm 1.
Let us name this scheduler as M-FixSch. The running time of
this heuristic is easily proven as follows.
k
is from (5) or explicit SINR feedback. Jkis the
k,jkin (5). Without loss
Proposition 1. Algorithm 1, which is named as M-FixSch, has
a complexity of O(Z + V · M).

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Algorithm 1 Fixed M stream scheduler (M-FixSch) [1]
1: Initialization:
2: j∗
stream for that beamformer
3:
4: Execution:
5: i?= argmax1≤i≤V
6: K?= Vi?
7: return K?and j∗
k= argmax0≤j≤JkφTX
k,j for k ∈ F
? j = 0 means no
?
k∈Vilog?1 + φTX
k,j∗
k
?
k
? j∗
kare for Algorithm 2
Proof: It first searches the stream with the largest SINR
for each beamformer
code set that yields the largest sum throughput?O(MV )?on
The computational complexity of M-FixSch is thus ex-
ponentially less than OptSch. Moreover, it has a desirable
feature that it actually achieves the optimality in terms of sum
throughput if |K?| = M.
However, if sum throughput is mainly limited by nonzero
interference, it is possible that |K?| < M. In this case, it
is probable that a fewer scheduled streams would experience
less sum interference and per stream transmit power would
be larger, which together results in higher SINRs of served
streams.
?O(Z)?
on Line 2 and then finds the
Line 5.
Algorithm 2 Adaptive scheduler with individual SNRs (Adp-
Sch)
1: Initialization:
2: R?= 0, K?= ∅
3: for k ∈ F do jk = 1 if ∃ stream with beamformer k and
otherwise jk= 0
4: find j∗
5:
6: Execution:
7: for all code set i = 1,··· ,V do
8:
K = Vi\ {k : jk= 0}
9:
while jk?= j∗
10:
R∗= 0, K∗= ∅
11:
while |K| ?= ∅ do
12:
13:
if R > R∗then
14:
R∗= R, K∗= K
15:
end if
16:
17:
end while
18:
if R∗> R?then
19:
R?= R∗, K?= K∗
20:
end if
21:
K = Vi\ {k : jk= 0}
22:
increase jk: k = argmaxk?∈K:jk??=j∗
23:
end while
24: end for
25: return K?
kby M-FixSch
kfor k ∈ Vi do ? M stream-selection begins
? Stream deselection begins
R =?
remove k?= argmaxk∈K?
k∈Klog?1 + φTX
k,jk(K)?
n∈K\{k}γn from K
k?
?
n∈K\{k?}γn
B. Adaptive Scheduler with Individual SNRs (AdpSch)
Observe that the sum throughput in (6) is mainly contributed
by two factors: (i) the SINRs of the selected spatial streams
and (ii) the number of the streams. To obtain the largest SINR,
a stream has to have strong signal power γkas well as small
sum interference?
both factors, AdpSch, which is described in Algorithm 2,
performs a two step procedure: for each code set, it does (i)
selection of M stream-combination and (ii) subset selection
of that stream combination. The two step procedure is iterated
until a search is performed on a sufficiently large space.
1) M stream-selection: The first step is to select M or less
spatial streams for a given set if the number of streams in the
set is less than M. Without loss of generality, we assume there
are always M streams in the rest of the discussion.
M stream-selection is from the streams with the largest indi-
vidual SNRs to those with smaller ones. For each beamformer
k in a code set, the spatial stream with the largest individual
information SNR γk is found, forming M stream-selection.
After performing stream subset selection, the stream of the
beamformer with the largest sum interference is replaced with
the stream of the next largest information SNR, which has the
same beamformer, and so forth.
Then the central question in M stream-selection is when the
algorithm should cease evolving to a different set of spatial
streams. The scheduler stops evolution of stream-selection if
it has considered all streams of which individual information
SNRs are larger than or equal to those of streams selected by
M-FixSch. This rule is based on the fact that the streams of
which jkis less than or equal to j∗
and larger sum interference than those that would have been
scheduled by M-FixSch.
2) Stream subset selection: The second step is to choose
a subset of the M streams given by the M stream-selection.
Instead of computing all possible subsets, the one with the
largest sum interference is iteratively deselected from the
streams, which is done on Line 16. The subset that yields
the largest sum throughput is chosen after all streams are
deselected.
In sum, the algorithm reveals a low complexity which can
be proven as follows:
n∈K\{k}γnwhere K is a set of streams.
Instead of seeking a set of streams simultaneously satisfying
khave stronger signal power
Proposition 2. AdpSch described in Algorithm 2 has a com-
plexity of O(Z · max{M2,logZ}).
Proof: The adaptive scheduler performs as follows: M
stream-selection (Sc), stream deselection (Dc) for each code
set (V sets) and summation to obtain sum throughput (M).
Let us denote the complexity as (Sc· V · Dc· M).
First of all, all spatial streams should be sorted in a
descending order and classified to corresponding codewords,
which takes Z logZ time. For M stream-selection, it is not
difficult to observe that Sc· V depends on the distribution
of spatial streams over a codebook and is maximized when
all spatial streams are mapped to the codewords in a single
code set. This can be proven by the fact that the first M
stream-selection on a code set takes M while subsequent
selection takes only one spatial stream for replacement. Thus,
Sc·V ≤ Z −M +1. In addition, Dc≤ M where the equality
holds when all codewords in a code set have at least one spatial
stream.

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In sum, the complexity is
Sc· V · Dc· M + Z logZ
≤ (Z − M + 1)Dc· M + Z logZ
≤ (Z − M + 1)M2+ Z logZ
≤ Z · M2+ Z log = O?Z · max{M2,logZ}?,
which completes the proof.
In terms of the number of receivers, the complexity of
the proposed adaptive scheduler is exponentially less than
O?M2Z?required by the exhaustive search, and comparative
M is usually a small number). Specifically, for M = 2,
the complexity is O(Z · logZ) unless Z ≤ 16. In practical
scenarios where M > 2 and Z ≤ 29, AdpSch runs in
O(Z ·M2), which is linear to the number of receivers. It also
turns out by Proposition 2 that AdpSch has a desirable feature
that the complexity stays the same even with the increase of
the codebook size; in fact, it decreases as V increases due to
the first M stream-selection for each set, which always takes
M streams if any.
(7)
to that of M-FixSch, which is O(Z + V · M) (note that
Algorithm 3 Any fixed m stream scheduler (Any-FixSch)
1: Initialization:
2: j∗
3: Wi = predetermined subset of beamformers such that Wi ⊂ Vi
and |Wi| = m
4:
5: Execution:
6: i?= argmax1≤i≤V
over a subset Wi
7: K?= Wi?
8: return K?
k= argmax0≤j≤JkφTX
k,j for k ∈ F
?
k∈Wilog?1 + φTX
k,j∗
k
?
? Summation is
C. Any-FixSch: Generalized M-FixSch
The adaptive scheduler in the previous subsection demon-
strates that even suboptimal adaptive scheduler requires sub-
stantial knowledge on γnto make decisions on the number of
streams and choices on which streams will be served. This
tells us that as long as such decisions are instantaneously
and adaptively made at the transmitter, the information should
be fed back to the transmitter, which is a prohibitively large
amount.
Inspired by this fact and simplicity of M-FixSch, Any-
FixSch is proposed, which generalizes M-FixSch to arbitrary
m streams cases and achieves part of sum throughput gains
by adaptive scheduling. Since Any-FixSch is nothing but to
schedule m instead of M streams, the complexity decreases
from that of M-FixSch, O(Z +V ·M) to O(Z +V ·m). It is
described in Algorithm 3.
By comparison between Algorithms 1 and 3, two differences
are observed. On Line 3 of Algorithm 3, predetermined subsets
Wi⊂ Viare used as a filter to strain off M −m beamformers.
Another is the summation on Line 6. It sums up throughput
of a subset of spatial streams in a code set.
How can Any-FixSch avoid making the expensive adaptive
decisions at the transmitter? Any-FixSch presumes that a
TABLE I
SUMMARY OF SPATIAL STREAM SCHEDULERS FOR OBLF
Running time
O(M · 2Z)
O(Z · max{M2,logZ})
O(Z + V · m)
O(Z + V · M)
Feedback rate
OptSch
AdpSch
Any-FixSch
M-FixSch
B · M + log2
?V · M?
B + log2
?V · M?
certain number of streams are sufficient to be scheduled,
and the number is m. This requires only the same amount
of feedback as M-FixSch and other user schedulers in the
literature do [5], [6].
As a result, the performance of Any-FixSch heavily depends
on the choice of m. Ideally, it should be m = |K?| for every
instance, which is possible only when the transmitter is fed
more information on receiver channels. Instead, it is assumed
that the transmitter in our system keeps tracking E?|K?|?by
FixSch is not a stand-alone scheduler. It is also considerable to
track |K?| by increase and decrease of m over time according
to resulting sum throughput (see similar type of tracking for a
MAC protocol rate adaptation in [7]), which is left as future
work.
In Any-FixSch, the m codewords that will be used at the
transmitter are predetermined by Wion Line 3, which does not
necessarily yield the largest SINR. However, fixing them does
not require a receiver to send back either information on which
codewords are used for the SINR or multiple SINR values
for different combinations of codewords in use. As the SINR
reconstruction by (5) in AdpSch, this adaptation also needs a
substantial amount of feedback, which should be avoided to
keep the feedback rate low.
intermittently running either OptSch or AdpSch. Thus, Any-
D. Feedback Rate for Spatial Stream Schedulers
In Table I, the computational performance of all schedulers
discussed in this section are summarized. In addition, feedback
rates for the schedulers are also listed. When OptSch and
AdpSch are used, the transmitter needs to have information
on SINRs of all possible combinations of spatial streams.
The amount of this information for feedback is substantially
reduced by SINR decomposition in (5); the rate drops from
B · (2M− 1) to B · M bits where B is the number of bits to
represent a (real) scalar.
Compared to high feedback rates for OptSch and AdpSch,
M-FixSch and Any-FixSch have the same, low feedback rate;
the codeword index is sent back by log(V ·M) bits. Additional
B bits are for the SINR.
If Any-FixSch is used in a combination with either OptSch
or AdpSch, the frequency of running a high feedback rate
scheduler determines the feedback rate of Any-FixSch. In
Table I, it is basically assumed that the frequency is small
enough to be ignored for the overall feedback rate.
IV. SIMULATION RESULTS
We carry out Monte-Carlo simulations on the sum through-
put. A Rayleigh fading channel is assumed. The scalar feed-

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0204060 80100120140
0
0.5
1
1.5
2
2.5
3
M=8, Total SNR=10 dB
Number of Receivers
Throughput Gain over OBLF with M−FixSch
V=64, OptSch
V=64, AdpSch−Aggr
V=64, AdpSch
V=64, Any−FixSch
V=4, OptSch
V=4, AdpSch−Aggr
V=4, AdpSch
V=4, Any−FixSch
Fig. 1.
where M = 8 and SNR=10 dB.
Throughput gain over OBLF with respect to the number of receivers
051015 202530
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
M=8, Total SNR=10 dB
SNR (dB)
Throughput Gain over OBLF with M−FixSch
V=64, OptSch
V=64, AdpSch−Aggr
V=64, AdpSch
V=64, Any−FixSch
V=4, OptSch
V=4, AdpSch−Aggr
V=4, AdpSch
V=4, Any−FixSch
Fig. 2.
and Z = 8.
Throughput gain over OBLF with respect to SNR where M = 8
back corresponding to γn is not quantized. Any-FixSch is
assumed to obtain E?|K?|?
is the throughput of a new scheme and R is that of OBLF
with M-FixSch.
Figs. 1 and 2 show the sum throughput gain by using
spatial stream schedulers. One major trend in Fig. 1 is that
the gain exponentially diminishes as the number of receivers
increases. With a large number of receivers, the streams chosen
by OBLF with FixSch do not interfere with each other much,
which results in |K?| ≈ M. Interestingly, the gain increases as
feedback rates increase from V = 4 to V = 64; recall that the
interference mitigation by spatial stream schedulers are from
the subtraction of highly interfering spatial streams, which is
done independent of a feedback rate. The logarithmic gain
observed in Fig. 2 over SNR shows that the system comes
out of the interference-limited regime thanks to the adaptive
schedulers.
by infrequently running OptSch.
Throughput gain is computed by (Rnew−R)/R where Rnew
When the number of receivers is small, the gain by AdpSch
depicted in Fig. 2 is close to the exhaustive search. This
gap becomes larger as the number of receivers increases as
shown in Fig. 1, which is the result of the design of AdpSch;
the diversity of spatial streams makes the ignored cases in
two steps of AdpSch more probable, thus worsening the sub-
optimality.
In designing AdpSch, it has been conjectured that the
evolution of M stream-selection beyond j∗
helpful. To verify this, the performance of AdpSch with the
evolution to all spatial streams in the M stream-combination
step is depicted by the name of AdpSch-Aggr. It is noticeable
that the conjecture is indeed true in all cases in Figs. 1 and 2.
While AdpSch requires a high feedback rate as discussed in
Section III-D, Any-FixSch, which requires the same feedback
rate as M-FixSch performs close to OptSch and AdpSch in
many cases, which are observed in Fig. 2. Moreover, in Fig. 1,
it is seen that its performance even exceeds that of OBLF-
AdpSch when multiuser diversity is largely induced and the
feedback rate is high (V = 64). This is, however, more due to
the worsened sub-optimality of AdpSch with large multiuser
diversity. Nevertheless, in a practical system, it is expected
to have feedback from only a subset of receivers, thus a few
receivers regime being more important.
kwould be rarely
V. CONCLUSION
In this paper, we observe the significance of the interfer-
ence in a limited-feedback based MIMO broadcast channel,
proposing one way of countermeasure, adaptive scheduling.
It chooses only a subset of all streams that it may schedule
together, considering mutual interference caused by limited
feedback. AdpSch exploits a large set of information to make
instantaneous and adaptive decisions at the transmitter and
Any-FixSch is designed to avoid such decision makings. Both
show a linear complexity to the number of receivers. As
future work, the tradeoff between the overall feedback rate and
frequency of running either OptSch or AdpSch for estimating
E?|K?|?will be studied.
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