Antenna Combining for the MIMO Downlink Channel
Antenna Combining for the MIMO Downlink
Channel
Nihar Jindal
University of Minnesota, Department of ECE
Minneapolis, MN 55455, USA
Email: nihar@umn.edu
Abstract
A multiple antenna downlink channel where limited channel feedback is available to the transmitter is considered.
In a vector downlink channel (single antenna at each receiver), the transmit antenna array can be used to transmit
separate data streams to multiple receivers only if the transmitter has very accurate channel knowledge, i.e., if there
is high-rate channel feedback from each receiver. In this work it is shown that channel feedback requirements can
be significantly reduced if each receiver has a small number of antennas and appropriately combines its antenna
outputs. A combining method that minimizes channel quantization error at each receiver, and thereby minimizes
multi-user interference, is proposed and analyzed. This technique is shown to outperform traditional techniques
such as maximum-ratio combining because minimization of interference power is more critical than maximization
of signal power in the multiple antenna downlink. Analysis is provided to quantify the feedback savings, and the
technique is seen to work well with user selection and is also robust to receiver estimation error.
I. INTRODUCTION
Multi-user MIMO techniques such as zero-forcing beamforming allow for simultaneous transmission of multiple
data streams even when each receiver (mobile) has only a single antenna, but very accurate channel state information
(CSI) is generally required at the transmitter in order to utilize such techniques. In the practically motivated finite
rate feedback model, each mobile feeds back a finite number of bits describing its channel realization at the
beginning of each block or frame. In the vector downlink channel (multiple transmit antennas, single antenna at
each receiver), the feedback bits are determined by quantizing the channel vector to one of 2B quantization vectors.
While a relatively small number of feedback bits suffice to obtain near-perfect CSIT performance in a point-to-
point vector/MISO (multiple-input, single-output) channel [1], considerably more feedback is required in a vector
downlink channel. If zero-forcing beamforming (ZFBF) is used, the feedback rate must be scaled with the number
of transmit antennas as well as SNR in order to achieve rates close to perfect CSIT systems [2]. In such a system
the transmitter emits multiple beams and uses its channel knowledge to select beamforming vectors such that nulls
are created at certain users. Inaccurate CSI leads to inaccurate nulling and thus translates directly into multi-user
interference and reduced SINR/throughput.
In this paper we consider the MIMO downlink channel, in which the transmitter and each mobile have multiple
antennas (M transmit antennas, N antennas per mobile), in the same limited feedback setting. We propose a receive
antenna combining technique, dubbed quantization-based combining (QBC), that converts the MIMO downlink into
a vector downlink in such a way that the system is able to operate with reduced channel feedback. Each mobile
linearly combines its N antenna outputs and thereby creates a single antenna channel. The resulting vector channel
is quantized and fed back, and transmission is then performed as in a normal vector downlink channel.
With QBC the combiner weights are chosen on the basis of both the channel and the vector quantization codebook
to produce the effective single antenna channel that can be quantized most accurately. On the other hand, traditional
combining techniques such as the maximum-ratio based technique that is optimal for point-to-point MIMO channels
with limited channel feedback [3] or direct quantization of the maximum eigenmode are aimed towards maximization
of received signal power but generally do not minimize channel quantization error. Since channel quantization error is
so critical in the MIMO downlink channel, quantization-based combining leads to better performance by minimizing
quantization error (i.e., interference power) possibly at the expense of channel (i.e., signal) power.
One way to view the advantage of QBC is through its reduced feedback requirements relative to the vector
downlink channel. In [2] it is shown that scaling (per mobile) feedback as B = M−1
PdB , where P represents the
http://arXiv.org/abs/0704.1308v2
SNR, suffices to maintain a maximum gap of 3 dB (equivalent to 1 bps/Hz per mobile) between perfect CSIT and
limited feedback performance in a vector downlink channel employing ZFBF. With QBC, our analysis shows that
the same throughput (3 dB away from a vector downlink with perfect CSIT) can be achieved if feedback is scaled
at the slower rate of B ≈ M−N
PdB . In other words, QBC allows a MIMO downlink to mimic vector downlink
performance with reduced channel feedback.
Alternatively, QBC can be thought of as an effective method to utilize multiple receive antennas in a downlink
channel in the presence of limited channel feedback. Although it is possible to send multiple streams to each mobile
if receive combining is not performed, this requires even more feedback from each mobile than a single-stream
approach. In addition, QBC has the advantage that the transmitter need not be aware of the number of receive
antennas being used.
The remainder of this paper is organized as follows: In Section II we introduce the system model and some
preliminaries. In Section III we describe a simple antenna selection method that leads directly into Section IV
where the much more powerful quantization-based combining technique is described in detail. In Section V we
analyze the throughput and feedback requirements of QBC. In Section VI we compare QBC to alternative MIMO
downlink techniques, and finally we conclude in Section VII.
II. SYSTEM MODEL AND PRELIMINARIES
We consider a K mobile (receiver) downlink channel in which the transmitter (access point) has M antennas,
and each of the mobiles has N antennas. The received signal at the i-th antenna is given by:
yi = h
i x + ni, i = 1, . . . , NK (1)
where h1,h2, . . . ,hKN are the channel vectors (with hi ∈ CM×1) describing the KN receive antennas, x ∈ CM×1
is the transmitted vector, and n1, . . . ,nNK are independent complex Gaussian noise terms with unit variance. The
k-th mobile has access to y(k−1)N+1, . . . , yNk. The input must satisfy a power constraint of P , i.e. E[||x||2] ≤ P .
We use Hk to denote the concatenation of the k-th mobile’s channels, i.e. Hk = [h(k−1)N+1 · · ·hNk]. We consider
a block fading channel with iid Rayleigh fading from block to block, i.e., the channel coefficients are iid complex
Gaussian with unit variance. Each of the mobiles is assumed to have perfect knowledge of its own channel Hi,
although we analyze the effect of relaxing this assumption in Section V-C. In this work we study only the ergodic
capacity, or the long-term average throughput. Furthermore, we only consider systems for which N < M because
QBC is not very useful if N ≥ M ; this point is briefly discussed in Section IV.
A. Finite Rate Feedback Model
In the finite rate feedback model, each mobile quantizes its channel to B bits and feeds back the bits perfectly
and instantaneously to the transmitter at the beginning of each block [3][4]. Vector quantization is performed using
a codebook C of 2B M -dimensional unit norm vectors C , {w1, . . . ,w2B}, and each mobile quantizes its channel
to the quantization vector that forms the minimum angle to it [3] [4]:
ĥk = arg min
w=w1,...,w2B
sin2 (∠(hk,w)) . (2)
For analytical tractability, we study systems using random vector quantization (RVQ) in which each of the 2B
quantization vectors is independently chosen from the isotropic distribution on the M -dimensional unit sphere and
where each mobile uses an independently generated codebook [5]. We analyze performance averaged over random
codebooks; similar to Shannon’s random coding argument, there always exists at least one quantization codebook
that performs as well as the ensemble average.
B. Zero-Forcing Beamforming
After receiving the quantization indices from each of the mobiles, the AP can use zero-forcing beamforming
(ZFBF) to transmit data to up to M users. For simplicity let us consider the N = 1 scenario, where the channels are
the vectors h1, . . . ,hM . When ZFBF is used, the transmitted signal is defined as x =
k=1 xkvk, where each xk
is a scalar (chosen complex Gaussian) intended for the k-th mobile, and vk ∈ CM is the k-th mobile’s BF vector.
If there are M mobiles (randomly selected), the beamforming vectors v1, . . . ,vM are chosen as the normalized
rows of the matrix [ĥ1 · · · ĥM ]−1, i.e., they satisfy ||vk|| = 1 for all k and ĥHk vj = 0 for all j 6= k. If all multi-user
interference is treated as additional noise and equal power loading is used, the resulting SINR at the k-th receiver
is given by:
SINRk =
|hHk vk|2
j 6=k
. (3)
The coefficient that determines the amount of interference received at mobile k from the beam intended for mobile
j, |hH
vj |2, is easily seen to be an increasing function of mobile k’s quantization error.
In the above expression we have assumed that M mobiles are randomly selected for transmission and that equal
power is allocated to each mobile. However, the throughput of zero-forcing based MIMO downlink channels can
be significantly increased by transmitting to an intelligently selected subset of mobiles [6]. In order to maximize
throughput, users with nearly orthogonal channels and with large channel magnitudes are selected, and waterfilling
can be performed across the channels of the selected users. In [7] a low-complexity greedy algorithm that selects
users and performs waterfilling is proposed. If this algorithm is used, a zero-forcing based system can come quite
close to the true sum capacity of the MIMO downlink, even for a moderate number of users.
C. MIMO Downlink with Single Antenna Mobiles
In [2] the vector downlink channel (N = 1) is analyzed assuming that equal power ZFBF is performed without
user selection on the basis of finite rate feedback (with RVQ). The basic result of [2] is that:
RFB(P ) ≥ RCSIT (P ) − log2
1 + P · E
∠(ĥk,hk)
where RFB(P ) and RCSIT (P ) are the ergodic per-user throughput with feedback and with perfect CSIT, respec-
tively, and the quantity E
∠(ĥk,hk)
is the expected quantization error. The expected quantization error
can be accurately upper bounded by 2−
M−1 and therefore the throughput loss due to limited feedback is upper
bounded by log2
1 + P · 2−
, which is an increasing function of the SNR P . If the number of feedback bits
(per mobile) is scaled with P according to:
B = (M − 1) log2 P ≈
M − 1
PdB ,
then the difference between RFB(P ) and RCSIT (P ) is upper bounded by 1 bps/Hz at all SNR’s, or equivalently
the power gap is at most 3 dB. As the remainder of the paper shows, quantization-based combining significantly
reduces the quantization error (more precisely, it increases the exponential rate at which quantization error goes to
zero as B is increased) and therefore decreases the rate at which B must be increased as a function of SNR.
III. ANTENNA SELECTION FOR REDUCED QUANTIZATION ERROR
In this section we describe a simple antenna selection method that reduces channel quantization error. Description
of this technique is primarily included for expository reasons, because the simple concept of antenna selection
naturally extends to the more complex (and powerful) QBC technique. In point-to-point MIMO, antenna selection
corresponds to choosing the receive antenna with the largest channel gain, while in the MIMO downlink the receive
antenna that can be vector quantized with minimal angular error is selected. Mobile 1, which has channel matrix
H1 = [h1 · · ·hN ] and a single quantization codebook consisting of 2B quantization vectors w1, . . . ,w2B , first
individually quantizes each of its N vector channels h1, . . . ,hN
ĝi = arg min
w=w1,...,w2B
sin2 (∠(hi,w)) i = 1, . . . , N, (5)
and then selects the antenna with the minimum quantization error:
j = arg min
i=1,...,N
sin2 (∠(hi, ĝi)) , (6)
and feeds back the quantization index corresponding to ĝj . The mobile uses only antenna j for reception, and thus
the system is effectively transformed into a vector downlink channel.
1,1γγγγ
2,1γγγγ
effy1
1,2γγγγ
2,2γγγγ
effy2
1,3γγγγ
2,3γγγγ
effy3
Fig. 1. Effective Channel for M = K = 3, N = 2 System
Due to the independence of the channel and quantization vectors, choosing the best of N channel quantizations
is statistically equivalent to quantizing a single vector channel using a codebook of size N · 2B . Therefore, antenna
selection effectively increases the quantization codebook size from 2B to N · 2B , and thus the system achieves the
same throughput as a vector downlink with B + log2 N feedback bits. Although not negligible, this advantage is
much smaller than that provided by quantization-based combining.
IV. QUANTIZATION-BASED COMBINING
In this section we describe the quantization-based combining (QBC) technique that reduces channel quantization
error by appropriately combining receive antenna outputs. We consider a linear combiner at each mobile, which
effectively converts each multiple antenna mobile into a single antenna receiver. The combiner structure for a 3
user channel with 3 transmit antennas (M = 3) and 2 antennas per mobile (N = 2) is shown in Fig. 1. Each
mobile linearly combines its N outputs, using appropriately chosen combiner weights, to produce a scalar output
(denoted by yeff
). The effective channel describing the channel from the transmit antenna array to the effective
output of the k-th mobile (yeff
) is simply a linear combination of the N vectors describing the N receive antennas.
After choosing combining weights the mobile quantizes the effective channel vector and feeds back the appropriate
quantization index. Only the effective channel output is used to receive data, and thus each mobile effectively has
only one antenna.
The key to the technique is to choose combiner weights that produce an effective channel that can be quantized
very accurately; such a choice must be made on the basis of both the channel vectors and the quantization codebook.
This is quite different from maximum ratio combining, where the combiner weights and quantization vector are
chosen such that received signal power is maximized but quantization error is generally not minimized. Note that
antenna selection corresponds to choosing the effective channel from the N columns of Hk, while QBC allows for
any linear combination of these N column vectors.
A. General Description
Let us consider the effective received signal at the first mobile for some choice of combiner weights, which we
denote as γ1 = (γ1,1, . . . , γ1,N ). In order to maintain a noise variance of one, the combiner weights are constrained
to have unit norm: ||γ1|| = 1. The (scalar) combiner output, denoted yeff1 , is:
yeff1 =
γH1,i(h
i x + ni) =
γH1,ih
γH1,ink
= (heff1 )
x + n,
where n =
i=1 γ
1,ini is unit variance complex Gaussian because |γ1| = 1. The effective channel vector heff1 is
simply a linear combination of the vectors h1, . . . ,hN : h
i=1 γ1,ihi = H1γ1. Since γ1 can be any unit norm
vector, heff1 can be in any direction in the N -dimensional subspace spanned by h1, . . . ,hN , i.e., in span(H1).
Because quantization error is so critical to performance, the objective is to choose combiner weights that yield
an effective channel that can be quantized with minimal error. The error corresponding to effective channel heff1 is
l=1,...,2B
∠(heff1 ,wl)
. (7)
1By well known properties of iid Rayleigh fading, the matrix H1 is full rank with probability one [8].
Therefore, the optimal choice of the effective channel is the solution to:
l=1,...,2B
∠(heff1 ,wl)
, (8)
where heff1 is allowed to be in any direction in span(H1). Once the optimal effective channel is determined, the
combiner weights γ1 can be determined through a simple pseudo-inverse operation.
Since the expression for the optimum effective channel given in (8) consists of two minimizations, without loss
of optimality the order of the minimization can be switched to give:
l=1,...,2B
∠(heff1 ,wl)
, (9)
For each quantization vector wl, the inner minimization finds the effective channel vector in span(H1) that forms
the minimum angle with wl. By basic geometric principles, the minimizing h
1 is the projection of wl on span(H1).
The solution to the inner minimization in (9) is therefore the sine squared of the angle between wl and its projection
on span(H1), which is referred to as the angle between wl and the subspace
2. As a result, the best quantization
vector, i.e., the solution of (9), is the vector that forms the smallest angle between itself and span(H1). The optimal
effective channel is the (scaled) projection of this particular quantization vector onto span(H1).
In order to perform quantization, the angle between each quantization vector and span(H1) must be computed.
If q1, . . . ,qN form an orthonormal basis for span(H1) and Q1 , [q1 · · ·qN ], then sin2(∠(w, span(H1))) =
1 − ||QH1 w||2. Therefore, mobile 1’s quantized channel, denoted ĥ1, is:
ĥ1 = arg min
w=w1,...,w2B
|∠(w, span(H1))| = arg max
w=w1,...,w2B
||QH1 w||2. (10)
Once the quantization vector has been selected, it only remains to choose the combiner weights. The projection
of ĥ1 on span(H1), which is equal to Q1Q
1 ĥ1, is scaled by its norm to produce the unit norm vector s
1 . The
direction specified by sproj1 has the minimum quantization error amongst all directions in span(H1), and therefore
the effective channel should be chosen in this direction. First we find the vector u1 ∈ CN such that H1u1 = sproj1 ,
and then scale to get γ1. Since s
1 is in span(H1), u1 is uniquely determined by the pseudo-inverse of H1:
1 , (11)
and the combiner weight vector γ1 is the normalized version of u1: γ =
||u1|| . The quantization procedure is
illustrated for a N = 2 channel in Fig. 2. In the figure the span of the two channel vectors is shown along with
the quantization vector h1, its projection on the channel subspace, and the effective channel.
B. Algorithm Summary
We now summarize the quantization-based combining procedure performed at the k-th mobile:
1) Find an orthonormal basis, denoted q1, . . . ,qN , for span(Hk) and define Qk , [q1 · · ·qN ].
2) Find the quantization vector closest to the channel subspace:
ĥk = arg max
w=w1,...,w2B
||QHk w||2. (12)
3) Determine the direction of the effective channel by projecting ĥk onto span(Hk).
||QkQHk ĥk||
. (13)
4) Compute the combiner weight vector γk:
. (14)
2If the number of mobile antennas is equal to the number of transmit antennas (N = M ), the channel vectors span CM with probability
one. Therefore, each quantization vector has zero angle with the channel subspace and as a result the solution to the inner minimization
in (9) is trivially zero for each wl. Thus, performing quantization with the sole objective of minimizing angular error (i.e., QBC) is not
meaningful when N = M and is therefore not studied here.
Fig. 2. Quantization procedure for a two antenna mobile
Each mobile performs these steps, feeds back the index of its quantized channel ĥk, and then linearly combines
its N received signals using vector γk to produce its effective channel output y
= (heff
)Hx+n with heff
= Hkγk.
Note that the transmitter need not be aware of the number of receive antennas or of the details of this procedure
because the downlink channel appears to be a single receive antenna channel from the transmitter’s perspective;
this clearly eases the implementation burden of QBC.
V. THROUGHPUT ANALYSIS
Quantization-based combining converts the MIMO downlink channel into a vector downlink with channel vectors
heff1 , . . . ,h
and channel quantizations ĥi · · · ĥK . We first derive the statistics of the effective vector channel, then
analyze throughput for ZFBF with equal power loading and no user selection, and finally quantify the effect of
receiver estimation error.
A. Channel Statistics
We first determine the distribution of the quantization error and the effective channel vectors with respect to both
the random channels and random quantization codebooks.
Lemma 1: The quantization error sin2(∠(ĥk,h
)), is the minimum of 2B independent beta (M−N,N) random
variables.
Proof: If the columns of M×N matrix Qk form an orthonormal basis for span(Hk), then cos2 (∠(wl, span(Hk)) =
||QHk wl||2 for any quantization vector. Since the basis vectors and quantization vectors are isotropically chosen and
are independent, this quantity is the squared norm of the projection of a random unit norm vector in CM onto a
random N -dimensional subspace, which is described by the beta distribution with parameters N and M − N [9].
By the properties of the beta distribution, sin2 (∠(wl, span(Hk)) = 1−cos2 (∠(wl, span(Hk)) is beta (M −N,N).
Finally, the independence of the quantization and channel vectors implies independence of the 2B random variables.
Lemma 2: The normalized effective channels h
||heff1 ||
, . . . ,
||heff
|| are iid isotropic vectors in C
Proof: From the earlier description of QBC, note that
||heff
|| = s
, which is the projection of the best
quantization vector onto span(Hk). Since each quantization vector is chosen isotropically, its projection is isotropi-
cally distributed within the subspace. Furthermore, the best quantization vector is chosen based solely on the angle
between the quantization vector and its projection. Thus sproj
is isotropically distributed in span(Hk). Since this
subspace is also isotropically distributed, the vector sproj
is isotropically distributed in CM . Finally, the independence
of the quantization and channel vectors from mobile to mobile implies independence of the effective channel
directions.
Lemma 3: The quantity ||heff
||2 is χ2
2(M−N+1).
Proof: Using the notation from Section IV-A, the norm of the effective channel is given by:
||heffk ||2 = ||Hkγk||2 = ||Hk
||uk||
||2 = 1
||uk||2
||Hkuk||2 =
||sproj
||uk||2
||uk||2
, (15)
where we have used the definitions heff
= Hkγk and γk =
||uk|| , and the fact that uk satisfies Hkuk = s
Therefore, in order to characterize the norm of the effective channel it is sufficient to characterize 1||uk||2 . The N -
dimensional vector uk is the set of coefficients that allows s
, the normalized projection of the chosen quantization
vector, to be expressed as a linear combination of the columns of Hk (i.e., the channel vectors). Because s
is isotropically distributed in span(Hk) (Lemma 2), if we change coordinates to any (N -dimensional) basis for
span(Hk) we can assume without loss of generality that the projection of the quantization vector is [1 0 · · · 0]T .
Therefore, the distribution of 1||uk||2 is the same as the distribution of
[(HHk Hk)
. Since the N×N matrix HH
is Wishart distributed with M degrees of freedom, this quantity is well-known to be χ2
2(M−N+1); see [10] for a
proof.
The norm of the effective channel has the same distribution as that of a (M −N +1)-dimensional random vector
instead of a M -dimensional vector. An arbitrary linear combination (with unit norm) of the N channel vectors
would result in another iid complex Gaussian M -dimensional vector, whose squared norm is χ22M , but the weights
defining the effective channel are not arbitrary due to the inverse operation.
B. Sum Rate Performance Relative to Perfect CSIT
After receiving the quantization indices from each of the mobiles, a simple transmission option is to perform
equal-power ZFBF based on the channel quantizations (as described in Section II-B). If K = M or K > M and
M users are randomly selected, the resulting SINR at the k-th mobile is given by:
SINRk =
|(heff
)Hvk|2
j 6=k
|(heff
)Hvj|2
. (16)
The ergodic sum rate achieved by QBC, denoted RQBC(P ), is therefore given by:
RQBC(P ) = EH,W
|(heff
)Hvk|2
j 6=k
|(heff
)Hvj |2
where the expectation is taken with respect to the fading and the random quantization codebooks.
In order to study the benefit of QBC we compare RQBC(P ) to the sum rate achieved using zero-forcing
beamforming on the basis of perfect CSIT in an M transmit antenna vector downlink channel (single receive
antenna), denoted RZF−CSIT (P ). We use the vector downlink with perfect CSIT as the benchmark because QBC
converts the system into a vector downlink, and the rates achieved by QBC cannot exceed RZF−CSIT (P ) (even as
B → ∞). We later describe how this metric can easily be translated into a comparison between RQBC(P ) and the
sum rate achievable with linear precoding (i.e., block diagonalization) in an N receive antenna MIMO downlink
channel with CSIT.
In a vector downlink with perfect CSIT, the BF vectors (denoted vZF,k) can be chosen perfectly orthogonal to
all other channels. Thus, the SNR of each user is as given in (3) with zero interference terms in the denominator
and the resulting average rate is:
RZF−CSIT (P ) = EH
|hHk vZF,k|2
Following the procedure in [2], the rate gap ∆R(P ) is defined as the difference between the per-user throughput
achieved with perfect CSIT and with feedback-based QBC:
∆R(P ) , RZF−CSIT (P ) − RQBC(P ). (17)
Similar to Theorem 1 of [2], we can upper bound this throughput loss:
Theorem 1: The per-user throughput loss is upper bounded by:
∆R(P ) ≤
l=M−N+1
log2 e + log2
M− N+ 1
E[sin2(∠(ĥk,h
k ))]
Proof: See Appendix.
The first term in the expression is the throughput loss due to the reduced norm (Lemma 3) of the effective channel,
while the second (more significant) term, which is an increasing function of P , is due to quantization error. In
order to quantify this rate gap, the expected quantization error needs to be bounded. By Lemma 1, the quantization
error is the minimum of 2B iid beta(M − N,N) RV’s. Furthermore, a general result on ordered statistics applied
to beta RV’s gives [9, Chapter 4.I.B]:
E[sin2(∠(ĥk,h
k ))] ≤ F
where FX(x) is the inverse of the CDF of a beta (M − N,N) random variable, which is:
FX(x) =
N−1−i
xM−N+i(1 − x)N−1+i ≈
xM−N ,
where the approximation is the result of keeping only the lowest order x term and dropping (1 − x) terms; this is
valid for small values of x. Using this we get the following approximation:
E[sin2(∠(ĥk,h
k ))] ≈ 2
. (18)
The accuracy of this approximation is later verified by our numerical results. Plugging this approximation into the
upper bound in Theorem 1 we get:
∆R(P ) ≈
l=M−N+1
log2 e + log2
1+P ·
M−N+1
If B is fixed, quantization error causes the system to become interference-limited as the SNR is increased (see [2,
Theorem 2] for a formal proof when N = 1). However, if B is scaled with the SNR P such that the quantization
error decreases as 1
, the rate gap in (19) can be kept constant and the full multiplexing gain (M ) is achieved. In
order to determine this scaling, we set the approximation of ∆R(P ) in (19) equal to a rate constant log2 b and solve
for B as a function of P . Thus, a per-mobile rate loss of at most log2 b (relative to RZF−CSIT (P )) is maintained
if B is scaled as:
BN ≈ (M − N) log2 P − (M − N) log2 c − (M − N) log2
M−N+1
− log2
≈ M − N
PdB − (M − N) log2 c − (M − N) log2
M−N+1
− log2
, (20)
where c = b · e−(
l=M−N+1
) − 1. Note that a per user rate gap of log2 b = 1 bps/Hz is equivalent to a 3 dB power
gap in the sum rate curves.
As discussed in Section II-C, scaling feedback in a single receive antenna downlink as B1 =
PdB maintains
a 3 dB gap from perfect CSIT throughput. Feedback must also be increased linearly if QBC is used, but the slope of
this increase is M−1
when mobiles have only a single antenna compared to a slope of M−N
for antenna combining.
If we compute the difference between the N = 1 feedback load and the QBC feedback load, we can quantify how
much less feedback is required to achieve the same throughput (3 dB away from a vector downlink channel with
perfect CSIT) if QBC is used with N antennas/mobile:
∆QBC(N) = B1 − BN ≈
N − 1
PdB + log2
− (N − 1) log2 e.
The sum rate of a 6 transmit antenna downlink channel (M = 6) is plotted in Fig. 3. The perfect CSIT zero-
forcing curve is plotted along with the rates achieved using finite rate feedback with B scaled according to (20) for
N = 1, 2 and 3. For N = 2 and N = 3 QBC is performed and the fact that the throughput curves are approximately
3 dB away from the perfect CSIT curve verify the accuracy of the approximations used to derive the feedback
scaling expression in (20). In this system, the feedback savings at 20 dB are 7 and 12 bits, respectively, for 2 and
3 receive antennas. All numerical results in the paper are generated using the method described in Appendix II.
It is also important to compare QBC throughput to the throughput of a MIMO downlink channel with N antennas
per mobile. The most meaningful comparison is to the rate achievable with block diagonalization (BD) [11] without
0 5 10 15 20 25
SNR (dB)
ZF−CSIT
(N=1)
(N=1,2,3)
BD−CSIT
(N=3)
BD−CSIT
(N=2)
~ 3 dB
Fig. 3. Sum rate of M = K = 6 downlink channel
user selection and with equal power loading. In this case, M
mobiles are transmitted to (with N data streams per
mobile). In [12] it is shown that the BD sum rate is
∆BD−ZF (N) = (log2 e)
N − j
larger than RZF−CSIT (P ) at asymptotically high SNR, and that this offset is accurate even for moderate SNR’s.
This can be translated to a power offset by multiplying by 3
to give 3 log2 e
dB, which equates to 2.16
dB and 3.61 dB for N = 2 and N = 3. Therefore, the rate offset between QBC and BD with CSIT is the sum of
∆R(P ) (equation 17) and ∆BD−ZF (N). In Fig. 3 the BD sum rate curves are plotted, and their shifts relative to
ZF-CSIT are seen to follow the predicted power gaps.
C. Effect of Receiver Estimation Error
Although the analysis until now has assumed perfect CSI at the mobiles, a practical system always has some
level of receiver error. We consider the scenario where a shared pilot sequence is used to train the mobiles. If βM
downlink pilots are used (β ≥ 1 pilots per transmit antenna), channel estimation at the k-th mobile is performed on
the basis of observation Gk =
βPHk +nk. The MMSE estimate of Hk is Ĝk =
Gk, and the true channel
matrix can be written as the sum of the MMSE estimate and independent estimation error:
Hk = Ĝk + ek, (21)
where ek is white Gaussian noise, independent of the estimate Ĝk, with per-component variance (1+βP )
−1. After
computing the channel estimate Ĝk, the mobile performs QBC on the basis of the estimate Ĝk to determine the
combining vector γk. As a result, the quantization vector ĥk very accurately quantizes the vector Ĝkγk, which is
the mobile’s estimate of the effective channel output, while the actual effective channel is given by heff
= Hkγk.
For simplicity we assume that coherent communication is possible, and therefore the long-term average throughput
is again E[log2(1 + SINRk)] where the same expression for SINR given in (16) applies
3. The general throughput
analysis in Section V still applies, and in particular, the rate gap upper bound given in Theorem 1 still holds
if the expected quantization error takes into account the effect of receiver noise. As shown in Appendix III, the
approximate rate loss with receiver error is:
∆R(P ) ≈ log2 e
l=M−N+1
+ log2
1+P ·
M−N+1
. (22)
3We have effectively assumed that each mobile can estimate the phase and SINR at the effective channel output. In practice this could be
accomplished via a second round of pilots as described in [13].
0 5 10 15 20 25 30
SNR (dB)
Perfect CSIT & CSIR
Combining & Perfect
RX Error: Beta = 2
RX Error: Beta = 1
Fig. 4. Combining with Imperfect CSIR: M = 4, N = 2, K = 4, B scaled with SNR
Comparing this expression to (19) we see that estimation error leads only to the introduction of an additional 1
term.
If feedback is scaled according to (20) the rate loss is log2(b+β
−1) rather than log2(b). In Figure 4 the throughput
of a 4 mobile system with M = 4 and N = 2 is plotted for perfect CSIT/CSIR and for QBC performed on the
basis of perfect CSIR (β = ∞) and imperfect CSIR for β = 1 and β = 2. Estimation error causes non-negligible
degradation, but the loss decreases rather quickly with β (which can be increased at a reasonable resource cost
because pilots are shared).
VI. PERFORMANCE COMPARISONS
In this section we compare the throughput of QBC to other receive combining techniques and to limited feedback-
based block diagonalization4. For all results on receiving combining, the user selection algorithm of [7] is applied
assuming limited feedback (B bits) regarding the direction of the effective channel and perfect knowledge of the
effective channel norm5. We first describe these alternative approaches and then discuss some numerical results.
A. Alternate Combining Techniques
The optimal receive combining technique for a point-to-point MIMO channel in a limited feedback setting is to
select the quantization vector that maximizes received power [3]:
ĥk = arg max
w=w1,...,w2B
||HHk w||2. (23)
Because this method roughly corresponds to maximum ratio combining, it is referred to as MRC. If BF vector w
is used by the transmitter, received power is maximized by choosing γ =
w|| [3], which yields h
= Hkγk =
wk|| . When B is not very small, with high probability the quantization vector that maximizes ||H
k w||2 is the
vector that is closest to the eigenvector corresponding to the maximum eigenvalue of HkH
. To see this, consider
the maximization of ||HH
w|| when w is constrained to have unit norm but need not be selected from a finite
codebook. This corresponds to the classical definition of the matrix norm, and the optimizing w is in the direction
of the maximum singular value of Hk. When B is not too small, the quantization error is very small and as a
result the solution to (23) is extremely close to ||Hk||2. As a result, selecting the quantization vector according to
the criteria in (23) is roughly equivalent to directly finding the quantization vector that is closest to the direction
of the maximum singular value of Hk.
4 It should be noted that comparisons with block diagonalization are somewhat rough because systems that perform BD on the basis of
limited feedback and that employ user/stream selection have not yet been extensively studied in the literature, to the best of our knowledge.
As a result, it may be possible to improve upon the BD systems we use here as the point of comparison.
5Although the rate gap upper bound derived in Theorem 1 only rigorously applies to systems with equal power loading and random
selection of M mobiles, the bound can be used to reasonably approximate the throughput degradation due to limited feedback even when
user selection is performed. See [14] for a further discussion of the effect of limited feedback on systems employing user selection.
2 4 6 8 10 12 14
Feedback Bits (B)
Approximation
Approximation
Ant. Selection
Max Eig.
Fig. 5. Quantization Error for Different Combining Techniques (M = 4, N = 2)
Effective Channel Norm Quantization Error
Single RX Antenna (N = 1) χ22M 2
−B/(M−1)
Antenna Selection χ22M 2
−(B+log2 N)/(M−1)
MRC ≈ max eigenvalue 2−B/(M−1)
Max Eigenvector max eigenvalue 2−B/(M−1)
QBC χ22(M−N+1) 2
−B/(M−N)
TABLE I
SUMMARY OF COMBINING TECHNIQUES
The maximum singular value of Hk can be directly quantized if the mobile first selects the combiner weights γk
such that the effective channel heff
= Hkγk is in the direction of the maximum singular value, which corresponds
to selecting γk equal to the eigenvector corresponding to the maximum eigenvalue of the N × N matrix HHk Hk,
and then finds the quantization vector closest to heff
. The effective channel norm satisfies ||heff
||2 = ||Hk||2, which
can be reasonably approximated as a scaled version of a χ22MN random variable [15]. Therefore the norm of the
effective channel is large, but notice that the quantization procedure reduces to standard vector quantization, for
which the error is roughly 2−
M−1 .
In Figure 5, numerically computed values of the quantization error (log2(E[sin
2(∠(heff
, ĥk))]) are shown for
QBC, antenna selection, MRC (corresponding to equation 23), and direct quantization of the maximum eigenvector,
along with approximation 2−
M−1 as well as the approximation from (18), for a M = 4, N = 2 channel. Note that
the error of QBC is very well approximated by (18), and the exponential rate of decrease of the other techniques
are all well approximated by 2−
M−1 .
Each combining technique transforms the MIMO downlink into a vector downlink with a modified channel norm
and quantization error. These techniques are summarized in Table I. The key point is that only QBC changes
the exponent of the quantization error6, which determines the rate at which feedback increases with SNR. When
comparing these techniques note that the complexity of QBC and MRC are essentially the same: QBC and MRC
require computation of ||QHk w||2 and ||HHk w||2, respectively.
B. Block Diagonalization
An alternative manner in which multiple receive antennas can be used is to extend the linear precoding structure
of ZFBF to allow for transmission of multiple data streams to each mobile. Block diagonalization (BD) selects
6An improvement over QBC is to choose the quantization vector and combining weights that maximize the expected received SINR (the
true SINR depends on the BF vectors, which are unknown to the mobile). This extension of QBC, which will surely outperform QBC and
MRC, has been under investigation by other researchers since the initial submission of this manuscript and the results will be published
shortly [16].
0 5 10 15 20
SNR (dB)
BD (2 users)
Ant. Selection
Single Antenna
Fig. 6. Different Combining Techniques: M = 4, N = 2, K = 4, B scaled with SNR
precoding matrices such multi-user interference is eliminated at each receiver, similar to ZFBF. In order to select
appropriate precoding matrices, the transmitter must know the N -dimensional subspace spanned by each mobile
channel Hk. Thus an appropriate feedback strategy is to have each mobile quantize and feedback its channel
subspace. The effect of limited feedback in this setting (assuming there are M
mobiles and equal power loading
across users and streams is performed) was studied in [17]. In order to achieve a bounded rate loss relative to a
perfect CSIT (BD) system, feedback (per mobile) needs to scale approximately as N(M − N) log2 P . Thus, the
aggregate feedback load summed over M
mobiles is approximately M(M − N) log2 P , which is (approximately)
the same as the aggregate feedback in a QBC system in which each of the M mobiles uses B ≈ (M −N) log2 P .
Thus, there is a rough equivalence between QBC and BD in terms of feedback scaling, and this is later confirmed
by our numerical results.
It is also possible to perform user and stream selection when BD is used, and [18] presents an extension of the
algorithm of [7] to the multiple receive antenna setting (referred to as maximum eigenmode transmission, or MET).
In essence, MET treats each mobile’s N eigenmodes as a different single antenna receiver and selects eigenmodes
in a greedy fashion using the approach of [7]. Thus, in a limited feedback setting a reasonable strategy is to have
each user separately quantize the directions of its N eigenvectors and also feed back the corresponding eigenvalues.
C. Numerical Results
In Figures 6 and 7 throughput curves are shown for a 4 transmit antenna, 2 receive antenna (M = 4, N = 2)
system with K = 4 mobiles. Sum rate is plotted for three different combining techniques (QBC, antenna selection,
and MRC) and for a vector downlink channel (N = 1); the BD curves are discussed in later paragraphs. In Fig.
6, B (per mobile) is scaled according to (20), i.e., roughly as (M − N) log2 P , while in Fig. 7 each mobile uses
10 bits of feedback. As expected, the throughput of antenna selection, MRC, and the single antenna system all lag
behind QBC in Fig. 6, particularly at high SNR. This is because the (M −N) log2 P scaling of feedback is simply
not sufficient to maintain good performance if these techniques are used. To be more precise, the quantization
error goes to zero slower than 1
which corresponds to interference power that increases with SNR, and thus a
reduction in the slope (i.e., multiplexing gain) of these curves. In Fig. 7, MRC outperforms QBC for SNR less
than approximately 12 dB because signal power is more important than quantization error (i.e., interference power),
i.e., the system is not yet interference-limited. However, at higher SNR’s QBC outperforms MRC because of the
increased importance of quantization error.
Figures 6 and 7 also include plots of the throughput of a BD system. In this system, 2 of the 4 users are randomly
selected to feedback subspace information, and equal power BD with no selection is used to send 2 streams to each
of these mobiles, for a total of 4 streams. In order to equalize the aggregate feedback load, each of the 2 users
is allocated double the feedback budget of the combining-based systems; this corresponds to using two times the
scaling of (20) in Fig. 6 and 20 bits per mobile in Fig. 7. BD performs slightly better than QBC in both figures,
but we later see that this advantage is lost for larger K.
0 5 10 15 20
SNR (dB)
Single Antenna
Ant. SelectionMRC
BD (2 users)
Fig. 7. Different Combining Techniques: M = 4, N = 2, K = 4, B = 10
Figures 8 displays throughput for a 4 transmit antenna, 2 receive antenna (M = 4, N = 2) system at 10 dB
against K, the number of mobiles. Capacity refers to the sum capacity of the system (with CSIT), MET-CSIT is the
throughput achieved using the MET algorithm on the basis of CSIT[18], and ZF-CSIT is the throughput of a vector
downlink with CSIT and user selection [7]. Below these are four limited feedback curves for 10 bits of feedback
per mobile. The first three, QBC, MRC, and antenna selection, correspond to different combining techniques, while
MET-FB corresponds to performing MET on the basis of 5 bit quantization of each eigenmode (10 bits total
feedback per mobile). QBC achieves significantly higher throughput than MRC or antenna selection, particularly
for larger values of K. The ZF-CSIT curve is shown because it serves as an upper bound on the performance of
QBC, and the gap between the two is quite reasonable even for B = 10. MET-FB is seen to perform extremely
poorly: this is not too surprising because the MET algorithm is likely to only choose the strongest eigenmode
of a few users [18], and thus half of the feedback is essentially wasted on quantization of each user’s weakest
eigenmode. This motivates dedicating all 10 bits to quantization of the strongest eigenmode, but note that this
essentially corresponds to MRC, which is outperformed by QBC. The huge gap between MET-CSIT and MET-FB
indicates that MET has the potential to provide excellent performance, but extremely high levels of feedback may
be necessary to realize MET’s potential.
Finally, Figure 9 shows throughput versus number of users K for a 6 transmit antenna (M = 6) channel with
either 1 or 2 receive antennas. Sum capacity for N = 1 and N = 2 is plotted, along with the sum rate of a
perfect-CSIT TDMA system in which only the receiver with the largest point-to-point capacity is selected for
transmission. The ZF and QBC curves correspond to systems with user selection and either single receive antennas
or quantization-based combining, respectively, for feedback levels of 10, 15, and 20 bits per mobile. For each
feedback level, an additional receive antenna with QBC provides a significant throughput gain relative to a single
receive antenna system. Furthermore, QBC significantly outperforms TDMA (N = 2) for B = 15 or B = 20, and
provides an advantage over TDMA for B = 10 when the number of users is sufficiently large. Note, however, that
there is a significant gap between QBC and N = 2 capacity even when 20 bits of feedback are used; this indicates
that there may be room for significant improvement beyond QBC.
VII. CONCLUSION
The performance of multi-user MIMO techniques such as zero-forcing beamforming critically depend on the
accuracy of the channel state information provided to the transmitter. In this paper, we have shown that receive
antenna combining can be used to reduce channel quantization error in limited feedback MIMO downlink chan-
nels, and thus significantly reduce channel feedback requirements. Unlike traditional maximum-ratio combining
techniques that maximize received signal power, the proposed quantization-based combining technique minimizes
quantization error, which translates into minimization of multi-user interference power.
Antenna combining is just one method by which multiple receive antennas can be used in the MIMO downlink. It
is also possible to transmit multiple streams to each mobile, or to use receive antennas for interference cancellation
0 20 40 60 80 100
Users
Capacity
MET−CSIT
ZF−CSIT (N=1)
Ant. Selection
MET− FB
Fig. 8. Combining and User Selection: M = 4, N = 2, B = 10
0 20 40 60 80 100 120
Users
Capacity (N=2)
Capacity (N=1)
QBC (N=2)
B=10,15,20
ZF (N=1)
B=10,15,20
N=1,2
Fig. 9. Combining and User Selection: M = 6, N = 1, 2
if the structure of the transmitted signal is known to the mobile. It remains to be seen which of these techniques
is most beneficial in practical wireless systems when channel feedback resources and complexity requirements are
carefully accounted for.
APPENDIX I
PROOF OF THEOREM 1
Plugging the rate expressions into the definition of ∆(P ), we have ∆(P ) = ∆a + ∆b where
∆a = EH
1 + ρ|hHk vZF,k|2
− EH,W
log2
ρ|(heffk )Hvj|2
∆b = EH,W
log2
j 6=k
ρ|(heffk )Hvj |2
where ρ , P
. To upper bound ∆a, we define normalized vectors h̃k = hk/||hk|| and h̃effk = h
/||heff
||, and note
that the norm and directions of hk and of h
are independent. Using this we have:
log2
ρ|(heffk )Hvj |2
 ≥ EH,W
1 + ρ|(heffk )Hvk|2
= EH,W
1 + ρ||heffk ||
2|h̃eff
1 + ρXβ||hk||2|h̃k
vZF,k|2
, (24)
where Xβ is β(M −N + 1, N − 1). Since the BF vector vZF,k is chosen orthogonal to the (M − 1) other channel
vectors {hj}j 6=k, each of which is an iid isotropic vector, it is isotropic and is independent of h̃k. By Lemma 2 the
same is also true of vk and h̃
, and therefore we can substitute |h̃k
vZF,k|2 for |(heffk )Hvk|2. Finally, note that
the product Xβ||hk||2 is χ22(M−N+1) because ||hk||
2 is χ22M , and therefore Xβ ||hk||2 and ||heffk ||2 have the same
distribution. Using (24) we get:
∆a ≤ EH
1 + ρ||hk||2|h̃k
vZF,k|2
1 + ρXβ||hk||2|h̃k
vZF,k|2
≤ −E [log2 (Xβ)] = log2 e
l=M−N+1
where we have used log2 (Xβ) = log2
2(M−N+1)
and results from [8] to to compute E [log2 (Xβ)].
Finally, we upper bound ∆b using Jensen’s inequality:
∆b ≤ log2
1 + E
j 6=k
ρ|(heffk )Hvj |2
= log2
1 + ρ(M − 1)E
||(heffk )||2
|(h̃effk)Hvj |2
= log2
1 + ρ(M − 1)(M − N + 1)E
|(h̃effk)Hvj |2
= log2
1 + ρ(M − N + 1)E
h̃effk,hk
where the final step uses Lemma 2 of [2] to get E
|(h̃effk)Hvj|2
h̃effk,hk
APPENDIX II
GENERATION OF NUMERICAL RESULTS
Rather than performing brute force simulation of RVQ, which becomes infeasible for B larger than 15 or 20,
the statistics of RVQ can be exploited to efficiently and exactly emulate the quantization process:
1) Draw a realization of the quantization error Z according to its known CDF (Lemma 1).
2) Draw a realization of the corresponding quantization vector according to:
ĥk =
1 − Z
where u is isotropic in span(Hk), s is isotropic in the nullspace of span(Hk), with u, s independent.
These steps exactly emulate step 2 of QBC. The same procedure can also be used to emulate antenna selection,
quantization of the maximum eigenvector, and no combining (N = 1). Because the CDF of the quantization error
is not known for MRC, MRC results are generated using brute force RVQ.
APPENDIX III
RATE GAP WITH RECEIVER ESTIMATION ERROR
We bound the rate gap using the technique of [13]. We first restate the result of Theorem 1 in terms of the
interference terms E
|(heff
)Hvj |2
∆R ≤ log2 e
l=M−N+1
+ log2
1 + P
M − 1
|(heffk )
vj |2
. (25)
Using the representation of the channel matrix given in (21), we can write the interference term as:
(heffk )
vj = (Hkγk)
Ĝkγk
vj + (ekγk)
The first term in the sum is statistically identical to the interference term when there is perfect CSIR, while the
second term represents the additional interference due to the receiver estimation error. Because the noise and the
channel estimate are each zero-mean and are independent we have:
|(heffk )
vj |2
Ĝkγk
∣(ekγk)
The first term comes from the perfect CSIR analysis and is equal to the product of 1
M−1 and the expected quantization
error with perfect CSIR. Because γk and vj are each unit norm and ek is independent of these two vectors, the
quantity (ekγk)
vj is (zero-mean) complex Gaussian with variance (1 + βP )
−1, which is less than (1 + βP )−1.
We finally reach (22) by using the approximation for quantization error from (18) and plugging into (25), and
noting that (1 + βP )−1 ≈ (βP )−1.
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	Introduction
	System Model and Preliminaries
	Finite Rate Feedback Model
	Zero-Forcing Beamforming
	MIMO Downlink with Single Antenna Mobiles
	Antenna Selection for Reduced Quantization Error
	Quantization-Based Combining
	General Description
	Algorithm Summary
	Throughput Analysis
	Channel Statistics
	Sum Rate Performance Relative to Perfect CSIT
	Effect of Receiver Estimation Error
	Performance Comparisons
	Alternate Combining Techniques
	Block Diagonalization
	Numerical Results
	Conclusion
	Appendix I: Proof of Theorem ??
	Appendix II: Generation of Numerical Results
	Appendix III: Rate Gap with Receiver Estimation Error
	References
