Capacity of a Multiple-Antenna Fading Channel with a Quantized Precoding
  Matrix
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 1
Capacity of a Multiple-Antenna Fading Channel
With a Quantized Precoding Matrix
Wiroonsak Santipach, Member, IEEE, and Michael L. Honig, Fellow, IEEE
Abstract—Given a multiple-input multiple-output (MIMO)
channel, feedback from the receiver can be used to specify
a transmit precoding matrix, which selectively activates the
strongest channel modes. Here we analyze the performance of
Random Vector Quantization (RVQ), in which the precoding ma-
trix is selected from a random codebook containing independent,
isotropically distributed entries. We assume that channel elements
are i.i.d. and known to the receiver, which relays the optimal
(rate-maximizing) precoder codebook index to the transmitter
using B bits. We first derive the large system capacity of
beamforming (rank-one precoding matrix) as a function of B,
where large system refers to the limit as B and the number
of transmit and receive antennas all go to infinity with fixed
ratios. RVQ for beamforming is asymptotically optimal, i.e., no
other quantization scheme can achieve a larger asymptotic rate.
We subsequently consider a precoding matrix with arbitrary
rank, and approximate the asymptotic RVQ performance with
optimal and linear receivers (matched filter and Minimum Mean
Squared Error (MMSE)). Numerical examples show that these
approximations accurately predict the performance of finite-size
systems of interest. Given a target spectral efficiency, numerical
examples show that the amount of feedback required by the
linear MMSE receiver is only slightly more than that required
by the optimal receiver, whereas the matched filter can require
significantly more feedback.
Index Terms—Beamforming, large system analysis, limited
feedback, Multi-Input Multi-Output (MIMO), precoding, vector
quantization.
I. INTRODUCTION
G IVEN a multi-input multi-output (MIMO) channel, pro-viding channel information at the transmitter can in-
crease the achievable rate and simplify the coder and decoder.
Namely, this channel information can specify a precoding
matrix, which aligns the transmitted signal along the strongest
channel modes (i.e., singular vectors corresponding to the
Manuscript recieved December 22, 2006; revised July 17, 2008. This work
was supported by the U.S. Army Research Office under grant DAAD19-99-1-
0288 and the National Science Foundation under grant CCR-0310809, and was
presented in part at IEEE Military Communications (MILCOM), Boston, MA,
USA, October 2003, IEEE International Symposium on Information Theory
(ISIT), Chicago, IL, USA, June 2004, and IEEE International Symposium on
Spread Spectrum Techniques and Applications (ISSSTA), Sydney, Australia,
August 2004.
W. Santipach was with the Department of Electrical Engineering and
Computer Science; Northwestern University, Evanston, IL 60208 USA. He
is currently with the Department of Electrical Engineering; Faculty of En-
gineering; Kasetsart University, Bangkok, 10900 Thailand (email: wiroon-
sak.s@ku.ac.th).
M. L. Honig is with the Department of Electrical Engineering and Com-
puter Science; Northwestern University, Evanston, IL 60208 USA (email:
mh@eecs.northwestern.edu).
Communicated by H. Boche, Associate Editor for Communications.
Digital Object Identifier 10.1109/TIT.2008.2011437
largest singular values). In practice, the precoding matrix must
be quantized at the receiver, and relayed to the transmitter
via a feedback channel. The corresponding achievable rate is
therefore limited by the accuracy of the quantizer.
The design and performance of quantized precoding matri-
ces for multi-input single-output (MISO) and MIMO channels
has been considered in numerous references, including [1]–
[12]. In those references, and in this paper, the channel is
assumed to be stationary, known at the receiver, and the perfor-
mance is evaluated as a function of the number of quantization
bits B. (This is in contrast with other work, which models
estimation error at the receiver, but does not explicitly account
for quantization error (e.g., [13], [14]), and which assumes a
time-varying channel with feedback of second-order statistics
[15]–[18].) Optimization of vector quantization codebooks is
discussed in [1]–[3], [5], [6] for beamforming, and in [4],
[7], [12] for MIMO channels with precoding matrices that
provide multiplexing gain (i.e., have rank larger than one). It
is shown in [2], [3] that this optimization can be interpreted
as maximizing the minimum distance between points in a
Grassmannian space. (See also [9].) The performance of this
class of Grassmannian codebooks is also studied in [8]–[10].
In this paper, we evaluate the performance of a Random
Vector Quantization (RVQ) scheme for the precoding matrix.
Namely, given B feedback bits, the precoding matrix is
selected from a random codebook containing 2B matrices,
which are independent and isotropically distributed. RVQ has
been analyzed in other source coding contexts (e.g., see [19]
and the related discussion in [20]), and achieves the rate-
distortion bound for ergodic Gaussian sources. This work is
motivated by prior work [21] in which RVQ is considered
for signature quantization in Code-Division Multiple Access
(CDMA). In that scenario, limited feedback is used to select a
signature for a particular user, which maximizes the received
Signal-to-Interference-Plus-Noise-Ratio (SINR). RVQ has the
attractive properties of being tractable and asymptotically
optimal. Namely, in [21] the received SINR with RVQ is
evaluated in the asymptotic (large system) limit as processing
gain, number of users, and feedback bits all tend to infinity
with fixed ratios. Furthermore, it is shown that no other
quantization scheme can achieve a larger asymptotic SINR.
Here we assume an i.i.d. block Rayleigh fading channel
model with independent channel gains, and take ergodic
capacity as the performance criterion. The receiver relays B
bits to the transmitter (per codeword) via a reliable feedback
channel (i.e., no feedback errors) with no delay. We start by
evaluating the capacity of MISO and MIMO channels with a
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2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009
quantized beamformer, i.e., rank-one precoding matrix. Our
results are asymptotic as the number of transmit antennas Nt
and feedback bits B both tend to infinity with fixed B/Nt
(feedback bits per degree of freedom). For the MIMO channel
the number of receive antennas Nr also tends to infinity
in proportion with Nt and B. The asymptotic expressions
accurately predict the performance of finite-size systems of
interest as a function of normalized feedback and background
Signal-to-Noise Ratio (SNR). In analogy with the optimality
result shown in [21], RVQ is also asymptotically optimal in
this scenario, i.e., no other quantization scheme can achieve
a larger asymptotic rate. Furthermore, numerical examples for
small Nt show that RVQ performance averaged over code-
books is essentially the same as that obtained from codebooks
optimized via the Lloyd-Max algorithm [1], [5], [6]. (See
also the numerical examples in [22], which compare RVQ
performance with the optimized (Grassmannian) codebooks
in [3].)
We then consider quantization of a precoding matrix with
arbitrary rank. Namely, a rank K precoding matrix multiplexes
K independent streams of transmitted information symbols
onto the Nt transmit antennas. In that case, the capacity with
limited feedback is approximated in the limit as B, Nt, Nr,
and K all tend to infinity with fixed ratios K/Nt, Nr/Nt,
and B/N2r . That is, the number of feedback bits again scales
linearly with the number of degrees of freedom, which is pro-
portional to N2r . Although our results for beamforming suggest
that RVQ is also asymptotically optimal in this scenario, this
remains an open question.
The asymptotic results for a precoder matrix with arbitrary
rank K can be used to determine the normalized rank, or
multiplexing gain K/Nt, which maximizes the capacity. This
optimized rank in general depends on the normalized feedback,
the ratio of antennas Nt/Nr, and the SNR. For example, if
Nt/Nr ≥ 1 and the SNR is sufficiently large, then as the
feedback increases from zero to infinity, the optimized rank
decreases from one to Nr/Nt. Numerical results are presented,
which illustrate the effect of normalized rank on achievable
rate, and also show that the asymptotic results accurately
predict simulated results for finite-size systems of interest.
We also evaluate the performance of RVQ with linear
receivers (i.e., the matched filter and linear Minimum Mean
Squared Error (MMSE) receivers), and compare their perfor-
mance with the optimal (capacity-achieving) receiver. With the
optimal precoding matrix, corresponding to infinite feedback,
both linear receivers are optimal. With limited feedback the
two linear receivers are simpler than the optimal receiver,
but require more feedback to achieve a target rate. Numerical
results show that this additional feedback required by the linear
MMSE receiver is quite small, whereas the additional feedback
required by the matched filter can be significant (e.g., about
one bit per precoding matrix element).
In addition to quantizing the optimal precoding matrix,
power for each data stream can also be optimized, quantized,
and fed back to the transmitter (e.g., see [23], [24]). Asymptot-
ically, the amount of feedback required to specify the power is
negligible compared to the feedback required for the precoding
matrix. Furthermore, uniform power over the set of activated
channel typically performs close to the optimal (water-filling)
performance [8]. We therefore only consider quantization of
the precoding matrix.
Other related work on RVQ for MIMO channels has been
presented in [25]–[27]. Namely, exact expressions for the
ergodic capacity with beamforming and RVQ for a finite-size
MISO channel are derived in [25]. The performance of RVQ
for precoding over a broadcast MIMO channel is analyzed
in [26], [27]. A closely related random beamforming scheme
for the multiuser MIMO broadcast channel was previously
presented in [28]. In that work, the growth in sum capacity is
characterized asymptotically as the number of users becomes
large with a fixed number of antennas. (Random beamforming
was previously proposed in [29], although there the main focus
is to improve fairness among users.)
The paper is organized as follows. Section II describes
the channel model, Section III considers the capacity of
beamforming with limited feedback, and sections IV and V
examine the capacity of a quantized precoding matrix with
optimal and linear receivers, respectively. Derivations of the
main results are given in the appendices.
II. CHANNEL MODEL
We consider a point-to-point, flat Rayleigh fading channel
with Nt transmit antennas and Nr receive antennas. Let x =
[xk] be a K×1 vector of transmitted symbols with covariance
matrix IK , where IK is the K ×K identity matrix, and K is
the number of independent data streams. The received Nr × 1
vector is given by
HV x+ n (1)
where H = [hnr ,nt ] is an Nr × Nt channel matrix, V =
[v1 v2 . . . vK ] is an Nt ×K precoding matrix, and n is a
complex Gaussian noise Nr×1 vector with covariance matrix
σ2nINr . Assuming rich scattering and Rayleigh fading, the
elements of H are independent, and the channel coefficient be-
tween the ntth transmit antenna and the nrth receive antenna,
hnr,nt , is a circularly symmetric complex Gaussian random
variable with zero mean and unit variance (E[|hnr,nt |2] = 1).
We assume i.i.d. block fading, i.e., the channel is static
within a fading block, and the channels across blocks are
independent. The ergodic capacity is achieved by coding the
transmitted symbols across an infinitely large number of fading
blocks. With perfect channel knowledge at the receiver and
a given precoding matrix V , the ergodic capacity is the
mutual information between x and y with a complex Gaussian
distributed input, averaged over the channel, given by
I(x;y) = EH
log det
where ρ = 1/σ2n is the background SNR. We wish to
specify the precoding matrix V that maximizes the mutual
information, subject to a power constraint ‖vk‖ ≤ 1, for
1 ≤ k ≤ K .
With unlimited feedback, the columns of the optimal pre-
coding matrix, which maximizes (2), are eigenvectors of the
channel covariance matrix H†H . With B feedback bits per
SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 3
fading block, we can specify the precoding matrix from a
quantization set or codebook V = {V1, · · · ,V2B} known
a priori to both the transmitter and receiver. The receiver
chooses the Vj that maximizes the sum mutual information,
and relays the corresponding index back to the transmitter.
Of course, the performance (ergodic capacity) depends on the
codebook V .
III. BEAMFORMING WITH LIMITED FEEDBACK
We start with a rank-one precoding matrix, corresponding
to a single data stream (K = 1). In that case, the precoding
matrix is specified by an Nt×1 beamforming vector v, which
ideally corresponds to the strongest channel mode. That is, the
optimal v, which maximizes the ergodic capacity in (2), is the
eigenvector of H†H corresponding to the largest eigenvalue.
This vector is computed at the receiver and a quantized version
is relayed back to the transmitter.
Let V = {v1, . . . ,v2B} denote the quantization codebook
for v, given B feedback bits. Optimization of this codebook
has been considered in [2], [3], [6] with outage capacity and
ergodic capacity as performance metrics. The performance of
an optimized codebook is difficult to evaluate exactly, and
is approximated in [2], [3], [6], [9]–[11]. Here we consider
RVQ in which v1, · · · ,v2B are independent, isotropically
distributed random vectors, each with unit norm. This is
motivated by the observation that given a channel matrix H
with i.i.d. elements, the eigenvectors of H†H are isotropi-
cally distributed [30], hence the codebook entries should be
uniformly distributed over the space of beamforming vectors.
A. MISO Channel
We first consider a MISO channel, corresponding to a single
receive antenna (Nr = 1). In that case, H is an Nt×1 channel
vector, which we denote as h. The optimal beamformer, which
maximizes the mutual information in (2), is the normalized
channel vector h/‖h‖ and the corresponding mutual informa-
tion is Eh[log(1 + ρh
†h)]. The receiver selects the quantized
precoding vector to maximize the mutual information, i.e.,
v̂ = arg max
1≤j≤2B
Ij = log(1 + ρ|h†vj |2)
and the corresponding achievable rate is
INtrvq , max
1≤j≤2B
Ij . (4)
where the superscript Nt denotes the system size. The achiev-
able rate depends on the codebook V and the channel vector
h, and is therefore random. Rather than averaging INtrvq over
V and h to find the ergodic capacity, we instead evaluate the
limiting performance as Nt and B tend to infinity with fixed
B̄ = B/Nt (feedback bits per transmit antenna). In this limit,
INtrvq converges to a deterministic constant. This is illustrated
in Fig. 1, which shows the pdf of |h†v̂|2/‖h‖2 for different
Nt with no feedback (B̄ = 0), and for RVQ with B̄ = 2. The
figure shows that convergence of the pdf to a point mass is
faster with feedback than without.
As Nt → ∞, (h†h)/Nt → 1 almost surely, so that
log(1 + ρh†h)− log(ρNt) → 0. That is, with perfect channel
0 0.2 0.4 0.6 0.8 1
Pdf for |h†v̂j|
2/ ‖h‖2 with RVQ codebook
 = 25
 = 15
B̄ = 0 B̄ = 2
Fig. 1. pdf of |h†v̂|2/‖h‖2 with RVQ for different values of Nt.
knowledge at the transmitter, the ergodic capacity increases as
log(ρNt). With finite feedback there is a rate loss, which is
defined as
rvq = I
rvq − log(ρNt). (5)
For finite Nt, I
rvq is random; however, in the large system
limit I△rvq converges to a deterministic constant.
Theorem 1. As (Nt, B) → ∞ with fixed B̄ = B/Nt, the rate
difference I△rvq converges in the mean square sense to
I△rvq = log(1− 2−B̄). (6)
The proof is given in Appendix A. For B̄ > 0, the rate loss
due to finite feedback is a constant. As B̄ → 0, this rate loss
tends to infinity, since with B̄ = 0, the capacity tends to a
constant as Nt → ∞, whereas the capacity grows as logNt
for B̄ > 0. Of course, as B̄ → ∞ (unlimited feedback), the
rate loss vanishes.
RVQ is asymptotically optimal in the following sense.
Suppose that {VNt} is an arbitrary sequence of codebooks
for the beamforming vector where
VNt =
2 , . . . ,v
is the codebook for a particular Nt and ‖vNtj ‖2 = 1 for each
j. The associated rate is given by
IVNt = max
1≤j≤2B
log(1 + ρ|h†vNtj |
2) (8)
and the rate difference I△VNt
= IVNt − log(ρNt).
Theorem 2. For any sequence of codebooks {VNt},
lim sup
(Nt,B)→∞
] ≤ I△rvq. (9)
The proof is given in Appendix B.
Although the optimality of RVQ holds only in the large
system limit, numerical results in Section III-C show that for
finite-size systems of interest RVQ performs essentially the
same as optimized quantization codebooks.
4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009
B. Multi-Input Multi-Output (MIMO) Channel
We now consider quantized beamforming for a MIMO
channel, i.e., with multiple transmit and receive antennas.
Taking the rank K = 1 maximizes the diversity gain [31],
but the corresponding capacity grows only as logNt instead
of linearly with Nt, which is the case when K grows pro-
portionally with Nt. (This is true with both unlimited and
limited feedback, assuming a fixed number of feedback bits
per precoder element.) Also, a beamformer is significantly less
complex than a matrix precoder with K > 1, and requires less
feedback to specify.
We again consider an RVQ codebook V with 2B indepen-
dent unit-norm vectors, where each vector is uniformly dis-
tributed over the Nt-dimensional unit sphere. The achievable
rate is EH [I
rvq ], where
INtrvq = EV
1≤j≤2B
1 + ρ‖Hvj‖2
log(1 + ρ max
1≤j≤2B
‖Hvj‖2)
. (11)
As for the MISO channel, with unlimited feedback the achiev-
able rate increases as log(ρNt). We again define the rate
difference due to quantization as
rvq , I
rvq − log(ρNt) = EV
where
Hvj . (13)
Evaluating the expectation in (12) is difficult for finite Nt,
Nr, and B, so that we again resort to a large system analysis.
Namely, we let Nt, Nr, and B each tend to infinity with fixed
B̄ = B/Nt and N̄r = Nr/Nt. For each Nt andNr the channel
matrix H is chosen as the Nr × Nt upper-left corner of a
matrix H̄ with an infinite number of rows and columns, and
with i.i.d. complex Gaussian entries.
The received power in this large system limit is given by
γ∞rvq = lim
(Nt,Nr,B)→∞
1≤j≤2B
where convergence to the deterministic limit can be shown in
the mean square sense. Conditioned on H̄ , the γj’s are i.i.d.
since the beamforming vectors vj are i.i.d., and applying [32,
Theorem 2.1.2], it can be shown that
γ∞rvq = lim
(Nt,Nr,B)→∞
1− 2−B
where Fγ|H̄(·) is the cdf of γj given H̄ . Analogous results
for the interference power in CDMA with quantized signatures
have been presented in [21], so that we omit the proofs of
(14) and (15). Note that N̄r ≤ γ∞rvq ≤ (1+
2, where the
lower and upper bounds correspond to B̄ = 0 and B̄ = ∞,
respectively. That is, (1+
2 is the asymptotic maximum
eigenvalue of the channel covariance matrix 1
H†H [33].
The asymptotic rate difference is given by
I∆rvq = lim
(Nt,Nr,B)→∞
I∆rvq = log(γ
rvq) (16)
The limit in (15) can be explicitly evaluated, and is inde-
pendent of the channel realization H̄ .
Theorem 3. For 0 ≤ B̄ ≤ B̄∗, γ∞rvq satisfies
γ∞rvq
rvq = 2−B̄
and for B̄ ≥ B̄∗,
γ∞rvq = (1 +
2 − exp
N̄r log(N̄r)
− (N̄r − 1) log(1 +
N̄r) +
N̄r − B̄ log(2)
where
B̄∗ =
log(2)
N̄r log
. (19)
The proof is given in Appendix C and is motivated by an
analogous result for CDMA, presented in [34]. As stated in
Theorem 3, γ∞rvq depends only on B̄ and N̄r. Letting N̄r → 0
gives the the asymptotic capacity of the MISO channel with
RVQ. As for the MISO channel, RVQ is asymptotically
optimal.
Theorem 4. As (Nt, Nr, B) → ∞ with fixed N̄r = Nr/Nt
and B̄ = B/Nt,
lim sup
(Nt,Nr,B)→∞
INtVNt
− log(ρNt) ≤ I∆rvq (20)
for any sequence of codebooks {VNt}.
The proof is similar to the proof of Theorem 2 in [21] and
is therefore omitted.
C. Numerical Results
Figs. 2 and 3 show I△rvq for MISO and MIMO channels,
respectively, with beamforming and RVQ versus normalized
feedback bits (B̄) with ρ = 5 and 10 dB. Also shown for
comparison are achievable rates with a quantization code-
book optimized via the Lloyd-Max algorithm [1], [5], [6],
and the capacity with perfect beamforming, corresponding to
unlimited feedback. The results for RVQ are averaged over
codebook realizations, and are essentially the same as those
shown for the optimized Lloyd-Max codebooks. For the MISO
channel, the asymptotic capacity (6) accurately predicts the
simulated results shown even with a relatively small number
of transmit antennas (Nt = 3 and 6). For the MIMO results
N̄r = 1.5, and simulation results are shown for 4 × 6 and
16 × 24 channels. The asymptotic results accurately predict
the performance for the larger channel, and are somewhat less
accurate for the smaller channel.
Comparing finite feedback with perfect beamforming, the
results show that one feedback bit per complex entry (B̄ = 1)
provides more than 50% of the potential gain due to feedback.
For both the MISO and MIMO examples shown, the perfect
beamforming capacity is nearly achieved with two feedback
bits per complex coefficient.
SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 5
0 0.5 1 1.5 2 2.5 3
MISO; SNR = 5 dB
RVQ, simulation w/ N
 = 3, N
RVQ, simulation w/ N
 = 6, N
Optimal beamforming
RVQ, large system
Lloyd−Max, simulation w/ N
 = 6, N
Lloyd−Max, simulation w/ N
 = 3, N
Fig. 2. Asymptotic and simulated rate differences versus feedback bits for
a MISO channel with beamforming.
0 0.5 1 1.5 2 2.5 3 3.5 4
 = 1.5; SNR = 10 dB
Optimal Beamforming
RVQ, simulation w/ N
 = 4, N
RVQ, simulation w/ N
 = 16, N
 = 24
RVQ, large system
Lloyd−Max, simulation w/ N
 = 4, N
Lloyd−Max, simulation w/ N
 = 16, N
 = 24
Fig. 3. Asymptotic and simulated rate differences versus feedback bits for
a MIMO channel with beamforming (N̄r = 1.5).
IV. PRECODING MATRIX WITH ARBITRARY RANK
In this section we consider the performance of a single-user
MIMO channel with precoding matrix V having rank K > 1.
We wish to determine the asymptotic capacity with RVQ as
in the previous section. Here we consider the large system
limit as (Nt, Nr, B,K) → ∞ with fixed ratios N̄r = Nr/Nt,
B̂ = B/N2r , and K̄ = K/Nt. That is, we scale the rank
of the precoding matrix with Nt. The number of feedback
bits is normalized by N2r , instead of Nr, since the feedback
must scale linearly with degrees of freedom (in this case the
number of channel elements NrNt).) Given a fixed number
of feedback bits per channel coefficient, the capacity grows
linearly with the number of antennas (Nt or Nr).
Given a rank K ≤ Nt, the precoding matrix is chosen from
the RVQ set
V = {Vj , 1 ≤ j ≤ 2B}, (21)
where the entries are independent Nt × K random unitary
matrices, i.e., V †j Vj = IK . This codebook is an extension of
the RVQ codebook for beamforming. Letting
JNrj =
log det
INr +
, (22)
the receiver again selects the quantized precoding matrix,
which maximizes the mutual information
V̂ = arg max
1≤j≤2B
JNrj . (23)
For finite Nr, we define
INrrvq = EV [ max
1≤j≤2B
JNrj |H̄ ] (24)
log det
INr +
HV̂ V̂
and the average sum mutual information per receive antenna
with B feedback bits is then EH [I
rvq ].
Here the power allocation over channel modes is “on-
off”. Namely, active modes are assigned equal powers. This
simplifies the analysis, and it has been observed that the
additional gain due to an optimal power allocation (water
pouring) is quite small [8].
Since the entries of the RVQ codebook are i.i.d., the mutual
informations JNrj , j = 1, . . . , 2
B, are also i.i.d. for a given H .
In principle, the large system limit of INrrvq can be evaluated,
in analogy with (15), given the cdf of JNrj given H , denoted
as FJ;Nr|H . This cdf appears to be difficult to determine in
closed-form for general (Nr, Nt,K), so that we are unable to
derive the exact asymptotic capacity with RVQ. Still, we can
provide an accurate approximation for this large system limit.
Before presenting this approximation, we first compare the
capacity with no channel information at the transmitter (B̂ =
0) to the capacity with perfect channel information (B̂ = ∞).
If B̂ = 0, then the optimal transmit covariance matrix
V V † = INt and K = Nt [35]. That is, all channel modes
are allocated equal power. As (Nt, Nr) → ∞ with fixed
N̄r = Nr/Nt, the capacity per receive antenna is given by
log det
INr +
log (1 + ρλ) g(λ) dλ
= Irvq(B̂ = 0) (27)
where convergence is in the almost sure sense, and g(λ) is the
asymptotic probability density function for a randomly chosen
eigenvalue of 1
HH†, and is given by [33]
g(λ) =
(λ − a)(b− λ)
2πλN̄r
for a ≤ λ ≤ b, (28)
and b =
for N̄r ≤ 1. The integral in (26) has been evaluated in [36],
which gives the closed-form expression
Irvq(B̂ = 0) = log ρy +
1− N̄r
6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009
where
1 + N̄r +
1 + N̄r +
− 4N̄r
 (31)
1 + N̄r +
1 + N̄r +
− 4N̄r
 . (32)
If B̂ = ∞, then the K columns of the optimal V are the
eigenvectors of the channel covariance matrix corresponding
to the K largest eigenvalues. As (Nt, Nr, B) → ∞, we have
Irvq(B̂ = ∞) =
g(λ) dλ (33)
where η satisfies
g(λ) dλ = min{1,
} (34)
for N̄r ≤ 1. We emphasize that this corresponds to a uniform
allocation of power over the set of K active eigenvectors. (This
result has also been presented in [8].) The rank of the optimal
V , or optimal multiplexing gain, is at most min{Nt, Nr} and
can be obtained by differentiating (33) with respect to K̄. It
can be verified that Irvq(B̂ = 0) ≤ Irvq(B̂ = ∞).
To illustrate the increase in capacity with feedback, in Fig. 4
we plot the rate ratio Irvq(B̂ = ∞)/Irvq(B̂ = 0) versus SNR
for different values of N̄r, where Irvq(B̂ = ∞) is optimized
over rank K . For large SNR ρ, we can expand
Irvq(B̂ = 0) = log(ρ) + o(log(ρ)) (35)
Irvq(B̂ = ∞) = log(ρ)
g(λ) dλ+ o(log(ρ)). (36)
Therefore
Irvq(B̂ = ∞)
Irvq(B̂ = 0)
g(λ) dλ = min{1, K̄
} (37)
which implies that the optimal rank K∗ = min{Nt, Nr}, and
the corresponding asymptotic rate ratio is one. The increase
in achievable rate from feedback is small in this case, since
for large SNRs, the transmitter excites all channel modes, and
the uniform power allocation asymptotically gives the same
capacity as water pouring. Of course, although the increase in
rate is small, feedback can simplify coding and decoding.
For small ρ, we can expand log(1+ρλ) and log(1+ρλ/K̄)
in Taylor series. Taking ρ→ 0 gives
Irvq(B̂ = ∞)
Irvq(B̂ = 0)
λg(λ) dλ ≤
λg(λ) dλ
g(λ) dλ
g(λ) dλ
g(λ) dλ
where b is the asymptotic maximum eigenvalue given by (29).
The inequality in (38) follows from (34), which implies K̄ ≥
g(λ) dλ. Note that (39) corresponds to allocating all
transmission power to the strongest channel mode, which is
known to maximize capacity at low SNRs. The maximal rate
ratio (39) can also be obtained from Theorem 3.
The rate increase due to feedback is substantial when N̄r
is small, and the rate ratio tends to infinity as N̄r → 0. This
is because the channel becomes a MISO channel, in which
case the capacity is a constant with B̄ = 0 and increases as
log(ρNt) with B̄ = ∞.
−10 −5 0 5 10 15 20 25
ρ (dB)
 = 0.1
 = 0.5
 = 0.75
Fig. 4. The rate ratio Irvq(B̂ = ∞)/Irvq(B̂ = 0) versus SNR (dB) for
various values of N̄r .
To evaluate the asymptotic capacity with arbitrary B̂, we
approximate JNrj given H̄ as a Gaussian random variable.
This is motivated by the fact that JNrj is Gaussian in the
large system limit [37], since HV is i.i.d. Conditioning on H
introduces dependence among the elements of HV ; however,
numerical examples indicate that the Gaussian assumption
is still valid for large Nt and Nr. Alternatively, if we do
not condition on H , then the rates {JNrj } are dependent.
Application of the results from extreme statistics, assuming
the rates {JNrj } are independent, gives an upper bound on the
asymptotic achievable rate (e.g., see the proof of Theorem 2
in [21]). This is illustrated by subsequent numerical results.
Evaluating the large system limit of INrrvq , assuming that the
cdf of JNrj is Gaussian, gives the approximate rate
Ĩrvq = µJ + σJ
2B̂ log 2 (40)
independent of the channel realization, where µJ and σ
are the asymptotic mean of JNrj , and variance of N
respectively. The derivation of (40) is a straightforward exten-
sion of [32, Sec. 2.3.2] and is not shown here. As B̂ → 0,
this approximation becomes exact. However, as B̂ → ∞,
the approximate rate Ĩrvq → ∞, whereas the actual rate
Irvq(B̂ = ∞) is finite, and can be computed from (33) and
(34). This is because JNrj is bounded for all Nr, whereas a
Gaussian random variable can assume arbitrarily large values.
Therefore the Gaussian approximation gives an inaccurate
estimate of Irvq for large B̂. (This implies that we should
approximate Irvq as min{Ĩrvq, Irvq(B̂ = ∞)}.)
SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 7
The asymptotic mean and variance of JNrj are computed in
Appendix D. The asymptotic mean is given by
ρ− N̄r
+ log
1 + ρ− N̄r
where
2N̄rρ
− 4K̄
The asymptotic variance is approximated for 0 ≤ K̄ = N̄r ≤ 1
and small SNR (ρ ≤ −5 dB) as
σ2J ≈ ρ2(1 − N̄r). (43)
The asymptotic variance for moderate SNRs and normalized
rank K̄ 6= N̄r can be computed easily via numerical simula-
tion.1
In contrast with the beamforming results in the preceding
section, we are unable to show that RVQ is asymptotically
optimal when the precoding matrix has arbitrary rank. The
corresponding argument for beamforming relies on the evalu-
ation of the asymptotic rate difference I∆rvq. Since here we
are unable to evaluate Irvq exactly, we cannot apply that
argument. Nevertheless, numerical results have indicated that
the performance of RVQ matches that of optimized codebooks
(e.g., see [22]).
Fig. 5 shows Ĩrvq with normalized rank K̄ = N̄r versus
B̂ for ρ = −5, 0, 5 dB and N̄r = 0.5. The dashed lines
show the unlimited feedback capacity Irvq(B̂ = ∞), which is
computed from (33) with optimized K̄. The asymptotic rate
with RVQ is computed from (40), where σJ for ρ = −5 dB is
approximated by (43), and σJ is determined from simulation
with Nt = 20 for ρ = 0 and 5 dB. Also shown in Fig. 5 are
simulation results for Irvq with Nt = 8 and Nr = 4. Because
the size of the RVQ codebook increases exponentially with
B̂, it is difficult to generate simulation results for moderate to
large values of B̂. Hence simulation results are shown only
for B̂ ≤ 0.8. The asymptotic results accurately approximate
the simulated results shown. The accuracy increases as the
feedback B̂ decreases.
Since Ĩrvq is a function of both rank K̄ and feedback B̂, for
a given B̂, we can select K̄ to maximize Ĩrvq. Fig. 6 shows
mutual information per receive antenna versus normalized
rank from (40) with N̄r = 0.2, ρ = 5 dB, and different
values of B̂. (σJ is obtained from numerical simulations.)
The maximal rates are attained at K̄ = 1, 0.3, and 0.2 for
B̂ = 0, 0.5, and 2, respectively. In general, the optimal rank
is approximately N̄r for large enough B̂ and SNR. The results
in Fig. 6 indicate that taking K̄ = N̄r achieves near-optimal
performance, independent of B̂ when B̂ > 0. As B̂ increases,
the rate increases and the difference between the rate with
optimized rank and full-rank (K̄ = 1) also increases. For the
1We note that the simulation needed to compute this variance is much
simpler than the simulation, which would be required to obtain the RVQ rate
directly, especially with a moderate to large number of feedback bits.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
 = 0.5
SNR = −5 dB
SNR = 0 dB
SNR = 5 dB
 (B/N2
 = ∞)
 w/ (K = N
Sim. I
 = 8)
Fig. 5. Sum mutual information per receive antenna with RVQ and an optimal
receiver versus normalized feedback. The asymptotic approximation is shown
along with Monte Carlo simulation results for an 8× 4 channel.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
 = 0.2; N
 = 10; SNR = 5 dB
2 = 0
2 = 0.5
2 = 2
Fig. 6. Mutual information per receive antenna versus normalized rank with
different normalized feedback. Discrete points correspond to simulation with
Nt = 10.
example shown, the rate increase from selecting the optimal
rank is as high as 50% when B̂ = 2.
V. QUANTIZED PRECODING WITH LINEAR RECEIVERS
In this section we evaluate the performance of a quantized
precoding matrix with linear receivers (matched filter and
MMSE), and compare with the performance of the opti-
mal receiver. As B̂ → ∞, the optimal precoding matrix
eliminates the cross-coupling among channel modes, and the
optimal receiver becomes the linear matched filter. Hence the
corresponding achievable rates should be the same in this
limit. However, for finite B̂ the optimal receiver is expected
to perform better than the linear receiver. Given a target
rate, increasing the feedback therefore enables a reduction in
receiver complexity.
We again assume that there are K independent data streams,
which are multiplexed by the linear precoder onto Nt transmit
antennas. To detect the transmitted symbols in data stream k,
the received signal y is passed through the Nr × 1 receive
8 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009
filter ck. The matched filter is given by
Hvk (44)
where vk is the kth column of the precoding matrix V , and
the linear MMSE filter is given by
† + σ2nINr
Hvk. (45)
The SINR at the output of the linear filter ck is
SINRk =
|c†kHvk|2
i6=k Hviv
† +Kσ2nINr
. (46)
Of course, the interference among data streams can signifi-
cantly decrease the channel capacity.
The performance measure is again mutual information be-
tween the transmitted symbol xk and the output of the filter
ck, denoted by x̂k. In what follows, we assume independent
coders and decoders for each data stream. Assuming that the
interference plus noise at the output of the linear filter has a
Gaussian distribution, which is true in the large system limit to
be considered, the sum mutual information of all data streams
per receive antenna is given by
RNr =
I(xk, x̂k) (47)
log(1 + γk). (48)
where γk is the SINR for the kth data stream. Given a channel
matrix H , the sum rate RNr depends on the precoding matrix
V . We are interested in maximizing RNr subject to the power
constraint ‖vk‖ ≤ 1, ∀k, assuming that the power is allocated
equally across streams.
Given the codebook of precoding matrices V = {Vj , 1 ≤
j ≤ 2B}, the receiver selects the precoding matrix
V̂ = arg max
1≤j≤2B
RNr(Vj). (49)
We again consider RVQ, in which the Vj’s are i.i.d. unitary
matrices.
A. Matched filter
Substituting (44) into (46), the SINR at the output of the
matched filter is given by
γk;mf =
†Hvk)
Kσ2n(v
†Hvk) +
i=1,i6=k |v
†Hvi|2
where subscript k denotes the kth data stream. The average
sum rate per receive antenna is given by
1≤j≤2B
{RNrmf (Vj) =
log(1 + γk;mf)}
where the expectation is over the channel matrix and code-
book. Since the pdf of RNrmf is unknown for finite (Nt, Nr,K),
we are unable to evaluate (51). Motivated by the central limit
theorem,2 in what follows we approximate the cdf of RNrmf as
2The terms in the sum in (51) are not i.i.d., which prevents a direct
application of the central limit theorem.
0 0.2 0.4 0.6 0.8 1
 (nat)
PDF for R
 = 10; N
 = 1; K/N
 = 0.3; SNR = 5dB
Empirical pdf
Gaussian approximation
Fig. 7. Comparison of the empirical pdf for Rmf;Nr with the Gaussian
approximation.
Gaussian. The mean is taken to be the asymptotic limit
µmf = lim
(Nt,Nr,K)→∞
RNrmf =
K̄(1 + σ2n)
This limit follows from the fact that γk;mf converges almost
surely to [K̄(1 + σ2n)/N̄r]
−1 as (Nt, Nr,K) → ∞ with fixed
N̄r and K̄ . As for the optimal receiver,
N2r var[R
mf |Vj ] → σ
mf (53)
where σ2mf can be easily obtained by numerical simulations.
The accuracy of the Gaussian approximation for RNrmf is
illustrated in Fig. 7, which compares the empirical pdf with the
Gaussian approximation for Nr = 10, N̄r = 1, K/Nr = 0.3
and SNR = 5dB. The difference between the empirical and
asymptotic means vanishes as (Nt, Nr,K) → ∞.
We wish to apply the theory of extreme order statistics [32]
to evaluate the large system limit
Rrvq;mf = lim
(Nt,Nr,K,B)→∞
[ max
1≤j≤2B
RNrmf (Vj)|V ]. (54)
Given V , the sum rates {RNrmf (V1), . . . , R
mf (V2B )} are identi-
cally distributed. However, the RNrmf (Vj)’s are not independent
since each depends on H . This makes an exact calculation
of the asymptotic rate difficult. Nevertheless, for a small
number of entries in the codebook (small B), assuming that
the rates for a given codebook are independent leads to
an accurate approximation. We therefore replace the rates
RNrmf (Vj), j = 1, · · · , 2
B , with i.i.d. Gaussian variables with
mean µmf and variance σ
r . In analogy with the analysis
of the optimal receiver in the preceding section, this gives the
approximate asymptotic rate
R̃rvq;mf = µmf + σmf
2B̂ log 2. (55)
Numerical results, to be presented, show that this asymptotic
approximation is very accurate for small to moderate values
of normalized feedback B̂. As B̂ → 0, this approximation
becomes exact. However, as B̂ → ∞, R̃rvq;mf → ∞, whereas
Rrvq;mf with B̂ = ∞ is the same as the asymptotic rate with
SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 9
RVQ and an optimal receiver, given by (33) and (34). Hence
Rrvq;mf with B̂ = ∞ is finite. As for the analysis of the optimal
receiver, this discrepancy is again due to the fact that the cdf
of RNrmf , which has compact support, is being approximated
by a Gaussian cdf with infinite support, and also because the
dependence among the sum rates RNrmf (Vj) is being ignored.
B. MMSE receiver
Substituting (45) into (46) gives the SINR at the output of
MMSE receiver for the kth symbol stream
γk;mmse = v
† +Kσ2nINr
As for the matched filter receiver, given a codebook V , we
approximate the pdf of the instantaneous sum rate
RNrmmse =
log(1 + γk;mmse) (57)
as a Gaussian pdf with mean
µmmse = lim
(Nt,Nr)→∞
RNrmmse =
log(1 + γmmse) (58)
where the large system SINR is given by [30]
γmmse =
1− K̄/N̄r
(1 − K̄/N̄r)2
1 + K̄/N̄r
As for the matched filter, the asymptotic variance σ2mmse can
be obtained via numerical simulation.
In analogy with (55), the asymptotic rate with RVQ and the
MMSE receiver is given by
Rrvq;mmse ≈ R̃rvq;mmse = µmmse + σmmse
2B̂ log 2. (60)
As for the matched filter receiver, when B̂ is large, R̃mmse
over-estimates Rmmse. For B̂ = ∞, Rmmse = Rmf = Irvq
with the optimal receiver, given by (33).
C. Numerical Results
Fig. 8 compares the approximation for asymptotic RVQ
performance with a matched filter receiver from (55) with
simulated results for Nt = 12, N̄r = 0.75, K/Nt = 1/2, and
SNR = 5 dB. Also shown for comparison are the asymptotic
rate for RVQ with an optimal receiver, derived in Section IV,
the water-filling capacity (B̂ = ∞), and the rate achieved with
a scalar quantizer for each coefficient.3 For the case shown,
the analytical approximation gives an accurate estimate of the
performance of the finite size system with limited feedback.
The capacity with the water-filling power allocation is only
slightly greater than that achieved with the on-off power allo-
cation. The optimal receiver requires B̂ ≈ 0.6 bit/dimension
3For the scalar quantization results the available bits are spread evenly
over the corresponding fraction of precoding coefficients. The remaining
coefficients are set to one.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
 = 0.75; K/N
 = 0.5; SNR = 5 dB
Water−filling capacity (large sys.)
RVQ w/ Opt Rx. 
(large sys.)   
RVQ w/ MF Rx. 
(large sys.)  
Scalar quantization (N
 = 12)
RVQ w/ MF Rx. 
 = 12)    
Fig. 8. Sum rate per receive antenna versus normalized feedback bits with a
matched filter receiver. Results are shown for RVQ (asymptotic and Nt = 12)
and scalar quantization. Also shown are results for the optimal receiver, and
the water-filling capacity with infinite feedback.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
 = 0.75; K/N
 = 0.5; SNR = 5 dB
RVQ w/ MMSE Rx. 
 = 12)      
Scalar quantization (N
 = 12)
RVQ w/ MMSE Rx. 
(large sys.)    
Water−filling capacity (large sys.)
RVQ w/ Opt Rx. (large sys.)
Fig. 9. Sum rate per receive antenna versus normalized feedback bits with a
linear MMSE receiver. Results are shown for RVQ (asymptotic and Nt = 12)
and scalar quantization. Also shown are results for the optimal receiver, and
the water-filling capacity with infinite feedback.
to achieve the capacity corresponding to unlimited feedback
(33), whereas the matched filter requires 1.2 feedback bits per
dimension to reach that capacity. For other target rates, these
curves illustrate the trade-off between feedback and receiver
complexity.
Fig. 9 shows the same set of results as those shown in Fig. 8,
but with an MMSE receiver. These results show that for the
parameters selected, the MMSE receiver performs nearly as
well as the optimal receiver, and requires substantially less
feedback than the matched filter to achieve a target rate.
Again the asymptotic approximation accurately predicts the
performance of a system with a relatively small number of
antennas.
VI. CONCLUSIONS
We have studied the capacity of single-user MISO and
MIMO fading channels with limited feedback. The feedback
10 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009
specifies a transmit precoding matrix, which can be optimized
for a given channel realization. We first considered the perfor-
mance with a rank-one precoding matrix (beamformer), and
showed that the RVQ codebook is asymptotically optimal.
Exact expressions for the asymptotic mutual information for
MISO and MIMO channels were presented, and reveal how
much feedback is required to achieve a desired performance.
For the cases considered, one feedback bit for each precoder
coefficient can achieve close to the water-filling capacity.
Perhaps more important than the increase in capacity provided
by this feedback is the associated simplification in the coding
and decoding schemes that can achieve a rate close to capacity.
The performance of a precoding matrix with rank K > 1
was also evaluated with RVQ. Although numerical examples
and our beamforming results (K = 1) suggest that RVQ is
also asymptotically optimal in this case, proving this is an
open problem. To compute the asymptotic achievable rate for
RVQ with both optimal and linear receivers, the achievable
rate with a random channel and fixed precoding matrix is
approximated as a Gaussian random variable. The asymptotic
rate then depends on the asymptotic mean and variance of this
random variable. Although the asymptotic variance appears to
be difficult to compute analytically, it can be easily obtained
by simulation. Numerical results have shown that the resulting
approximation accurately estimates the achievable rate with
limited feedback for finite-size systems of interest.
Numerical examples comparing the performance of opti-
mal and linear receivers have shown that the linear MMSE
receiver requires little additional feedback, relative to the
optimal receiver, to achieve a target rate close to the water-
filling capacity. The matched filter requires significantly more
feedback than the MMSE receiver (more than 0.5 bit per
degree of freedom for the cases shown). At low feedback rates
the achievable rate with RVQ is generally much greater than
that associated with scalar quantization. Of course, this comes
at a price of high complexity, since the receiver is assumed
to compute the performance metric for every entry in the
codebook. Other reduced complexity schemes for quantizing
a beamforming vector are presented in [21], [38], [39].
Key assumptions for our results are that the channel is
stationary and known at the receiver, and that the channel
elements are i.i.d. Depending on user mobility and associated
Doppler shifts, the channel may change too fast to allow reli-
able channel estimation and feedback. In that case, feedback of
channel statistics, as proposed in [15]–[18], [40], can exploit
correlation among channel elements. The design of quanti-
zation codebooks for precoders, which takes correlation into
account, is addressed in [40]. The effect of channel estimation
error on the performance of limited feedback beamforming
with finite coherence time (i.e., block fading) is presented in
[41], [42].
We have also assumed that the channel gains are not
frequency-selective. Limited feedback schemes for frequency-
selective scalar channels are discussed in [43], and could
be combined with the quantization schemes considered here.
Finally, the approach presented here for a single-user MIMO
channel can also be applied to multi-user models. Quantization
of beamformers for the MIMO downlink have been considered
in [26]–[28]. In that scenario the potential capacity gain due
to feedback is generally much more than for the single-user
channel considered here. The benefits of limited feedback for
related models (e.g., frequency-selective MIMO downlink) are
currently being studied.
APPENDIX
A. Proof of Theorem 1
Given h, the receiver selects the quantized beamforming
vector v̂ to maximize the instantaneous rate in (3). Since log
is monotonically increasing, the quantized beamforming vector
is given by
v̂ = arg max
1≤j≤2B
Yj = |v†jh|
2/‖h‖2
. (61)
Since the codebook entries vj , j = 1, · · · , 2B, are i.i.d., the
Yj’s, given h, are also i.i.d. with the cdf [25]
FY |h(y) = 1− (1 − y)Nt−1, 0 ≤ y ≤ 1. (62)
We wish to determine the distribution of maxj Yj given h.
From [32, Theorem 2.1.2] it follows that
maxj Yj − an
D−→ Y (63)
where Y is a Weibull random variable having distribution
Hγ(x) =
1, x ≥ 0
exp(−(−x)γ), x < 0 , (64)
D denotes convergence in distribution, and an and bn are nor-
malizing sequences, where n = 2B. Specifically, the theorem
requires that ω(FY |h) = sup{y : FY |h(y) < 1} be finite,
and that the distribution function F ∗
Y |h(y) = FY |h(ω(FY |h)−
1/y), y > 0 satisfies, for all y > 0,
1− F ∗
Y |h(ty)
1− F ∗
Y |h(t)
= y−γ (65)
where the constant γ > 0.
Substituting the expression for FY |h in (62) into (65), where
ω(FY |h) = 1, gives
1− F ∗
Y |h(ty)
1− F ∗
Y |h(t)
= lim
(1/(ty))
(1/t)
Nt−1 (66)
= y−(Nt−1) (67)
so that [32, Theorem 2.1.2] applies when Nt > 1. Further-
more, the normalizing constants are given by
an = ω(FY |h) = 1 (68)
bn = ω(FY |h)− inf
y : 1− FY |h(y) ≤
= 1− F−1
. (70)
SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 11
To take the limit as Nt → ∞, we will assume that the
channel vector h contains the first Nt elements of an infinite-
length i.i.d. complex Gaussian vector h̄. Rearranging terms in
(63) and taking the large system limit gives
(Nt,n)→∞
EV [max
Yj |h̄] (71)
= lim
(Nt,n)→∞
an + bnE[Y] (72)
= 1− lim
(Nt,n)→∞
Nt − 1
= 1− lim
(Nt,B)→∞
Nt−1 (74)
= 1− 2−B̄ (75)
where the gamma function Γ(z) =
tz−1e−t dt and we
have used the fact that E[Y] = −Γ(1− 1/(Nt − 1)) [44].
From (63), as (n,Nt) → ∞,
var[max
Yj |h̄]− b2nvar[Y] → 0. (76)
Since var[Y] = Γ(1−2/(Nt−1))−Γ2(1−1/(Nt−1)) → 0, it
follows that var[maxj Yj |h] → 0. This establishes that given
Yj → 1− 2−B̄ (77)
in the mean square sense. The asymptotic rate difference is
given by
I△rvq = lim
(Nt,B)→∞
log(1 + ρ‖h‖2 max
1≤j≤2B
− log(ρNt)
= lim
(Nt,B)→∞
‖h‖2 max
1≤j≤2B
= log(1− 2−B̄) (80)
in the mean square sense, since ‖h‖2/Nt → 1 almost surely.
B. Proof of Theorem 2
The rate difference associated with codebook VNt is
= max
1≤j≤2B
|h†vNtj |
= log
+ max
1≤j≤2B
|h†vNtj |
. (82)
Taking expectation of the rate difference with respect to h and
applying Jensen’s inequality, we obtain
] ≤ log
|h†v̂Nt |2]
= log
‖h‖2]E[µ]
where the optimal beamforming vector
Nt = arg max
1≤j≤2B
|h†vNtj |
2, (85)
µ = |h†v̂Nt |2/‖h‖2, and (84) follows from the fact that ‖h‖2
and µ are independent [25].
We now derive an upper bound for E[|h†v̂Nt |2/‖h‖2] .
From (30) in [2] we have
Pr{|h†v̂Nt |2 > γs | ‖h‖2 = γ}
1, 0 ≤ s < s∗
2B(1− s)Nt−1, s∗ ≤ s ≤ 1
where
s∗ = 1− 2−
Nt−1 . (87)
Since the right-hand side of (86) is independent of γ, averaging
over γ gives
Pr{µ > s} ≤
1, 0 ≤ s < s∗
2B(1− s)Nt−1, s∗ ≤ s ≤ 1 . (88)
Integrating by parts, we have that
E[µ] =
Pr{µ > x} dx (89)
Pr{µ > x} dx+
Pr{µ > x} dx. (90)
Substituting (88) into (90) and evaluating both integrals gives
E[µ] ≤ 1− 2−
Nt−1 +
Nt−1 . (91)
Substituting E[‖h‖2] = Nt and (91) into (84) gives
] ≤ log
1− 2−
Nt−1 +
Nt−1 +
and taking the large system limit gives
(Nt,B)→∞
] ≤ log(1− 2−B̄). (93)
Theorem 1 states that RVQ achieves this upper bound, and
therefore upper bounds the asymptotic rate difference corre-
sponding to any quantization scheme.
C. Proof of Theorem 3
We first prove the theorem for N̄r ≥ 1. Let z = F−1γ|H̄(1−
2−B). Rearranging (15) gives
(Nt,Nr)→∞
z→γ∞rvq
1− Fγ|H̄(z)
Nt = 2−B̄. (94)
Next, we derive upper and lower bounds for the left-hand side
of (94) and show that they are the same. The derivation of the
upper bound is motivated by the evaluation of a similar bound
for CDMA signature optimization in [34]. That is,
1− Fγ|H̄(z) = Pr
γj > z| H̄
vj > z|Λ,U
where γj =
†Hvj and we have applied the singular
value decomposition 1
H†H = UΛU† , where U is an
Nt × Nt unitary matrix, Λ = diag{λ1, · · · , λNt}, and the
eigenvalues are ordered as λ1 ≥ λ2 ≥ · · · ≥ λNt . Also, wj
12 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009
is an Nt × 1 vector with independent, circularly symmetric,
zero-mean and unit-variance Gaussian elements. Both vj and
wj/‖wj‖ are isotropically distributed, i.e., Uwj/‖wj‖ and
wj/‖wj‖ have the same distribution, so that
1− Fγ|H̄(z) = 1− Fγ|Λ(z) (98)
i=1 λiw
i=1 w
(z − λi)w2i > 0
(100)
(z − λi)w2i > 0
, ∀ρ > 0
(101)
(z − λi)w2i
(102)
where {wi} are elements of wj . (We omit the index j to
simplify the notation.) Applying Markov’s inequality and the
independence of the wi’s gives
1− Fγ|Λ(z) ≤ E{wi}
(z − λi)w2i
(103)
−ρ(z − λi)w2i
(104)
exp {−ρ(z − λi)x} e−xdx (105)
exp {−(1 + ρ(z − λi))x} dx (106)
1 + ρ(z − λi)
(107)
= exp
log(1 + ρ(z − λi))
(108)
when 1 + ρ(z − λi) > 0 for all i, or ρ < 1/(λ1 − z). Taking
the large system limit, we obtain
(Nt,Nr)→∞
z→γ∞rvq
1− Fγ|Λ(z)
Nt ≤ exp{−Φ(γ∞rvq, ρ)} (109)
for 0 < ρ < 1
−γ∞rvq
, where
Φ(γ∞rvq, ρ) ,
log(1 + ρ(γ∞rvq − λ))g(λ)dλ, (110)
g(λ) is given by (28)-(29), and λ∞max = lim(Nt,Nr)→∞ λ1 =
2. To tighten the upper bound, we minimize (109)
with respect to ρ, i.e.,
(Nt,Nr)→∞
z→γ∞rvq
1− Fγ|Λ(z)
Nt ≤ exp{−Φ(γ∞rvq, ρ∗)} (111)
where
ρ∗ = arg max
0<ρ< 1
−γ∞rvq
Φ(γ∞rvq, ρ). (112)
A similar expression for RVQ performance when used to
quantize signatures for CDMA is derived in [34].
To derive the lower bound, we use a change of measure.
(A similar approach was used in [45, Section 1.2].) Let yi ,
(λi − z)w2i , which is a scaled exponential random variable
with cdf Fi(·). We define the new distribution
Gi(x) ,
Mi(ρ∗)
∗y dFi(y), (113)
so that
∗)dGi(x) = e
ρ∗xdFi(x), (114)
where ρ∗ is given in (112), and the moment generating
function for yi is
Mi(θ) , E[e
θyi] (115)
eθ(λi−z)xe−x dx (116)
1 + θ(z − λi)
. (117)
Applying the change of measure (114), we have
1− Fγ|Λ(z)
= Pr{
yi > 0} (118)
· · ·
yi > 0] dF1(y1) · · · dFNt(yNt) (119)
· · ·
yi > 0]e
∗y1dF1(y1) (120)
· · · eρ
∗yNtdFNt(yNt)
· · ·
yi > 0]e
yi dG1(y1)
(121)
· · · dGNt(yNt)
where
1[x > 0] =
1 : x > 0
0 : x ≤ 0 . (122)
For any ǫ > 0,
1− Fγ|Λ(z)
· · ·
1[ǫNt ≥
yi > 0]e
(123)
dG1(y1) · · ·dGNt(yNt)
∗)e−ρ
· · ·
1[ǫNt ≥
yi > 0] (124)
dG1(y1) · · ·dGNt(yNt)
∗)e−ρ
∗ǫNt Pr{ǫNt ≥
ỹi > 0} (125)
SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 13
where the ỹi’s are independent random variables with cdf
Gi(·), and the second inequality follows since ρ∗ > 0. To
determine the probability on the right-hand side of (125), we
first compute the mean of ỹi,
y dGi(y) (126)
Mi(ρ∗)
∗ydFi(y) (127)
= (1 + ρ∗(z − λi))
(λi − z)xeρ
∗(λi−z)xe−x dx
(128)
λi − z
1 + ρ∗(z − λi)
. (129)
Therefore
λi − z
1 + ρ∗(z − λi)
(130)
and the asymptotic mean
m∞ = lim
(Nt,Nr)→∞
z→γ∞rvq
mi (131)
λ− γ∞rvq
1 + ρ∗(γ∞rvq − λ)
g(λ) dλ <∞. (132)
Similarly, since ỹi is exponentially distributed, the variance of
ỹi is
σ2i =
λi − z
1 + ρ∗(z − λi)
<∞ (133)
and the asymptotic variance
σ2∞ = lim
(Nt,Nr)→∞
z→γ∞rvq
σ2i (134)
λ− γ∞rvq
1 + ρ∗(γ∞rvq − λ)
g(λ) dλ <∞. (135)
Both the asymptotic mean and variance are finite.
Since the ỹi’s are independent with finite mean and variance,
the central limit theorem implies that the cdf for
i=1 ỹi −
i=1mi
i=1 σ
(136)
converges to a Gaussian cdf with zero mean and unit variance.
Therefore we have
Pr{0 <
ỹi ≤ ǫNt}
= Pr{−
NtaNt < T ≤ −
NtaNt +
} (137)
= FT (−
NtaNt +
)− FT (−
NtaNt) (138)
where FT (·) is the cdf for T and
aNt ,
i=1mi/Nt
i=1 σ
i /Nt
, (139)
bNt ,
σ2i /Nt → σ∞. (140)
Let φ(·) denote a Gaussian cdf with zero mean and unit
variance. We can rewrite (138) as
Pr{0 <
ỹi ≤ ǫNt} = φ(−
NtaNt +
NtaNt) + ζNt − ξNt
(141)
where
ζNt , FT (−
NtaNt +
− φ(−
NtaNt +
), (142)
ξNt , FT (−
NtaNt)− φ(−
NtaNt). (143)
Applying the Berry-Esséen theorem [46], we can bound
both ζNt and ξNt for large Nt as
|ζNt |, |ξNt | ≤
(144)
where C is a positive constant that depends on the variance
and third moment of ỹ. Similar to the mean and variance, we
can show that the third moment is also finite.
We can now evaluate
NtaNt +
)− φ(−
NtaNt)
NtaNt+
NtaNt
2/2 dt (145)
e−Nt(ant−ǫ/bNt)
. (146)
Substituting (144) and (146) into (141), we have
Pr{0 <
ỹi ≤ ǫNt} = O(1/
Nt) (147)
Taking the large system limit and applying L’Hopital’s rule,
it follows that
(Nt,Nr)→∞
z→γ∞rvq
[Pr{0 <
ỹi ≤ ǫNt}]
Nt = 1. (148)
Taking the Ntth root and large system limit on both sides of
(125) gives
(Nt,Nr)→∞
z→γ∞rvq
1− Fγ|Λ(z)
Nt ≥ exp{−Φ(γ∞rvq, ρ∗)} (149)
where we use (148) and let ǫ→ 0.
14 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009
The lower bound in (149) is exactly the upper bound (111).
Therefore,
(Nt,Nr)→∞
z→γ∞rvq
1− Fγ|Λ(z)
Nt = exp{−Φ(γ∞rvq, ρ∗)} (150)
= 2−B̄ (151)
and the asymptotic RVQ received power satisfies the fixed-
point equation
Φ(γ∞rvq, ρ
∗) = B̄ log(2) (152)
where ρ∗ is given by (112). The goal of the rest of the proof
is to simplify (152).
To determine ρ∗, we first compute
∂Φ(γ∞rvq, ρ)
γ∞rvq − λ
1 + (γ∞rvq − λ)ρ
g(λ) dλ (153)
+ γ∞rvq
︸ ︷︷ ︸
g(λ) dλ (154)
SΛ(y) (155)
where SΛ(·) is the Stieltjés Transform of the asymptotic
eigenvalue distribution of Λ. Setting the derivative to zero and
solving for ρ gives
SΛ(y) = −ρ =
γ∞rvq − y
(156)
as the only valid solution. Substituting the expression for SΛ
given in [33] into (156) gives
(−1 + N̄r − y)±
y2 − 2(N̄r + 1)y + (N̄r − 1)2
γ∞rvq − y
(157)
which simplifies to the quadratic equation
(N̄r − γ∞rvq)y2 + [γ∞rvq + N̄rγ∞rvq + (γ∞rvq)2]y = 0. (158)
Solving for y gives y = 0 or y = γ∞rvq[1 + 1/(γ
rvq − N̄r)],
or equivalently, ρ = −1/γ∞rvq or ρ = (γ∞rvq − N̄r)/γ∞rvq. Since
ρ > 0, we must have
γ∞rvq − N̄r
γ∞rvq
. (159)
Since
∂2Φ(γ∞rvq, ρ)
λ− γ∞rvq
1 + ρ∗(γ∞rvq − λ)
g(λ) dλ (160)
< 0, (161)
therefore ρ∗ achieves a maximum.
By also evaluating Φ(γ∞rvq, ρ) at the boundary points ρ = 0
and ρ = 1/(λ∞max − γ∞rvq), we have
γ∞rvq−N̄r
γ∞rvq
, N̄r ≤ γ∞rvq ≤ N̄r +
N̄r)2−γ∞rvq
, N̄r +
N̄r ≤ γ∞rvq < (1 +
(162)
To evaluate Φ(γ∞rvq, ρ
∗), we re-write (110) as
Φ(γ∞rvq, ρ
log(ρ∗) + log
+ γ∞rvq
g(λ) dλ (163)
= log(ρ∗) +
+ γ∞rvq
g(λ) dλ. (164)
To evaluate the integral in (164), we apply the following
Lemma.
Lemma 1. For x ≥ (1 +
Θ(x) ,
log(x− λ)g(λ) dλ (165)
= log(w(x)) +
N̄ru(x)− (N̄r − 1) log
(166)
where
w(x) =
(x− 1− N̄r) +
(x− 1− N̄r)2 − 4N̄r
, (167)
u(x) =
(x− 1− N̄r)−
(x− 1− N̄r)2 − 4N̄r
. (168)
The proof of this Lemma is similar to that given in [36]
and is therefore omitted here.
For N̄r +
N̄r ≤ γ∞rvq < (1 +
2, we substitute ρ∗ =
[(1 +
2 − γ∞rvq]−1 into (164) to obtain
Φ(γ∞rvq, [(1 +
2 − γ∞rvq]−1) (169)
= − log[(1 +
2 − γ∞rvq] + Θ
(170)
= − log[(1 +
2 − γ∞rvq] +
N̄r log(N̄r)
− (N̄r − 1) log(1 +
N̄r) +
N̄r (171)
= B̄ log(2). (172)
Solving for γ∞rvq gives (18). Taking γ
rvq = N̄r +
N̄r and
solving for B̄ gives B̄∗ in (19).
For N̄r ≤ γ∞rvq < N̄r+
N̄r, or 0 ≤ B̄ ≤ B̄∗, we substitute
γ∞rvq−N̄r
γ∞rvq
into (164) to obtain
γ∞rvq,
γ∞rvq − N̄r
γ∞rvq
= log(γ∞rvq − N̄r)− log(γ∞rvq)
γ∞rvq +
γ∞rvq
γ∞rvq − N̄r
(173)
To simplify (173), we let ψ , γ∞rvq − N̄r and re-write (173) as
ψ − N̄r,
ψ + N̄r
= log(ψ)− log(ψ + N̄r)
1 + N̄r + ψ +
(174)
After some manipulation we have
1 + N̄r + ψ +
, (175)
1 + N̄r + ψ +
, (176)
SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 15
1 + N̄r + ψ +
= log(N̄r)− log(ψ)
−(N̄r − 1) log
(177)
Substituting (177) into (174), we obtain
ψ − N̄r,
ψ + N̄r
= ψ − N̄r log
(178)
= γ∞rvq − N̄r − N̄r log(γ∞rvq) + N̄r log(N̄r). (179)
Setting this to B̄ log(2) and simplifying gives (17).
For N̄r < 1, the asymptotic eigenvalue density of
is given by
g(λ) = (1 − N̄r)δ(λ) +
(λ− a)(b − λ)
. (180)
where a and b are given by (28)-(29). Following the same
steps again from (110) gives (18) and (17). This completes
the proof of Theorem 3.
D. Derivation of (41)-(43)
To compute µJ , we first write
JNrj =
1 + ρ
(181)
where υk is the kth eigenvalue of Υ =
As (Nt, Nr,K) → ∞, the empirical eigenvalue distribution
converges to a deterministic function FΥ(t). The asymptotic
mean is given by
µJ = lim
(Nt,Nr,K)→∞
E[JNrj ] =
1 + ρ
dFΥ(t).
(182)
A similar integral has been evaluated in [36, Eq. (6)], and the
result can be directly applied to (182), giving (41).
To compute the variance, we express JNrj differently by
first performing the singular value decomposition H =
VHΣHU
, where VH is the Nr ×Nr left singular matrix,
UH is the Nt × Nr right singular matrix, and ΣH is an
Nr × Nr diagonal matrix. Here we assume that Nt ≥ Nr.
(The result for Nt < Nr can be shown by a similar approach.)
We therefore have
JNrj =
log det (INr + ρΛLj) (183)
log (1 + ρηi) (184)
where Λ = 1
, Lj = U
j UH , and ηi is the ith
eigenvalue of ΛLj . To compute var[J
j ], correlations be-
tween pairs of ηi’s are needed. Although the joint distribution
of eigenvalues is known, it is complicated, so that computing
the variance appears intractable.
To approximate the variance of JNrj , we substitute a Taylor
series expansion for log(1 + δx) into (184) to write
JNrj =
η2i +
η3i + . . . (185)
tr{ΛL} − ρ
tr{(ΛL)2}+ ρ
tr{(ΛL)3}
+ . . . (186)
for ρηmax < 1, where ηmax = maxi ηi, and is the maximum
eigenvalue of HV V †H†/K . Since HV is Nr×K and i.i.d.,
ηmax has asymptotic value (1 +
N̄r/K̄)
2. If K̄/N̄r = 1,
then the condition asymptotically becomes ρ < 1/4 (-6 dB).
Ignoring the terms of order ρ3 and higher, we can approximate
the variance of JNrj at low SNR as
var[JNrj ] ≈ ρ
tr{ΛL}
. (187)
Letting Λ = diag{λi} and lij denote the (i, j)th element of
L, the first term in (187) can be expanded as
var[tr{ΛL}|Λ] =
E[l2ii]− E2[lii]
λiλj (E[liiljj ]− E[lii]E[ljj ]) .
(188)
For a given UH and random unitary V with K = Nr, The-
orem 3 in [7] states that L has a multivariate beta distribution
with parameters Nr and Nt − Nr. (The distribution of L is
not known for general K .) From Theorem 2 in [47], we have
E[lii] =
Nr + 1
Nt + 2
(189)
E[l2ii] =
(Nr + 1)(Nr + 3)
(Nt + 2)(Nt + 4)
(190)
E[liiljj ] =
Nr(Nr + 1)(Nt + 4) + (Nr + 1)(Nt −Nr + 1)
(Nt + 1)(Nt + 2)(Nt + 4)
i 6= j, (191)
for 1 ≤ i, j ≤ Nr. Substituting (189)-(191) into (188) gives
var[tr{ΛL}|Λ] =
N̄2r (1 − N̄r) +O
(N̄r − 1)N̄3r +O
(192)
Taking expectation with respect to Λ, and the large system
limit, we have
EΛ (var[tr{ΛL}|Λ]) → 1− N̄r. (193)
Also, in the large system limit
λ2i →
t2dFΛ(t) =
(194)
λiλj →
tdFΛ(t)
(195)
16 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009
where FΛ(t) is the asymptotic distribution for the diagonal
elements of Λ or, equivalently, the asymptotic eigenvalue
distribution of HH†/Nr.
Substituting (193) into (187), we have
σ2J = lim
(Nr,Nt)→∞
N2r var[J
j ] (196)
≈ ρ2(1− N̄r). (197)
ACKNOWLEDGEMENT
The authors thank the anonymous reviewers for their de-
tailed comments and for pointing out mistakes in the proofs
of Theorems 2 and 3, which appeared in an earlier draft.
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Wiroonsak Santipach (S’00-M’06) received the B.S. (summa cum laude),
M.S., and Ph.D. degrees all in electrical engineering from Northwestern
University, Illinois, USA in 2000, 2001, and 2006, respectively.
He is currently a lecturer at the Department of Electrical Engineering,
Faculty of Engineering, Kasetsart University in Bangkok, Thailand. His
research interests are in wireless communications, and include performance
evaluation of CDMA and MIMO system.
Michael L. Honig (S’80-M’81-SM’92-F’97) received the B.S. degree in elec-
trical engineering from Stanford University in 1977, and the M.S. and Ph.D.
degrees in electrical engineering from the University of California, Berkeley,
in 1978 and 1981, respectively. He subsequently joined Bell Laboratories in
Holmdel, NJ, where he worked on local area networks and voiceband data
transmission. In 1983 he joined the Systems Principles Research Division
at Bellcore, where he worked on Digital Subscriber Lines and wireless
communications. Since the Fall of 1994, he has been with Northwestern
University where he is a Professor in the Electrical Engineering and Computer
Science Department. He has held visiting scholar positions at the Technical
University of Munich, Princeton University, the University of California,
Berkeley, Naval Research Laboratory (San Diego), and the University of
Sydney. He has also worked as a free-lance trombonist.
Dr. Honig has served as an editor for the IEEE Transactions on Information
Theory (1998-2000), the IEEE Transactions on Communications (1990-1995),
and was a guest editor for the European Transactions on Telecommunications
and Wireless Personal Communications. He has also served as a member
of the Digital Signal Processing Technical Committee for the IEEE Signal
Processing Society, and as a member of the Board of Governors for the
Information Theory Society (1997-2002). He is the recipient of a Humboldt
Research Award for Senior U.S. Scientists, and the co-recipient of the 2002
IEEE Communications Society and Information Theory Society Joint Paper
Award.
	Introduction
	Channel Model
	Beamforming with Limited Feedback
	MISO Channel
	Multi-Input Multi-Output (MIMO) Channel
	Numerical Results
	Precoding Matrix with Arbitrary Rank
	Quantized Precoding with Linear Receivers
	Matched filter
	MMSE receiver
	Numerical Results
	Conclusions
	Appendix
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
	Derivation of (??)-(??)
	References
	Biographies
	Wiroonsak Santipach
	Michael L. Honig
