Feedback Compression for Correlated Broadcast
Claude Simon∗, Ruben de Francisco†, Dirk T.M. Slock†, and Geert Leus∗
∗TU Delft, Fac. EEMCS, Mekelweg 4, 2628 CD Delft, The Netherlands
†Eur´ ecom Institute, 2229, Route des Crˆ etes, 06560 Valbonne Sophia-Antipolis, France
Abstract—In this paper we apply Predictive Vector Quanti-
zation (PVQ) to quantize time-correlated broadcast channels.
PVQ exploits the time-correlation of the channel to reduce the
quantization error of the channels, and thus to improve the sum
rate of the system. PVQ predicts the actual channel based on a
number of previous channels, and then quantizes the difference
between the prediction and the true channel. In this paper we
show how the corresponding codebooks can be designed, and we
present a prediction strategy. The performance of PVQ for a
broadcast system is depicted through numerical simulations.
Index Terms—Broadcast channels, vector quantization, time-
varying channels, multiuser channels, MIMO systems.
Space division multiple access (SDMA) has emerged in the
last years as an attractive transmission scheme for multiple-
input multiple-output (MIMO) broadcast channels , . It
has been shown to outperform time division multiple access
(TDMA) . The optimal SDMA scheme for the Gaussian
MIMO broadcast channel is dirty-paper coding (DPC) , ,
i.e., the rate region of DPC corresponds to the capacity region
of the channel. Unfortunately, DPC has a high computational
complexity, and is thus difficult to implement. However, zero-
forcing (ZF) beamforming has been lately shown  to reach
asymptotically the same performance as DPC for a high
number of users. Most existing SDMA schemes assume chan-
nel knowledge at the transmitter side. However, in general,
channel state information (CSI) knowledge is only available at
the receiver side, and must be fed back to the transmitter. The
feedback link is generally assumed to be bandwidth limited,
meaning that only a limited number of bits can be fed back
to the transmitter. The CSI must thus be quantized before it
can be fed back to the transmitter.
A low-complexity SDMA scheme that works with limited
feedback is opportunistic SDMA (OSDMA) . OSDMA is an
extension of opportunistic beamforming  to multiple users.
It uses a random set of orthonormal beamforming vectors at
the base station with NT antennas to simultaneously transmit
independent data streams to the NT users with the highest
signal-to-noise ratio (SINR). Several extensions of OSDMA
have been proposed lately ,  to incorporate larger sets
of beamforming vectors, and thus, to improve the performance
The research of the authors at TU Delft was supported in part by NWO-
STW under the VIDI program (DTC.6577)
The research of the authors at Eurecom was supported in part by the
Eurecom Institute, and by the national RNRT project OPUS.
for scenarios with a lower number of users. Even though
the OSDMA algorithms have a good performance for i.i.d.
channels, there exists, to the best of the authors’ knowledge,
no extension of these algorithms to exploit time-correlated
In this paper we present a scheme that uses Predictive
Vector Quantization (PVQ)  to exploit the correlation
between successive channel realizations in order to improve
the quantization, and thus to improve the sum rate of the
system. Further, our scheme does not make any assumptions
on the scheduling function and on the transmission strategy,
which allows for a high flexibility.
Notation: We use capital boldface letters to denote matrices,
and small boldface letters to denote vectors. E(·) denotes
expectation, |A| the cardinality of a set A, and ?a? the l2-
norm of a vector a. Im denotes the m × m identity matrix,
and ⊗ represents the Kronecker product.
II. SYSTEM MODEL
We assume the downlink of a flat-fading multiuser system,
where the base station is equipped with NT antennas, serving
K single-antenna users. Given a set S of NT users scheduled
for transmission, the corresponding data model using linear
beamforming at time instant n is
where yk∈ C is the received symbol of user k, hH
the channel vector of user k, wi∈ CNT×1the beamforming
vector for user i, and sithe data symbol transmitted to user i.
The noise nk∈ C is i.i.d., and zero mean circularly symmetric
complex Gaussian distributed with variance N0.
Although the proposed methods work for more general
channel models, we assume for simplicity that the different
channel vectors are i.i.d., and that the channel correlation is
separable in space and time:
k[n]wi[n]si[n] + nk[n]
k[n − m]) = Rρm
where R is the space-correlation matrix, and ρmis the time-
correlation function. In this work, we will mainly concentrate
on the time-correlation. Exploitation of the space-correlation
for limited feedback in broadcast channels has been studied
1-4244-1370-2/07/$25.00 ©2007 IEEE
The data is transmitted in a block-wise fashion. We assume
a data-rate limited feedback link that can feed back B bits at
the beginning of each block. Further, the feedback is assumed
to be instantaneous and error-free.
We assume that the receivers have achieved perfect CSI
through the use of pilots. The users then quantize the full
channel, or just the channel direction if the feedback of scalars
is permitted, to an element of a codebook C, and feed back
the according index to the base station. The base station then
decides, based on the received feedback, which set of users to
serve, and their corresponding beamforming vector.
The performance of the vector quantization (VQ) step can
be improved by taking the time correlation of the channel into
account. Vector quantizers with memory allow to quantize the
actual channel more efficiently, i.e., the quantization error of
VQ with memory is smaller than the quantization error of
VQ without memory for the same amount of feedback. Even
though there exists a large number of VQs with memory ,
we focus in this paper solely on predictive VQ (PVQ) since its
simplicity makes it a good candidate for practical systems. It
allows to exploit the correlation of the channel by considering
a variable number of previous channels, without an exponential
increase of the storage requirements for the codebooks as is
the case for finite-state vector quantizers.
A. Linear Beamforming
The most common linear beamforming schemes are transmit
matched filtering and zero-forcing (ZF) beamforming. Trans-
mit matched filtering uses the normalized channel vector as
beamforming vector. A scheme with a better performance is
ZF beamforming. It provides a good tradeoff between the high-
complexity schemes with good performance, e.g., DPC, and
schemes like matched filtering.
The different ZF beamforming vectors are calculated based
on the concatenated matrixˆH. The rows ofˆH consist of all
the quantized channelsˆhH
of the users from the set S. The
ZF beamforming vectors are then the normalized columns of
the pseudo-inverse ofˆH.
B. User Selection
The optimal set of active users scheduled for transmission,
denoted as S∗, is selected to maximize the sum rate by an
extensive search over all possible combinations of users
where the signal-to-interference-and-noise ratio (SINR) is cal-
log2(1 + SINRk)
andˆhkis the quantized CSI known to the transmitter.
III. PREDICTIVE VECTOR QUANTIZATION
This section gives an overview of PVQ and its application
to channel quantization of broadcast channels. For simplicity
reasons, we omit the user index here.
PVQ starts by estimating the actual channel h[n] based on
the m previously quantized channelsˆh[n−i],i = 1...m , at
both the base station and the users, resulting in
˜h[n] = P(ˆh[n − 1],ˆh[n − 2],...,ˆh[n − m])
where P(·) denotes the prediction function. The users, who
have full CSI knowledge, then calculate the true error e[n]
between the estimated channel˜h[n] and the true channel h[n]:
e[n] = h[n] −˜h[n]
The error is quantized by finding the entry in the quantization
codebook C with the smallest Euclidean distance to the true
eQ[n] = argmin
?e[n] − c?2.
The quantized error eQ[n] is fed back to the base station, and
the quantized channel at time instant n is then computed as
ˆh[n] =˜h[n] + eQ[n].
The challenge of PVQ is to design the codebook and the
A. Codebook Design
A popular approach to design a codebook for PVQ is the
open-loop approach . It does not have an iterative nature,
and it relies on the assumption that the quantized channels
are a good approximation of the real channels. The codebook
design assumes that the prediction function is known, and it
uses regular VQ without memory on a training set T , where
the different elements of the training set T are the ideal
prediction errors calculated as
eideal[n] = h[n] − P(h[n − 1],h[n − 2],...,h[n − m]). (9)
The application of a memoryless VQ is possible since the
prediction step in (9) removes, in the ideal case, the time
correlation between the channels at different time instances.
Note that the ideal prediction error eideal[n] differs from the
true error e[n] in (6). The true error is calculated as a function
of the previously quantized channels, and thus depends on
the quantization codebook. Using the ideal prediction error
to design the codebooks removes this dependence, hence the
name open-loop approach. Iterative designs, i.e., closed-loop
approaches , only provide a minor gain.
The most common algorithm to design codebooks is the
generalized Lloyd algorithm (GLA) . It is a descent
algorithm , i.e., it reduces the average distortion of the
codebook with every iteration. However, the GLA is not
guaranteed to find the global optimal codebook for non-convex
distortion functions , since it may get trapped in a local
A more robust approach to find good codebooks is a Monte-
Carlo based codebook design . This approach gener-
ates random codebooks, estimates their performance through
Monte-Carlo simulations, and finally keeps the codebook with
the best performance. Even though this approach works well
for small codebooks, it becomes computational expensive for
The optimal design aims at finding a codebook that maxi-
mizes the overal sum rate of the system . However, this
design objective is computationally complex, and it depends
on all the components of the system, e.g., the number of users,
the selected beamforming strategy, the selection function.
To reduce the computational complexity, we focus instead
on codebooks which minimize the average Euclidean distance
between the ideal prediction error, and the quantized prediction
E(?eideal[n] − eideal,Q[n]?2)
eideal,Q[n] = argmin
?eideal[n] − c?2.
B. Prediction Function
The other crucial part in designing the PVQ is the prediction
function. A common technique for PVQ  is vector linear
Based on the previous m known channel vectors we want
to predict the actual vector h[n] using the coefficient matrices
?h[n] = −
Ajh[n − j]
The goal is to minimize the average mean square prediction er-
ror. Using the orthogonality principle, the coefficient matrices
can be derived from
j = 1,...,m
where Rijis the channel correlation matrix
Rij= E(h[n − i]h[n − j]H).
Stacking (13) in matrix form as
the coefficient matrices Ajcan now be found through simple
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Number of Users
PVQ (ZF, f DTf = 3)
PVQ (ZF, f DTf = 3⋅10−1)
PVQ (ZF, f DTf = 3⋅10−2)
Perfect CSI (ZF)
The sum rate for different number of users. (NT= 2, and SNR =
For the channel model presented in Section II, we have that
Rij= Rj−i= Rρj−i. In that case, (15) becomes
If R is assumed diagonal, it is clear that this equation can be
solved for every channel entry separately.
Fig. 1 depicts the sum rate of PVQ with ZF beamforming,
and of OSDMA-LF . We assume a base station with
NT= 2 antennas, K users with SNR = 10 dB, and a data rate
limited feedback link (B = 3 bits). The channel is modeled
through (2) with R = INTand ρm= J0(2πfDTfm) where
J0 is the Bessel function of zeroth-order, fD the Doppler
spread, and Tf the frame length (Jakes’ model ). Thus,
we simply have to simulate different products fDTf. The
algorithm predicts the actual channel based on the last m = 3
channels using polynomial extrapolation of order p = m − 1.
The initial channels are assumed to be known perfectly, which
can be approximated by starting the algorithm with a high-
resolution memoryless VQ. In order to make a fair comparison
to OSDMA-LF possible, we enforce the feedback limitation,
i.e., no scalar SINR feedback is allowed. Thus, we also have
to quantize the SINR feedback of the OSDMA-LF scheme.
The SINR codebook is generated with the GLA using the
mean squared error as distortion function. We simulate all
the possible bit-distributions between SINR quantization and
beamforming indexing, and finally choose the distribution
which results in the highest SINR . We see in Fig. 1
how the performance of PVQ with ZF improves for higher
fDTfvalues, i.e., for scenarios with a higher time correlation
between the channels. Simulations depicting the performance
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Number of Users
Binit = 9 bit
Binit = 6 bit
Binit = 3 bit
(NT= 2, Tf= 10−2s, fD= 30 Hz, B = 3 bits, and SNR = 10 dB)
Influence of the size of the initialization codebook on the sum rate.
of CSI quantization for spatially correlated channels can be
found in .
Fig. 2 shows the influence of the initial quantization of the
first m channels on the average sum rate. The plot compares
the scenario where perfect CSI of the first m channels is
available to scenarios where the first m channels have been
quantized. We see how the sum rate increases after the first
frame for larger codebooks. However, the importance of the
quantization of the first m channels degrades over time, and
all the schemes would converge to the same sum rate after a
We depicted through numerical simulations the benefits of
using PVQ for time-correlated channels. PVQ uses a simple
prediction step to remove the correlation between the channel
to be quantized and the previous channels. This allows to
improve the performance of the quantization step.
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