Cone Codebook for Limited Feedback Beamforming over Correlated MIMO Channels
Yating Wu
1
, Yidong Cui
2
and Fei Yu
3
1. School of Communication and Information Engineering
Shanghai University
2. School of Software Engineering
Beijing University of Posts and Telecommunications
3. School of Electronic and Information Engineering
Shenzhen Polytechnic
Abstract—A new beamforming codebook design is proposed
to reduce the feedback rate of quantitized beamforming
MIMO systems over temporally correlated channels. Based
on a first-order Gauss-Markov fading model, we develop an
adaptive cone codebook that can fold or expand to adapt
to the interframe temporal correlations of the channel and
rotate to track the time-varying channel. Through Monte Carlo
simulations, we demonstrate that the proposed cone codebook
based limited feedback scheme outperforms the conventional
scheme with fixed codebook and compresses the feedback bits
without degrading the system performance.
Keywords— MIMO, limited feedback, codebook, beamforming.
I. INTRODUCTION
Multiple-input multiple-output (MIMO) technology offers
significant gains in capacity and robustness of the wireless
communication systems [1][2]. Channel state information
(CSI) at the transmitter helps adjust the transmitted signals to
exploit the benefits offered by MIMO channels, such as spatial
diversity and multiplexing gains. In systems where the forward
and reverse channels are not reciprocal, such as frequency
division duplexing (FDD) systems, the CSI is required to be
fed back from the receiver to the transmitter. Limited feedback,
which quantizes the CSI and sends it from the receiver to
transmitter over a limited-rate feedback channel, is of great
importance and has been an active research area recently [3].
One critical issue of the limited feedback beamforming
systems is the beamforming codebook design problem. For
independent and identically distributed (i.i.d.) channels, the
optimal beamforming vector has been shown to be uniformly
distributed on the unit sphere. Therefore, the codebook design
problem has been reduced to maximizing the minimum dis-
tance between codewords, which is shown to be equivalent
to the Grassmannian line packing problem [4][5]. Grassman-
nian beamforming, although optimal for flat Rayleigh fading
channels, fails to take advantage of the temporal correlation
in time-selective fading scenarios.
Recently, some efforts have been devoted to limited feed-
back for temporally correlated MIMO channels. In [6], the
authors present a differential feedback scheme where only
the channel variation is quantized instead of the whole chan-
nel subspace. The rotation codebook which characterizes the
channel variation is constructed as a function of the temporal
correlation statistics. While in [7], the feedback overhead
is reduced by truncating the transition probabilities of the
channel. Specifically, the channel is modeled as a finite-
state Markov chain and the feedback bits are compressed by
ignoring transitions between Markov states that occur with
small probabilities. However, this method incurs an additional
storage cost at both mobile and the base station, and the
compression ratio decreases with the codebook size. A low-
rate scalar quantization scheme is proposed in [8][9]. By
quantization in the new parameter domain of the channel,
only one bit per parameter is required for CSI feedback.
Nevertheless, the number of the channel parameters affects
the feedback efficiency. In this paper, we propose a limited
feedback scheme based on a simple spherical cone codebook
for beamforming in correlated MIMO channels. The main
objective of the cone codebook design is to improve the limited
feedback performance, or equivalently, to reduce the required
feedback rates by exploiting the temporal correlation. The
proposed cone codebook can fold or expand to a certain degree
according to the Doppler frequency and adaptively rotate its
central axis to track the time-varying channel.
The rest of the paper is organized as follows. Section
II provides the system and signal model. In Section III,
we present the new limited feedback scheme based on the
proposed cone codebook construction algorithm. Monte Carlo
simulations for the proposed scheme are presented in Section
IV. Finally, section V concludes the paper.
We use the following notations. Matrices are denoted by
boldcase capital letters such as H and vectors by bold lower
case letters such as h. (·)
H
denotes the conjugate transpose
of a vector or a matrix and k·k denotes the vector two-norm.
E[·] refers to the expectation operator. C
M
denotes the M -
dimension complex vector space.
II. SYSTEM DESCRIPTION AND SIGNAL MODEL
We consider a limited feedback MIMO system employing
transmit beamforming and receive combining with M transmit
antennas and N receive antennas, as depicted in Fig. 1. The
received signal after combining is given by
y = z
H
Hw
o
s + z
H
n (1)
where s is the transmitted data symbol. w
o
and z are the
beamforming and combining vectors, respectively. Without
loss of generality, we set kzk = 1 and kw
o
k = 1. The additive