ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 2, pp. 89–97.
Pleiades Publishing, Inc., 2011.
Original Russian Text
B.S. Tsybakov, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 2, pp. 7–16.
On a Model and Capacity of MIMO Channels
B. S. Tsybakov
Qualcomm Inc. San Diego, USA
University of California, San Diego, USA
Received July 31, 2007; in ﬁnal form, June 14, 2010
Abstract—A model of a MIMO fading channel is considered which does not assume that
the discrete-time axis is divided into long intervals with a constant channel. It assumes only
that there are possible states (or subchannels) of the channels, and at each moment both the
transmitter and receiver know the channel state. The average input power in the subchannels
can be diﬀerent and not equal to the average input power over the full time axis. We obtain a
lower bound for the capacity of the considered vector channel. Also, we consider a vector channel
with one transmitting and two or more receiving antennas. We obtain optimum distributions
of the average power over the subchannels and lower bounds for the channel capacity.
There are many papers that used the variable-channel model for ﬁnding the fading channel
capacity (see, for example, a survey [1, Section II/C/1]). According to the variable-channel model,
the channel works for a relatively long time as a constant memoryless channel (i.e., a channel without
fading), and during this time both the transmitter and receiver know the channel parameters.
Long time intervals follow one after another. “Relatively long time” means the following: by
using optimal coding for a constant channel with known parameters in this long time interval, it
is possible to achieve in this interval a transmission rate that is very close to the capacity of a
constant channel. In every such long time interval, the channel is still Gaussian, and the channel is
ergodic over the full time axis in the considered variable-channel model. The average input-signal
power in the model is the same in each long time interval, and it equals the average input-signal
power over the full time axis.
In the present paper we consider a diﬀerent model of a fading channel. The model is more
general than the variable-channel model, since it does not assume that the discrete-time axis is
divided into long intervals with constant channels. It assumes only that there are possible states
of the channel, and in each time moment both the transmitter and receiver know the channel state.
The model does not require the channel staying in a given state for a long time interval. We
assume only that if we consider the time moments when the channel is in a given state, over these
moments the channel can be interpreted as a constant channel with parameters known to both the
transmitter and receiver. Thus, there are diﬀerent constant channels, called subchannels.Asin
the variable-channel model, the channel in the considered model must be ergodic, but the average
input-signal power in the subchannels can be diﬀerent and not equal to the average input-signal
power over the full time axis. A fading channel changing its input-signal power depending on a
state from a ﬁnite set is used in the CDMA wireless system (see [2, Ch. 10]).
A similar “reasonable model for a slow varying channel” is presented in  for a speciﬁc case of
a Gaussian nonvector channel with an inﬁnite set of states. That paper contains a deﬁnition of a