ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 3, pp. 183–196.
Pleiades Publishing, Inc., 2006.
Original Russian Text
B.S. Tsybakov, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 3, pp. 21–36.
A Diﬀerent Approach to Finding the Capacity
of a Gaussian Vector Channel
B. S. Tsybakov
Institute for Information Transmission Problems, RAS, Moscow
Qualcomm Inc., San Diego, USA
University of California, San Diego, USA
Received October 20, 2005; in ﬁnal form, March 28, 2006
Abstract—The paper considers a Gaussian multiple-input multiple-output (MIMO) discrete-
time vector channel with memory. The problem is to ﬁnd the capacity of such a channel.
It is known that the capacity of Gaussian vector channels with memory was given in . In the
present paper, we show a diﬀerent approach, which uses another deﬁnition of the capacity. For a
channel with n = 2 inputs and outputs, this approach gives an expression for the capacity which
is diﬀerent from that in . The paper shows what the dependence of input signal components
should be to give this capacity. A multidimensional water-ﬁlling interpretation works for the
optimum vector input signal power distribution but cannot work for the description of the input
component dependences. For the case of n ≥ 3 inputs and outputs, we give a lower bound on
the channel capacity.
The paper considers a discrete-time Gaussian vector channel with memory. It gives an expression
for the capacity of such channel with two inputs and two outputs and a lower bound on the capacity
of a channel with an arbitrary number of inputs and outputs. The capacity of Gaussian vector
channels with memory was given in ; here it is denoted by
C. The method of ﬁnding the capacity
in  was to prove that transmission with a small error probability is possible if and only if the
information transmission rate is less than
C. In the current paper we show a diﬀerent approach
to ﬁnding the capacity. Here the capacity, denoted by C, is found as the maximum of mutual
information rate between the channel input (which satisﬁes a given power constraint) and output.
Our consideration gives an expression for the capacity diﬀerent from that in .
To present our results, we need to introduce some notation.
We consider the Gaussian vector channel with memory
with the power constraint
(t) · ξ(t)] ≤ P, (2)
where ξ(t), t =0± 1, ±2,..., is a vector input signal with n components,