  The capacity of the point-to-point vector Gaussian channel under the peak
power constraint is not known in general. This paper considers a simpler
scenario in which the input signal vector is forced to have a constant envelope
(or norm). The capacity-achieving distribution for the non-identity $2\times 2$
MIMO channel when the input vector lies on a circle in $\mathbb{R}^2$ is
obtained and is shown to have a finite number of mass points on the circle.
Subsequently, it is shown that the degrees of freedom (DoF) of a full-rank $n$
by $n$ channel with constant envelope signaling is $n-1$ and it can be achieved
by a uniform distribution over the surface of the hypersphere whose radius is
defined by the constant envelope. Finally, for the 2 by 2 channel, the power
allocation scheme of the constant envelope signaling is compared with that of
the conventional case, in which the constraint is on the average transmitted
power. It is observed that when the condition number of the channel is close to
one, both schemes have a similar trend while this is not the case as the
condition number grows.
