On the Multiplexing Gain of Discrete-Time MIMO Phase Noise Channels

The capacity of a point-to-point discrete-time multi-input-multiple-output (MIMO) channel with phase uncertainty (MIMO phase noise channel) is still open. As a matter of fact, even the pre-log (multiplexing gain) of the capacity in the high signal-to-noise ratio (SNR) regime is unknown in general. We make some progress in this direction for two classes of such channels. With phase noise on the individual paths of the channel (model A), we show that the multiplexing gain is 1/2, which implies that the capacity does not scale with the channel dimension at high SNR. With phase noise at both the input and output of the channel (model B), the multiplexing gain is upper-bounded by 1/2 min{n
_t,
(n
_r
- 2)
⁺
+ 1}, and lower-bounded by 1/2 min {n
_t
, ⌊n
_r
+1/2⌋}, where n
_t
and n
_r
are the number of transmit and receive antennas, respectively. The multiplexing gain is enhanced to 1/2 min{n
_t
, n
_r
} without receive phase noise, and to 1/2 min {2n
_t
-1, n
_r
} without transmit phase noise. In all the cases of model B, the multiplexing gain scales linearly with min {n
_t
, n
_r
}. Our main results rely on the derivation of non-trivial upper and lower bounds on the capacity of such channels.