Cutoff Rate of MIMO Systems in Rayleigh Fading Channels With Imperfect CSIR and Finite Frame Error Probability

Using the random coding argument, in this paper, we derive the effective cutoff rate
R

_E
of multiple-input-multiple-output (MIMO) space-time (ST) codes in Rayleigh fast-fading channels with imperfect channel state information at the receiver (CSIR). In contrast to conventional cutoff rate analysis, where the codeword length
N
is infinity and the frame error probability P
_f
is implicitly zero, we loosen the definition to include the more realistic case of finite
N
and finite
P

_f
. Using the tight upper and lower bounds on the pairwise error probability (PEP) in the work of Li and Kam, we are able to derive in turn tight lower and upper bounds on the cutoff rate of MIMO ST coding systems. The results are very general and can be applied to any linear modulation scheme, like
M
-ary phase-shift keying (MPSK) and
M
-ary quadrature-amplitude modulation (MQAM). Numerically, we found that for the two-transmit and two-receive antenna configuration, the nonasymptotic segments of our cutoff rate upper bounds for four, 16, and 64 MQAM actually coincide with the ergodic capacity curve. Furthermore, we found that the performance of the Smart-Greedy codes proposed in the seminal work of Tarokh falls within the range predicted by our cutoff rate bounds. Finally, we show in this paper that for MIMO systems employing pilot-symbol-assisted channel estimation, the asymptotic cutoff rate no longer linearly increases with the number of transmit antennas. Instead, for a normalized fade rate of
fD
and a signal constellation of size
M
, the maximum effective cutoff rate is (8
f

_D
)
^-1
log
₂

M
, which is achieved when the number of transmit antennas is the integer closest to 1/(4
f

_D
) . The result suggests that with a large antenna array and high user mobility, a more bandwidth-efficient channel-estimation strategy is desired.