High-speed binary data transmission over the additive band-limited Gaussian channel

The transmission of the linear sum of
m
biphase modulated signals in a time interval
T
is considered for an additive Gaussian white noise channel of bandwidth
W
. Previous analyses consider the case where
m \leq n \equiv 2WT
. In this paper, the probability of error is derived for
m > n
, a situation which arises when the channel bandwidth is insufficient to support the data rate. Two distinct problems are considered. In the first, termed uncoded transmission, all
m
signals are independently biphase modulated. It is shown, for this case, that if the channel signal-to-noise ratio increases linearly with
n
, the error probability can be made to go to zero approximately exponentially in
n
for any value of
m/n
. In the second problem, termed ceded transmission, only
k < m
of the signals are independently modulated. (The remaining
(m - k)
signals carry redundant information.) By using a suboptimum receiver, it is shown that for a fixed channel signal-to-noise ratio the error probability goes to
0
exponentially in
n
if
k/n
is less than some number
C^{\ast}
. For high signal-to-noise ratio,
C^{\ast}
is greater than
1
, a situation which could not occur if
m \leq n
.