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

The transmission of the linear sum of biphase modulated signals in a time interval is considered for an additive Gaussian white noise channel of bandwidth . Previous analyses consider the case where . In this paper, the probability of error is derived for , 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 signals are independently biphase modulated. It is shown, for this case, that if the channel signal-to-noise ratio increases linearly with , the error probability can be made to go to zero approximately exponentially in for any value of . In the second problem, termed ceded transmission, only of the signals are independently modulated. (The remaining 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 exponentially in if is less than some number . For high signal-to-noise ratio, is greater than , a situation which could not occur if .