Summary: This module presents an analysis of Antipodal Signaling, beginning with rectangular pulses and then extending the analysis to arbitrary pulse shapes and receiver filters. It mirrors the analog module, but deals specifically with the process as seen within a digital receiver implementation. The effects of intersymbol interference (ISI) on the bit-error rate will be included in a subsequent module.

Antipodal signaling uses a single pulse shape to represents bit values of 0 and 1. Often, but not always, the basic pulse waveform will have a positive mean value. This waveform represents a 1 and the negative of the same pulse shape represents a 0. For rectangular pulse shaping, this would be represented as

Suppose we send the waveform

This sampled signal is
typically passed through a "matched filter" prior to "detection."
The matched filter for the rectangular pulse shape is given by
*sufficient statistic* for the determination of the bit that was sent, that is, it has not lost any information contained in the original received signal pertaining to a "best estimate" of which bit was sent.
The matched filter for rectangular pulse shaping is equivalent to summation over

The probability of error of an antipodal signal sampled in AWGN is given by

This well-known formula for the probability of bit error holds when a single, isolated bit is transmitted. It also holds when a continuous stream of bits are transmitted provided that neighboring bits have no effect on the value of μ for each bit as it is being detected.