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 stst corresponding to a single bit with the value of the bit (0 or 1) unknown at the receiver. Often we are concerned with the detection of this signal in the presence of additive white, Guassian noise (AWGN). The noisy signal rt = st + nt rt=st+nt is sampled near the front end of the digital receiver, producing samples rn rn taken every TsTs seconds. Assuming that the sampling clock is synchronized with the symbol clock (an assumption that has little effect for TsTbTsTb), then each sample will have a mean value given by

The variance of the sample is given by

This sampled signal is typically passed through a "matched filter" prior to "detection." The matched filter for the rectangular pulse shape is given by hn= s1-nhn=s1-n where s1ns1n is the sampled version of the analog signal s1 t s1t. . Then the output of the matched filter sn =hn *rnsn=hn*rn contains a 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 S= Tb Ts S=Tb Ts samples. After application of the matched filter, the signal sampled at "sample time" SS has a mean value of

and the noise variance of that sample is given byAlthough the mean is sensitive to observing the sample at precisely sample-time SS, the noise variance is not.

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

Substituting the values above, we find that the optimal bit-error probability for binary antipodal signaling 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.

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This work is licensed by Christopher Schmitz under a Creative Commons Attribution License (CC-BY 2.0), and is an Open Educational Resource.

Last edited by Christopher Schmitz on Jul 17, 2009 8:54 am -0500.