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Analysis of Antipodal Signaling (Analog Receiver)

Module by: Christopher Schmitz. E-mail the author

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. 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

s1 t = + Eb Tb ; 0 ≤ t ≤ Tb s1t=+ Eb Tb ;0≤t≤Tb
(1)
s0 t = - Eb Tb ; 0 ≤ t ≤ Tb s0t=- Eb Tb ;0≤t≤Tb
(2)

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 typically passed through a "matched filter" prior to "detection."

The matched filter for the rectangular pulse shape is given by ht= s1-tht=s1-t. Then the output of the matched filter st =ht *rtst=ht*rt 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 integration over TbTb seconds. After application of the matched filter, the signal sampled at time TbTb has a mean value of

μ=± Eb Tb μ=± Eb Tb
(3)
and the noise variance of that sample is given by
σ2= No Tb 2 σ2= No Tb 2
(4)
Although the mean is sensitive to sampling at precisely TbTb, the noise variance is not. The probability of error of an antipodal signal sampled in AWGN is given by
Pe= Q |μ| σ Pe=Q |μ| σ
(5)
Substituting the values above, we find that the optimal bit-error probability for binary antipodal signaling is given by
Pe= Q 2 Eb No Pe=Q 2 Eb No
(6)

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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