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Analysis of Antipodal Signaling (Digital 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. 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

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

sn =1Ts 0 Ts ( ± Eb Tb )dt = ± Eb Tb sn=1Ts0 Ts(± Eb Tb )dt=± Eb Tb
(3)
The variance of the sample is given by
σ2 =1Ts2 No Ts 2 = No 2Ts σ2=1Ts2 No Ts 2 = No 2Ts
(4)

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

μ = 0 S-1 ± Eb Tb = ±S Eb Tb = ± Eb Tb Ts2 μ=0 S-1± Eb Tb =±S Eb Tb =± Eb Tb Ts2
(5)
and the noise variance of that sample is given by
σ2 = S No 2Ts = Tb Ts No 2Ts = No Tb 2Ts2 σ2=S No 2Ts = Tb Ts No 2Ts = No Tb 2Ts2
(6)
Although 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

Pe= Q |μ| σ Pe=Q |μ| σ
(7)
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
(8)

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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