Digital Communications: Fundamentals and Applications, 3rd Edition

Appendix 8A: The Sum of Log-Likelihood Ratios

Following are the algebraic details yielding the results shown in Equation (8.12), rewritten below:

L (d_{1}) ⊞ L (d_{2}) \underset{=}{Δ} L (d_{1} \oplus d_{2}) = {log}_{e} (\frac{e^{L (d_{1})} + e^{L (d_{2})}}{1 + e^{L (d_{1})} e^{L (d_{1})}}) (8A.1)

We start with a likelihood ratio of the APP that a data bit equals +1 compared to the APP that it equals –1. Since the logarithm of this likelihood ratio, denoted L(d), has been conveniently taken to the base e, it can be expressed as

L (d) = {log}_{e} [\frac{P (d = + 1)}{P (d = - 1)}] = {log}_{e} [\frac{P (d = + 1)}{1 - P (d = + 1)}] (8A.2)

so that

e^{L (d)} = [\frac{P (d = + 1)}{1 - P (d = + 1)}] (8A.3)

Solving for P(d = +1) we obtain

e^{L (d)} - e^{L (d)} \times P (d = + 1) = P (d = + 1) (8A.4)

e^{L (d)} = P (d = + 1) \times [1 + e^{L (d)}] (8A.5)

and

P (d = + 1) = \frac{e^{L (d)}}{1 + e^{L (d)}} (8A.6)

Observe from Equation 8A.6 that

P (d = - 1) = 1 - P (d = + 1) = 1 - \frac{e^{L (d)}}{1 + e^{L (d)}} = \frac{1}{1 + e^{L (}}

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Digital Communications: Fundamentals and Applications, 3rd Edition by Fredric J. Harris, Bernard Sklar

Appendix 8A: The Sum of Log-Likelihood Ratios

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