34) By taking the logarithm of the above likelihood ratio, a useful metric known as Log-Likelihood Ratio (LLR) is obtained, which is a real-valued number representing a soft decision output of a detector. (9-35) where L(x)d) is the LLR of test statistics X obtained by measurement of the channel output X under the condition that either d = +1 or d=-1 may have been transmitted and L(d) denotes the a priori LLR of the data bit d. Introduction of a decoder would improve the reliability of the above decision making process. For a systematic code, it can be shown that the LLR (soft output) out of the decoder can be written as Loutput(d) = Linput(d) + Le (d) where Linput(d) is the LLR of the data bit output of the detector (input to the decoder) and Le(d) represents additional information that is obtained from the decoding process. The output sequence of a systematic decoder consists of values representing data and parity bits. Thus, the decoder LLR can be decomposed into a data component that is associated with the detector measurement and the extrinsic component that is represented by the decoder contribution due to parity. The soft decision Loutput (d) is a real number that provides a hard decision, well as the reliability of that decision. The sign of Loutput (d) denotes the hard decision, i.e., for positive values of Loutput (d) decide d = +1 and for negative values of Loutput (d) decide d = -1. The magnitude of Loutput (d) denotes the reliability of the decision [12-14]. Turbo decoding relies on the exchange of probabilistic information between the two soft-input soft-output decoders shown in Figure 9-41. The extrinsic information is the result of the decoder's estimation of bit d, but not taking its own input into account. It is precisely this extrinsic information that is exchanged iteratively between the two soft-decision decoders during the decoding process. Subtracting the decoder's input from its output prevents the decoder from acting as a positive feedback amplifier and introduces stability in the feedback