Discretization of Continuous Valued Random Variables. In the analysis of the coding theorem, we need to consider discrete valued random variables at the legitimate receivers. We adapt following the technique outlined in [36, pp. 50-51]. • The channel estimates hˆBA and hˆAB are discretized as follows. Let us define: I1 = {−j∆1, −(j − 1)∆1, . . . , (j − 1)∆1, j∆1}, (55) where j is an integer and we select ∆1 = √1 . We find elements in I1 closest to the real and imaginary parts of hˆAB and denote ΣhˆAB Σ as the resulting quantization. We define ΣhˆBAΣ j in a similar fashion. • We discretize xA(t) and xB(t) to [xA(t)]k and [xB(t)]k respectively, whose real and imaginary parts take values in the set I2 = {−k∆2, −(k − 1)∆2, . . . , (k − 1)∆2, k∆2}, (56) Σ where k is an integer and ∆2 = √1 . We select [xA(t)]k to be the closest such point to xA(t) with |[xA(t)]k| ≤ |xA(t)|. Thus we have that E Σ|[xA(t)]k| ≤ E[|xA(t)|2] ≤ P2. We define [xB(t)]k in an analogous manner. • We also discretize the channel output at each legitimate receiver. Note that from (1), we have yB,k(t) = hAB(t)[xA(t)]k + nAB(t). Note that yB,k(t) is continuous valued even through [xA(t)]k is discrete. We discretize yB,k(t) to [yB,k(t)]l whose real and imaginary parts take values over the interval I3 = {−l∆3, −(l − 1)∆3, . . . , (l − 1)∆3, l∆3} (57) l where l is an integer and ∆3 = √1 . AB Σh Σ Note that since we use scalar quantization and the sequences hˆK and hˆK are sampled i.i.d. the sequences ΣhˆK Σj and BA AB ˆK BA j
Discretization of Continuous Valued Random Variables. In the analysis of the coding theorem, we need to consider discrete valued random variables at the legitimate receivers. We adapt following the technique outlined in [36, pp. 50-51]. • The channel estimates hˆBA and hˆAB are discretized as follows. Let us define: I1 = {−j∆1, −(j − 1)∆1, . . . , (j − 1)∆1, j∆1}, (55) j where j is an integer and we select ∆1 = √1 . We find elements in I1 closest to the real and imaginary parts of hˆAB and denote hhˆAB i as the resulting quantization. We define hhˆBAi in a similar fashion. • We discretize xA(t) and xB(t) to [xA(t)]k and [xB(t)]k respectively, whose real and imaginary parts take values in the set I2 = {−k∆2, −(k − 1)∆2, . . . , (k − 1)∆2, k∆2}, (56) where k is an integer and ∆2 = √1 . We select [xA(t)]k to be the closest such point to xA(t) with |[xA(t)]k| ≤
