In the paper, the m chosen according to Eq. (21) makes \frac{1-\bar{α_m}}{\bar{α_m}} close to 2σ^2, which makes \bar{α_m} approximately equal to 1/(1+2σ^2).
However, the KL divergence in Eq. (19) approaches 0 when \bar{α_m} = 1/(1+σ^2), which mismatches with the \bar{α_m} using m chosen according to Eq. (21). As a consequence, the m obtained by Eq. (21) cannot guarantee the KL divergence converge to 0.
I wonder if the coefficient of noise term should be 2σ^2 rather than σ^2 in Proposition 1, Eqs. (4), (11) (12) and so on. In AWGN case for the convenience of analysis, the complex symbol to be transmitted x_c has average energy of unity, and the complex channel noise has average energy of 2σ^2. The real part of the received signal has average energy of (0.5 + σ^2), thus after normalization and reshape, the y_r ∈ R^{2k} has unity average energy, of which 1/(1+2σ^2) belongs to the signal and (2σ^2)/(1+2σ^2) belongs to noise.
In the paper, the m chosen according to Eq. (21) makes \frac{1-\bar{α_m}}{\bar{α_m}} close to 2σ^2, which makes \bar{α_m} approximately equal to 1/(1+2σ^2). However, the KL divergence in Eq. (19) approaches 0 when \bar{α_m} = 1/(1+σ^2), which mismatches with the \bar{α_m} using m chosen according to Eq. (21). As a consequence, the m obtained by Eq. (21) cannot guarantee the KL divergence converge to 0. I wonder if the coefficient of noise term should be 2σ^2 rather than σ^2 in Proposition 1, Eqs. (4), (11) (12) and so on. In AWGN case for the convenience of analysis, the complex symbol to be transmitted x_c has average energy of unity, and the complex channel noise has average energy of 2σ^2. The real part of the received signal has average energy of (0.5 + σ^2), thus after normalization and reshape, the y_r ∈ R^{2k} has unity average energy, of which 1/(1+2σ^2) belongs to the signal and (2σ^2)/(1+2σ^2) belongs to noise.