Asymptotic Modulation Efficiency (AME)

For the ME in (11), the AME can be calculated by

η∞ = lim γ→∞ 2

Q −1

Q((1− √ ζ)γ)+Q((1+ √ ζ)γ) 2

2 γ . (14)

With (13) and (14), it shows that AME indicates how fast ESNR is approaching SNR when γ → ∞. This can be expressed by η∞ = ∂η(γ) ∂γ |γ=∞ . (15)

The AME for QPSK/QPSK hierarchical modulation can also be found in Fig. 8, where they are the points when SNR becomes larger and larger. The AME of a hierarchical modulation only depends on its signal constellation design and is independent to the background noise.