Magnitude of the backward-map phase shift in practical encoders

Determine whether practical encoders with real-linear, non-holomorphic backward maps can produce a large phase shift rg c] in the descent condition for scalar channel aging.

Background

The scalar conditional-mean analysis assumes a complex-linear local backward map, but practical encoders may use real parameters or non-holomorphic normalization and therefore induce only a real-linear map. In that setting, the descent criterion depends not only on the channel correlation coefficient $\rho$, but also on the complex quantity $c=\|u_P\|^2+j\langle u_P,v_P\rangle_{\mathbb R}$, whose phase shifts the boundary for mean descent.

The paper reports that the tested real-linear example has a very small phase, $|\arg c|\le 0.11^\circ$, but leaves unresolved whether practical encoder architectures can generate substantially larger values.