Characterize residual chirp effects under nonzero-offset localized allocation

Characterize how the additional linear phase contributed by the modulator chirp rate c_2 affects the DAFT-spread affine frequency division multiplexing transmitter when localized allocation starts at a nonzero bin k_0\neq 0.

Background

The paper reduces the interaction between the precoder and modulator chirp rates to a residual chirp rate \beta=c_2-\delta_2 for single-user, zero-offset localized allocation, where the allocation occupies the first M bins. Under this assumption, the two quadratic chirp phases combine directly and the resulting system can be analyzed through a unitary circulant spreading kernel.

For an allocation beginning at bin k_0\neq0, the modulator chirp rate c_2 contributes an additional linear phase that is not incorporated into the paper’s residual-chirp reduction. The authors explicitly defer analysis of this case, leaving unresolved how the added phase modifies the transmitter representation and potentially the PAPR–BER behavior.