Prerequisites & Notation

Before You Begin

Chapters 4-9 built OTFS under the idealizing assumption of integer delay-Doppler paths. Real channels rarely oblige. This chapter confronts the gap: physical Doppler shifts generically do not align to the DD grid, creating inter-Doppler interference that degrades detection and estimation. We quantify the gap and develop mitigations.

Discrete DD input-output relation(Review OTFS Ch. 4)
Self-check: Can you state the integer-Doppler form $Y_{DD}[\ell, k] = \sum_i h_i\,X_{DD}[(\ell-\ell_i)\bmod M, (k-k_i)\bmod N] + W_{DD}$ ?
OTFS detection: MMSE, MP, LCD(Review OTFS Ch. 8)
Self-check: Can you identify which detectors degrade gracefully under model mismatch?
DD diversity theorem(Review OTFS Ch. 9)
Self-check: Do you recall that the diversity order is $\min(P, k_{\max})$ under fractional Doppler?
Embedded pilot channel estimation(Review OTFS Ch. 7)
Self-check: Can you explain why integer-Doppler assumption is convenient for threshold-based path detection?
Basis expansion models (BEM) for time-varying channels(Review FSI Ch. 11)
Self-check: Have you seen the complex-exponential BEM for approximating $h(t)$ over a block?

Notation for This Chapter

Symbols introduced in this chapter.

Symbol	Meaning	Introduced
$\epsilon_i$	Fractional Doppler offset: $\nu_i T = k_i^{\text{int}} + \epsilon_i$ , $\|\epsilon_i\| < 1/2$	s01
$\epsilon_i^{(\tau)}$	Fractional delay offset: $\tau_i W = \ell_i^{\text{int}} + \epsilon_i^{(\tau)}$	s01
$K_{\text{IDI}}(\epsilon)$	Inter-Doppler interference kernel (sinc-like function of fractional offset)	s02
$\rho_{\text{IDI}}$	IDI power fraction: energy leaked to neighboring Doppler bins	s02
$Q_{\text{BEM}}$	Number of basis functions in the BEM approximation	s03
$M_{\text{os}}, N_{\text{os}}$	Oversampled delay and Doppler dimensions: $M_{\text{os}} = \alpha M, N_{\text{os}} = \beta N$	s04
$w[k]$	Doppler-axis window (e.g., Hamming, Blackman) for sidelobe suppression	s03

← Ch 9 Integer vs Fractional Doppler