Chapter Summary

Chapter 10 Summary: OFDM and OTFS Sensing

Key Points

• 1.

  OFDM sensing model: OFDM pilots serve as the probing signal. After data-symbol compensation, the 2D-FFT across subcarriers (range) and symbols (Doppler) produces the range-Doppler map. The OFDM channel matrix IS the sensing operator $\mathbf{A}$ --- a partial 2D Fourier matrix with Kronecker structure.

• 2.

  Resolution limits: Range resolution $\Delta R = c/(2W)$ depends on total bandwidth; Doppler resolution $\Delta v = \lambda/(2 M T_{\mathrm{sym}})$ depends on the CPI length. The cyclic prefix imposes a hard limit on maximum unambiguous range: $R_{\max} = c T_{\mathrm{cp}} / 2$.

• 3.

  OFDM ambiguity function: A product of two Dirichlet kernels with $-13.2$ dB sidelobes. Random data symbols create a near-ideal thumbtack (separable range-Doppler). Windowing reduces sidelobes at the cost of wider main lobe.

• 4.

  ICI at high Doppler: When $\nu / \Delta f \gtrsim 0.1$, inter-carrier interference degrades OFDM sensing quadratically. This is the fundamental limitation motivating OTFS.

• 5.

  OTFS modulation: Operates in the delay-Doppler domain via the Zak/symplectic Fourier transform. The delay-Doppler channel has exactly $K$ non-zero entries for $K$ targets, regardless of velocity. Implemented as a pre/post-processing layer on OFDM hardware.

• 6.

  OTFS sensing matrix: Block-circulant structure enabling $O(N \log N)$ matrix-vector products via 2D-FFT. Fractional Doppler causes leakage that must be handled by oversampling or parametric estimation.

• 7.

  Waveform comparison: FMCW (dedicated sensing, low PAPR, no data), OFDM (ISAC-native, ICI at high Doppler), OTFS (ISAC-native, robust to high Doppler), PMCW (low PAPR, code-dependent sidelobes). The waveform determines the structure of $\mathbf{A}$.

• 8.

  Golden thread: The channel estimation problem in OFDM/OTFS is mathematically identical to the radar imaging problem --- both recover parameters of complex sinusoids from structured measurements. The waveform choice determines which structure $\mathbf{A}$ has, and this structure determines what reconstruction algorithms are efficient.