OTFS as the Natural ISAC Waveform

Define two functions on the DD grid:

• Data grid $X_{DD}[\ell, k]$: QAM symbols transmitted as OTFS.
• Target-scene grid $\mathcal{T}_{DD}[\ell, k]$: indicator function of target presence at DD cell $(\ell, k)$ (1 if target, 0 if no target); or equivalently $|h_i|^2$ values at target positions.

At the receiver, both grids are observed on the same DD signal: $$Y_{DD}[\ell, k] \;=\; h \star\star X_{DD}[\ell, k] \;+\; W_{DD}[\ell, k].$$ Sensing: estimate the DD-kernel $h(\tau, \nu)$ (positions of non-zero entries). Data detection: recover $X_{DD}$.

Both problems operate on the same observation using different likelihood models. OTFS's structural innovation is that they are duals of each other: the matrix-inversion that recovers data is the adjoint of the correlation that estimates targets. In practice, a joint MAP estimator handles both tasks with $\sim 2\times$ the complexity of either alone.

For OTFS-ISAC with random QAM data and a single target at parameters $\Theta = (\tau, \nu)$, the Fisher information matrix $J(\Theta)$ averaged over data realizations is approximately diagonal: $$J(\Theta) \;\approx\; \rho\,\text{diag}(c_\tau^2,\,c_\nu^2), \qquad c_\tau = 2W/c,\,c_\nu = 2f_0/c.$$ The resulting CRLB for range and velocity estimation: $$\sigma_R^2 \;\geq\; \frac{c^2}{8\pi^2 W^2 \rho}, \qquad \sigma_v^2 \;\geq\; \frac{c^2}{8\pi^2 T^2 f_0^{2} \rho}.$$ Same as dedicated radar — the data randomness does not hurt the sensing task asymptotically.