Range and Velocity Resolution

For a pulse with bandwidth $W$, the autocorrelation $\int s(t) s^*(t - \tau)dt$ has first null at $\tau = 1/W$.

Target at range $R$: round-trip delay $\tau = 2R/c$. Two targets resolved if $|\tau_1 - \tau_2| > 1/W$, i.e., $|R_1 - R_2| > c/(2W) = \Delta R$. $\blacksquare$

From the ambiguity theorem: first null in $\nu$ at $\nu = 1/T$. Velocity resolution: $\Delta v = c \Delta\nu/(2 f_0) = c/(2 T f_0)$.

Single OFDM symbol duration $T_{\text{sym}} = 1/\Delta f$. Doppler resolution: $1/T_{\text{sym}} = \Delta f$. At $\Delta f = 30$ kHz: $\Delta\nu = 30$ kHz. Velocity resolution: $c \cdot 30{,}000/(2f_0)$. At $f_0 = 5$ GHz: $\Delta v = 900$ m/s. Useless.

$N_{\text{PRI}}$ pulses at interval $T_{\text{PRI}}$. Doppler resolution: $1/(N_{\text{PRI}} T_{\text{PRI}})$. For typical $N_{\text{PRI}} = 16, T_{\text{PRI}} = 100\,\mu$s: $\Delta\nu = 625$ Hz. At 5 GHz: $\Delta v = 18.8$ m/s. Worse than OTFS with same dwell ($T = 1.6$ ms): $\Delta\nu = 625$ Hz — same!

Given the same time-bandwidth product, OTFS and OFDM-pulse- Doppler achieve the same Doppler resolution. The difference is that OTFS does this while carrying data, enabling ISAC. OFDM-pulse-Doppler requires a dedicated radar waveform with no data capacity. $\blacksquare$

For a single-target observation in noise: $F_{\tau\tau} = \rho \iint |d A_s / d\tau|^2 |_{\tau=0, \nu=0}$ $F_{\nu\nu} = \rho \iint |d A_s / d\nu|^2 |_{\tau=0, \nu=0}$ $F_{\tau\nu} = \rho \text{Re}(\int (d A_s/d\tau)(d A_s/d\nu)^*)$

OTFS ambiguity is separable: $A_s = \text{sinc}(W\tau)\text{sinc}(T\nu)$. Fisher matrix is diagonal: $F_{\tau\nu} = 0$. Diagonal entries: $F_{\tau\tau} \propto \rho W^2$, $F_{\nu\nu} \propto \rho T^2$.

$\sigma_\tau^2 \geq 1/F_{\tau\tau} = c_1/(\rho W^2)$. $\sigma_\nu^2 \geq 1/F_{\nu\nu} = c_2/(\rho T^2)$. Translate to range and velocity via $\tau = 2R/c$ and $\nu = 2vf_0/c$: the claimed CRLB. $\blacksquare$