Chapter Summary

Chapter Summary

Key Points

• 1.

  The ambiguity function is the radar signature of a waveform. Defined as $A_s(\tau, \nu) = \int s(t) s^*(t-\tau) e^{-j 2\pi\nu t}\,dt$, it characterizes the matched-filter output for hypothesized $(\tau, \nu)$ offsets. Its shape — and in particular its sidelobe structure — determines radar resolution and ambiguity.

• 2.

  Woodward's volume theorem is the fundamental constraint. $\iint |A_s|^2\,d\tau\,d\nu = E_s^2$: the ambiguity volume is fixed by waveform energy. Narrow main lobes require sidelobe energy to be distributed thinly. Different waveforms position themselves differently on this volume-conserving constraint.

• 3.

  OTFS has the thumbtack ambiguity. For the OTFS transmit signal, $A_s(\tau, \nu) \approx \text{sinc}(W\tau)\,\text{sinc}(T\nu)$ — a sharp peak at the origin with first nulls at $(1/W, 1/T)$. Sidelobes at $-13$ dB (rectangular) or $-43$ dB (Hamming). This is ideal for joint range-velocity estimation.

• 4.

  Range resolution is $c/(2W)$; velocity resolution is $c/(2T f_0)$. These information-theoretic limits apply to any waveform. Both OTFS and OFDM-pulse-Doppler achieve them at equal $(W, T)$ resources. OFDM single-symbol cannot resolve velocity at all ($\Delta v = 900$ m/s for 5G NR).

• 5.

  OTFS achieves the CRLB with a single coherent frame. For a single target at SNR $\rho$: $\sigma_R^2 \geq c^2/(8\pi^2 W^2 \rho)$ and $\sigma_v^2 \geq c^2/(8\pi^2 T^2 f_0^{2} \rho)$. OTFS's thumbtack ambiguity gives a diagonal Fisher information matrix — maximum estimation efficiency.

• 6.

  Unambiguous region = DD grid. $R_{\max} = cM/(2W)$, $v_{\max} = cN/(4Tf_0)$. $MN$ resolution cells cover the unambiguous $(R, v)$ plane. Oversampling by $\beta$ extends the region by $\beta$ at the cost of $\beta$-fold more compute. For terrestrial ISAC: $\beta = 1$ suffices; for LEO: $\beta \geq 16$.

• 7.

  OTFS is the natural ISAC waveform. At equal $(W, T)$ resources, OTFS matches OFDM-pulse-Doppler's radar performance while carrying data continuously. OFDM requires dedicated radar mode (sacrificing data) to achieve comparable Doppler resolution. The structural difference: OTFS's signal space is the DD grid — the native radar basis. OFDM's is the TF grid — the wrong basis for Doppler.