Ferkans — Interactive Telecom Tutor

Where Resolution Breaks

A radar waveform's resolution and accuracy (Sections 3-4) apply only within its unambiguous region — the range-Doppler area where targets can be unambiguously located. Beyond this region, the ambiguity function repeats (aliasing) and two different physical targets produce the same measurement. This section defines the unambiguous region for OTFS and explains its connection to the DD grid extent.

Definition:
Unambiguous Range and Velocity

The unambiguous range $R_{\max}$ is the maximum range at which a target can be detected without ambiguity (i.e., without wrapping to another bin). For OTFS: $R_{\max} \;=\; \frac{c M}{2 W} \;=\; \frac{c}{2 \Delta f}\,\text{(per OFDM symbol)} \cdot \frac{M}{N}\,\text{(for full frame)}.$

The unambiguous velocity $v_{\max}$ is the maximum velocity that can be detected without Doppler aliasing: $v_{\max} \;=\; \frac{c N}{4 T f_0} \;=\; \frac{c}{4 T_{\text{sym}} f_0}.$ (Factor 2 from needing both positive and negative velocities representable.)

$R_{\max}$ is the product of range resolution $\Delta R$ and the number of delay bins $M$ . $v_{\max}$ is the velocity resolution $\Delta v$ times the half-number of Doppler bins $N/2$ . The product $R_{\max} v_{\max}$ defines the unambiguous area.

Theorem: Unambiguous Area = MN $\times$ Resolution Cell

The unambiguous range-Doppler area is $R_{\max} \cdot 2 v_{\max} \;=\; \Delta R \cdot \Delta v \cdot MN.$ There are $MN$ resolution cells — one for each DD grid cell — within the unambiguous region.

For an OTFS frame with $(M, N) = (512, 16)$ at $W = 20$ MHz, $T = 3.2$ ms, $f_0 = 5$ GHz:

$\Delta R = 7.5$ m, $\Delta v = 9.4$ m/s.
$R_{\max} = 512 \cdot 7.5 = 3.84$ km.
$v_{\max} = 8 \cdot 9.4 = 75$ m/s (up to this velocity, unambiguous).

The unambiguous region is the "grid" — each cell is one resolution cell. Outside the grid, the ambiguity function folds (aliases) and targets become ambiguous with targets in the grid's interior. This is the radar analog of the Nyquist unambiguous region.

For automotive applications: $R_{\max} = 3.84$ km (covers typical 200 m sensing range), $v_{\max} = 75$ m/s (covers typical vehicle speeds < 60 m/s). Unambiguous for all realistic automotive scenarios.

For LEO satellite radar: $v_{\max}$ may be insufficient (orbital velocities $\sim 7{,}500$ m/s). Oversampling ( $\beta > 1$ ) extends the unambiguous region at the cost of more grid cells.

Proof

Unambiguous Range

DD grid spans $M$ delay bins of spacing $\Delta\tau = 1/W$ . Total delay extent: $M/W$ . Range: $R \in [0, cM/(2W)] = [0, R_{\max}]$ .

Unambiguous Doppler

DD grid spans $N$ Doppler bins of spacing $\Delta\nu = 1/T$ . Positive Doppler: $N/2$ bins. Max Doppler: $N/(2T)$ . Velocity: $v \in [-cN/(4Tf_0), cN/(4Tf_0)]$ .

Area

Area = $R_{\max} \cdot 2 v_{\max} = cM/(2W) \cdot cN/(2Tf_0) = (c^2 MN)/(4 W T f_0)$ . Also: $\Delta R \cdot \Delta v \cdot MN = (c/(2W)) \cdot (c/(2Tf_0)) \cdot MN$ . Match. $\blacksquare$

Key Takeaway

The unambiguous region is the DD grid. $MN$ resolution cells, each of size $\Delta R \times \Delta v$ . The OTFS waveform's capacity (data) is $MN$ QAM symbols; the radar's unambiguous capacity is $MN$ range-Doppler cells. The same grid serves both — the fundamental ISAC unification.

Example: Unambiguous Region for Automotive ISAC

For automotive ISAC at 77 GHz with $W = 100$ MHz, $T = 3$ ms, $(M, N) = (150, 14)$ (5G-aligned), compute the unambiguous region and assess its adequacy for typical driving scenarios.

Solution

Resolutions

$\Delta R = 1.5$ m. $\Delta v = 0.65$ m/s.

Unambiguous bounds

$R_{\max} = M \cdot \Delta R = 225$ m. $v_{\max} = N/2 \cdot \Delta v = 4.6$ m/s.

Automotive assessment

Range 225 m: covers typical highway braking distance (< 100 m at 130 km/h). Adequate.
Velocity $\pm 4.6$ m/s: not adequate. Highway speeds up to $\pm 40$ m/s.

Fix: increase $N$

For $v_{\max} = 50$ m/s at same resolution: $N/2 = 50/0.65 = 77$ . So $N = 154$ needed. Frame duration: $T = N T_{\text{sym}} = 154 \cdot 33\,\mu$ s = 5.1 ms. Slightly longer frame. Acceptable.

Resulting parameters

$(M, N) = (150, 154)$ , $T = 5$ ms, grid $MN = 23{,}100$ cells. Unambiguous range: 225 m, velocity: $\pm 50$ m/s. Data rate: $\sim 23{,}000/5\text{ms} \cdot 2 = 9.2$ Mbps QPSK. Fully usable for automotive ISAC.

Theorem: Oversampling Extends the Unambiguous Region

With Doppler oversampling factor $\beta$ , the unambiguous velocity becomes $v_{\max}^{(\beta)} \;=\; \beta \cdot v_{\max} \;=\; \frac{c \beta N}{4 T f_0}.$ The velocity resolution shrinks by the same factor: $\Delta v_{(\beta)} = \Delta v/\beta$ .

So oversampling extends the unambiguous velocity and improves the resolution. The cost is a $\beta$ -times larger grid (DD cells $\beta MN$ ) and thus $\beta$ -times the detection compute.

Oversampling in the Doppler direction adds more Doppler bins at finer spacing. Both effects — wider range and finer resolution — come from the simple fact that the oversampled grid has $\beta$ times more cells in Doppler.

For LEO satellite radar at $v = 7500$ m/s and $f_0 = 10$ GHz: $v_{\max}$ with $\beta = 1$ , $T = 4$ ms is only $60$ m/s — totally inadequate. With $\beta = 128$ oversampling: $v_{\max} = 7680$ m/s — just barely covers orbital velocity. At LEO, $\beta \geq 16$ is often necessary to cover the velocity range of interest.

Proof

Oversampled grid

$N_{\text{os}} = \beta N$ Doppler bins. Bin spacing: $\Delta\nu_{\text{os}} = 1/(\beta T)$ .

Maximum Doppler

$\nu_{\max} = N_{\text{os}}/(2 T) = \beta N/(2T)$ . Velocity: $v_{\max}^{(\beta)} = c \nu_{\max}/(2 f_0) = \beta c N /(4 T f_0) = \beta v_{\max}$ . $\blacksquare$

⚠️Engineering Note

Designing the Unambiguous Region for Your Application

Design principle: unambiguous region must cover the target scene's extent.

Automotive (urban): $R \leq 100$ m, $|v| \leq 50$ m/s. Need $M \geq 100/\Delta R$ , $N \geq 50/(\Delta v/2)$ .
Automotive (highway): $R \leq 200$ m, $|v| \leq 80$ m/s. Larger $M, N$ required.
Indoor (gesture/health): $R \leq 10$ m, $|v| \leq 5$ m/s. Modest grid sufficient.
Aerial (UAV): $R \leq 1$ km, $|v| \leq 50$ m/s. Medium-to-large grid.
LEO satellite: $R \leq 1000$ km, $|v| \leq 7{,}500$ m/s. Very large oversampled grid. $\beta = 16$ - $128$ typical.

For most terrestrial ISAC (automotive, indoor, UAV), 5G NR numerology parameters give adequate $R_{\max}, v_{\max}$ without oversampling. LEO and extreme-Doppler scenarios need explicit oversampling.

Practical Constraints

•
Unambiguous region set by DD grid size $(M, N)$
•
Oversampling $\beta$ extends region by factor $\beta$
•
Terrestrial ISAC: $\beta = 1$ usually sufficient

Common Mistake: Velocity Aliasing Creates Ghost Targets

Mistake:

Designing an ISAC system for $v_{\max}$ less than the target scene's maximum velocity. Targets moving faster than $v_{\max}$ produce Doppler shifts that alias into false lower-velocity bins — "ghost targets" appear in the ambiguity surface.

Correction:

Always size $N$ so $v_{\max} \geq v_{\text{scene,max}}$ . For automotive: $v_{\max} = 80$ m/s is a reasonable operational choice. For LEO: use oversampling $\beta \geq 16$ . Aliasing cannot be "corrected" by the detector — the information is lost.

Diagnostic: if the detector finds targets clustered at the velocity extreme, suspect aliasing; check whether fast targets in the scene are folded to slower-apparent velocities.

Why This Matters: To Chapter 12: Putting It All Together

Chapter 11 establishes OTFS's radar-sensing properties: thumbtack ambiguity, ideal range-velocity resolution, matched CRLB, unambiguous region coincides with the DD grid. Chapter 12 takes the next step: Integrated Sensing and Communication with OTFS. The Yuan-Schober-Caire and Gaudio-Kobayashi-Caire CommIT contributions propose OTFS as the 6G ISAC waveform and develop joint estimation and detection algorithms that exploit the DD geometry both for data decoding and for target localization. The ambiguity-function framework of this chapter is the technical foundation.

The Unambiguous Sensing Region