The OTFS Ambiguity Function

The Thumbtack Ambiguity

The OTFS transmit waveform (Chapter 6) is a weighted sum of delay-Doppler-shifted copies of a prototype pulse gtx(t)g_{tx}(t), with the DD grid carrying data symbols. This structure β€” the Gabor expansion on the critical DD lattice β€” has a specific ambiguity function that is fundamentally different from a single pulse's ambiguity.

The point is that OTFS's ambiguity function is approximately separable in Ο„\tau and Ξ½\nu: a sharp peak at the origin that falls off rapidly in both directions, with sidelobes distributed evenly across the ambiguity plane. This "thumbtack" shape is the essential property that makes OTFS a natural ISAC waveform.

Theorem: OTFS Ambiguity Function (Main-Lobe Form)

For the OTFS transmit signal s(t)=1MNβˆ‘β„“,kXDD[β„“,k] gtx(tβˆ’β„“Ts) ej2Ο€kΞ”f(tβˆ’β„“Ts)s(t) = \frac{1}{\sqrt{MN}}\sum_{\ell, k} X_{DD}[\ell, k]\,g_{tx}(t - \ell T_s)\,e^{j 2\pi k \Delta f(t - \ell T_s)}, the ambiguity function at small offsets (Ο„,Ξ½)(\tau, \nu) (within one DD cell: βˆ£Ο„βˆ£β‰€Ξ”Ο„=1/W|\tau| \leq \Delta\tau = 1/W, βˆ£Ξ½βˆ£β‰€Ξ”Ξ½=1/T|\nu| \leq \Delta\nu = 1/T) is As(Ο„,Ξ½)β€…β€Šβ‰ˆβ€…β€ŠAg(Ο„,Ξ½) sinc(WΟ„) sinc(TΞ½),A_s(\tau, \nu) \;\approx\; A_g(\tau, \nu)\,\text{sinc}(W\tau)\,\text{sinc}(T\nu), where AgA_g is the prototype pulse's ambiguity function (constant for a rectangular pulse at short offsets).

The sinc(WΟ„) sinc(TΞ½)\text{sinc}(W\tau)\,\text{sinc}(T\nu) factor is the thumbtack shape: sharp peak at the origin, first nulls at (Ο„,Ξ½)=(Β±1/W,Β±1/T)(\tau, \nu) = (\pm 1/W, \pm 1/T), sidelobes at βˆ’13-13 dB (rectangular) or lower (windowed).

The OTFS waveform sums MNMN pulses with approximately independent phases (from random QAM data). The coherent sum at (Ο„,Ξ½)=(0,0)(\tau, \nu) = (0, 0) gives a peak of MN\sqrt{MN}; at non-zero (Ο„,Ξ½)(\tau, \nu), the phases decorrelate and the sum gives approximately MN\sqrt{MN} noise-like magnitude β€” a 1/MN1/\sqrt{MN} suppression factor. This is the "thumbtack" shape.

In the frequency domain: the OTFS signal's spectrum is spread across [0,W][0, W] and the signal occupies duration TT β€” both uniformly. The ambiguity function is therefore concentrated at the origin in both dimensions.

Proof
Autocorrelation of Gabor expansion

s(t)sβˆ—(tβˆ’Ο„)=1MNβˆ‘β„“,kβˆ‘β„“β€²,kβ€²X[β„“,k]Xβˆ—[β„“β€²,kβ€²] gtx(tβˆ’β„“Ts) gtxβˆ—(tβˆ’Ο„βˆ’β„“β€²Ts) ej2Ο€kΞ”f(tβˆ’β„“Ts)βˆ’j2Ο€kβ€²Ξ”f(tβˆ’Ο„βˆ’β„“β€²Ts)s(t) s^*(t - \tau) = \frac{1}{MN}\sum_{\ell, k}\sum_{\ell', k'} X[\ell, k] X^*[\ell', k']\,g_{tx}(t - \ell T_s)\,g_{tx}^*(t - \tau - \ell' T_s)\,e^{j 2\pi k \Delta f(t - \ell T_s) - j 2\pi k'\Delta f(t - \tau - \ell' T_s)}.

Expectation over data

For random i.i.d. QAM data with E[X]=0\mathbb{E}[X] = 0, E[XXβˆ—]=Ξ΄\mathbb{E}[X X^*] = \delta: E[s(t)sβˆ—(tβˆ’Ο„)]=1MNβˆ‘β„“,k∣gtx(tβˆ’β„“Ts)∣2 ej2Ο€kΞ”fΟ„\mathbb{E}[s(t) s^*(t-\tau)] = \frac{1}{MN}\sum_{\ell, k} |g_{tx}(t - \ell T_s)|^2\,e^{j 2\pi k \Delta f\tau}.

Sum over $k$

βˆ‘k=0Nβˆ’1ej2Ο€kΞ”fΟ„\sum_{k=0}^{N-1} e^{j 2\pi k \Delta f\tau} is a Dirichlet kernel in Ο„\tau, concentrated at Ο„=0\tau = 0 with width 1/(NΞ”f)=1/W1/(N\Delta f) = 1/W.

Ambiguity function

Multiply by eβˆ’j2πνte^{-j 2\pi \nu t} and integrate in tt: As(Ο„,Ξ½)∝sinc(WΟ„)β‹…sinc(TΞ½)β‹…(prototypeΒ pulseΒ correction)A_s(\tau, \nu) \propto \text{sinc}(W\tau) \cdot \text{sinc}(T\nu) \cdot (\text{prototype pulse correction}). The product of the two sincs gives the thumbtack shape. β– \blacksquare

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Key Takeaway

OTFS has the ideal thumbtack ambiguity function. Sharp peak at (0,0)(0, 0), first nulls at (Β±1/W,Β±1/T)(\pm 1/W, \pm 1/T), sidelobes at βˆ’13-13 dB or lower. This ambiguity shape gives OTFS simultaneous range resolution Ξ”R=c/(2W)\Delta R = c/(2W) and velocity resolution Ξ”v=c/(2Tf0)\Delta v = c/(2 T f_0) β€” both achieved with a single waveform. The thumbtack is the signature that distinguishes OTFS radar from OFDM radar.

OTFS Ambiguity Function (Thumbtack)

Plot ∣As(Ο„,Ξ½)∣2|A_s(\tau, \nu)|^2 for the OTFS waveform as a 2D heatmap over (Ο„,Ξ½)(\tau, \nu) in units of DD grid cells. Observe the sharp main lobe at the origin and sidelobes distributed evenly across the ambiguity plane. Compare with the OFDM rectangular-pulse ambiguity from Β§1.

Parameters
64
16

Theorem: Peak Sidelobe Level of OTFS Ambiguity

For the OTFS waveform with rectangular prototype pulse and no windowing, the peak sidelobe level (PSL) of the ambiguity function is PSLOTFSβ€…β€Šβ‰ˆβ€…β€Šβˆ’13.3Β dBΒ belowΒ mainΒ lobe.\text{PSL}_{\text{OTFS}} \;\approx\; -13.3 \text{ dB below main lobe}. This is the Dirichlet-kernel sidelobe level, common to all uniformly-weighted waveforms. With Hamming windowing: PSLOTFSβ‰ˆβˆ’43\text{PSL}_{\text{OTFS}} \approx -43 dB. With Blackman: PSLOTFSβ‰ˆβˆ’58\text{PSL}_{\text{OTFS}} \approx -58 dB.

For comparison: OFDM rectangular pulse ambiguity has PSL along the Doppler axis of βˆ’13.3-13.3 dB (same formula) but a very broad main lobe (no Doppler suppression). The thumbtack shape is what matters: OTFS's PSL is localized around the main lobe, not spread across an entire axis.

PSL determines the "detection floor" for secondary targets. At βˆ’13-13 dB: secondary targets 13 dB weaker than the primary are resolved. At βˆ’43-43 dB (Hamming): 43 dB weaker targets are resolved β€” sufficient for most practical radar scenes.

Proof
Sidelobe envelope

The sinc2\text{sinc}^2 factor in the ambiguity function has first sidelobe at sinc2(3/2)/sinc2(0)=0.045\text{sinc}^2(3/2) / \text{sinc}^2(0) = 0.045, =βˆ’13.3= -13.3 dB.

Windowed case

Windowing reduces sidelobes per Harris (1978): Hamming reduces to βˆ’43-43 dB, Blackman to βˆ’58-58 dB. Main-lobe width grows by Ξ±ENBW\alpha_{\text{ENBW}} factor.

Implication

For unwindowed OTFS: multiple strong targets within 13 dB of each other may not be resolvable by a single ambiguity peak search. For windowed OTFS with Hamming: 40+ dB dynamic range, typical for automotive radar and ISAC applications. β– \blacksquare

OTFS Thumbtack vs OFDM Ridge Ambiguity

Side-by-side visualization of the OTFS and OFDM ambiguity functions in (Ο„,Ξ½)(\tau, \nu). OTFS shows the thumbtack: sharp peak at the origin. OFDM single-symbol shows the ridge: sharp in delay, wide plateau in Doppler. This is the structural difference that makes OTFS the natural ISAC waveform.

OTFS vs OFDM Ambiguity: 3D Perspective

OTFS vs OFDM Ambiguity: 3D Perspective
3D perspective views of ∣As(Ο„,Ξ½)∣2|A_s(\tau, \nu)|^2 for OTFS (left) and OFDM rectangular pulse (right). OTFS: sharp thumbtack centered at origin, sidelobes evenly distributed. OFDM: sharp ridge along Ο„=0\tau = 0, extending across the entire Doppler axis β€” unable to resolve Doppler. The thumbtack vs ridge distinction is the fundamental structural difference between the two waveforms as radar signals.

Example: Computing the OTFS Ambiguity Numerically

An OTFS system has M=64M = 64, N=16N = 16, W=20W = 20 MHz, T=3.2T = 3.2 ms. Compute the main-lobe width in Ο„\tau and Ξ½\nu, and the peak sidelobe level, both unwindowed.

Solution
Main-lobe widths

Ο„\tau-direction: first null at Ο„=1/W=50\tau = 1/W = 50 ns. Ξ½\nu-direction: first null at Ξ½=1/T=312\nu = 1/T = 312 Hz.

Resolutions

Ξ”R=c/(2W)=1.5Γ—108/(2β‹…2Γ—107)=7.5\Delta R = c/(2W) = 1.5 \times 10^8/(2 \cdot 2 \times 10^7) = 7.5 m. Ξ”v=cΔν/(2f0)=3Γ—108β‹…312/(2β‹…5Γ—109)=9.36\Delta v = c \Delta\nu/(2f_0) = 3\times 10^8 \cdot 312/(2 \cdot 5 \times 10^9) = 9.36 m/s (assuming f0=5f_0 = 5 GHz).

PSL

Unwindowed: βˆ’13-13 dB. Hamming windowed: βˆ’43-43 dB, at cost of 36% wider main lobes (range res 1010 m, velocity res 1212 m/s).

Implication

The waveform simultaneously achieves 7.5 m range resolution and ∼10\sim 10 m/s velocity resolution β€” suitable for automotive ISAC where pedestrians (1.4 m/s) to trucks (30 m/s) need resolution.

Ambiguity Is the Waveform; Detection Is the Algorithm

The ambiguity function characterizes the waveform β€” what the matched filter sees at various (Ο„,Ξ½)(\tau, \nu) hypotheses. It is the raw radar signature.

Detection combines the ambiguity function with target-scene hypotheses: for a single target, peak-pick the ambiguity surface; for multiple targets, apply CFAR (Constant False Alarm Rate) or compressed-sensing super-resolution.

OTFS's thumbtack ambiguity gives excellent peak-pick performance but still requires sophisticated algorithms for scenes with densely-packed targets (urban automotive radar, RF imaging). The detection complexity is largely orthogonal to the ambiguity shape β€” OTFS wins on ambiguity shape; the detection algorithms (Chapter 13) are the same general methods as in OFDM-radar.

Historical Note: From Woodward to OTFS-Radar

1953 - 2024

Philip Woodward introduced the ambiguity function in 1953 in his classic monograph Probability and Information Theory, with Applications to Radar. His insight β€” that a fixed signal ambiguity volume forces a trade between range and velocity resolution β€” has shaped radar waveform design ever since. Different waveforms are positioned at different vertices of the Woodward uncertainty polytope: pulse trains (thumbtack), LFM/chirps (skewed ridge), barker codes (thumbtack at short TWTW), and most recently OTFS (thumbtack at large TWTW).

OTFS-radar as an explicit concept emerged in 2020 via the Gaudio-Kobayashi-Caire paper (IEEE TWC 2020), which established the thumbtack ambiguity of OTFS and framed OTFS as a natural ISAC waveform. The Yuan-Schober-Caire ISAC tutorial (IEEE ComMag 2024) Part III extended this to the full joint sensing-comms framework. Both are CommIT contributions treated in detail in Chapter 12.

Woodward's 1953 insight is as sharp today as when it was written. OTFS does not violate it; OTFS is positioned at the Woodward vertex most useful for joint range-velocity estimation over the communications time-bandwidth product. This is the concrete content of the thumbtack-ambiguity claim.

The Ambiguity FunctionRange and Velocity Resolution