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Why the Ambiguity Function Matters

In radar, a transmitted waveform $s(t)$ probes the environment; the reflected signal is a sum of scaled, delayed, Doppler-shifted copies of $s(t)$ . The receiver correlates the received signal with $s(t)$ at hypothesized $(\tau, \nu)$ — a matched filter. The output of this correlator is called the ambiguity function $A_s(\tau, \nu)$ . It is the single most important quantity in radar waveform design: its shape determines the achievable resolution, side-lobe level, and ambiguity structure of the range-Doppler map.

The point is that different waveforms have different ambiguity functions. OTFS's ambiguity is a thumbtack shape (sharp peak, low sidelobes in all directions) — structurally ideal for joint range-velocity estimation. OFDM's ambiguity is a ridge (sharp in range, flat in Doppler) — good for range but poor for velocity. This chapter quantifies these shapes.

Definition:
Ambiguity Function

For a waveform $s(t)$ of duration $T$ and energy $E_s = \int|s|^2dt$ , the ambiguity function is $A_s(\tau, \nu) \;=\; \int_{-\infty}^{\infty} s(t)\,s^*(t - \tau)\,e^{-j 2\pi \nu t}\,dt.$ $A_s(0, 0) = E_s$ (peak at zero delay, zero Doppler). $|A_s(\tau, \nu)|^2$ is the ambiguity surface — the matched-filter output when the target is at $(\tau, \nu)$ and we correlate against hypothesis $(0, 0)$ . Equivalently, it is the matched-filter response to a target at the origin when we test for a target at $(\tau, \nu)$ .

Two conventions differ in the sign of the exponent: $e^{-j 2\pi \nu t}$ (radar-engineering convention, used here) vs $e^{+j 2\pi \nu t}$ (signal-processing convention). Results are equivalent up to a conjugate; check the sign convention when cross-referencing.

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Theorem: Key Properties of the Ambiguity Function

The ambiguity function $A_s(\tau, \nu)$ satisfies:

Peak at origin: $|A_s(\tau, \nu)| \leq |A_s(0, 0)| = E_s$ , with equality iff $(\tau, \nu) = (0, 0)$ .
Volume invariance (Woodward's theorem): $\iint |A_s(\tau, \nu)|^2\,d\tau\,d\nu = E_s^2$ . The total volume under the squared ambiguity surface is fixed.
Symmetry: $|A_s(-\tau, -\nu)| = |A_s(\tau, \nu)|$ .
Scaling: dilating $s(t) \to s(t/a)$ gives $A_{s_a}(\tau, \nu) = A_s(\tau/a, a\nu)$ .

Woodward's theorem (property 2) is the fundamental constraint: the ambiguity function's volume is fixed by the waveform's energy and time-bandwidth product. Narrow main lobes come at the cost of sidelobe energy that cannot be eliminated.

Waveform design is a zero-sum game in the ambiguity volume. A thumbtack ambiguity (concentrated peak) requires spreading sidelobe energy thinly across a large area. A ridge (concentrated in one dimension) pushes all sidelobe energy into the orthogonal direction — localizing that dimension's resolution but sacrificing the other.

OTFS and OFDM make different choices: OTFS spreads sidelobes evenly (thumbtack), OFDM concentrates them along Doppler (ridge). Neither can have both a narrow main lobe and uniformly low sidelobes — Woodward's theorem forbids it. Practical systems manage sidelobes through signal design and post-processing (e.g., windowing for rejection of known sidelobe lobes).

Proof

Cauchy-Schwarz

$|A_s(\tau, \nu)|^2 = |\int s s^* e^{-j2\pi\nu t}dt|^2 \leq (\int |s|^2)(\int |s|^2) = E_s^2$ . Equality iff the two factors align: $s(t) s^*(t-\tau) e^{-j2\pi\nu t}$ is constant. This requires $\tau = 0, \nu = 0$ .

Volume property

Applying Parseval in $(\tau, \nu)$ : $\iint |A_s|^2 d\tau d\nu = \iint |\tilde{A_s}|^2$ where $\tilde{A_s}$ is the 2D Fourier transform. Direct computation: $\tilde{A_s}(t, f) = |s(t) S(f)|^2$ , so $\iint |A_s|^2 = \int|s|^2 \int|S|^2 = E_s^2$ .

Symmetry

Change variables: $A_s(-\tau, -\nu) = \int s(t) s^*(t+\tau) e^{j 2\pi\nu t}dt$ . Substituting $t' = t + \tau$ : $A_s(-\tau, -\nu) = e^{-j 2\pi\nu\tau}\int s^*(t') s(t'-\tau)e^{-j 2\pi\nu t'}dt'$ which up to conjugation matches $A_s(\tau, \nu)$ .

Scaling

Direct substitution. $\blacksquare$

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Key Takeaway

The ambiguity function's shape is the waveform's radar signature. Woodward's volume constraint forces a trade: narrow main lobe + narrow sidelobe support requires a compact, spread ambiguity function (thumbtack). Wide main lobe + narrow sidelobe support requires a stretched function (ridge). OTFS's structure produces the thumbtack; OFDM produces the ridge. This is the structural reason OTFS is a better ISAC candidate than OFDM.

Example: Ambiguity of a Rectangular Pulse

Compute the ambiguity function of a rectangular pulse $s(t) = (1/\sqrt{T})\mathbf{1}_{[0, T)}(t)$ at $(\tau, \nu)$ . Discuss its shape.

Solution

Evaluate the integral

For $|\tau| < T$ : $A_s(\tau, \nu) = \frac{1}{T}\int_{|\tau|/2}^{T - |\tau|/2} e^{-j 2\pi \nu t}\,dt = \frac{T - |\tau|}{T}\,e^{-j\pi\nu(T - |\tau|)}\,\text{sinc}(\nu(T - |\tau|))$ .

Magnitude

$|A_s(\tau, \nu)|^2 = \left(\frac{T - |\tau|}{T}\right)^2 \text{sinc}^2(\nu(T - |\tau|))$ .

Shape

At $\nu = 0$ : triangle in $\tau$ from $-T$ to $T$ , peak at $\tau = 0$ . At $\tau = 0$ : sinc-squared in $\nu$ , first null at $\nu = 1/T$ . Overall: triangular-sinc-squared ridge along the Doppler axis.

Implication

A single pulse has good range resolution (triangle is narrow at short $\tau$ ) but poor Doppler resolution (sinc-squared is spread over $1/T$ ). This is the OFDM-radar baseline: a ridge along Doppler.

Ambiguity of a Rectangular Pulse

Plot $|A_s(\tau, \nu)|^2$ for the rectangular pulse as a 2D heatmap. Observe: narrow peak along $\tau = 0$ (good range resolution), broad sinc-squared ridge along $\nu$ (poor Doppler resolution). This is the ambiguity structure of OFDM-radar (per OFDM symbol). Scroll to see both range and Doppler axes.

Parameters

Pulse duration

T

(µs)2

Bandwidth

W

(MHz)10

Theorem: Time-Bandwidth Product and Ambiguity Area

The area of the ambiguity function's main lobe (where $|A_s(\tau, \nu)|^2 \geq E_s^2/2$ , i.e., $-3$ dB) is bounded by $A_{\text{main}} \;\geq\; \frac{1}{TW},$ where $TW$ is the time-bandwidth product. Equality is achieved by the Gaussian pulse; other waveforms have $A_{\text{main}} \geq 1/(TW)$ .

In terms of resolution: range resolution is $\Delta R = c/(2W)$ , velocity resolution is $\Delta v = c/(2 T f_0)$ , regardless of waveform. The waveform shape determines how sharply these are achieved — the main-lobe quality — but not the theoretical limit.

The time-bandwidth product $TW$ determines how finely one can resolve $(\tau, \nu)$ — this is the radar analog of the signal space dimensionality. A larger $TW$ means finer resolution, but also more "cells" in the ambiguity function that must be populated by sidelobe energy.

For a typical OTFS frame at $W = 20$ MHz, $T = 4$ ms: $TW = 8 \times 10^4$ — many cells. The ambiguity surface is inherently high-dimensional.

Proof

Fourier uncertainty

$A_s(\tau, 0) =$ autocorrelation of $s$ . Its effective width is $\geq 1/W$ (Fourier uncertainty, since $s$ has bandwidth $W$ ). Similarly, $A_s(0, \nu)$ has effective width $\geq 1/T$ (since $s$ has duration $T$ ).

Product

Main-lobe area at the origin: $\geq (1/W) \cdot (1/T) = 1/(TW)$ . By Woodward's volume theorem, no pulse can have a smaller main lobe without trading sidelobe area.

Achievable by Gaussian

The Gaussian pulse $s(t) \propto e^{-t^2/(2\sigma^2)}$ saturates the bound with $\sigma = 1/(W\sqrt{2\pi})$ . Its ambiguity is Gaussian in both $\tau$ and $\nu$ , with minimum joint uncertainty. $\blacksquare$

Resolution vs Accuracy vs Ambiguity

Three radar terms are often confused:

Resolution: $\Delta R, \Delta v$ — minimum separation for two targets to be distinguishable. Determined by main-lobe width.
Accuracy: CRLB — minimum variance of the $(\tau, \nu)$ estimate. Determined by main-lobe shape and SNR.
Ambiguity: the $(\tau, \nu)$ range within which the target lies — determined by the first large sidelobe in the ambiguity function.

Ideal radar waveform: narrow main lobe (good resolution/accuracy), low sidelobes (unambiguous detection), and matched to target scene parameters. OTFS's thumbtack ambiguity achieves the first two simultaneously.

🔧Engineering Note

Why OTFS as Radar

Historically, radar and communications were separate: radar uses short pulses or chirps for fine range-Doppler; communications uses OFDM or CP-OFDM for data efficiency. Merging them required one waveform to serve both goals — the motivation for ISAC (Integrated Sensing and Communication).

OFDM-radar was proposed in the 2010s: it uses the OFDM data symbols' output as a radar waveform. Good range (via bandwidth), poor Doppler (ridge ambiguity). Compromises.

OTFS-radar, emerging around 2020, has the thumbtack ambiguity (this chapter) and thus simultaneously resolves range and velocity — with the same waveform that carries data. This is the fundamental reason OTFS is the canonical ISAC candidate for 6G (Chapter 12).

The key intuition: OTFS's data is placed on the DD grid itself, which is the "correct" basis for radar. OFDM's data is in the TF grid, which is the wrong basis for radar. This basis mismatch is what gives OFDM the ridge ambiguity; OTFS avoids it by signaling on the radar-native grid.

The Ambiguity Function