ambiguityfunction

The ambiguity function is a two-dimensional function that characterizes how a waveform correlates with delayed and Doppler-shifted versions of itself (or of another waveform). It is widely used in radar, sonar, and wireless communications to assess range (delay) and velocity (Doppler) resolution and tolerance.

For a finite-energy, baseband signal s(t), the auto-ambiguity function is defined as

A(τ, ν) = ∫_{-∞}^{∞} s(t) s*(t − τ) e^{-j 2π ν t} dt,

where τ is the time delay and ν is the Doppler frequency shift. A cross-ambiguity function for two signals x(t) and y(t) is

A_xy(τ, ν) = ∫ x(t) y*(t − τ) e^{-j 2π ν t} dt.

Key properties include:

- A(0, 0) equals the energy of the signal, ∥s∥^2.

- A(τ, 0) is the autocorrelation as a function of delay; A(0, ν) is the Fourier transform of the instantaneous power.

- For complex-valued s(t), A(−τ, −ν) = A*(τ, ν).

- The magnitude |A(τ, ν)| indicates how well the received signal matches a delayed and Doppler-shifted version of the reference; the peak occurs at the true delay and Doppler for a matched waveform.

- The ambiguity function’s shape along τ and ν relates to range and velocity resolution and Doppler tolerance; sidelobes reflect interference from unresolved delays or Doppler shifts.

Relation to other representations: the ambiguity function is connected to time-frequency representations and can be transformed to and from forms like the Wigner distribution. Practical use includes evaluating waveform designs for desired resolution, Doppler resilience, and clutter rejection in radar and related systems.