Ambiguity function
It represents the distortion of a returned pulse due to the receiver matched filter[1] (commonly, but not exclusively, used in pulse compression radar) of the return from a moving target.
The ambiguity function is defined by the properties of the pulse and of the filter, and not any particular target scenario.
Many definitions of the ambiguity function exist; some are restricted to narrowband signals and others are suitable to describe the delay and Doppler relationship of wideband signals.
A more concise way of representing the ambiguity function consists of examining the one-dimensional zero-delay and zero-Doppler "cuts"; that is,
The matched filter output as a function of time (the signal one would observe in a radar system) is a Doppler cut, with the constant frequency given by the target's Doppler shift:
Pulse-Doppler radar equipment sends out a series of radio frequency pulses.
If the waves reflect off a single object, the detector will see a signal which, in the simplest case, is a copy of the original pulse but delayed by a certain time
Unfortunately, this procedure may yield false positives, i.e. wrong values
The ambiguity function plays a key role in the field of time–frequency signal processing,[3] as it is related to the Wigner–Ville distribution by a 2-dimensional Fourier transform.
This relationship is fundamental to the formulation of other time–frequency distributions: the bilinear time–frequency distributions are obtained by a 2-dimensional filtering in the ambiguity domain (that is, the ambiguity function of the signal).
When the wave speed in the medium is sufficiently faster than the target speed, as is common with radar, this compression in frequency is closely approximated by a shift in frequency Δf = fc*v/c (known as the doppler shift).
For a narrow band signal, this approximation results in the narrowband ambiguity function given above, which can be computed efficiently by making use of the FFT algorithm.
This is not usually desirable (if a target has any Doppler shift from an unknown velocity it will disappear from the radar picture), but if Doppler processing is independently performed, knowledge of the precise Doppler frequency allows ranging without interference from any other targets which are not also moving at exactly the same velocity.
Approximations exist, however, and noise-like signals such as binary phase-shift keyed waveforms using maximal-length sequences are the best known performers in this regard.
[8] (1) Maximum value (2) Symmetry about the origin (3) Volume invariance (4) Modulation by a linear FM signal (5) Frequency energy spectrum (6) Upper bounds for
However, if the measured pulse has a frequency offset due to Doppler shift, the matched filter output is distorted into a sinc function.
The greater the Doppler shift, the smaller the peak of the resulting sinc, and the more difficult it is to detect the target.
[citation needed] In general, the square pulse is not a desirable waveform from a pulse compression standpoint, because the autocorrelation function is too short in amplitude, making it difficult to detect targets in noise, and too wide in time, making it difficult to discern multiple overlapping targets.
It has the advantage of greater bandwidth while keeping the pulse duration short and envelope constant.
Thus, an uncompensated Doppler shift changes the target's apparent range; this phenomenon is called range-Doppler coupling.
The ambiguity function can be extended to multistatic radars, which comprise multiple non-colocated transmitters and/or receivers (and can include bistatic radar as a special case).
For these types of radar, the simple linear relationship between time and range that exists in the monostatic case no longer applies, and is instead dependent on the specific geometry – i.e. the relative location of transmitter(s), receiver(s) and target.
Therefore, the multistatic ambiguity function is mostly usefully defined as a function of two- or three-dimensional position and velocity vectors for a given multistatic geometry and transmitted waveform.
Just as the monostatic ambiguity function is naturally derived from the matched filter, the multistatic ambiguity function is derived from the corresponding optimal multistatic detector – i.e. that which maximizes the probability of detection given a fixed probability of false alarm through joint processing of the signals at all receivers.
The nature of this detection algorithm depends on whether or not the target fluctuations observed by each bistatic pair within the multistatic system are mutually correlated.
If so, the optimal detector performs phase coherent summation of received signals which can result in very high target location accuracy.
[10] If not, the optimal detector performs incoherent summation of received signals which gives diversity gain.
[11] An ambiguity function plane can be viewed as a combination of an infinite number of radial lines.
Each radial line can be viewed as the fractional Fourier transform of a stationary random process.
The Ambiguity function (AF) is the operators that are related to the WDF.
