Ambiguity function

In pulsed radar and sonar signal processing, an ambiguity function is a two-dimensional function of propagation delay and Doppler frequency , . It represents the distortion of a returned pulse due to the receiver matched filter (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. Often the definition of the ambiguity function is given as the magnitude squared of other definitions (Weiss). For a given complex baseband pulse , the narrowband ambiguity function is given by where denotes the complex conjugate and is the imaginary unit. Note that for zero Doppler shift (), this reduces to the autocorrelation of . A more concise way of representing the ambiguity function consists of examining the one-dimensional zero-delay and zero-Doppler "cuts"; that is, and respectively. 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. Each pulse has a certain shape (waveform)—how long the pulse is, what its frequency is, whether the frequency changes during the pulse, and so on. 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 —related to the object's distance—and shifted by a certain frequency —related to the object's velocity (Doppler shift). If the original emitted pulse waveform is , then the detected signal (neglecting noise, attenuation, and distortion, and wideband corrections) will be: The detected signal will never be exactly equal to any because of noise.