page_title: Ambiguity function
section_title: Square pulse
hierarchical_section_title: Ambiguity function / Square pulse
caption_attribution_description: English: Ambiguity function plot for a length 20 square pulse.
context_page_description: 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 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.
  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 a time is a Doppler cut, with constant frequency given by the target's Doppler shift.
context_section_description: Consider a simple square pulse of duration  and
  amplitude :
  where  is the Heaviside step function. The
  matched filter output is given by the autocorrelation of the pulse, which is a triangular pulse of height  and
  duration  (the zero-Doppler cut). 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.
  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.
caption: Ambiguity function for a square pulse
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