Editing Ambiguity function (section)

== Square pulse ==

[[File:Square pulse ambiguity function 2.png|280px|thumb|right|Ambiguity function for a square pulse]]

Consider a simple square pulse of duration <math>\tau</math> and
amplitude <math>A</math>:

:<math>A (u(t)-u(t-\tau)) \, </math>

where <math>u(t)</math> is the [[Heaviside step function]]. The
matched filter output is given by the [[autocorrelation]] of the pulse, which is a triangular pulse of height <math>\tau^2 A^2</math> and
duration <math>2 \tau</math> (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. {{citation_needed|date=October 2019}}

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.