page_title: Pulse compression
section_title: Cross-correlation between the transmitted and the received signal
hierarchical_section_title: Pulse compression / Pulse compression by linear frequency modulation (or chirping) / Cross-correlation between the transmitted and the received signal
caption_attribution_description: Français&#160;: Signal original chirpé (porteuse 10 Hz, modulation 16 Hertz, durée 1 s) et deux échos (amplitude 0,5 et 0,3, déphasage 3s et 5s)
context_page_description: Pulse compression is a signal processing technique commonly used by radar, sonar and echography to increase the range resolution as well as the signal to noise ratio. This is achieved by modulating the transmitted pulse and then correlating the received signal with the transmitted pulse.
context_section_description: As for the "simple" pulse, let us compute the cross-correlation between the transmitted and the received signal. To simplify things, we shall consider that the chirp is not written as it is given above, but in this alternate form (the final result will be the same):
  Since this cross-correlation is equal (save for the  attenuation factor), to the autocorrelation function of , this is what we consider:
  It can be shown that the autocorrelation function of  is:
  The maximum of the autocorrelation function of  is reached at 0. Around 0, this function behaves as the sinc (or cardinal sine) term, defined here as . The −3 dB temporal width of that cardinal sine is more or less equal to . Everything happens as if, after matched filtering, we had the resolution that would have been reached with a simple pulse of duration . For the common values of ,  is smaller than , hence the pulse compression name.
  Since the cardinal sine can have annoying sidelobes, a common practice is to filter the result by a window (Hamming, Hann, etc.). In practice, this can be done at the same time as the adapted filtering by multiplying the reference chirp with the filter. The result will be a signal with a slightly lower maximum amplitude, but the sidelobes will be filtered out, which is more important.
   
   
caption: Before matched filtering
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