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Coherent integration gain is the effect obtained by increasing the length of time during which a coherent signal is observed. An example is the 3 dB gain in SNR experienced by doubling the observation time of a CW signal (most easily viewed by looking at spectral estimates of increasing lengths -- 128 samples to 256 samples to 512 samples, etc). To obtain coherent integration gain, the signal must itself be coherent (i.e., occupy a small fraction of the analysis bandwidth).
Non-coherent integration gain is the effect obtained by averaging together signal estimates taken during successive time slices, each having the same, fixed length. An example is the averaging of spectral estimates by performing point-by-point summation of the power in identical frequency bins, which smooths out the noise variance, but does not actually increase the SNR itself. A similar effect is observed when averaging together successive radar power returns point-by-point in identical range bins.
See also Dynamic Range