Blind Deconvolution - Mathematical Concept

Mathematical Concept

Suppose we have a signal transmitted through a channel. The channel can usually be modeled as a linear system, so the receptor receives a convolution of the original signal with the impulse response of the channel. If we want to reverse the effect of the channel, to obtain the original signal, we must process the received signal by a second linear system, inverting the response of the channel. This system is called an equalizer.

If we are given the original signal, we can use a supervising technique, such as finding a Wiener filter, but without it, we can still explore what we do know about it to attempt its recovery. For example, we can filter the received signal to obtain the desired spectral power density. This is what happens, for example, when the original signal is known to have no auto correlation, and we "whiten" the received signal.

Whitening usually leaves some phase distortion in the results. Most blind deconvolution techniques use higher-order statistics of the signals, and permit the correction of such phase distortions. We can optimize the equalizer to obtain a signal with a PSF approximating what we know about the original PDF.