A method for processing a received, modulated pulse (i.e. waveform) that requires
predictive deconvolution to resolve a scatterer from noise and other scatterers
includes receiving a return signal; obtaining L+(2M-1)(N-1) samples y of the return
signal, where y(l)={tilde over (x)}T(l)s+v(l); applying RMMSE estimation
to each successive N samples to obtain initial impulse response estimates [{circumflex
over (x)}1{-(M-1)(N-1)}, . . . , {circumflex over (x)}1{-1},
{circumflex over (x)}1{0}, . . . , {circumflex over (x)}1{L-1},
{circumflex over (x)}1{L}, . . . , {circumflex over (x)}1{L-1+(M-1)(N-1)}];
computing power estimates {circumflex over ()}1(l)=|{circumflex
over (x)}1(l)|2 for l=-(M-1)(N-1), . . . , L-1+(M-1)(N-1);
computing MMSE filters according to w(l)=(l)(C(l)+R)-1s, where
(l)=|x(l)|2 is the power of x(l), and R=E[v(l)vH(l)]
is the noise covariance matrix; applying the MMSE filters to y to obtain [{circumflex
over (x)}2{-(M-2)(N-1)}, . . . , {circumflex over (x)}2{-1},
{circumflex over (x)}2{0}, . . . , {circumflex over (x)}2{L-1},
{circumflex over (x)}2{L}, . . . , {circumflex over (x)}2{L-1+(M-2)(N-1)}];
and repeating (d)-(f) for subsequent reiterative stages until a desired length-L
range window is reached, thereby resolving the scatterer from noise and other scatterers.
The RMMSE predictive deconvolution approach provides high-fidelity impulse response
estimation. The RMMSE estimator can reiteratively estimate the MMSE filter for
each specific impulse response coefficient by mitigating the interference from
neighboring coefficients that is a result of the temporal (i.e. spatial) extent
of the transmitted waveform. The result is a robust estimator that adaptively eliminates
the spatial ambiguities that occur when a fixed receiver filter is used.