FIR perfect signal reconstruction from multiple convolutions: minimum deconvolver orders

Given p co-prime finite impulse response (FIR) filters h/sub i//spl isin/R(m/sub h/), it well known that there exist q/sub i//spl isin/R(m/sub q/) such that /spl Sigma//sub i/h/sub i/*q/sub i/=/spl delta/. Importantly, this enables signal recovery from its convolutions with p/spl ges/2 co-prime FIR filters by FIR filtering. We show that such q/sub i/ exist almost surely if and only if m/sub q//spl ges/[(m/sub h/-1)/(p-1)], where [x] is the smallest integer greater than or equal to x. The results also provide conditions for full rank of certain key matrices arising in the blind multichannel deconvolution problem.