FIR perfect signal reconstruction from multiple convolutions: minimum deconvolver orders

Given p co-prime finite impulse response (FIR) filters h_i ∈R(m_h), it well known that there exist q _i∈R(m_q) such that Σ_ih_i*q_i=δ. Importantly, this enables signal recovery from its convolutions with p⩾2 co-prime FIR filters by FIR filtering. We show that such q_i exist almost surely if and only if m_q⩾[(m_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