We considered the least squares error (LSE) and mean square error (MSE) optimum best delay deconvolution filter design. We presented the partition-based finite impulse response-infinite impulse response (FIR-IIR) filters where the channel zeros are partitioned into two regions. The first region is composed of the selected channel zeros inside the unit circle, and the second region is composed of the remaining channel zeros outside the first region. Two methods for obtaining partitions are proposed, namely, optimum partitioning and ring-based partitioning.,, Resulting FIR-IIR filters are compared with the FIR and FIR-IIR unit circle best delay inverse filters in terms of their LSE. It is shown that the partition-based FIR-IIR filter performances are about 4-5 dB better than their FIR counterpart for the same complexity. Then, starting with the best delay FIR Wiener deconvolution filter formulation, we extended our results for the design of FIR-IIR MSE optimum deconvolution filters. We have compared the optimum-partitioning and ring-based FIR-IIR inverse filters with the FIR and IIR Wiener deconvolution filters. It is shown that FIR-IIR best delay inverse filters are less sensitive to the estimation errors compared with the IIR Wiener filters, and they perform better than the FIR Wiener filters. Furthermore, they are always causal and stable, making them suitable for real-time implementations.