This paper considers finite-time input reconstruction for discrete-time linear time-invariant systems in the case where the initial condition is unknown. There are three main results. First, a specific construction of finite-impulse-response (FIR) delayed left inverse with the minimal delay for systems with zero nonzero zeros is presented. Next, it is shown that, in the presence of an arbitrary unknown initial condition, finite-time input reconstruction is possible using a delayed left inverse H if and only if H is FIR. Finally, it is shown that a transfer function with full column normal rank has an FIR delayed left inverse with the minimal delay if and only if the system has zero nonzero zeros.