We analyze the problem of blind deconvolution, which involves simultaneously estimating the input signal and blur kernel. We propose the deterministic subspace assumption, which is commonly used in practice, and provide some theoretical results. We derive a tight sufficient condition for the identifiability of the signal and convolution kernels, which is only violated on a set of Lebesgue measure zero. Additionally, we also analyze the blind deconvolution problem with modulated inputs. When the signal is sparse and the blur kernel has a short support, we present some related algorithms with optimal sampling complexity.