The problem of estimating the states of a linear system with quantized measurements is investigated. By introducing the concept of the so-called grid function, it is shown that the quantization effects may be interpreted as a modification of the nonquantized Bayesian posterior distribution. A new filter is obtained by approximating the grid function with a certain class of functions. An explicit filter representation resembling Gaussian sum filters is derived. It is shown that the grid function may also serve as a tool for characterizing well-known filters. We conclude by comparing the filter structure and performance with these well-known filters.