The problem of signal restoration in the presence of observation space noise has been tackled extensively. However, restoration of degraded signals in the presence of signal space noise leads to considerable complexity because it becomes difficult to distinguish between the original signal and the noise. In this paper, a partial projection filter has been devised for the restoration of signals degraded by both the signal space and the observation space noises. A closed form of the proposed filter has been derived and its performance has been verified experimentally.

The partial projection filter gives optimal signal restoration in the presence of both the signal space and the observation space noises. In this paper, the filter has been characterized from the point of view of its signal restoration and noise suppression capabilities. The filter is shown to suppress the noise component in the restored signal while retaining the signal component, thus maximizing the signal-to-noise ratio. Further, a digital implementation of the filter is presented in matrix form in contrast to its original operator based derivation, for practical applications.

The partial projection filter (PTPF) for a given observation operator provides an optimal signal restoration in the presence of both the signal space and observation space noises. However, restoration error by the filter still depends on the observation operator which consists of measurement and sampling processes. In this paper, we determine a sampling operator which minimizes the restoration error by the PTPF. We see that under some assumptions about noise statistics, the restoration error by the PTPF is divided into two terms corresponding to the error arising from the signal space noise and that from the observation space noise. It has been found that although the restoration error due to the signal space noise is independent of the sampling operator, the restoration error arising from the observation space noise can arbitrarily be decreased by increasing the number of sample points in the proposed sampling operator. An illustrative example of optimal sampling in the trigonometric polynomial space is also given.