A novel technique for PDF estimation using DSP methods

This paper deals with estimating the pdf and joint pdf of a scalar and vector valued random variable using multirate signal processing methods. We estimate the pdf using a histogram and smoothen it out by passing the histogram through a multirate system comprising of a low pass filter and decimator. Such a multirate processing reduces variance of the histogram and can be readily generalized to the vector case i.e. the joint pdf. The low pass filter is designed to remove the high frequency components in the characteristic function of the histogram and hence smoothen it out. After passing through the decimator, we interpolate and use a low pass filter to remove the replicas generated. This low pass filter is the biorthogonal partner of earlier filter. The combination of (low pass filter1→ decimator) and (interpolator → low pass filter2) is chosen so that the output is minimum mean square estimate of the original sampled pdf. The combination of (low pass filter1→ decimator) and (interpolator → low pass filter2) is a linear time varying system and the design can be carried out using the standard orthogonal projection theory. The condition of biorthogonal partner guarantees unbiasedness of the pdf estimate.