Optimal Filtering with Multirate Observations

This paper considers the problem of optimal filtering for several channels of observations occurring at different sampling rates. The problem is formulated in general terms with arbitrary sampling rates for the observations. The set of observation sequences (channels) are jointly cyclostationary and the resulting optimal filters are linear and periodically timevarying (LPTV). A formula for the period of the system is given and the explicit forms of the Wiener-Hopf equations and the estimation error variance are derived. Simulation results are presented for estimation of a signal with known correlation function in noise using two observation sequences with sampling rates differing from that of the underlying signal to be estimated. I. THE MULTIRATE SYSTEM In this paper we consider the problem of optimal filtering where the observation sequences are sampled at different rates. This problem occurs, for example, when two or more sensors with different resolutions are observing the same underlying phenomena. A typical model for M observation sequences in this case is shown in Fig. 1 where d n is the signal of interest, i n represents the impulse response of linear time-invariant filters for output i , i n represents the independent noise sequences and i x n represents the observation sequences reduced in rate by decimation with factors i K . Fig. 1. Multirate Model with Multiple Observation Sequences The basic form of the optimal filter is shown in Fig. 2. Fig. 2. Multirate Optimal Filter As will be shown, k i h n represents the coefficients of M linear periodically time-varying (LPTV) filters. Their outputs are added to produce the estimate d̂ n at the full rate. A simplified version of this problem has been addressed in [1] and [2] where two sequences are observed, one at the full rate and the other decimated by a factor of 2. A least squares formulation of the problem is reported in [3], where the decimation factor of the second observation sequence is arbitrary. In this paper we consider the general case where the M sampling rates each may be different from that of the quantity to be estimated and different from each other by arbitrary factors. This problem is closely related to the problem of estimation for cyclostationary random processes considered in [4]. However we avoid the matrix spectral factorization problem cited there by considering the use of FIR filters. Such FIR filters are often preferred in practice since they are simple to implement and avoid possible problems of instability associated with recursive filters. Fig. 3 shows an example of a 2-signal multirate system where the decimation factors for the 1 1 [ ] x m and 2 2 [ ] x m sequences are 2 and 3, respectively, and the filter orders are 1 4 P and 2 2 P . This figure will be referred to throughout the paper in discussing the estimation equations. By inspection, the periodicity of the signals and LPTV filters in this particular system shown is six. d n 1 1 x m