Autoregression and irregular sampling: Filtering

Abstract

Two linear stochastic processes may be separated using the Wiener filter. This takes on a particularly simple form if one process is from a known autoregressive (AR) model, and the other has unknown spectral characteristics (and one assumes its covariance matrix to be a scalar multiple of the identity matrix). This is because the inverse of the covariance matrix C is essentially the square of the prediction error filter matrix F , i.e. C −1∝ F † F . However, if one does not know the relative strengths of the two signals to be separated, one has to estimate their power (variance) ratio. We address this problem. Then we show how to generalise the filter to cope with data that are observed at irregular time intervals. In fact, one can achieve this by replacing the elements of F with time-varying prediction coefficients. The derivation of these coefficients, established by the author in previous work on autoregressive spectral analysis of irregularly sampled data, uses the theory of linear continuous-time stochastic processes, and is presented here. One of the most difficult areas in irregular sampling is that the spectrogram of irregular samples is a smeared version of the underlying spectrum. Therefore the dynamic range is much reduced and one can only see the strongest spectral features in the spectrogram. By filtering out the stronger signals in the time domain and reassessing the spectrum, one may identify the weaker ones. Test results confirm this.