Shaping frequency-dependent time resolution when estimating spectral properties with parametric methods

The problem of tracking time-varying properties of a signal is studied. The somewhat contradictory notion of "time-varying spectrum" and how to estimate the "current" spectrum in an on-line fashion is discussed. The traditional concepts and relations between time and frequency resolution are crucial for this problem. We introduce two definitions for the time resolution of filters, essentially measuring the effective number of past data that are used to form the estimate. In, for example, wavelet transform techniques, frequency-dependent time resolutions are used so that fewer data are used at higher frequencies, thus enabling faster tracking of high-frequency components (at the price of worse frequency resolution). The main contribution of the paper is to show how this same feature can be introduced when estimating spectra via a time-varying, autoregressive model of the signal. This is achieved by a special choice of nominal covariance matrix for the underlying parameter changes.