A model for an ARMA process split in sub-bands

Multi-rate digital processing [1] is today a well-established topic, extensively applied in communications, image and audio industry and other areas, for signal coding, adaptive or statistical processing etc. A special class of discrete random processes [2] are those obtained by passing white-noise through a linear digital filter—called Moving-Average (MA) for an Finite-Impulse-Response (FIR) filter, Autoregressive (AR) for an all-pole filter or Autoregressive-MovingAverage (ARMA) in the general case. Many applications in speech and audio employ these processes as models. This is particularly true for audio restoration [3], where ARMA and AR processes play an important role in description of signals. In this context, model-based processing in sub-bands would profit from a similar description of the sub-band signals, if it was possible. This is the motivation behind this work. This work uses polyphase decomposition of systems [1] and power spectral density of random signals [2] to get a simple model that describes sub-band signals resulting from the analysis of an ARMA process by a decimating filter bank. The special case of an AR process passing through an octave-band filter bank is also detailed, leading to some useful interpretations. Applications of these results to audio modeling and restoration will appear in future works [4]. Practical simulations yield good results. The paper is organized as follows. Section 2 covers the general case of the ARMA process divided in bands, Section 3 derives and discusses the case of the AR process divided in octaves and Section 4 summarizes the results.