Statistical Signal Extraction and Filtering: A Partial Survey

In common parlance, a filter is a device for removing solids or suspended par- ticles from liquids. In the late 17th century, the term began to be used by the natural philosophers in a manner that gave expression to their understanding of the nature of light. It was recognised that white light is a compound of coloured lights of di!ering wavelengths. A coloured glass was seen as a device that selectively transmits some of the light, corresponding to a range of wave- lengths, while blocking the remainder. Therefore, it was described as an optical filter. A direct analogy with light led engineers, in the early 20th century, to talk of electronic filters. Electronic filters are constructed from capacitors, resistors and inductors. A circuit in which a voltage signal passes through an inductor, or in which a capacitor provides a path to earth, imposes less attenuation on low-frequency signals than on high-frequency signals. Therefore, it constitutes a lowpass filter. If the signal passes through a capacitor, or has a path to earth through an inductor, then the circuit imposes less attenuation on high- frequency signals than on low-frequency signals, and it constitutes highpass filter. In these examples, one is imagining a stream or a current flowing continu- ously through the filter. The notion of a filter seems inappropriate to statisti- cal time-series analysis, where the data are a sequence of discrete observations. However, over a period of half a century at least, there has been a gradual shift in electronic technology from analogue devices, which are naturally analysed in terms of continuous time, to digital devices, which are best described in terms of events occurring at discrete points in time. In the process, the terminology of electronic filtering has made a transition from the analogue to the digital domain; and electronic filtering has come to be known as signal processing. Given the increasing commonality between digital signal processing and statistical time-series analysis, there are compelling reasons for why the two disciplines should share a common terminology, and this is what has transpired. Such is the convergence of these disciplines that, nowadays, their adherents contribute often to the same academic journals and they can be found at the same conferences. Nevertheless, considerable di!erences remain, both of emphasis and of con- ceptualisation. In particular, statisticians tend to operate principally within the time domain, in which their discretely sampled data naturally reside, whereas engineers, who are familiar with harmonic motions and oscillating currents, feel at home in the frequency domain. Our own account of linear filtering and