Frequency Wavenumber Processing
In signal processing, extraction of a signal buried in noise has been a primary goal of lasting interest. A digital fi lter is often employed to modify the spectrum of a signal in some prescribed manner. Notably, the proposed fi lter is designed to possess unit response in the spectral region where the desired signal is residing and low response where the undesired signal and noise are residing. This strategy will work only when the spectrum of the desired signal does not overlap (or only partially overlap) with the spectrum of the undesired signal and noise. This is also true in wavefi eld processing. Spectrum shaping is necessary whenever the aim is to enhance certain types of wavefi elds and suppress other types of unwanted wavefi elds, often termed as noise or interference. Such a selective enhancement is possible on the basis of spectral differences in the frequency wavenumber (ω-k) domain. For example, it is possible, using a digital fi lter, to enhance the wavefi eld traveling at a speed different from that of the noise or the interference, as their spectra lie on different radial lines in the (ω-k) plane. Other situations where spectral shaping is required are (a) prediction of wavefi elds, (b) interpolation of wavefi elds, and (c) extrapolation of wavefi eld into space where the fi eld could not be measured. In this chapter, we shall consider the design of pass fi lters, specially useful in wavefi eld processing such as a fan and a quadrant fi lter. When signal and noise are overlapping in the frequency wavenumber domain, simple pass fi lters are inadequate. Optimum fi lters such as Wiener fi lters are required. We cover this topic in some depth in view of its importance. Next, we introduce the concept of noise cancellation through prediction. In Chapter 6, we shall evaluate the effectiveness of some of the techniques described here.