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Fig. 3. Modern spectral (spatial) estimation using the maximum entropy method parametric estimator for direction-of-arrival estimation of three plane waves at 20o, 40o, 55o arrival angles; (a) Synthesized 16-element array temporal measurements (signal-to-noise = 0 dB) with true source locations; (b) Maximum-entropy-method spatial spectral estimation results with estimated arrivals at 20.4o, 40.0o, 53.6o generated from an ensemble of 256 realizations.
Finally, we use this extracted model to develop an explicit model-based processor (MBP) by developing a set of har- monic equations (ODE) for a sinusoid in noise and construct the MBP based on these relations. The results are shown in Fig. 2e. So we see that once we have defined the acoustical problem, and assessed the a priori information including the underlying phenomenology, then we can proceed up the staircase and exit any time we are satisfied with the result. This is the “bottoms-up” approach.7
Next we choose to investigate this approach in more detail by selecting some processing areas with an accompa- nying set of acoustical applications. Here we illustrate not only the fundamental approach to problem solving, but also to observe some of the popular processing paradigms avail- able to the acoustician for analysis and information extrac- tion. Our aim is to define a specific problem that represents a class of problems and then show some of the potential sig- nal processing solutions available, demonstrating them through simulation or experiments.
Step 1: Simple spectral estimation techniques
Classical spectral analysis is a very powerful example of a set of tools that have evolved in signal processing especial- ly in acoustics. Here a raw measurement is “transformed” to the spectral or Fourier domain for analysis. Modern tech- niques of spectral estimation can be considered both “black or gray-box” processors and even physics-based processors
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depending on the underlying application. We call the black-
box/gray-box methods “parametric processors,” since they employ a variety of underlying model sets to achieve their enhancement and improved spectral estimation. Thus, the parametric spectral estimator consists of a processor employed to estimate the parameters of the underlying
model set and then perform a power spectrum estimate using this model.
Modern spectral analysis techniques easily extrapolate to the space-time domain as long as we assume that the incom- ing wave front is separable in space (array) and time (or fre- quency). In this context, we can consider a measurement array as a spatial sampler of the arriving wave front. If we fur- ther assume that the temporal portion of the wave is restrict- ed to a narrow frequency band, then we collapse its temporal response to a single frequency line (in the Fourier space) that can also be considered a parameter. So we see that estimating the arrival angle in the case of a planar wave front or the source location in the case of a spherical wave front can be considered a problem of “spatial” spectral estimation and all of the usual modern techniques (with some restrictions)
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sensor array to convey these ideas using a concrete example.
Spectral (spatial) estimation for direction-of-arrival
Suppose a 16-element linear array with acoustic sensors uniformly spaced at 2.5m is impinged upon by a set of plane waves generated from three (3) sources emanating from 20o, 40o, and 55o incidence angles. The temporal frequency is
apply to the array signal processing problem as well.
In acoustics, a large set of problems reduce to array pro- cessing or spatial spectral estimation in this context. Such problems as ocean acoustic (sonar) signal processing for tar- get direction-of-arrival (DOA) estimation or localization fall into this category along with ultrasonic NDE and biomedical
Clearly, seismic array processing, of which most of these ideas evolve, is a root application of arrays for epicenter location and velocity estimation. With this information at hand, let us consider a simple example of a plane wave impinging on a
processing searching for flaws or abnormalities.
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