Active acoustical sensing should utilize signal modalities with sufficient degrees of freedom to identify range, range rate, and those higher derivatives necessary to identify object rotation, as well as potential spectral properties which could be used for signature analysis of specific objects in a field. In addition, the specular nature of acoustic relfection from smooth objects demands signal processing which has substantial dynamic range along with good noise immunity. It is also shown that time delay spectrometry (TDS) is a modality which meets these requirements. The principles of TDS are developed as they apply to its use as a phase-coherent spread-spectrum communication technique. A constant amplitude signal with programmable phase structure is used as acoustic stimulus. System response to that stimulus consists of an ensemble of delayed replicas with complex amplitude and phase modulation. This can be interpreted as a signal whose dimensionality is determined by the degrees of freedom in the programmable phase structure. Selective spatial filtering followed by mapping to a lower-dimensional space can bring the property of interest to a usable form. This includes the conventional time domain reponses (range, rate,…) and frequency domain responses (spectrum, cepstrum,…).

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Expressing time as a fourth space coordinate under the rule x4 = ict, where i is the imaginary operator, allows the interpretation of a complex velocity which is the four-dimensional gradient of a potential. The three-conventional space components of this gradient yield particle velocity, while the fourth component yields sound pressure divided by the product of density and sound speed c. In accordance with an energy theorem of Heyser [J. Audio Eng. Soc. 19, 902 (1971)], this four-dimensional velocity is interpreted as the local component of an energy function whose global support is represented by a complex Hilbert transform component. This leads to a total energy density which contains a term identifiable as the reactive component of an instantaneous complex intensity. Coinciding with the well-known expression for the time averaged intensity of monochromatic signal dependence, this instantaneous intensity applies to any time dependence and has a time average identical to the conventional expression. It is shown that, in the case where instantaneous reactive intensity vanishes, the instantaneous active intensity is numerically identical to the energy-time curve (ETC) which is now used in acoustical analysis.

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