Digital acoustics: processing wave fields in space and time using DSP tools

Abstract

I . I N T R O D U C T I O NIn the world of digital signal processing (DSP), there are three fundamental tools that have become the basis of every algorithm, system, and theory dealing with the processing of digital audio. Those are the Nyquist sampling theory, the Fourier transform, and digital filtering. We could add a fourth one -the short time Fourier transform -which generalizes the Fourier transform to account for non-stationary signals such as music and speech. These concepts are so embedded into the creative thinking of audio engineers and scientists that new ideas are often intuitively based on one (or more) of these fundamental tools. Digital audio coding, speech synthesis, and adaptive echo cancellation are great examples of complex systems built on the theories of sampling, Fourier analysis, and digital filtering.Fourier theory itself is built on some of the most basic tools of mathematics, such as vector spaces and integration theory (although harmonic analysis was not originally conceived this way by Joseph Fourier [1]). From an intuitive perspective, the Fourier transform can be seen as a change of representation obtained by projecting the input signal s (t) onto an orthogonal set of complex exponential functions ϕ(ω, t) = e − j ωt , given by S(ω) = R s (t)ϕ(ω, t)dt. The Fourier representation is useful for many types of signals, and is oftentimes the logical choice. As a consequence, many of the mathematical and computational tools available today for the purpose of DSP have been developed

The Ray Space Transform: A New Framework for Wave Field Processing

“…2) Performance measure: In order to validate the localization approach, we measure the performance with the localization error, defined as the distance between the actual source position and the estimated one: ˆ where r ˆJ is obtained through (34).…”

“…sources that are closer than r = 2Δ 2 /λ,r being the source distance, Δ the length of the array and λ the wavelength [33]. The representation proposed in [31], [32] enabled the authors to approximate the acoustic field produced by sources in the near field by far field components, which facilitates the estimation of the Direction-of-Arrival of acoustic sources and enables spatial filtering [34].…”