A NÁLISIS DE UN SISTEMA

A number of problems that can exploit the basic concepts proposed in this thesis give rise to an array of possible future research projects. A selected subset of these problems directly related to spatial audio are discussed below.

§7.2 Future Research 129

Design of Physical Structures for Beamforming

In Chapter 3, we have considered a source location estimator that exploits the diver- sity offered by scattering and reflections caused by a rigid body. It was shown that this method resulted in higher-resolution source location estimates using a smaller number of sensors than would typically be required. Further, in Chapter 2, it was suggested that this type of diversity could be modelled as a function of the normal vector to the surface of this rigid body. This implies that some structure could be synthesised to maximize the spatial diversity information obtained from a specified direction or region. Thus, the inverse problem of specifying a physical shape of an object that describes a desired diversity pattern will contribute to extending the practical applications of the method proposed in Chapter 3.

Exploiting Frequency Diversity for Source Separation

We can consider two main approaches to the problem of source separation; spatial beamforming and statistical analysis of speech signals. A limitation of the conven- tional beamforming approaches to the problem is the requirement of physically large sensor arrays and a large number of sensors. This requirement stems from the lim- ited spatial diversity of traditional array systems. Thus, the concept of the exploiting the frequency-domain diversity of the scatterer in Chapter 3 can be extended to high- resolution beamforming, e.g., eigenspace spatial beamforming. The localization and beamforming processes can therefore be used to enhance the source signal from the desired spatial regions, enabling a more robust application of the statistical speech separation methods.

Speaker Tracking in Under-Determined Systems

In Chapter 4, the binaural source localization problem was considered. It was found that a sound source could still be localized in an under-determined system (i.e., more than one active speaker in the binaural scenario), provided that the inter-subband correlation was at least partially known. This criteria is easily satisfied by audio signals such as speech, and is widely used to characterize individual speakers. Thus, the proposed method could be integrated to track, separate and enhance a desired source. A number of applications in automotive, virtual reality and robotic systems can also be envisaged.

Sparse Equalization for Robust Reverberation Control

The viability of the reverberation control mechanism described in Chapter 6 relies on the ability to accurately decompose the sound field coefficient in a region of interest, which in turn relies on the sensor positioning and operating frequency. In addition,

increasing the size of this region requires a larger number of measurement and re- production channels. However, the complexity of the acoustic environment remains the same, and therefore exploiting the sparsity of the acoustic environment becomes an attractive option for reverberant sound field control. Thus, the investigation of exploiting the sparsity of the reverberant image sources in space will contribute to improving the robustness of the algorithm described in Chapter 6 and enable more practical applications of sound field control in large spatial regions.

Appendix A

Signal Subspace Decomposition for

Direction of Arrival Estimation

In Chapters 3 and 4, we have considered the problem of source localization using an array of sensors placed on some rigid body, where the concepts of signal subspace decomposition could be applied to identify the source locations. In this context, the direction of arrival estimation of narrowband far field sources using a linear array represents the simplest source localization scenario that describes these concepts. The following discussion describes the application of the signal subspace technique known as MUSIC (MUltiple SIgnal Classification) [88], to the direction of arrival estimation problem.

A.1

Narrowband Signal Model

Consider a linear array with M uniformly spaced sensors illustrated in Figure A.1. In a sound field with Qnarrowband sources in the far field, the measured signal at themth_{(m=1, . . . ,M)sensor can be expressed as}

ym(ω,t) = Q

∑

q=1

Am(ω,θq)sq(ω,t) +nm(ω,t), (A.1)

where Am(ω,θq)is the sensor response to theqth (q=1, . . . ,Q)sourcesq(ω,t)in the

direction θq andnm(ω,t) is the noise at the sensor. Omitting the frequency depen-

dence ω, theM sensor signals can be expressed in the matrix form

y(t) =A(Θ)s(t) +n(t), (A.2) where y(t) =h y1(t) y2(t) · · · yM(t) iT (1×M) , 131

Figure A.1: DOA estimation of far field sources impinging on a linear array. s(t) =h s1(t) s2(t) · · · sQ(t) iT (1×Q) , and n(t) =h n1(t) n2(t) · · · nM(t) iT (1×M) .

A(Θ)is the array manifold matrix that describes the response of the sensors to the impinging sources. For a linear array with a uniform sensor spacing d, the array manifold can be expressed as

A(Θ) = h _a(θ1) a(θ2) · · · a(θQ) i (M×Q) =      

e−ik(0)dcosθ1 _e−ik(0)dcosθ2 · · · _e−ik(0)dcosθQ

e−ik(1)dcosθ1 _e−ik(1)dcosθ2 · · · _e−ik(1)dcosθQ

..

. ... . .. ...

e−ik(M−1)dcosθ1 _e−ik(M−1)dcosθ2 · · · _e−ik(M−1)dcosθQ

      (M×Q) , (A.3)

wherea(θq)is the steering vector in the directionθq.

A.2

Signal Subspace Decomposition

The correlation matrix of the sensor signalsy(t)can be expressed as

R,Eny(t)y(t)Ho=A(Θ)Ens(t)s(t)HoA(Θ)H+Enn(t)n(t)Ho (A.4) for uncorrelated s(t)andn(t), whereE_{·}represents the expectation operator over time. If we consider the individual sources to be uncorrelated andn(t)to be spatially and temporally white Gaussian noise, (A.4) can be simplified further as