Direction of arrival (DOA) estimation of multiple sources using sensor arrays has been an active problem in signal processing for decades. The geometry of the sensor array plays a major role in determining the source separation capability in sonar, radar, robotics and communications applications, where accurate localization is im- portant. Traditionally, the various linear, circular and spherical sensor arrays used for DOA estimation assume free field propagation conditions between the source and sensors. However, this may not be true in certain applications, due to the scattering and reflection of sound waves by the rigid body used as a sensor mount. These effects
are generally exacerbated as the shape and structure of the mounting object becomes more complex, but they could be considered as a form of spatial diversity contained in the frequency-domain of the channel transfer function. This chapter introduces a DOA estimation method based on signal subspace techniques that exploits the addi- tional diversity afforded by a sensor array mounted on a complex-shaped rigid body, such as an aircraft, submarine or robotic platform.
Broadband DOA estimation techniques can be broadly categorized into those based on cross-correlation analysis, high-resolution signal subspace techniques or a combination of the two. Cross-correlation-based techniques are typically described using simple array geometries in free field, yet are equally applicable to a sensor array mounted on a rigid body. The time difference of arrival (TDOA) at the sensors is inherently a function of source direction. Therefore, the TDOA at the peaks of the cross-correlation function can be used to identify the source directions of arrival. The Steered Response Power (SRP) [30] method is a popular multi-sensor implementa- tion of the Generalized Cross Correlation (GCC) method [54], that exploits the corre- lation between signals for DOA estimation. Although more successful multi-sensor variants of these algorithms have been developed [31], they are still fundamentally TDOA estimators, that map time delay to a source location. Hence, any diversity present in the frequency-domain of the channel transfer function remains unseen and unutilized by these DOA estimators.
Signal subspace techniques such as MUltiple Signal Classificiation (MUSIC) [88] and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) [87] are inherently narrowband methods of DOA estimation. The Coherent Signal Subspace (CSS) [101] was proposed in order to transform the broadband DOA esti- mation problem into a narrowband problem. This was achieved by transforming (fo- cussing) each subband of the broadband signal into a known narrowband frequency. The concept of the CSS has since been developed into a number of broadband DOA estimation techniques based on beamforming [58, 102, 103], modal decomposition [1, 99] and unitary focussing matrices [46, 47, 101]. Although uniform linear arrays (ULAs) are typically required for the DOA estimation algorithms based on ESPRIT, narrowband transformations that map arbitrary array geometries to ULAs [8, 27, 110] have been demonstrated. Hence, the broadband DOA estimators based on MUSIC and ESPRIT can theoretically be adapted to any array geometry, including a sen- sor array mounted on a complex-shaped rigid body. However, the complicated be- haviour of the channel transfer functions in the frequency-domain make them far more susceptible to imperfections in the frequency focussing process, which can re- sult in a reduction of the DOA estimation accuracy as described in Appendix B. As an
§3.1 Introduction 29
example, consider Wideband MUSIC [101], a MUSIC broadband DOA estimator that implements the CSS concept. The broadband signals are segmented into multiple frequencies, multiplied by a frequency focussing transformation, and the resulting focussed correlation matrices are then aggregated. For a sensor array mounted on a rigid scatterer, the channel transfer function behaves in a complicated fashion in the frequency-domain, and results in imperfections in the numerical calculation of the focussing transformations. Hence, the signal spaces of the focussed correlation ma- trices may not fully aligned with each other. Aggregating these matrices results in the gradual increase of the rank of the coherent signal subspace as additional frequency segments are included. This misalignment eventually leads to the disappearance of the noise subspace, which degrades the performance of the DOA estimator and introduces additional complexity to the noise subspace identification process.
In this context, the human auditory system represents an excellent example of a two sensor array mounted on a complex-shaped rigid body. The head and torso act as scattering objects, while the structures within the pinna produce reflections that act as multipath signals [40]. This results in direction specific changes to the phase and amplitude of a signal, collectively known as the Head-Related Transfer Function (HRTF). Perceptual studies in the past have found that the localization cues embed- ded in the HRTFs provide the necessary spatial diversity information for localization in 3-D [7, 12, 45, 70, 83, 93]. Further, these results suggest that frequency-domain diversity is a critical piece of information used to reduce the resource requirements of the 3-D DOA estimation problem, while improving resolution between adjacent source locations. However, works in the area are focussed on empirical modelling used for the special case of binaural sound source localization [48, 84, 108, 111].
In this chapter, we use the inspiration derived from biological localization mecha- nisms to propose a broadband DOA estimation technique for sensor arrays mounted on complex-shaped rigid bodies. In Section 3.2, we present the background theory on the subband representation of broadband signals, and show that each subband carries both directional and source information. This includes a frequency dependent carrier term, which must be removed if multiple subband signals are to be combined. Section 3.3 introduces the signal model, subband signal extraction and focussing pro- cesses, and describes how the subband signals can be combined to retain the spatial diversity information in frequency. Next, the channel transformation matrix is de- fined, and used to derive the requirements for the existence of a noise subspace. Section 3.4 describes the broadband DOA estimators for several DOA estimation sce- narios; an ideal scenario where sources are uncorrelated between subbands, the real- world equivalent of the ideal scenario and the DOA estimation of known sources.
x y z Sensor 1 Sensor 2 Sensor m Source q h1 ( Өq,t ) h2 ( Өq,t ) hm ( Өq,t )
Figure 3.1: Source-sensor channel impulse responses of a sensor array mounted on a complex-shaped rigid body.
The performance of our algorithm is compared with a correlation-based DOA esti- mation technique, SRP-PHAT, and a signal subspace technique, Wideband MUSIC. Section 3.5 briefly describes these algorithms, the performance measure for compar- ing the different algorithms and the simulation setup. Simulation results are dis- cussed in Section 3.7, and is followed by an analysis of the computational complexity in Section 3.8.