the pseudo-inverse of ⌫g,L (f). Given the estimated Ŝg(f) obtained using ⌫g,L (f), then

model-based sparse component analysis framework was established in Chapter 3 along with the three fundamental components. The first component is structured sparse representation which was elaborated in Chapter 4. The second component is compressive acoustic measurements which was characterized in Chapter 5 and the third component is model-based sparse recovery algorithm that we have studied in the previous Chapter 6. This framework assumed that the geometrical set-up of microphone array is known in advance. The recent studies presented in Section 6.4.3 demonstrate that the conventional microphone arrays are not an optimal design and the sparse recovery techniques yield higher performance using ad-hoc microphone topology. Hence, in this chapter we draw a generalization to our framework by formulating a unified structured sparse coding scheme for source-sensor localization and speech recovery. Having the source and sensors being localized, we elaborate on optimality of inverse filtering to perform speech separation and dereverberation. 7.1 Generalized Formulation We consider a scenario in which an unknown sound signal Sg(f) at frequency f emanates from an unknown location ⌫g in an enclosure and impinges on an array ofM microphones located at L = {µ1,µ2, · · ·,µM} on a 2-D plane. The room responseH(f,⌫g,µm) from the source location ⌫g to the location µm is known for each of theMmicrophone locations 1. The signal captured by a microphone located at µm would therefore be Xm(f) =H(f,⌫g,µm)Sg(f) and representing X (f) = [XT1 (f) · · ·XTM(f)]> and ⌫g,L (f) = [H(f,⌫g,µ1) · · ·H(f,⌫g,µM)]>, we can write X (f) =⌫g,L (f)Sg(f) (7.1) ⌫g,L (f) is also known as the array manifold vector which is specific to the source location ⌫g and the locations of the microphones atL = {µ1,µ2, · · ·,µM}. Our objective is to estimate the location of the source and sensors and recover the speech signal. 1. Please refer to Chapter 5 for an intricate discussion on characterization of the room acoustic channel