# Big-Delay Estimation for Speech Separation in Assisted Living Environments
## 3 The Elevatogram for Large Delay Estimation  
The Elevatogram was introduced in [5]. In previous contributions, it was demonstrated that in an anechoic condition without any environmental noise, the Elevatogram can estimate relative delays up to more than 500 samples, given a sampling frequency of, \(F_{\text{s}}=16\text{ kHz}\)[5; 8]. The TF representation of the mixtures, \(x_{1}[n]\) and \(x_{2}[n]\), in Equations (8) and (9), are given as \(\mathbf{X}_{1}[k,\tau]\) and \(\mathbf{X}_{2}[k,\tau]\), respectively. The Cross-Power Spectrum (CPS) is calculated as  
\[\mathbf{X}[k,\tau]=\mathbf{X}_{1}[k,\tau]\overline{\mathbf{X}}_{2}[k,\tau], \tag{17}\]  
where the complex conjugate of \(\mathbf{X}_{2}[k,\tau]\) is denoted as \(\overline{\mathbf{X}}_{2}[k,\tau]\). The normalized CPS contains the phase information for a given delay, \(\delta_{j}\), and is expressed as  
\[\Delta\phi[k,\tau]=\angle\mathbf{X}[k,\tau]=\angle\mathbf{X}_{1}[k,\tau]- \angle\mathbf{X}_{2}[k,\tau]+\Omega\delta_{j}+2\pi r, \tag{18}\]  
where \(r\in\mathbb{Z}\) comes from the modulo-\(2\pi\) phase ambiguity [72], and \(\Delta\phi[k,\tau]\in\mathbb{R}\) and \(\Omega=2\pi k/K\). The parameter, \(\Delta\phi[k,\tau]\), can be modeled as a wrapped scaled-Gaussian Mixture Model (scaled-GMM) or Von-Mises noise distribution [73; 74], because the phase is a circular variable that is bounded in the interval \((-\pi,\pi]\). The random variable \(\Delta\phi[k,\tau]\) is zero-mean. It has a periodicity of \(2\pi\) and is unimodal in one cycle [75] and i.i.d. A histogram of the \(k^{th}\) discrete frequency row of \(\Delta\phi[k,:]\) was generated in the Elevatogram in [5]. The bins of the histogram are given by the vector \([-\pi,\ldots,\pi]^{T}\), which has \(L\) elements, each corresponding to a quantization level, which are uniformly separated, and each bin is spaced by \(\Delta_{d}\). If the bins are indexed by \(\hat{\phi}\), then the set of TF bins contributing to any \(\hat{\phi}\), is obtained using  
\[I[\hat{\phi}]:=\{\{k,\tau\}:|\Delta\phi[k,\tau]-\hat{\phi}|<\Delta_{d}\}. \tag{19}\]The higher the cardinality of \(|I[\hat{\phi}]|\), the more intensely that bin \(\hat{\phi}\) has been activated in the histogram, \(\mathbf{h}_{k}^{T}\). The histogram is generated as follows,  
\[\mathbf{h}_{k}=\text{hist}(\Delta\phi[k,:],L)\in\mathbb{R}^{1\times L}. \tag{20}\]  
The Elevatogram [5] computes a histogram for each of the \(K\) frequencies. It generates a matrix, \(\mathbf{P}\in\mathbb{R}^{L\times K}\), which was also called the Elevatogram by the authors of [5], and is obtained by concatenation resulting in a phase-frequency matrix,  
\[\mathbf{P}=[\mathbf{h}_{1}^{T},\mathbf{h}_{2}^{T},\mathbf{h}_{3}^{T},\ldots \mathbf{h}_{K}^{T}]\in\mathbb{R}^{L\times K}. \tag{21}\]  
The slanted lines in the example shown in Figure 2a, which depict the distribution of phase with respect to frequency, represent a single line that undergoes phase wraparound at \(1,3,5\) and \(7\) kHz. The larger the delay, the more often this phase wraparound occurs. The location of the set of most significant collinear points forming a straight line in the phase-frequency domain was determined in [5], as shown in Figure 2b. This concept is analogous to the distance-angle accumulator in the Hough transform [76], which receives the highest activation. This value is marked in green and denoted as \(\phi_{\text{max}}\). The relative delay produced by the Elevatogram [5] is  
\[\delta_{j}=-\frac{K}{L}\tan\big{(}\phi_{\text{max}}\big{)}. \tag{22}\]  
In general, the phase wraparound issue prevents the IPD cues from being mapped to a unique delay, \(\delta_{j}\). Using the closed-form solution in Equation (22), the relative delay can be estimated unambiguously. This explains the success of the Elevatogram approach [5]. For better estimation, the base histogram matrix, \(\mathbf{P}\), was tiled by the author of [5] by concatenating tiles in a \(\mathcal{T}\times\mathcal{T}\) grid.