Separating Horn Output from Base Leakage

Let $M=256$ denote the number of impulse-response samples in each measured impulse response

Let $N=25$ denote the number of angles (-180:15:180) at which impulse-response measurements were taken

Denote the $M\times N$ impulse-response matrix by ${\mathbf{h}}$. Each column of ${\mathbf{h}}$ is an impulse response at some horn angle.

We model ${\mathbf{h}}$ as

$${\mathbf{h}}=$$   $$\mbox{${\bm \alpha}$}$$$$+$$   $$\mbox{${\bm \gamma}$}$$$$\cdot$$   diag$$(z^{-\tau_i}) + {\mathbf{e}}$$

where

• $\tau_i =$ arrival-time delay, in samples, for the horn output in the $i$th row
• Arrival times estimated by cross-correlation between $i$th impulse response and the same impulse response after converting it to minimum phase.
• Diagonal matrix diag$(z^{-\tau_i})$ denotes a shift operator which delays the $i$th column of $\mbox{${\bm \gamma}$}$ by $\tau_i$ samples.
• $\mbox{${\bm \gamma}$}$ contains the horn-output impulse response (without the base leakage) shifted to time zero
• Error matrix ${\mathbf{e}}$ is to be minimized in the least-squares sense

Each column of the matrix $\mbox{${\bm \alpha}$}$ contains a copy of the estimated horn-base leakage impulse-response:

   $$\mbox{${\bm \alpha}$}$$$$= \underline{a}\cdot\mathbf{1}^T$$

where $\mathbf{1}^T = [1,1,\dots,1]$.

The estimated angle-dependent impulse-responses in $\mbox{${\bm \gamma}$}$ are modeled as linear combinations of $K=5$ fixed impulse responses, viewed (loosely) as principal components:

   $$\mbox{${\bm \gamma}$}$$$$= {\mathbf{g}}\cdot {\mathbf{w}}$$

where ${\mathbf{g}}$ is the $M\times K$ orthonormal matrix of fixed filters (principal components), and ${\mathbf{w}}$ is a $K\times N$ matrix of weights, found in the usual way by a truncated singular value decomposition (SVD).