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Separating Horn Output from Base Leakage

Since Fig. 6 indicates the existence of fixed and angle-dependent components in the measured impulse responses, and since such angle-independent component is strongly suppressed by baffling in the cabinet enclosure, it is desirable to eliminate this fixed component from the measurements. For this purpose, an iterative algorithm was developed which models the two components separately.

Let $ M=256$ denote the number of impulse-response samples in each measured impulse response,and let $ N=25$ denote the number of angles (-180:15:180) at which impulse-response measurements were taken. We denote the $ M\times N$ impulse-response matrix by $ {\mathbf{h}}$. Each column of $ {\mathbf{h}}$ is an impulse response at some horn angle. (Figure 6 can be interpreted as a plot of the transpose of $ {\mathbf{h}}$.)

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

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

where $ \tau_i$ is the arrival-time delay, in samples, for the horn output in the $ i$th row (the delays clearly visible in Fig. 6 as a function of angle). These arrival times are estimated as the location of the peak in the cross-correlation between the $ i$th impulse response and the same impulse response after converting it to minimum phase [11]. The diagonal matrix diag$ (z^{-\tau_i})$ denotes a shift operator which delays the $ i$th column of $ \mbox{${\bm \gamma}$}$ by $ \tau_i$ samples. Thus, $ \mbox{${\bm \gamma}$}$ contains the horn-output impulse response (without the base leakage) shifted to time zero (i.e., the angle-dependent delay is removed). Finally, the 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:

   $\displaystyle \mbox{${\bm \alpha}$}$$\displaystyle = \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:

   $\displaystyle \mbox{${\bm \gamma}$}$$\displaystyle = {\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) [7].



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``Doppler Simulation and the Leslie'', by Julius O. Smith III, Stefania Serafin, Jonathan Abel, David P. Berners, Music 421 Handout, Spring 2002 .
Copyright © 2006-03-30 by Julius O. Smith III, Stefania Serafin, Jonathan Abel, David P. Berners
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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