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Figure 8 plots the $ K=5$ weighted principal components identified for the angle-dependent component of the horn radiativity. Each component is weighted by its corresponding singular value, thus visually indicating its importance. Also plotted using the same line type are the zero-lines for each principal component. Note in particular that the first (largest) principal component is entirely positive.

Figure 8: First 5 principal components weighted by their corresponding singular values. Each angle-dependent impulse response is modeled as a linear combination of these angle-independent impulse-response components.
\includegraphics[width=0.9\textwidth ]{eps/tscw}

Figure 9 shows the complete horn impulse-response model ( $ \mbox{${\bm \alpha}$}$$ +$   $ \mbox{${\bm \gamma}$}$$ \cdot$   diag$ (z^{-\tau_i})$ ), overlaid with the original raw data $ {\mathbf{h}}$ . We see that both the fixed base-leakage and the angle-dependent horn-output response are closely followed by the fitted model.

Figure 9: Overlay of measured (solid) and modeled (dotted) impulse-responses at multiples of 15 degrees.
\includegraphics[width=0.9\textwidth ]{eps/mmir}

Figure 10 shows the estimated impulse response of the base-leakage component $ \underline{a}(n)$ , and Fig.11 shows the modeled angle-dependent horn-output components $ \mbox{${\bm \gamma}$}$ delayed out to their natural arrival times.

Figure 10: Modeled base-leakage impulse-response (angle-independent).
\includegraphics[width=0.9\textwidth ]{eps/alpha}

Figure 11: Modeled horn-output impulse-responses at multiples of 15 degrees.
\includegraphics[width=0.9\textwidth ]{eps/dirc}

Figure 12 shows the average power response of the horn outputs. Also overlaid in that figure is the average response smoothed according to Bark frequency resolution [16]. This equalizer then becomes $ H_0(z)$ in Fig.[*]. The filters $ H_{0L}(z)$ and $ H_{0R}(z)$ in Fig.[*] are obtained by dividing the Bark-smoothed frequency-response at each angle by $ H_0(z)$ and designing a low-order recursive filter to provide that equalization dynamically as a function of horn angle. The impulse-response arrival times $ \tau_i$ determine where in the delay lines the filter-outputs are to be summed in Fig.[*].

Figure 12: Average angle-dependent amplitude response overlaid with Bark-smoothed response to be used as a fixed equalization applied to the source.
\includegraphics[width=0.9\textwidth ]{eps/lhorneq}

Figure 13 shows a spectrogram view of the angle-dependent amplitude responses of the horn with $ H_0(z)$ (Bark-smoothed curve in Fig.12) divided out. This angle-dependent, differential equalization is used to design the filters $ H_{0L}(z)$ and $ H_{0R}(z)$ in Fig.[*]. Note that below 12 Barks or so, the angle-dependence is primarily to decrease amplitude as the horn points away from the listener, with high frequencies decreasing somewhat faster with angle than low frequencies.

Figure 13: Angle-dependent amplitude response divided by Bark-smoothed average response to be used as the basis for design of time-varying, angle-dependent equalization to be applied after $ H_0(z)$ .
\includegraphics[width=0.9\textwidth ]{eps/lnrpsimage}

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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 © 2016-03-26 by Julius O. Smith III, Stefania Serafin, Jonathan Abel, David P. Berners
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University