Separating Horn Output from Base Leakage

Note that Fig.3.16 indicates the existence of fixed and angle-dependent components in the measured impulse responses. An iterative algorithm was developed to model the two components separately [473].

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 3.16 can be interpreted as a plot of the transpose of ${\mathbf{h}}$.)

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

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

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.3.16 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 [337]. 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:

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

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) [167].

Algorithm.

To start the separation algorithm, $\mbox{${\bm \gamma}$}$$_0$ is initialized to the zero-shifted impulse response data ${\mathbf{h}}\cdot$diag$(z^{\tau_i})$, ignoring the tails of the base-leakage they may contain. Then $\mbox{${\bm \alpha}$}$$_0$ is estimated as the mean of ${\mathbf{h}}-$$\mbox{${\bm \gamma}$}$$_0$diag$(z^{-\tau_i})$. This mean is then subtracted from ${\mathbf{h}}$ to produce ${\mathbf{b}}_1 = ({\mathbf{h}}-$   $\mbox{${\bm \alpha}$}$$_0)$diag$(z^{-\tau_i})$ which is then then converted to $\mbox{${\bm \gamma}$}$$_1 = {\mathbf{g}}_1 \cdot {\mathbf{w}}_1$ by a truncated SVD. A revised base-leakage estimate $\mbox{${\bm \alpha}$}$$_1$ is then formed as ${\mathbf{h}}-$$\mbox{${\bm \gamma}$}$$_1$diag$(z^{-\tau_i})$, and so on, until convergence is achieved.

Results.

Figure 3.18 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 3.18: 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 (from [473]).

Figure 3.19 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 3.19: Overlay of measured (solid) and modeled (dotted) impulse-responses at multiples of 15 degrees (from [473]).

Figure 3.20 shows the estimated impulse response of the base-leakage component $\underline{a}(n)$, and Fig.3.21 shows the modeled angle-dependent horn-output components $\mbox{${\bm \gamma}$}$ delayed out to their natural arrival times.

Figure 3.20: Modeled base-leakage impulse-response (angle-independent) (from [473]).

Figure 3.21: Modeled horn-output impulse-responses at multiples of 15 degrees (from [473]).

Figure 3.22 shows the average power response of the horn outputs. Also overlaid in that figure is the average response smoothed according to Bark frequency resolution [463]. This equalizer then becomes $H_0(z)$ in Fig.3.12. The filters $H_{0L}(z)$ and $H_{0R}(z)$ in Fig.3.12 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.3.12.

Figure 3.22: Average angle-dependent amplitude response overlaid with Bark-smoothed response to be used as a fixed equalization applied to the source (from [473]).

Figure 3.23 shows a spectrogram view of the angle-dependent amplitude responses of the horn with $H_0(z)$ (Bark-smoothed curve in Fig.3.22) divided out. This angle-dependent, differential equalization is used to design the filters $H_{0L}(z)$ and $H_{0R}(z)$ in Fig.3.12. 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 3.23: 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 $H_0(z)$ (from [473]).