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Digital Reverberation

Two quantities have been proposed as criteria for measuring the ``naturalness'' of synthetic reverberation: the time density and the frequency density [8]. A good reverberator should provide high values of both densities, thus giving smooth, dense time and frequency responses.

The frequency density $D_f$ is defined as the average number of resonances per Hertz. A general expression can be derived from the order of the system (5), assuming that all the poles are distinct and no cancellation occurs:

D_f = \frac{1}{F_s} \sum_{i=1}^{N}m_i
\end{displaymath} (32)

In real rooms, the frequency density increases at higher frequencies (as can be seen from (35) below).

In the prototype case where the delay lines all have the same length $m$, we have

D_f =\frac {N m}{F_s}
\end{displaymath} (33)

The time density $D_t$ is defined as the number of nonzero samples per second in the impulse response. In actual rooms, $D_t$ is an increasing function of time. In order to obtain dense reverberation after the early reflections (e.g., after 80 msec), it helps to use different delay lengths.

The actual positions of frequency peaks depend on the feedback matrix and the delay lengths. If the delay lengths are fixed, we can vary some time-frequency properties of the structure simply by varying the distribution of eigenvalues of the feedback matrix. The total length of the delay lines should be chosen in such a way that the frequency density, as determined by (32), is high enough. Then the matrix eigenvalues can be adjusted to avoid resonant peak clustering or other undesirable mode distributions.

It is interesting to discuss the effect of eigenvalues in the prototype case of equal delays. A uniform distribution of eigenvalues along the unit circle is optimum for the frequency response in the sense that it minimizes the maximum distance between peaks. However, it produces a highly repetitive time response. Conversely, clustering the eigenvalues around a point on the unit circle can be good for maximizing the length of time patterns, but the clustering of frequency peaks produces a poor reverberator amplitude response vs. frequency. We see from these considerations that there is a time-frequency tradeoff. This tradeoff can be addressed using circulant matrices.

Figure 3: Time and Frequency behaviors for two Circulant Feedback Delay Networks which differ only by a shift on the rows of the feedback matrix.
\begin{figure}{\large a)}\\
...i / 3} & 1 & e^{j\pi / 3}

A couple of examples of different eigenvalue distributions are given in Fig. Fig. 3. The matrix ${\bf A}_2$ used in Fig. Fig. 3b is simply obtained by a right circular shift of the rows of the matrix ${\bf A}_1$ which is given by the junction of equal-impedance waveguides and, as already stated, has eigenvalues only at $1$ and $-1$. We can express ${\bf A}_2$ as the product ${\bf A}_1 {\bf\Pi}$ where ${\bf\Pi}$ is the right-shift matrix

{\bf\Pi} = \left[ \begin{array}{lll}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0 \\
\end{array} \right] .
\end{displaymath} (34)

Both ${\bf A}_1$ and ${\bf\Pi}$ are circulant, therefore the eigenvalues of ${\bf A}_2$ are given by the collection of the element-wise products of the eigenvalues of ${\bf A}_1$ and the eigenvalues of ${\bf\Pi}$, which are the $N$-th complex roots of $1$ [3]. For clarity, we set all the delay lengths equal in the examples.

As a side comment, we notice that ${\bf\Pi}$ is the scattering matrix of the circulator, a circuit device which can be used to obtain the multiplication of one-port scattering parameters [18].

Figure 4: Zero positioning which gives a nearly flat low-frequency response for the CFDN of Fig. 3b.
...arge b)}\\

The shape of the frequency response depends also on the zeros which were discussed in section IV. In particular, Theorem 2 provides a way of setting the zeros exactly over the poles in the prototype equal-delay case. We anticipated in section IV that the way to choose the vectors ${\bf b}$ and ${\bf c}$ indicated in Theorem 2 can be useful for getting a flat amplitude response at low frequencies when the delay lengths are slightly varied from the prototype case. Fig. Fig. 4 depicts the time and frequency responses for the CFDN using the same feedback matrix as in Fig. Fig. 3b, having ${\bf b}^T=\left[1, 1, 1\right]$, ${\bf
c}^T=\left[0, -1, 1\right]$, and delay lengths $m=[16, 17, 15]$. As we can see from Fig. Fig. 4, we are able to get a nearly flat amplitude response at low frequencies without losing the reverberating character of the time response. We believe that this is a good alternative to allpass filters which tend to have degenerate impulse responses when the poles approach the unit circle.

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