FDN Reverberation

Feedback Delay Networks (FDN) were introduced earlier in §2.7. An example is shown in Fig.2.29 on page [*]. After a brief historical summary, this section will cover some practical considerations for the use of FDNs as reverberators.

History of FDNs for Artificial Reverberation

Feedback delay networks were first suggested for artificial reverberation by Gerzon [156], who proposed an ``orthogonal matrix feedback reverberation unit''. He noted that individual feedback comb filters yielded poor quality, but that several such filters could sound good when cross-coupled. An ``orthogonal matrix feedback'' around a parallel bank of delay lines was suggested as a means of obtaining maximally rich cross-coupling. He was especially concerned with good stereo spreading of the reverberation at a time when most artificial reverberators sought merely to decorrelate the reverberation in each output channel.

Later, and apparently independently, Stautner and Puckette [473] suggested a specific four-channel FDN reverberator and gave general stability conditions for the FDN. They proposed the feedback matrix

$\displaystyle \mathbf{A}= g\frac{1}{\sqrt{2}}
\left[\begin{array}{rrrr}
0 & 1 &...
... 0 \\
-1 & 0 & 0 & -1\\
1 & 0 & 0 & -1\\
0 & 1 & -1 & 0
\end{array}\right]
$

which can be understood as a permutation (with one row sign inversion) of a $ 4\times4$ Hadamard matrix (see §3.7.2).

More recently, Jot [217,216] developed a systematic FDN design methodology allowing largely independent setting of reverberation time in different frequency bands. Using Jot's methodology, FDN reverberators can be polished to a high degree of quality, and they are presently considered to be among the best choices for high-quality artificial reverberation.

Jot's early work was concerned only with single-input, single-output (SISO) reverberators. Later work [218] with Jullien and others at IRCAM was concerned also with spatializing the reverberation.

Figure 3.10: Feedback Delay Network (FDN) structure proposed for artificial reverberation by Jot.
\includegraphics[width=\twidth]{eps/FDNJot}

An example FDN reverberator using three delay lines is shown in Fig.3.10. It can be seen as an FDN (introduced in §2.7), plus an additional low-order filter $ E(z)$ applied to the non-direct signal. This filter is called a ``tonal correction'' filter by Jot, and it serves to equalize modal energy irrespective of the reverberation time in each band. In other words, if the decay time is made very short in some band, $ E(z)$ will have a large gain in that band so that the total energy in the band's impulse-response is unchanged. This is another example of orthogonalization of reverberation parameters: In this case, adjustments in reverberation time, in any frequency band, do not alter total signal energy in the impulse response in that band.


Choice of Lossless Feedback Matrix

As mentioned in §3.4, an ``ideal'' late reverberation impulse response should resemble exponentially decaying noise [314]. It is therefore useful when designing a reverberator to start with an infinite reverberation time (the ``lossless case'') and work on making the reverberator a good ``noise generator''. Such a starting point is ofen referred to as a lossless prototype [153,430]. Once smooth noise is heard in the impulse response of the lossless prototype, one can then work on obtaining the desired reverberation time in each frequency band (as will be discussed in §3.7.4 below).

In reverberators based on feedback delay networks (FDNs), the smoothness of the ``perceptually white noise'' generated by the impulse response of the lossless prototype is strongly affected by the choice of FDN feedback matrix as well as the (ideally mutually prime) delay-line lengths in the FDN (discussed further in §3.7.3 below). Following are some of the better known feedback-matrix choices.

Hadamard Matrix

A second-order Hadamard matrix may be defined by

$\displaystyle \mathbf{H}_2 \isdef
\frac{1}{\sqrt{2}}
\left[\begin{array}{rr}
1 & 1\\
-1 & 1
\end{array}\right],
$

with higher order Hadamard matrices defined by recursive embedding, e.g.,

$\displaystyle \mathbf{H}_4 \isdef
\frac{1}{\sqrt{2}}
\left[\begin{array}{rr}
\...
...}{rrrr}
1& 1& 1&1\\
-1& 1&-1&1\\
-1&-1& 1&1\\
1&-1&-1&1
\end{array}\right].
$

When $ n$ is a power of $ 4$, the Hadamard matrix $ \mathbf{H}_n$ of that order requires no multiplies in fixed-point arithmetic. An $ n\times
n$ Hadamard matrix has the maximum possible determinant of any $ n\times
n$ complex matrix containing elements which are bounded by $ 1$ in magnitude. This can be seen as an optimal mixing and scattering property of the matrix.

As of version 0.9.30, Faust's math.lib4.12contains a function called hadamard(n) for generating an $ n\times
n$ Hadamard matrix, where $ n$ must be a power of $ 2$. A Hadamard feedback matrix is used in the programming example reverb_designer.dsp (a configurable FDN reverberator) distributed with Faust.

A Hadamard feedback matrix is said to be used in the IRCAM Spatialisateur [218].


Householder Feedback Matrix

One choice of lossless feedback matrix $ \mathbf{A}_N$ for FDNs, especially nice in the $ 4\times4$ case, is a specific Householder reflection proposed by Jot [217]:

$\displaystyle \mathbf{A}_N = \mathbf{I}_N - \frac{2}{N}\uv_N\uv_N^T \protect$ (4.4)

where $ \uv_N^T = [1, 1, \dots, 1]$ can be interpreted as the specific vector about which the input vector is reflected in $ N$-dimensional space (followed by a sign inversion). More generally, the identity matrix $ \mathbf{I}_N$ can be replaced by any $ N\times N$ permutation matrix [153, p. 126].

It is interesting to note that when $ N$ is a power of 2, no multiplies are required [430]. For other $ N$, only one multiply is required (by $ 2/N$).

Another interesting property of the Householder reflection $ \mathbf{A}_N$ given by Eq.$ \,$(3.4) (and its permuted forms) is that an $ N\times N$ matrix-times-vector operation may be carried out with only $ 2N-1$ additions (by first forming $ \uv_N^T$ times the input vector, applying the scale factor $ 2/N$, and subtracting the result from the input vector). This is the same computation as physical wave scattering at a junction of identical waveguidesC.8).

An example implementation of a Householder FDN for $ N=3$ is shown in Fig.3.11. As observed by Jot [153, p. 216], this computation is equivalent to $ N$ parallel feedback comb filters with one new feedback path from the output to the input through a gain of $ -2/N$.

Figure 3.11: FDN using a Householder-reflection feedback matrix.
\includegraphics[width=0.7\twidth]{eps/householder1}

A nice feature of the Householder feedback matrix $ A_N$ is that for $ N\neq 2$, all entries in the matrix are nonzero. This means every delay line feeds back to every other delay line, thereby helping to maximize echo density as soon as possible.

Furthermore, for $ N=4$, all matrix entries have the same magnitude:

$\displaystyle \mathbf{A}_4 = \frac{1}{2}
\left[\begin{array}{rrrr}
1 & -1 & -1 ...
...
-1 & 1 & -1 & -1\\
-1 & -1 & 1 & -1\\
-1 & -1 & -1 & 1
\end{array}\right].
$

Only the $ N=4$ case is ``balanced'' in this way. For larger $ N$, the diagonal becomes larger than the off-diagonal elements, and as $ N$ becomes very large, the FDN approaches a bank of decoupled parallel comb filters.

Due to the elegant balance of the $ N=4$ Householder feedback matrix, Jot [216] proposes an $ N=16$ FDN based on an embedding of $ N=4$ feedback matrices:

$\displaystyle \mathbf{A}_{16} = \frac{1}{2}
\left[\begin{array}{rrrr}
\mathbf{A...
...\mathbf{A}_4 & -\mathbf{A}_4 & -\mathbf{A}_4 & \mathbf{A}_4
\end{array}\right]
$

Another method is to replace each of the four delay lines in an FDN(4) by a Gerzon vector allpass (see §2.8.5) which is $ 4\times4$ and contains four delay lines.


Householder Reflections

For completeness, this section derives the Householder reflection matrix from geometric considerations [451]. Let $ \mathbf{P}_{\underline{u}}$ denote the projection matrix which orthogonally projects vectors onto $ {\underline{u}}$, i.e.,

$\displaystyle \mathbf{P}_{\underline{u}}= \frac{\underline{u}\,\underline{u}^T}...
...frac{\underline{u}\,\underline{u}^T}{\left\Vert\,\underline{u}\,\right\Vert^2}
$

and

$\displaystyle \mathbf{P}_{\underline{u}}\, \underline{x}= \underline{u}\,\frac{...
...<\underline{u},\underline{x}\right>}{\left\Vert\,\underline{u}\,\right\Vert^2}
$

specifically projects $ \underline{x}$ onto $ \underline{u}$. Since the projection is orthogonal, we have

$\displaystyle \left<\underline{x}-\mathbf{P}_{\underline{u}}\underline{x},\unde...
...}-\mathbf{P}_{\underline{u}})\underline{x},\underline{u}\right>=\underline{0}.
$

We may interpret $ (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}$ as the difference vector between $ \underline{x}$ and $ \mathbf{P}_{\underline{u}}\underline{x}$, its orthogonal projection onto $ \underline{u}$, since

$\displaystyle (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}+ \mathbf{P}_{\underline{u}}\underline{x}= \underline{x}
$

and we have $ (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}\perp \mathbf{P}_{\underline{u}}\underline{x}$ by definition of the orthogonal projection. Consequently, the projection onto $ \underline{u}$ minus this difference vector gives a reflection of the vector $ \underline{x}$ about $ \underline{u}$:

$\displaystyle \underline{y}= \mathbf{P}_{\underline{u}}\underline{x}- (\mathbf{...
...line{u}})\underline{x}= (2\mathbf{P}_{\underline{u}}- \mathbf{I})\underline{x}
$

Thus, $ \underline{y}$ is obtained by reflecting $ \underline{x}$ about $ \underline{u}$--a so-called Householder reflection.


Most General Lossless Feedback Matrices

As shown in §C.15.3, an FDN feedback matrix $ \mathbf{A}_N$ is lossless if and only if its eigenvalues have modulus 1 and its $ N$ eigenvectors are linearly independent.

A unitary matrix $ Q$ is any (complex) matrix that is inverted by its own (conjugate) transpose:

$\displaystyle Q^{-1} = Q^H,
$

where $ Q^H$ denotes the Hermitian conjugate (i.e., the complex-conjugate transpose) of $ Q$. When $ Q$ is real (as opposed to complex), we may simply call it an orthogonal matrix, and we write $ Q^{-1} = Q^T$, where $ T$ denotes matrix transposition.

All unitary (and orthogonal) matrices have unit-modulus eigenvalues and linearly independent eigenvectors. As a result, when used as a feedback matrix in an FDN, the resulting FDN will be lossless (until the delay-line damping filters are inserted, as discussed in §3.7.4 below).


Triangular Feedback Matrices

An interesting class of feedback matrices, also explored by Jot [216], is that of triangular matrices. A basic fact from linear algebra is that triangular matrices (either lower or upper triangular) have all of their eigenvalues along the diagonal.4.13 For example, the matrix

$\displaystyle \mathbf{A}_3 = \left[\begin{array}{ccc}
\lambda_1 & 0 & 0\\ [2pt]
a & \lambda_2 & 0\\ [2pt]
b & c & \lambda_3
\end{array}\right]
$

is lower triangular, and its eigenvalues are $ (\lambda_1,
\lambda_2,\lambda_3)$ for all values of $ a$, $ b$, and $ c$.

It is important to note that not all triangular matrices are lossless. For example, consider

$\displaystyle \mathbf{A}_2 = \left[\begin{array}{cc} 1 & 0 \\ [2pt] 1 & 1 \end{array}\right]
$

It has two eigenvalues equal to 1, which looks lossless, but a quick calculation shows that there is only one eigenvector, $ [0,1]^T$. This happens because this matrix is a Jordan block of order 2 corresponding to the repeated eigenvalue $ \lambda=1$. A direct computation shows that

$\displaystyle \mathbf{A}_2^n = \left[\begin{array}{cc} 1 & 0 \\ [2pt] n & 1 \end{array}\right]
$

which is clearly not lossless.

One way to avoid ``coupled repeated poles'' of this nature is to use non-repeating eigenvalues. Another is to convert $ \mathbf{A}$ to Jordan canonical form by means of a similarity transformation, zero any off-diagonal elements, and transform back [329].


Choice of Delay Lengths

Following Schroeder's original insight, the delay line lengths in an FDN ($ M_i$ in Fig.3.10) are typically chosen to be mutually prime. That is, their prime factorizations contain no common factors. This rule maximizes the number of samples that the lossless reverberator prototype must be run before the impulse response repeats.

The delay lengths $ M_i$ should be chosen to ensure a sufficiently high mode density in all frequency bands. An insufficient mode density can be heard as ``ringing tones'' or an uneven amplitude modulation in the late reverberation impulse response.

Mean Free Path

A rough guide to the average delay-line length is the ``mean free path'' in the desired reverberant environment. The mean free path is defined as the average distance a ray of sound travels before it encounters an obstacle and reflects. An approximate value for the mean free path, due to Sabine, an early pioneer of statistical room acoustics, is

$\displaystyle {\overline d} = 4\frac{V}{S}\qquad\hbox{(mean free path)}
$

where $ V$ is the total volume of the room, and $ S$ is total surface area enclosing the room. This approximation requires the diffuse field assumption, i.e., that plane waves are traveling randomly in all directions [349,47] (see §3.2.1 for a simple construction). Normally, late reverberation satisfies this assumption well, away from open doors and windows, provided the room is not too ``dead''. Regarding each delay line as a mean-free-path delay, the average can be set to the mean free path by equating

$\displaystyle \frac{{\overline d}}{cT} = \frac{1}{N} \sum_{i=1}^N M_i
$

where $ c$ denotes sound speed and $ T$ denotes the sampling period. This number should be treated as a lower bound because in real rooms reflections are often diffuse, especially at high frequencies. In a diffuse reflection, a single incident plane wave reflects in many directions at once.


Mode Density Requirement

A guide for the sum of the delay-line lengths is the desired mode density. The sum of delay-line lengths $ M_i$ in a lossless FDN is simply the order of the system $ M$:

$\displaystyle M \isdef \sum_{i=1}^N M_i\qquad\hbox{(FDN order)}
$

The order increases slightly when lowpass filters are introduced after the delay lines to achieve a specific reverberation time at low and high frequencies (as described in the next subsection).

Since the order of a system equals the number of poles, we have that $ M$ is the number of poles on the unit circle in the lossless prototype. If the modes were uniformly distributed, the mode density would be $ M/f_s=MT$ modes per Hz. Schroeder [417] suggests that, for a reverberation time of 1 second, a mode density of 0.15 modes per Hz is adequate. Since the mode widths are inversely proportional to reverberation time, the mode density for a reverberation time of 2 seconds should be 0.3 modes per Hz, etc. In summary, for a sufficient mode density in the frequency domain, Schroeder's formula is

$\displaystyle M \geq 0.15 t_{60}f_s
$

For a sampling rate of 50 kHz and a reverberation time ($ t_{60}$) equal to 1 second, we obtain $ M\geq 7500$.


Prime Power Delay-Line Lengths

When the delay-line lengths need to be varied in real time, or interactively in a GUI, it is convenient to choose each delay-line length $ \hat{M}_i$ as an integer power of a distinct prime number $ p_i$ [457]:

$\displaystyle \hat{M}_i \isdefs p_i^{m_i}
$

where we call $ m_i\ge 1$ the ``multiplicity'' of the prime $ p_i$. With this choice, the delay-line lengths are always coprime (no factors in common other than $ 1$), and yet we can lengthen or shorten each delay line individually (by factors of $ p_i$) without affecting the mutually prime property.

Suppose we are initially given desired delay-line lengths $ M_i$ arranged in ascending order so that

$\displaystyle M_1 < M_2 < \cdots < M_N.
$

Then good prime-power approximations $ \hat{M}_i$ can be expected using the prime numbers in their natural order:

$\displaystyle p_i \in \{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,\ldots\}
$

Since $ \hat{M}_i=p_i^{m_i} \,\,\Rightarrow\,\,\log(\hat{M}_i) = m_i \log(p_i)$ (for any logarithmic base), an optimal (in some sense) choice of prime multiplicity $ m_i$ is

$\displaystyle m_i
\isdefs$   round$\displaystyle \left[\frac{\log(M_i)}{\log(p_i)}\right]
\isdefs \left\lfloor 0.5 + \frac{\log(M_i)}{\log(p_i)}\right\rfloor.
$

where $ M_i$ is the desired length in samples. That is, $ m_i$ can be simply obtained by rounding $ \log(M_i)/\log(p_i)$ to the nearest integer (max 1). The prime-power delay-line length approximation is then of course

$\displaystyle \hat{M}_i \isdefs p_i^{m_i},
$

and the multiplicative approximation error is bounded by $ p_i^{\pm1/2}$ (when $ M_i\ge\sqrt{p_i}$).

This prime-power length scheme is used to keep 16 delay lines both variable and mutually prime in Faust's reverb_designer.dsp programming example (via the function prime_power_delays in effect.lib).


Achieving Desired Reverberation Times

A lossless prototype reverberator, as in Fig.3.10 when $ g_i=1$, has all of its poles on the unit circle in the $ z$ plane, and its reverberation time is infinity. To set the reverberation time to a desired value, we need to move the poles slightly inside the unit circle. Furthermore, due to air absorption (§2.3B.7.15), we want the high-frequency poles to be more damped than the low-frequency poles [314]. As discussed in §2.3, this type of transformation can be obtained using the substitution

$\displaystyle z^{-1}\leftarrow G(z)z^{-1}, \protect$ (4.5)

where $ G(z)$ denotes the filtering per sample in the propagation medium (a lowpass filter with gain not exceeding 1 at all frequencies).4.14Thus, to set the FDN reverberation time to $ t_{60}(\omega)\isdeftext n_{60}(\omega)T$ at frequency $ \omega $, we want propagation through $ n_{60}$ samples to result in attenuation by $ 60$ dB, i.e.,

$\displaystyle \left[G(e^{j\omega T})\right]^{n_{60}(\omega)} \eqsp 0.001. \protect$ (4.6)

Solving for $ G$, the propagation attenuation per-sample, gives
$\displaystyle G(e^{j\omega T})$ $\displaystyle =\!$ $\displaystyle (0.001)^{\frac{1}{n_{60}(\omega)}}
\eqsp 10^{-3/n_{60}}
\eqsp \left(e^{\mbox{ln}(10)}\right)^{-3/n_{60}} \eqsp e^{-3\,\mbox{ln}(10)/n_{60}}$  
  $\displaystyle =\!$ $\displaystyle e^{-T/\tau(\omega)}
\protect$ (4.7)

The last form comes from $ t_{60}=3$ln$ (10)\tau\approx 6.91\tau$, where $ \tau $ denotes the time constant of decay (time to decay by $ 1/e$) [451], i.e.,

$\displaystyle e^{-t_{60}/\tau}=0.001 \;\;\Leftrightarrow\;\; t_{60}\eqsp -3$ln$\displaystyle (10)\tau. \protect$ (4.8)

Series expanding $ e^{-T/\tau(\omega)}$ and assuming $ n_{60}(\omega)\gg 7$ samples ( $ \tau(\omega)\gg T$ seconds) provides the practically useful approximation

\begin{eqnarray*}
e^{-T/\tau(\omega)}
&\!=\!& 1 - \frac{T}{\tau(\omega)} + \frac...
... \frac{3\mbox{ln}(10)}{n_{60}}
\approxs 1 - \frac{6.91}{n_{60}}.
\end{eqnarray*}

Conformal Map Interpretation of Damping Substitution

The relation $ G(e^{j\omega T})\approx e^{-T/\tau(\omega)}$ [Eq.$ \,$(3.7)] can be written down directly from $ z^{-1}\leftarrow G(z)z^{-1}$ [Eq.$ \,$(3.5)] by interpreting Eq.$ \,$(3.5) as an approximate conformal map [326] which takes each pole $ p_k=e^{j\omega_kT}$, say, from the unit circle to the point $ p'_k\approx G(e^{j\omega_kT})e^{-j\omega_kT}$. Thus, the new pole radius is approximately $ \vert p'_k\vert\approx\vert G(e^{j\omega_kT})\vert$, where the approximation is valid when $ G(z)$ is approximately constant between the new pole location and the unit circle. To see this, consider the partial fraction expansion [449] of a proper $ N$th-order lossless transfer function $ H(z)$ mapped to $ H'(z)\isdeftext H[z/G(z)]$:

$\displaystyle H'(z)
= \sum_{k=1}^N \frac{r_k}{1-p_kG(z)z^{-1}}
= \sum_{k=1}^N r_k\left[1+p_kG(z)z^{-1}+p_k^2G^2(z)z^{-2}+\cdots\right],
$

where $ p_k=\exp(j\omega_k T)$ denotes the $ k$th original pole on the unit circle. Then $ H'(z)$ has a pole at $ z'_k=p_kG(z'_k)$, which must be solved iteratively for $ z'_k$, in general, since $ G(z)$ can be a complicated function of $ z$. However, if $ G(z'_k)\approx G(p_k)$, which is typically true when damping digital waveguides for music applications, then $ z'_k\approx p_kG(p_k)=G[\exp(j\omega_k
T)]\exp(j\omega_k T)$. In other words, we can think of the pole $ p_k$ as moving from $ \exp(j\omega_k T)$ to near $ G[\exp(j\omega_k
T)]\exp(j\omega_k T)$, provided it doesn't move too far compared with the near-constant behavior of $ G(z)$. Another way to say it is that we need $ G(z)$ to be approximately the same at the new pole location and its initial location on the unit circle in the lossless prototype.

Happily, while we may not know precisely where our poles have moved as a result of introducing the per-sample damping filter $ G(z)$, the relation $ G^{n_{60}(\omega)}(e^{j\omega T})=0.001$ [Eq.$ \,$(3.6)] remains exact at every frequency by construction, as it is based only on the physical interpretation of each unit delay as a propagation delay for a plane wave across one sampling interval $ T$, during which (zero-phase) filtering by $ G(z)$ is assumed (§2.3). More generally, we can design minimum-phase filters for which $ \vert G^{n_{60}(\omega)}(e^{j\omega T})\vert=0.001$, and neglect the resulting phase dispersion.

In summary, we see that replacing $ z^{-1}$ by $ G(z)z^{-1}$ everywhere in the FDN lossless prototype (or any lossless LTI system for that matter) serves to move its poles away from the unit circle in the $ z$ plane onto some contour inside the unit circle that provides the desired decay time at each frequency.

A general design guideline for artificial reverberation applications [217] is that all pole radii in the reverberator should vary smoothly with frequency. This translates to $ G(z)$ having a smooth frequency response. To see why this is desired, consider momentarily the frequency-independent case in which we desire the same reverberation time at all frequencies (Fig.3.10 with real $ g_i\le 1$, as drawn). In this case, it is ideal for all of the poles to have this decay time. Otherwise, the late decay of the impulse response will be dominated by the poles having the largest magnitude, and it will be ``thinner'' than it was at the beginning of the response when all poles were contributing to the output. Only when all poles have the same magnitude will the late response maintain the same modal density throughout the decay.


Damping Filters for Reverberation Delay Lines

In an FDN, such as the one shown in Fig.3.10, delays $ z^{-1}$ appear in long delay-line chains $ z^{-M_i}$. Therefore, the filter needed at the output (or input) of the $ i$th delay line is $ G^{M_i}(z)$ (replace $ g_i$ by $ G^{M_i}(z)$ in Fig.3.10).4.15 However, because $ G(e^{j\omega T})$ is so close to $ 1$ in magnitude, and because it varies so weakly across the frequency axis, we can design a much lower-order filter $ H_i(z)\approx G^{M_i}(z)$ that provides the desired attenuation versus frequency to within psychoacoustic resolution. In fact, perfectly nice reverberators can be designed in which $ H_i(z)$ is merely first order for each $ i$ [314,217].


Delay-Line Damping Filter Design

Let $ t_{60}(\omega)$ denote the desired reverberation time at radian frequency $ \omega $, and let $ H_i(z)$ denote the transfer function of the lowpass filter to be placed in series with the $ i$th delay line which is $ M_i$ samples long. The problem we consider now is how to design these filters to yield the desired reverberation time. We will specify an ideal amplitude response for $ H_i(z)$ based on the desired reverberation time at each frequency, and then use conventional filter-design methods to obtain a low-order approximation to this ideal specification.

In accordance with Eq.$ \,$(3.6), the lowpass filter $ H_i(z)$ in series with a length $ M_i$ delay line should approximate

$\displaystyle H_i(z) = G^{M_i}(z)
$

which implies

$\displaystyle \left\vert H_i(e^{j\omega T})\right\vert^{\frac{t_{60}(\omega)}{M_iT}} = 0.001.
$

Taking $ 20\log_{10}$ of both sides gives

$\displaystyle 20 \log_{10}\left\vert H_i(e^{j\omega T})\right\vert = -60 \frac{M_i T}{t_{60}(\omega)}. \protect$ (4.9)

This is the same formula derived by Jot [217] using a somewhat different approach.

Now that we have specified the ideal delay-line filter $ H_i(e^{j\omega T})$ in terms of its amplitude response in dB, any number of filter-design methods can be used to find a low-order $ H_i(z)$ which provides a good approximation to satisfying Eq.$ \,$(3.9). Examples include the functions invfreqz and stmcb in Matlab. Since the variation in reverberation time is typically very smooth with respect to $ \omega $, the filters $ H_i(z)$ can be very low order.

First-Order Delay-Filter Design

The first-order case is very simple while enabling separate control of low-frequency and high-frequency reverberation times. For simplicity, let's specify $ t_{60}(0)$ and $ t_{60}(\pi/T)$, denoting the desired decay-time at dc ($ \omega=0$) and half the sampling rate ( $ \omega=\pi/T$). Then we have determined the coefficients of a one-pole filter:

$\displaystyle H_i(z) = \frac{g_i}{1-p_iz^{-1}}
$

The dc gain of this filter is $ H_i(1)=g_i/(1-p_i)$, while the gain at $ \omega=\pi/T$ is $ H_i(-1)=g_i/(1+p_i)$. From Eq.$ \,$(3.9) (or Eq.$ \,$(3.8)), we obtain two equations in two unknowns:

\begin{eqnarray*}
\frac{g_i}{1-p_i} &=& 10^{-3 M_i T / t_{60}(0)}
\eqsp e^{-M_iT...
...(\pi/T)}
\eqsp e^{-M_iT/\tau(\pi/T)} \isdefs R_\pi^{M_i}\\ [5pt]
\end{eqnarray*}

where $ D_i\isdeftext M_iT$ denotes the $ i$th delay-line length in seconds. These two equations are readily solved to yield

\begin{eqnarray*}
p_i &=& \frac{R_0^{M_i}-R_\pi^{M_i}}{R_0^{M_i}+R_\pi^{M_i}}\\ [5pt]
g_i &=& \frac{2R_0^{M_i}R_\pi^{M_i}}{R_0^{M_i}+R_\pi^{M_i}}
\end{eqnarray*}

The truncated series approximation

$\displaystyle R_\omega^{M_i} \isdefs e^{-\frac{M_iT}{\tau(\omega)}}
\approxs 1 ...
...\frac{6.91\,M_iT}{t_{60}(\omega)}
\isdefs 1 - \frac{6.91\,M_i}{n_{60}(\omega)}
$

has been found to work well in practical FDN reverberators.


Orthogonalized First-Order Delay-Filter Design

In [217], first-order delay-line filters of the form

$\displaystyle H_i(z) \eqsp g'_i \frac{1-p_i}{1-p_iz^{-1}}
$

are proposed. Clearly $ g_i=g'_i\cdot(1-p_i)$. This form has the advantage that the dc gain is always $ H_i(1)=g'_i$ for all (stable) values of $ p_i$. Therefore, we can set $ g'_i$ to give a desired reverberation time at dc, and not have to change it when $ p_i$ is varied to modulate the high-frequency decay rate. As in the previous section, from Eq.$ \,$(3.9), we obtain

$\displaystyle g'_i \eqsp 10^{-3 M_i T / t_{60}(0)}.
$

A calculation given in [217] arrives at

$\displaystyle p_i \eqsp \frac{\mbox{ln}(10)}{4}\log_{10}(g_i)\left(1-\frac{1}{\alpha^2}\right)
$

where

$\displaystyle \alpha \isdef \frac{t_{60}(\pi/T)}{t_{60}(0)} \protect$ (4.10)

denotes the ratio of reverberation time at half the sampling rate divided by the reverberation time at dc.4.16


Multiband Delay-Filter Design

In §3.7.5, we derived first-order FDN delay-line filters which can independently set the reverberation time at dc and at half the sampling rate. However, perceptual studies indicate that reverberation time should be independently adjustable in at least three frequency bands [217]. To provide this degree of control (and more), one can implement a multiband delay-line filter using a general-purpose filter bank [370,500]. The output, say, of each delay line is split into $ K$ bands, where $ K\ge 3$ is recommended, and then, from Eq.$ \,$(3.6), the gain in the $ k$th band for a length $ M_i$ delay-line can be set to

$\displaystyle G^{M_i}(e^{j\omega_kT})\eqsp 10^{-\frac{3M_i}{n_{60}(\omega_k)}} ...
...ln}(10)\,M_i}{n_{60}(\omega_k)} \approxs
1-\frac{6.91\,M_i}{n_{60}(\omega_k)}
$

to produce the desired decay-time in that band, where $ n_{60}(\omega)=t_{60}(\omega)/T$ denotes the desired 60-dB decay time in samples. Faust implementations of FDN reverberator along these lines are described in §3.7.9 below.


Spectral Coloration Equalizer

In the previous section, a ``graphical equalizer'' was used to set the reverberator decay time independently in each spectral band slice. While this gives much control over decay time, there is no control over the initial spectral gain in each band. Therefore, another good place for a graphical equalizer is at the reverberator input or output. Such an equalizer allows control of the initial spectral coloration of the reverberator. In the example of Fig.3.10, a spectral coloration equalizer is most efficiently applied to the input signal $ u(n)$, before entering the FDN (but after splitting off the direct signal to be scaled by $ d$ and added to the output), or the output of $ E(z)$ in Fig.3.10.


Tonal Correction Filter

Let $ h_k(n)$ denote the component of the impulse response arising from the $ k$th pole of the system. Then the energy associated with that pole is

$\displaystyle {\cal E}_k \eqsp \sum_{n=0}^\infty \left\vert h_k(n)\right\vert^2.
$

All other factors being equal, if the decay time of the mode is shortened by half, it follows that the total energy contributed by that mode to the impulse response is also reduced by half. To compensate for this effect, Jot introduced a tonal correction filter $ E(z)$ to be placed in series with the FDN, as shown in Fig.3.10.

In the case of the first-order delay-line filters discussed in §3.7.5, good tonal correction is given by the following one-zero filter:

$\displaystyle E(z) \eqsp \frac{ 1 - bz^{-1}}{1-b}
$

where

$\displaystyle b \eqsp \frac{1-\alpha}{1+\alpha}
$

and $ \alpha$ is defined in Eq.$ \,$(3.10).


FDNs as Digital Waveguide Networks

As discussed in §C.15, the FDN using a Householder-reflection feedback matrix $ \mathbf{A}_N = \mathbf{I}_N -
(2/N)\underline{u}\underline{u}^T$ is equivalent to a network of $ N$ digital waveguides intersecting at a single scattering junction [463,464,385]. The wave impedance in the $ i$th waveguide is simply $ \underline{u}[i]$, the $ i$th element of the axis-of-reflection vector $ \underline{u}$. The choice $ \underline{u}^T=[1,1,\dots,1]$ corresponds to all of the waveguides having the same impedance (the ``isotrophic junction'' case).


FDN Reverberators in Faust

The Faust example reverb_designer.dsp brings up a $ 16\times
16$ FDN reverberator in which the signal out of each delay line is split into five bands so that $ t_{60}(\omega_k)$ can be controlled independently in each band. The 16 delay-line lengths are distributed exponentially between a minimum and maximum length set by two min/max-length sliders, but rounded to the nearest integer-power of a distinct prime, as introduced above in §3.7.3). The FDN reverberator is implemented in Faust's effect.lib. The band-splitting is carried out by the filterbank function in Faust's filter.lib.

The Faust function filterbank(order,freqs) implements a filter bank having the needed properties using Butterworth lowpass/highpass band-splitting arranged in a dyadic tree (normally a good choice for audio filter banks). That is, the whole spectrum is split at the highest crossover frequency, the lowpass region is then split into two bands at the next crossover frequency down, and so on, splitting the lowpass band at each stage in the dyadic tree [455,500]. The number of poles in each Butterworth lowpass/highpass filter is specified by order, and freqs contains a list of desired crossover frequencies separating the bands. A certain amount of dispersion is also introduced, since the filter bank is causal and delay-equalized (so that the bands may be summed without phase cancellation artifacts at the band edges). Also note that the lower bands are effectively produced by higher order filters than the upper bands. When the reverberation time is longer than the dispersion delay, the dispersion should not be audible as such, although it can affect the ``sound'' of the reverberation. In general, however, artificial reverberators normally benefit from additional allpass dispersion.

Figure 3.12 shows the block diagram of a $ 4\times4$ FDN reverberator made from Faust's reverb_designer.dsp by changing 16 to 4. Figure 3.13 shows the Faust block diagram of the associated $ 4\times4$ Hamard matrix multiplication. As it shows, multiplication by a Hadamard matrix can be implemented (ignoring the normalizing scale factor) as a series of block sums and differences (often called butterflies or shufflers) in which the block size decreases by a factor of 2 each stage. Figures for the remaining components of the reverberator may be perused via the shell command faust2firefox reverb_designer.dsp followed by clicking on the blocks in the browser.

Figure 3.12: FDN reverberator implemented in the Faust example reverb_designer.dsp, but scaled down from order 16 to order 4.
\includegraphics[width=\twidth]{eps/fdnrev0}

Figure 3.13: Hadamard processing used in the Faust FDN reverberator.
\includegraphics[width=\twidth]{eps/hadamard4}


Zita-Rev1

A FOSS4.17 reverberator that combines elements of Schroeder (§3.5) and FDN reverberators (§3.7) is zita-rev1,4.18written in C++ for Linux systems by Fons Adriaensen. A Faust version of the zita-rev1 stereo-mode functionality is zita_rev1 in Faust's effect.lib. A high-level block diagram appears in Fig.3.14.

Figure 3.14: High-level block diagram of zita_rev1_engine() in Faust's effect.lib generated by the shell script faust2firefox.
\includegraphics[width=\twidth]{eps/zita-rev1}

The main high-level addition relative to an 8th-order FDN reverberator is the block labeled allpass_combs in Fig.3.14. This block inserts a Schroeder allpass comb filter (Fig.2.30) in series with each delay line. In zita-rev1 (as of this writing), the allpass-comb feedforward/feedback coefficients are all set to $ \pm 0.6$. The delay-line lengths and other details are readily found in the freely available source code (or by browsing the Faust-generated block diagram).

Zita-Rev1 Delay-Line Filters

In zita-rev1, the damping filter for each delay line consists of a low-shelf filter $ H_l(z)$ [449],4.19in series with a unique first-order lowpass filter $ H_h(z)$ that sets the high-frequency $ t_{60}$ to be half that of the middle-band at a particular frequency $ f_h$ (specified as ``HF Damping'' in the GUI). Since the filter $ H_h$ is constrained to be a lowpass, $ t_{60}(f)<t_{60}(f_h)$ for $ f>f_h$, i.e., the decay time gets shorter at higher frequencies.

Viewing the resulting damping filter $ H_d(z)=H_l(z)H_h(z)$ as a three-band filter bank3.7.5), let $ g_0$ and $ g_m$ denote the desired band gains at dc and ``middle frequencies'', respectively.4.20 Then the low shelf may be set for a desired dc-gain of $ g_0/g_m$, and its input (or output) signal multiplied by $ g_m$ to obtain the resulting filter

$\displaystyle H_l(z) \eqsp g_m + (g_0-g_m)\frac{1-p_l}{2}\frac{1+z^{-1}}{1-p_lz^{-1}},
$

where $ p_l$ denotes the (real) first-order lowpass pole, given by [449]

$\displaystyle p_l \isdefs \frac{1-\pi f_1T}{1+\pi f_1T},
$

where $ f_1$ specifies (in Hz) the crossover point between ``low'' and ``middle'' frequencies, and $ T$ denotes the sampling interval as usual.

The lowpass filter $ H_h(z)$ is also first order, and to provide half the middle-band $ t_{60}$ at the beginning of the ``high'' band, the lowpass should ``break'' to a gain of $ g_m$ at the ``HF Damping'' frequency $ f_h$ specified in the GUI. A unity-dc-gain one-pole lowpass has the form [449]

$\displaystyle H_h(z) = \frac{1-p_h}{1-p_hz^{-1}},
$

where the pole $ p_h$ must be found to give a gain of $ g_M$ at frequency $ f_h$:

$\displaystyle \left\vert H_h\left(e^{j2\pi f_hT}\right)\right\vert \eqsp
\left\vert\frac{1-p_h}{1-p_he^{-j2\pi f_hT}}\right\vert \eqsp g_M
$

Squaring and normalizing yields a quadratic equation of the form $ p_h^2 + b\,p_h +1=0$. Solving for $ p_h$ using the quadratic formula yields

$\displaystyle p_h \eqsp -\frac{b}{2} - \sqrt{\left(\frac{b}{2}\right)^2 - 1},
$

where

$\displaystyle -\frac{b}{2} \eqsp \frac{1-g_M^2\cos(2\pi f_h T)}{1-g_M^2} > 1,
$

and the unstable solution $ -b/2 + \sqrt{(b/2)^2 - 1} > 1$ is discarded. To ensure $ \vert g_M\vert<1$, the GUI must limit the middle-band $ t_{60}$ to finite values. (The upper limit is presently $ 8$ seconds for both low and middle frequencies.)


Further Extensions

Schroeder's original structures for artificial reverberation were comb filters and allpass filters made from two comb filters. Since then, they have been upgraded to include specific early reflections and per-sample air-absorption filtering (Moorer, Schroeder), precisely specified frequency dependent reverberation time (Jot), and a nearly independent factorization of ``coloration'' and ``duration'' aspects (Jot). The evolution from comb filters to feedback delay networks (Gerzon, Stautner, Puckette, Jot) can be seen as a means for obtaining greater richness of feedback, so that the diffuseness of the impulse response is greater than what is possible with parallel and/or series comb filters. In fact, an FDN can be seen as a richly cross-coupled bank of feedback comb filters whenever the diagonal of the feedback matrix is nonzero. The question then becomes what aspects of artificial reverberation have not yet been fully addressed?

Spatialization of Reverberant Reflections

While we did not go into the subject here, the early reflections should be spatialized by including a head-related transfer function (HRTF) on each tap of the early-reflection delay line [248].4.21

Some kind of spatialization may be needed also for the late reverberation. A true diffuse field3.2.1) consists of a sum of plane waves traveling in all directions in 3D space. Since we do not know how to achieve this effect using current systems for reverberation, the typical goal is to simply extract uncorrelated outputs from the reverberation network and feed them to the various output channels, as discussed in §3.5. However, this is not ideal, since the resulting sound field consists of wavefronts arriving from each of the speakers, and it is possible for the reverberation to sound like it is emanating from discrete speaker locations. It may be that spatialization of some kind can better fool the ear into believing the late reverberation is coming from all directions.


Distribution of Mode Frequencies

Another way in which current reverberation systems are ``artificial'' is the unnaturally uniform distribution of resonant modes with respect to frequency. Because Schroeder, FDN, and waveguide reverbs are all essentially a collection of $ N$ delay lines with feedback around them, the modes tend to be distributed as the superposition of the resonant modes of $ N$ feedback comb filters. Since a feedback comb filter has a nearly harmonic set of modes (see §2.6.2), aggregates of comb filters tend to provide a uniform modal density in the frequency domain. In real reverberant spaces, the mode density increases as frequency squared, so it should be verified that the uniform modes used in a reverberator are perceptually equivalent to the increasingly dense modes in nature. Another aspect of perception to consider is that frequency-domain perception of resonances actually decreases with frequency. To summarize, in nature the modes get denser with frequency, while in perception they are less resolved, and in current reverberation systems they stay more or less uniform with frequency; perhaps a uniform distribution is a good compromise between nature and perception?

At low frequencies, however, resonant modes are accurately perceived in reverberation as boosts, resonances, and cuts. They are analogous to early reflections in the time domain, and we could call them the ``early resonances.'' It is interesting that no system for artificial reverberation except waveguide mesh reverberation (of which the author is aware) explicitly attempts precise shaping of the low-frequency amplitude response of a desired reverberant space, at least not directly. The low-frequency response is shaped indirectly by the choice of early reflections, and the use of parallel comb-filter banks in Schroeder reverberators serves also to shape the low-frequency response significantly. However, it would be possible to add filters for shaping more carefully the low-frequency response. Perhaps a reason for this omission is that hall designers work very hard to eliminate any explicit resonances or antiresonances in the response of a room. If uneven resonance at low frequencies is always considered a defect, then designing for a maximally uniform mode distribution, as has been discussed for the high-frequency modes, would be ideal also at low frequencies. Quite the opposite situation exists when designing ``small-box reverberators'' to simulate musical instrument resonators [428,203]; there, the low-frequency modes impart a characteristic timbre on the low-frequency resonance of the instrument (see Fig.3.2).


Digital Waveguide Reverberators

It was mentioned in §3.7.8 above that FDNs can be formulated as special cases of Digital Waveguide Networks (DWN) (see Appendix C for a fuller development of DWNs). Specifically, an FDN is obtained from a DWN consisting of a single scattering junctionC.15). It follows that the DWN paradigm provides a more generalized framework in which to pursue further improvements of reverberation architecture. For example, when multiple FDNs are embedded within a single DWN, it becomes possible to richly cross-couple them in an energy-controlled manner in order to create richer recursive structures than either alone. General DWNs were proposed for artificial reverberation in [430,433].


The Digital Waveguide Mesh for Reverberation

A special case of digital waveguide networks known as the digital waveguide mesh has also been proposed for use in artificial reverberation systems [396,518].

As discussed in §2.4, a digital waveguide (bidirectional delay line) can be considered a computational acoustic model for traveling waves in opposite directions. A mesh of such waveguides in 2D or 3D can simulate waves traveling in any direction in the space. As an analogy, consider a tennis racket in which a rectilinear mesh of strings forms a pseudo-membrane.

A major advantage of the waveguide mesh for reverberation applications is that wavefronts are explicitly simulated in all directions, as in real reverberant spaces. Therefore, a true diffuse field can be developed in the late reverberation. Also, the echo density grows with time and the mode density grows with frequency in a natural manner for the 2D and 3D mesh. Finally, the low-frequency modes of the reverberant space can be simulated very precisely (for better or worse).

The computational cost of a waveguide mesh is made tractable relative to more conventional finite-difference simulations by (1) the use of multiply-free scattering junctions and (2) very coarse meshes. Use of a coarse mesh means that the ``physical modeling'' aspects of the mesh are only valid at low frequencies. As practical matter, this works out well because the ear cannot hear mode tuning errors at high frequencies. There is no error in the mode dampings in a lossless reverberator prototype, because the waveguide mesh is lossless by construction. Therefore, the only errors relative to an ideal simulation of a lossless membrane or space are (1) mode tuning error, and (2) finite band width (cut off at half the sampling rate). The tuning error can be understood as due to dispersion of the traveling waves in certain directions [518,399]. Much progress has been made on the problem of correcting this dispersion error in various mesh geometries (rectilinear, triangular, tetrahedral, etc.) [521,398,399].

See §C.14 for an introduction to the digital waveguide mesh and a few of its properties.


Time Varying Reverberators

In real rooms, thermal convention currents cause the propagation path delays to vary over time [58]. Therefore, for greater physical accuracy, the delay lines within a digital reverberator should vary over time. From a more practical perspective, time variation helps to break up and obscure unwanted repetition in the late reverberation impulse response [430,104].


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Delay-Line Interpolation
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Freeverb