circulant matrices

whose absorption depends on the spatial directions taken by these waves. For
instance, in a rectangular enclosure, axial waves are absorbed less than oblique
waves [63]. Therefore, neighbouring modes associated with different directions
can have different reverberation times. Actually, for commonly-found rooms hav-
ing irregularities in the geometry and in the materials, the response is close to
that of a room having diffusive walls, where the energy rapidly spreads among
the different modes. In these cases, we can find that the decay time is quite
uniform among the modes [50].

governs the stability of the whole structure. In particular, it is desirable to
start with a lossless prototype, i.e. a reference structure providing an endless,
flat decay. The reader interested in general matrix classes that might work as
prototypes is deferred to the literature [44, 84, 81, 39]. Here we only mention
the class of circulant matrices, having general form

The stability of a FDN is related to the magnitude of its eigenvalues, which
can be computed by the Discrete Fourier Transform of the first raw, in the
case of a circulant matrix. By keeping these eigenvalues on the unit circle (i.e.,
magnitude one) we ensure that the whole structure is stable and lossless. The
control over the angle of the eigenvalues can be translated into a direct control
over the degree of diffusion of the enclosure that is being simulated by the FDN.
The limiting cases are the diagonal matrix, corresponding to perfectly reflecting
walls, and the matrix whose rows are sequences of equal-magnitude numbers
and (pseudo-)randomly distributed signs [81].

Several authors suggested to use lengths in samples that are mutually coprime
numbers in order to minimize the collision of echoes in the impulse response.
However, if the FDN is linked to a physical and geometrical interpretation, as
it is done in the Ball-within-the-Box model [79], the delay lengths are derived
from the geometry of the room being simulated and the resulting digital reverb
quality is related to the quality of the actual room. In the case of a rectangular
room, a delay line will be associated to a harmonic series of normal modes,
all obtainable from a plane wave loop that bounces back and forth within the
enclosure.

If the impulse response of a target room is readily available, the most faithful re-
verberation method would be to convolve the input signal with such a response.
Direct convolution can be done by storing each sample of the impulse response
as a coefficient of an FIR filter whose input is the dry signal. Direct convolution
becomes easily impractical if the length of the target response exceeds small
fractions of a second, as it would translate into several hundreds of taps in the
filter structure. A solution is to perform the convolution block by block in the