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Practical considerations of time-varying feedback delay networks
Abstract
Feedback delay networks (FDNs) can be efficiently used to generate parametric artificial reverberation. Recently, the authors proposed a novel approach to time-varying FDNs by introducing a time-varying feedback matrix. The formulation of the time-varying feedback matrix was given in the complex eigenvalue domain, whereas this contribution specifies the requirements for real valued time-domain processing. In addition, the computational costs of different time-varying feedback matrices, which depend on the matrix type and modulation function, are discussed. In a performance evaluation, the proposed orthogonal matrix modulation is compared to a direct interpolation of the matrix entries.
... The theory of time-varying mixing matrices for artificial reverberation have been recently developed by the authors [16] and its design and efficient implementation have been discussed in [17]. The mixing matrix is designed to be unitary and energy-conserving at every time instance. ...
... A T (n)A(n) = I. In the following, we summarize the design of orthogonal time-varying mixing matrices as given in [17]. ...
... As proposed in [17], time-varying mixing matrices can be generated from ...
... The modification of systems based on state-space descriptions by feedback structures is a widely used method, e.g., in control theory [1,2], in the adjustment of boundary behavior of physical systems [13,14,29] and in the creation of artificial reverberation with feedback delay networks [31,32]. ...
... But, according to the real-time algorithms in [17,18,33], the system in (23) can be implemented in terms of 2nd order filters. Alternatively, the matrix powers in (23) can be performed efficiently on the eigenvalue decomposition of D d and U d [31,32]. The parameters γD, γU in (23) may be time varying for practical applications. ...
The attachment of feedback loops to physical or musical systems
enables a large variety of possibilities for the modification of the
system behavior. Feedback loops may enrich the echo density of
feedback delay networks (FDN), or enable the realization of complex
boundary conditions in physical simulation models for sound
synthesis. Inspired by control theory, a general feedback loop is attached
to a model of a vibrating membrane. The membrane model
is based on the modal expansion of an initial-boundary value problem
formulated in a state-space description. The possibilities of the
attached feedback loop are shown by three examples, namely by
the introduction of additional mode wise damping; modulation and
damping inspired by FDN feedback loops; time-varying modification
of the system behavior.
... The issue of stability is discussed in the context of accurate synthesis of sound-energy decay in multiple frequency bands. An attenuation filter is, in the case of this dissertation, realized as a time-invariant infiniteimpulse-response (IIR) filter, which requires all of its poles to lie within the unit circle in order to maintain stability [235,236,237,238]. If this condition is met in the overall magnitude of the attenuation filter (which can include several low-order filters), the IR of the reverberator will decay instead of being sustained or amplified. ...
In this dissertation, the discussion is centered around the sound energy decay in enclosed spaces. The work starts with the methods to predict the reverberation parameters, followed by the room impulse response measurement procedures, and ends with an analysis of techniques to digitally reproduce the sound decay.
The research on the reverberation in physical spaces was initiated when the first formula to calculate room's reverberation time emerged. Since then, finding an accurate and reliable method to predict reverberation has been an important area of acoustic research. This thesis presents a comprehensive comparison of the most commonly used reverberation time formulas, describes their applicability in various scenarios, and discusses their accuracy when compared to results of measurements. The common sources of uncertainty in reverberation time calculations, such as bias introduced by air absorption and error in sound absorption coefficient, are analyzed as well. The thesis shows that decreasing such uncertainties leads to a good prediction accuracy of Sabine and Eyring equations in diverse conditions regarding sound absorption distribution.
The measurement of the sound energy decay plays a crucial part in understanding the propagation of sound in physical spaces. Nowadays, numerous techniques to capture room impulse responses are available, each having its advantages and drawbacks. In this dissertation, the majority of commonly used measurement techniques are listed, whereas the exponential swept-sine is described in more detail. This work elaborates on the external factors that may impair the measurements and introduce error to their results, such as stationary and non-stationary noise, as well as time variance. The dissertation introduces Rule of Two, a method of detecting nonstationary disturbances in sweep measurements. It also shows the importance of using median as
a robust estimator in non-stationary noise detection.
Artificial reverberation is a popular sound effect, used to synthesize sound energy decay for the purpose of audio production. This dissertation offers an insight into artificial reverberation algorithms based on recursive structures. The filter design proposed in this work offers precise control over the decay rate while being efficient enough for real-time implementation. The thesis discusses the role of the delay lines and feedback matrix in achieving high echo density in feedback delay networks. It also shows that four velvet-noise sequences are sufficient to obtain smooth output in interleaved velvet noise reverberator. The thesis shows that the accuracy of reproduction increases the perceptual similarity between measured and synthesised impulse responses.
The insights collected in this dissertation offer insights into the intricacies of reverberation prediction, measurement and synthesis. The results allow for reliable estimation of parameters related to sound energy decay, and offer an improvement in the field of artificial reverberation.
... Further details on the implementation of MM and its computational complexity can be found in Ref. 32. Whereas FS is a single channel technique, which is applied independently to each channel, MM is inherently a multichannel technique, which introduces parameter dependencies between the channels as given in Eq. (15). ...
Various time-varying algorithms have been applied in multichannel sound systems to improve the system's stability and, among these, frequency shifting has been demonstrated to reach the maximum stability improvement achievable by time-variation in general. However, the modulation artifacts have been found to diminish the gain improvement unusable for a higher number of channels and high-quality applications such as music reproduction. This paper proposes alternatively time-varying mixing matrices, which is an efficient algorithm corresponding to symmetric up and down frequency shifting. It is shown with a statistical approach that time-varying mixing matrices can as well achieve maximum stability improvement for a higher number of channels. A listening test demonstrates the improved quality of time-varying mixing matrices over frequency shifting.
The mixing matrix of a Feedback Delay Network (FDN) reverberator is used to control the mixing time and echo density profile. In this work, we investigate the effect of the mixing matrix on the modes (poles) of the FDN with the goal of using this information to better design the various FDN parameters. We find the modal decomposition of delay network reverberators using a state space formulation, showing how modes of the system can be extracted by eigenvalue decomposition of the state transition matrix. These modes, and subsequently the FDN parameters, can be designed to mimic the modes in an actual room. We introduce a parameterized orthonormal mixing matrix which can be continuously varied from identity to Hadamard. We also study how continuously varying diffusion in the mixing matrix affects the damping and frequency of these modes. We observe that modes approach each other in damping and then deflect in frequency as the mixing matrix changes from identity to Hadamard. We also quantify the perceptual effect of increasing mixing by calculating the normalized echo density (NED) of the FDN impulse responses over time.
Time-varying components are used in some multichannel sound systems
designed for the enhancement of room acoustics. Time-variation can
usefully reduce the risk of producing ringing tones and improve
stability margins, provided that any modulation artefacts are inaudible.
Frequency-shifting is one form of time-variation which provides the best
case improvement in loop gain, and for which the single channel
stability limit has been derived. This paper determines the stability
limit for multiple channel systems with frequency-shifting by
generalizing the previous single-channel analysis. It is shown that the
improvement in stability due to frequency-shifting reduces with the
number of channels. Simulations are presented to verify the theory. The
stability limits are also compared with those of time-invariant systems,
and preliminary subjective assessments are carried out to indicate
useable loop gains with frequency-shifting.
Feedback delay networks are widely used for simulating the diffuse part of reverberation in a room. We present particular choices of the feedback coefficients, namely Galois sequences arranged in a circulant matrix, to produce a maximum echo density in the time response. These specific sets of coefficients give implementations having a low number of multipliers, and the resulting circuit can be efficiently pipelined. The resulting networks are compared with other efficient implementations.
The feedback delay network (FDN) has been proposed for digital
reverberation, The digital waveguide network (DWN) is also proposed with
similar advantages. This paper notes that the commonly used FDN with an
N×N orthogonal feedback matrix is isomorphic to a normalized
digital waveguide network consisting of one scattering junction joining
N reflectively terminated branches. Generalizations of FDNs and DWNs are
discussed. The general case of a lossless FDN feedback matrix is shown
to be any matrix having unit-modulus eigenvalues and linearly
independent eigenvectors. A special class of FDNs using circulant
matrices is proposed. These structures can be efficiently implemented
and allow control of the time and frequency behavior. Applications of
circulant feedback delay networks in audio signal processing are
discussed
This paper presents a new time-variant reverberation algorithm that can be used in reverberation enhancement systems. In these systems, acoustical feedback is always present and time variance can be used to obtain more gain before instability (GBI). The presented time-variant reverberation algorithm is analyzed and results of a practical GBI test are presented. The proposed reverberation algorithm has been used successfully with an electro-acoustically enhanced rehearsal room. This particular application is briefly overviewed and other possible applications are discussed.
The problem of the bounded-input/bounded-output stability of time-varying recursive filters is discussed. While simple, well-known criteria exist for the stability of time-invariant filters, guaranteeing stability when the filter coefficients are allowed to vary is much more difficult. Better insight into the causes for instability can be gained by using the state-space representation of the filter and examining the singular values of the state transition matrix. Two simple criteria based on the state transition matrix can be derived that guarantee the stability of the time-varying filter. Moreover, in the second-order case the singular values of this matrix provide a useful estimate of the maximum and average signal gains that result from the modification of the filter coefficients. These estimates can be used in practice to keep a time-varying filter from blowing up. It is also shown that some filter topologies are better suited to time-varying filtering than others, and a few techniques are presented that can be used to stabilize an otherwise unstable time-varying filter.
This paper introduces a time-variant reverberation algorithm as an extension of the feedback delay network (FDN). By modulating the feedback matrix nearly continuously over time, a complex pattern of concurrent amplitude modulations of the feedback paths evolves. Due to its complexity, the modulation produces less likely perceivable artifacts and the time-variation helps to increase the liveliness of the reverberation tail. A listening test, which has been conducted, confirms that the perceived quality of the reverberation tail can be enhanced by the feedback matrix modulation. In contrast to the prior art time-varying allpass FDNs, it is shown that unitary feedback matrix modulation is guaranteed to be stable. Analytical constraints on the pole locations of the FDN help to describe the modulation effect in depth. Further, techniques and conditions for continuous feedback matrix modulation are presented.
In reverberation enhancement systems (RESs), sound is constantly fed back from multiple microphones to multiple loudspeakers to enhance reverberation artificially in the target room. This contribution shows that such a system can be understood as an extended feedback delay network (FDN). A tuning process, similar to that of the FDN is presented, allowing arbitrary frequency-dependent reverberation elongation. The cross-talk between the loudspeakers and the microphones leads to comb filtering and isolated ringing modes in the RES, which produce undesired metallic and rough sounds. To mitigate these undesired effects, a cross-talk cancellation system is integrated in the RES. In a simulation example, the benefits of cross-talk cancellation is evaluated.
Some linear time-varying (LTV) components used to control feedback in sound systems were tested experimentally in real-time simulators and rooms with and without external reverberation. Gain before instability (GBI) was measured in single channels employing frequency shifting (FS), phase modulation (PM), and delay modulation (DM) implemented on a digital signal processor. FS performed according to the established theory. For PM GBI increased almost monotonically with modulation index beta, except for cases with large loop gain irregularities which displayed a reduced GBI for values of beta that corresponded to low carrier suppression. Also, GBI was practically independent of the modulation frequency f(m) already from 0.5 Hz even when this was much lower than the correlation distance of the loop gain transfer function. Rooms with different reverberation times gave different initial (time-invariant) GBI values but these differences decreased by the use of modulation. The GBI increase was larger for cases with external reverberation than for cases without clue to increased loop gain irregularity, and the GBI results depended on f(m). Since the possible GBI increase is determined by the initial GBI, LTV system performance should be measured in terms of GBI and not GBI increase alone. Robustness increased by equalizing the loop gain before employing LTV components. DM gave little protection for low frequencies but was efficient at high frequencies. (C) 1999 Acoustical Society of America. [S0001-4966(99)03407-4].
The first artificial reverberation algorithms were proposed in the early 1960s, and new, improved algorithms are published regularly. These algorithms have been widely used in music production since the 1970s, and now find applications in new fields, such as game audio. This overview article provides a unified review of the various approaches to digital artificial reverberation. The three main categories have been delay networks, convolution-based algorithms, and physical room models. Delay-network and convolution techniques have been competing in popularity in the music technology field, and are often employed to produce a desired perceptual or artistic effect. In applications including virtual reality, predictive acoustic modeling, and computer-aided design of acoustic spaces, accuracy is desired, and physical models have been mainly used, although, due to their computational complexity, they are currently mainly used for simplified geometries or to generate reverberation impulse responses for use with a convolution method. With the increase of computing power, all these approaches will be available in real time. A recent trend in audio technology is the emulation of analog artificial reverberation units, such as spring reverberators, using signal processing algorithms. As a case study we present an improved parametric model for a spring reverberation unit.
Artificial reverberator topologies making use of all-pass filters in a feedback loop are popular, but have lacked accurate control of decay time and energy level. This paper reviews a general theory of artificial reverberators based on Unitary-Feedback Delay Networks (UFDN), which allow accurate control of the decay time at multiple frequencies in such topologies. We describe the design of an efficient reverberator making use of chains of elementary filters, called "absorbent all-pass filters", in a feedback loop. We show how, in this particular topology, the late reverberant energy level can be controlled independently of the other control parameters. This reverberator uses the I3DL2 control parameters, which have been designed as a standard interface for controlling reverberators in interactive 3D audio.
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