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On Lossless Feedback Delay Networks
Abstract
Lossless Feedback Delay Networks (FDNs) are commonly used as a design prototype for artificial reverberation algorithms. The lossless property is dependent on the feedback matrix, which connects the output of a set of delays to their inputs, and the lengths of the delays. Both, unitary and triangular feedback matrices are known to constitute lossless FDNs, however, the most general class of lossless feedback matrices has not been identified. In this contribution, it is shown that the FDN is lossless for any set of delays, if all irreducible components of the feedback matrix are diagonally similar to a unitary matrix. The necessity of the generalized class of feedback matrices is demonstrated by examples of FDN designs proposed in literature.
... The delays, m , are typically chosen as co-prime of each other, so as to reduce the number of overlapping echoes and increase the echo density [23]. The design of the feedback matrix, A , starts from a lossless prototype, usually an orthogonal matrix such as Hadamard or Householder matrix, which have been shown to ensure (critical) stability regardless of the delays, a property defined by Schlecht and Habets as unilosslessness [8]. Losses are then incorporated by multiplying the unilossless matrix by a diagonal matrix of scalars designed to achieve a pre-set reverberation time, T 60 . ...
... In prior work [18], frequencyindependent homogeneous decay has been modeled by parameterizing A as the product of a unilossless matrix U and a diagonal matrix Ŵ(m) = diag(γ 1 , ..., γ N ) = diag(γ m 1 , ..., γ m N ) containing a delay-dependent absorption coefficient for each delay line, where γ ∈ (0, 1) is a constant gain-per-sample parameter. The feedback matrix is thus expressed as We let U be an orthogonal matrix, satisfying the unitary condition for unilosslessness [8]. To ensure this property, U is further parameterized by means of W ∈ R N ×N that, at each iteration, yields [18] where W Tr is the upper triangular part of W , and exp(·) is the matrix exponential. ...
... Namely, we found that, while closely matching the desired EDC, the IR of an FDN trained with = 0 , i.e., using only L EDC , tends to exhibit an unrealistic echo distribution compared to the RIRs of real-life environments. In this section, we compare the results of our differentiable FDN trained without Soft EDP regularization with those presented in Section 5.6 obtained using the composite loss function in (8). For conciseness, we limit our analysis to the Hallway RIR (h270); results obtained with other RIRs are comparable to what is shown below. ...
Over the past few decades, extensive research has been devoted to the design of artificial reverberation algorithms aimed at emulating the room acoustics of physical environments. Despite significant advancements, automatic parameter tuning of delay-network models remains an open challenge. We introduce a novel method for finding the parameters of a feedback delay network (FDN) such that its output renders target attributes of a measured room impulse response. The proposed approach involves the implementation of a differentiable FDN with trainable delay lines, which, for the first time, allows us to simultaneously learn each and every delay-network parameter via backpropagation. The iterative optimization process seeks to minimize a perceptually motivated time-domain loss function incorporating differentiable terms accounting for energy decay and echo density. Through experimental validation, we show that the proposed method yields time-invariant frequency-independent FDNs capable of closely matching the desired acoustical characteristics and outperforms existing methods based on genetic algorithms and analytical FDN design.
... The feedback matrix A is often chosen to have unimodular eigenvalues and linearly independent eigenvectors [22]. This, however, is not enough to ensure that all the system poles of the resulting FDN lie on the unit circle [23]. A feedback matrix that guarantees critical stability regardless of the choice of delays m is said to be unilossless [23]. ...
... This, however, is not enough to ensure that all the system poles of the resulting FDN lie on the unit circle [23]. A feedback matrix that guarantees critical stability regardless of the choice of delays m is said to be unilossless [23]. Notably, any orthogonal matrix is unilossless [23]. ...
... A feedback matrix that guarantees critical stability regardless of the choice of delays m is said to be unilossless [23]. Notably, any orthogonal matrix is unilossless [23]. As such, Hadamard, Householder, and circulant matrices are widely used [19]. ...
Differentiable machine learning techniques have recently proved effective for finding the parameters of Feedback Delay Networks (FDNs) so that their output matches desired perceptual qualities of target room impulse responses. However, we show that existing methods tend to fail at modeling the frequency-dependent behavior of sound energy decay that characterizes real-world environments unless properly trained. In this paper, we introduce a novel perceptual loss function based on the mel-scale energy decay relief, which generalizes the well-known time-domain energy decay curve to multiple frequency bands. We also augment the prototype FDN by incorporating differentiable wideband attenua-tion and output filters, and train them via backpropagation along with the other model parameters. The proposed approach improves upon existing strategies for designing and training differentiable FDNs, making it more suitable for audio processing applications where realistic and controllable artificial reverberation is desirable, such as gaming, music production, and virtual reality.
... Feedback delay networks are composed of delay lines in parallel, which are connected through a feedback matrix (or mixing matrix), which is unitary to conserve system energy [1]. Schlecht and Habets expanded the set of matrices that preserve energy in FDNs to unilossless matrices [2]. The input signal traverses through the delay lines and the mixing matrix, building echo density over time. ...
... This gives us independent control over the intraand inter-group mixing characteristics. To preserve energy in the system, the coupled mixing matrix needs to be unilossless [2]. Examples of such lossless matrices are orthonormal or unitary matrices [1], [2]. ...
... To preserve energy in the system, the coupled mixing matrix needs to be unilossless [2]. Examples of such lossless matrices are orthonormal or unitary matrices [1], [2]. ...
Feedback Delay Networks are one of the most popular and efficient means of generating artificial reverberation. Recently, we proposed the Grouped Feedback Delay Network (GFDN), which couples multiple FDNs while maintaining system stability. The GFDN can be used to model reverberation in coupled spaces that exhibit multi-stage decay. The block feedback matrix determines the inter- and intra-group coupling. In this paper, we expand on the design of the block feedback matrix to include frequency-dependent coupling among the various FDN groups. We show how paraunitary feedback matrices can be designed to emulate diffraction at the aperture connecting rooms. Several methods for the construction of nearly paraunitary matrices are investigated. The proposed method supports the efficient rendering of virtual acoustics for complex room topologies in games and XR applications.
... (2.44). This desired matrix property is referred to as unilosslessness in the literature (Schlecht and Habets, 2017). In practice, this is equivalent to having all poles of the transfer function eq. ...
... M T M = I). In fact, even though Schlecht and Habets (2017) identified the complete class of unilossless matrices (i.e. feedback matrices which guarantee the filter stability independently from the delay values), only two subclasses are, in fact, easy to identify and construct, which are (i) unitary matrices, and (ii) triangular matrices with unimodular diagonal values. ...
... Hence, output signals from the "main" filters (g a→a,l (z)) represent the power that "stays" within room a. Furthermore, in order to ensure stability of each FDN when these are uncoupled and no dissipation is available, matrices M aa are chosen to be unitary (Stautner and Puckette, 1982;Jot, 1992;Schlecht and Habets, 2017). ...
This thesis takes place within the RASPUTIN project and focuses on the development, evaluation and use of immersive acoustic virtual reality simulation tools for the purpose of helping blind individuals prepare in-situ navigations in unfamiliar reverberant environments. While several assistive tools, such as sensory substitution devices, can provide spatial information during navigation, an alternative approach is to devise a real-time room acoustic simulation and auralization engine for use by blind individuals at home to enable them to virtually navigate in unfamiliar environments under controlled circumstances, hence building mental representations of these spaces prior to in-situ navigation. In this thesis, I tackle three aspects of this subject. The first part focuses on efficient simulations and auralizations of coupled volumes, which occur in many buildings of interest for navigation preparation (e.g. city halls, hospitals, or museums) and whose simulation and auralization can be challenging. The second part focuses on the individualization of head related transfer functions, which is a necessary step in providing individualized and convincing auditory experiences. Finally, the last part investigates some aspects of the space cognition following use of different learning paradigms, such as tactile maps.
... The correspondence between building blocks, their types, the entries of L(g, ξ), and the value of ξ to represent each structure with the generalized structure are shown in Table 1. L(g, ξ) is unilossless [53] ∀ξ = 0 since it is diagonally similar to U(g) by P = diag(1, 1/ξ). However it is only unitary for ξ = 1. ...
... Experimental results in [31] show that this structure preserves energy down to numerical precision. Schlecht introduced the idea of rewriting different digital reverb structures as FDNs [8,60] so that, e.g., they can be studied using Laroche's time-varying stability criteria [61] or concepts like unilosslessness [53]. ...
... This was shown for the particular case of a 2-mult. (type II, ξ = 1/D g ) Schroeder allpass filter in [53]. In sum, A is unilossless but only unitary for ξ = 1. ...
... Feedback delay networks (FDNs) are a computationally efficient structure for artificial reverberation with a mature theoretical foundation [6,7,8,9], see Fig. 1a. In its early form, Schroeder and Logan [10] pursued colorless reverberation by concatenating delay-line-based allpass filters. ...
... To generate colorless reverberation with FDNs, additional recommendations on the actual choice of parameters, namely the feedback matrix and delay line lengths, were proposed, including co- prime delays [10], non-degenerate delay distributions [11], sufficient number of modes [10,7,5], dense lossless feedback matrices [8,12,9], and diffusion filters [13,14,15]. Despite many such recommendations, large FDNs can have hundreds of tuning parameters. ...
... The feedback matrix is the product of the unilossless matrix U , e.g., an orthogonal matrix [9], and the diagonal gain matrix Γ ...
A perceptual study revealing a novel connection between modal properties of feedback delay networks (FDNs) and colorless reverberation is presented. The coloration of the reverberation tail is quantified by the modal excitation distribution derived from the modal decomposition of the FDN. A homogeneously decaying all-pass FDN is designed to be colorless such that the corresponding narrow modal excitation distribution leads to a high perceived modal density. Synthetic modal excitation distributions are generated to match modal excitations of FDNs. Three listening tests were conducted to demonstrate the correlation between the modal excitation distribution and the perceived degree of coloration. A fourth test shows a significant reduction of coloration by the col-orless FDN compared to other FDN designs. The novel connection of modal excitation, allpass FDNs, and perceived coloration presents a beneficial design criterion for colorless artificial reverberation .
... FDNs generalize the well-known state space representation by replacing single time steps with different vector time steps, see Fig. 1. FDNs have well-established system properties such as losslessness and stability [5,6], decay control [7,8], impulse response density [9,10], and, modal distribution [11]. SISO allpass FDNs can be composed from simple allpass filters in series [2,12] or by nesting [13]. ...
... The numerator is a matrix-valued expression with Q m,A,B,C,D pzq " D detpP pzqq`C adjpP pzqq B, (6) where adjpAq denotes the adjugate of A [11]. The FDN system poles λ i , where 1 ď i ď N, are the roots of the generalized characteristic polynomial (GCP) p m,A pzq in (4). ...
... To demonstrate system properties of an FDN independent from delays m, we have earlier developed a representation of p m,A pzq based on the principal minors of A [6,19]. This representation is also useful to derive the uniallpass property of FDNs. ...
In the 1960s, Schroeder and Logan introduced delay line-based allpass filters, which are still popular due to their computational efficiency and versatile applicability in artificial reverberation, decorrelation, and dispersive system design. In this work, we extend the theory of allpass systems to any arbitrary connection of delay lines, namely feedback delay networks (FDNs). We present a characterization of uniallpass FDNs, i.e., FDNs, which are allpass for an arbitrary choice of delays. Further, we develop a solution to the completion problem, i.e., given an FDN feedback matrix to determine the remaining gain parameters such that the FDN is allpass. Particularly useful for the completion problem are feedback matrices, which yield a homogeneous decay of all system modes. Finally, we apply the uniallpass characterization to previous FDN designs, namely, Schroeder's series allpass and Gardner's nested allpass for single-input, single-output systems, and, Poletti's unitary reverberator for multi-input, multi-output systems and demonstrate the significant extension of the design space.
... FDNs can have single or multiple input and output channels distributed by the input, output, and direct gains. Further, FDNs have well-established system properties such as losslessness and stability [5,6], decay control [7,8], impulse response density [9,10], and, modal distribution [11]. Compared to general high-order allpass filters [12], FDNs are sparse filters, which are less flexible, but more computationally efficient. ...
... The numerator is a matrix-valued expression with Q m,A,B,C,D pzq " D detpP pzqq`C adjpP pzqq B, (6) where adjpAq denotes the adjugate of A [11]. The FDN system poles λ i , where 1 ď i ď N, are the roots of the generalized characteristic polynomial (GCP) p m,A pzq in (4). ...
... To demonstrate system properties of an FDN independent from delays m, we have earlier developed a representation of p m,A pzq based on the principal minors of A [6,20]. This representation is also useful to derive the uniallpass property of FDNs. ...
In the 1960s, Schroeder and Logan introduced delay-based allpass filters, which are still popular due to their computational efficiency and versatile applicability in artificial reverberation, decorrelation, and dispersive system design. In this work, we extend the theory of allpass systems to any arbitrary connection of delay lines, namely feedback delay networks (FDNs). We present a complete characterization of uniallpass FDNs, i.e., FDNs, which are allpass for an arbitrary choice of delays. Further, we develop a solution to the completion problem, i.e., given an FDN feedback matrix to determine the remaining gain parameters such that the FDN is allpass. Particularly useful for the completion problem are feedback matrices, which yield a homogeneous decay of all system modes. Finally, we apply the uniallpass characterization to previous FDN designs, namely, Schroeder's series allpass and Gardner's nested allpass for single-input, single-output systems, and, Poletti's unitary reverberator for multi-input, multi-output systems and demonstrate the significant extension of the design space.
... The structure of the feedback matrix A strongly influences the sound of a FDN, therefore its design is one of the main tasks when a FDN structure is derived. The question, whether a feedback matrix leads to "good reverberation" or not spread a high amount of literature based on matrix theory [80,184,187,188]. ...
... As shown in (5.4), (5.5) and in [184,185,187], the equations which describe a FDN have a strong relation to state space descriptions, where the outputs s i (k) of the delay lines serve as "system states" (see Sec. 4.8.3). Inspecting the transfer function of a FDN in (5.6) confirms this connection as it shows a strong relation to the resolvent operator which has been described in (4.74) in Sec. ...
... If the product of A d and D d is diagonal (which is given by the restrictions above), unitary matrices U d ensure the stability of the closed loop system. As the design of unitary matrices is a well known task in the context of feedback delay networks, it is suitable to design U d directly in the z-domain [15,80,187,188]. In the case of commutative matrices in (5.51), the design of the matrix U in the s-domain can be restricted to skew-hermetian matrices. ...
Most real-world systems are distributed parameter systems. This means that their dynamics depend not only on their temporal, but also on their spatial behaviour. Particularly, their parameters are not concentrated, but distributed over their spatial volume. Well-investigated examples include the sound of a guitar string, which depends on its spatial oscillation on the guitar, or an electrical transmission line, whose resistance is dependent on its length. Distributed parameter systems also occur naturally in the human body, where the transport of particles through a blood vessel is influenced by its spatial and temporal properties. An abstraction of the real world delivers a comprehensive mathematical description of distributed parameter systems in terms of initial-boundary value problems, which are derived by the first principles of physics. Initial-boundary value problems describe the dynamics of a distributed parameter system by partial differential equations, where its temporal initial state and its spatial boundary behaviour is modelled by suitable initial and boundary conditions. The analysis of natural existing distributed parameter systems as well as the design of synthetic distributed parameter systems are very important tools that may be employed to analyse the dynamics of particle transport in the human body from a communications point of view, or to create a digital guitar synthesizer. Therefore, it is indispensable to obtain suitable models to simulate the spatio-temporal dynamics of distributed parameter systems. The main challenge is to choose a suitable modelling technique which leads to a model that meets the predefined requirements. In the literature, a considerable number of different modelling techniques is found, each with its own advantages and disadvantages. These modelling techniques may be roughly divided into two different categories: numerical methods and analytical methods. Most numerical methods apply a suitable discretization rule to a set of partial differential equations. They lead to powerful simulation algorithms which are capable of simulating spatially complex physical problems in a very accurate way. However, these methods often have a very high computational complexity and provide only little insight into the influence of parameters on the output signal. In contrast to that, analytical methods try to find an explicit solution of an initial-boundary value problem before a discrete algorithm is established. Most analytical methods are based on well-investigated techniques from mathematics and systems theory. The derived models allow to establish a relationship between input and output variables of a system in terms of its parameters. Furthermore, they can lead to low-complexity algorithms with real-time capability by the application of a convenient discretization method. However, the elegance of analytical methods decreases for non-linear systems. Therefore, they are mostly applied to distributed parameter systems which can be described mathematically by linear initial-boundary value problems. The choice of a suitable modelling technique thus depends strongly on the distributed parameter system to be modelled and the requirements on the simulation model. If exact numerical results of a spatially complex distributed parameter system are required, numerical methods are preferable. However, if an exact closed-form description is necessary – for the analysis of a distributed parameter system, for example – analytical methods are advantageous. The modelling procedure used in this thesis is the Functional Transformation Method. This procedure contains diverse functional transformations, i.e., a Laplace and a Sturm-Liouville transformation. Finally, a model is formulated in terms of multidimensional transfer functions. The functional transformation method belongs to the class of analytical modelling techniques. But as most other analytical methods, the functional transformation method is not suitable for complex spatial shapes, non-linear distributed parameter systems and for complex boundary behaviour. Then the method loses its elegance and no explicit solution is obtained as numerical evaluations have to be involved. Nevertheless, analytical methods are a desired approach for the modelling of distributed parameter systems. Therefore, this thesis marks a starting point in overcoming some of the previously mentioned problems of analytical modelling techniques, i.e., of the functional transformation method. By developing suitable extensions it is possible to derive an explicit model of a distributed parameter system which includes the influence of complex boundary behaviour. Although the procedure of the functional transformation method is already formalised, the first goal of this dissertation is to improve its formulation. As an extension, an operator-based version of the involved Sturm-Liouville transformation is incorporated into the functional transformation method. Applying this extension variant to an initial-boundary value problem, a multidimensional state space description is obtained as a solution, which constitutes the model of the underlying distributed parameter system. This formulation as a state space description exhibits several advantages: it constitutes a unified solution of the functional transformation method and allows its analysis and modification by concepts from control and systems theory. The formulation of the simulation model in terms of a state space description is the basis for the second goal of this dissertation. The functional transformation method is extended by adapting concepts from control theory to incorporate the influence of complex boundary behaviour by the design of feedback loops. First, the complex boundary behaviour is separated from the system and it is modelled with a generic simple boundary behaviour which defines the open loop system. The complex boundary behaviour is used to design a feedback matrix which is attached to the simple model to form the closed loop system that fulfils the desired complex boundary behaviour. In particular, the feedback matrix shifts the eigenvalues of the open loop system into a position where they fulfil the complex boundary behaviour. With the developed concept it is possible to model distributed parameter systems with complex boundary behaviour in an explicit form. The same concept can be used to incorporate other physical effects into the model of a distributed parameter system. Furthermore, the concept allows to model interconnected systems, which builds the basis for a block-based modelling approach of interconnected distributed parameter systems. Applying the developed techniques to specific problems from different fields of application, their validity is confirmed in the third part of this dissertation. Specifically, the techniques are employed to model musical systems, electrical transmission lines and biological systems in the context of molecular communications. Within these applications, the developed methods are used to incorporate complex boundary behaviour and physical effects. In addition, general system modifications to change the timbre of a musical system are shown. Furthermore, two biological systems are interconnected by the design of a connection matrix. Where possible, the modelling results are compared to numerical simulations or measurements. All considered problems show that the developed concepts are suitable for the modelling of distributed parameter systems and constitute a meaningful extension to the functional transformation method.
... An FFDN is lossless if A(z) is paraunitary, i.e., A(z −1 ) H A(z) = I, where I is the identity matrix and · H denotes the complex conjugate transpose [25]. Although also non-paraunitary FFMs may yield lossless FFDNs [15,26], here we focus on paraunitary FFMs only. Paraunitary matrices are particularly useful as they are closed under multiplication, i.e., if A(z) and B(z) are paraunitary, then A(z)B(z) is paraunitary as well [27]. ...
... The deflation term may be interpreted as a penalty term if two eigenvalues approach each other too closely and guarantee that all eigenvalues reached are unique. For numerically stable evaluation of Newton correction in (26), the left side is evaluated for |z| ≤ 1 and the right side otherwise [32]. For the FFM, we give the explicit form of both Newton correction terms. ...
... Similar to (26), we have ...
Feedback delay networks (FDNs) are recursive filters, which are widely used for artificial reverberation and decorrelation. One central challenge in the design of FDNs is the generation of sufficient echo density in the impulse response without compromising the computational efficiency. In a previous contribution, we have demonstrated that the echo density of an FDN can be increased by introducing so-called delay feedback matrices where each matrix entry is a scalar gain and a delay. In this contribution, we generalize the feedback matrix to arbitrary lossless filter feedback matrices (FFMs). As a special case, we propose the velvet feedback matrix, which can create dense impulse responses at a minimal computational cost. Further, FFMs can be used to emulate the scattering effects of non-specular reflections. We demonstrate the effectiveness of FFMs in terms of echo density and modal distribution.
... Therefore, the computational complexity of the FDN scales with the number of delay lines and not the system order. FDNs are a popular choice for artificial reverberation applications particularly because of the favorable relation between FDN size and system order [6][7][8][9]. ...
... However, the bound may be arbitrarily loose. For instance, the maximum singular value of a triangular matrix max σ(A) may be arbitrarily large while all system poles lie on the unit circle [9]. For large delays m however, (23) shows that the pole magnitudes tend to be close to the unit circle. ...
... In particular, if all singular values are 1, which is equivalent to A being unitary, i.e., A H A = I, all system poles lie on the unit circle regardless of the delays m. Such an FDN is called lossless, and represents an important special case [9]. ...
Feedback delay networks (FDNs) belong to a general class of recursive filters which are widely used in sound synthesis and physical modeling applications. We present a numerical technique to compute the modal decomposition of the FDN transfer function. The proposed pole finding algorithm is based on the Ehrlich-Aberth iteration for matrix polynomials and has improved computational performance of up to three orders of magnitude compared to a scalar polynomial root finder. The computational performance is further improved by bounds on the pole location and an approximate iteration step. We demonstrate how explicit knowledge of the FDN's modal behavior facilitates analysis and improvements for artificial reverberation. The statistical distribution of mode frequency and residue magnitudes demonstrate that relatively few modes contribute a large portion of impulse response energy.
... Because of the self-similarity in (23), the magnitude responses of the band filters can be used as basis functions to approximate the magnitude response of the overall graphic equalizer. The filters are sampled at a K × 1 vector of control frequencies ωP spaced logarithmically over the complete frequency range, where K ≥ L. The resulting interaction matrix is B = 20 log 10 [g ′ , H 1,g ′ (ωP), . . . ...
... Case study for target reverberation time given in Section 4.1. The dashed lines indicate the approximation response resulting from(28), whereas the solid lines indicate the actual filter response which may differ because of violations of the self-similarity property(23). Both unconstrained methods MLSUnc and TLSUnc has large deviations between the approximation and actual filter response. ...
... In contrast to lossless feedback matrices used for example in[7], unilossless feedback matrices are lossless for all possible delays. For more information, the reader is referred to[23]. ...
The reverberation time is one of the most prominent acoustical qualities of a physical room. Therefore, it is crucial that artifi- cial reverberation algorithms match a specified target reverberation time accurately. In feedback delay networks, a popular framework for modeling room acoustics, the reverberation time is determined by combining delay and attenuation filters such that the frequency- dependent attenuation response is proportional to the delay length and by this complying to a global attenuation-per-second. How- ever, only few details are available on the attenuation filter design as the approximation errors of the filter design are often regarded negligible. In this work, we demonstrate that the error of the filter approximation propagates in a non-linear fashion to the resulting reverberation time possibly causing large deviation from the speci- fied target. For the special case of a proportional graphic equalizer, we propose a non-linear least squares solution and demonstrate the improved accuracy with a Monte Carlo simulation.
... Nowadays, one of the most widely used approaches in artificial reverberation is the feedback delay network (FDN), a system that generalizes the parallel comb-filter structure by interconnecting delays via a feedback matrix [3,4,5]. In FDNs, a commonly used approach is to first design a lossless prototype [6] to then achieve the desired frequency-dependent decay with attenuation filters [7,8]. However, a common bane of systems utilizing comb filters is sound coloration [1]. ...
... However, under this condition, evaluating HpzM q on the unitary circle becomes unfeasible, as the discrete generalized characteristic polynomial ppzM q " detpDmpzM q´1´Aq becomes singular and non-invertible. To avoid instabilities, we use a homogeneous FDN where A is parameterized according to (6), and γ is set at initialization to a value lower than one and kept constant during optimization. The value of γ used during optimization is chosen by examining the connection between the mean damping factor δ, used in room acoustics, and the mean spacing of resonance frequencies ∆f . ...
Artificial reverberation algorithms often suffer from spectral coloration, usually in the form of metallic ringing, which impairs the perceived quality of sound. This paper proposes a method to reduce the coloration in the feedback delay network (FDN), a popular artificial reverberation algorithm. An optimization framework is employed entailing a differentiable FDN to learn a set of parameters decreasing coloration. The optimization objective is to minimize the spectral loss to obtain a flat magnitude response, with an additional temporal loss term to control the sparseness of the impulse response. The objective evaluation of the method shows a favorable narrower distribution of modal excitation while retaining the impulse response density. The subjective evaluation demonstrates that the proposed method lowers perceptual coloration of late reverberation, and also shows that the suggested optimization improves sound quality for small FDN sizes. The method proposed in this work constitutes an improvement in the design of accurate and high-quality artificial reverberation, simultaneously offering computational savings.
... An FDN is lossless if Apzq is paraunitary, i.e., Apz -1 q H Apzq " I, where I is the identity matrix and¨H denotes the complex conjugate transpose [31]. For real scalar matrices A, the FDN is lossless if A is orthogonal, i.e., A J A " I. However also non-orthogonal feedback matrices may yield lossless FDNs [32,33], and we give some examples in Section 3. ...
... For this reason, we provide the randomOrthogonal function, which samples the space of all orthogonal matrices uniformly. The orthogonal matrices can be diagonally similar such that the lossless property is retained (see diagonallyEquivalent) [32]. The reverse process is less trivial, i.e., determining whether a matrix A is a diagonally similar to an orthogonal matrix. ...
Feedback delay networks (FDNs) are recursive filters, which are widely used for artificial reverberation and decorrelation. While there exists a vast literature on a wide variety of reverb topologies, this work aims to provide a unifying framework to design and analyze delay-based reverberators. To this end, we present the Feedback Delay Network Toolbox (FDNTB), a collection of the MAT-LAB functions and example scripts. The FDNTB includes various representations of FDNs and corresponding translation functions. Further, it provides a selection of special feedback matrices, topologies, and attenuation filters. In particular, more advanced algorithms such as modal decomposition, time-varying matrices, and filter feedback matrices are readily accessible. Furthermore, our toolbox contains several additional FDN designs. Providing MATLAB code under a GNU-GPL 3.0 license and including illustrative examples, we aim to foster research and education in the field of audio processing.
... Various algorithms are used to produce artificial reverberation, with the Feedback Delay Network (FDN) being currently among the most popular ones [5][6][7]. The first objective in designing an FDN is to make it lossless. ...
... An FDN is a comb filter structure with multiple delay lines interconnected by a feedback matrix [5]. When designing FDNs, the first step is to make it lossless, ensuring that the energy will not decay for any possible type of delay [7]. The frequency-dependent reverberation time can then be implemented by inserting an attenuation filter at the beginning or at the end of each delay line. ...
Artificial reverberation algorithms generally imitate the frequency-dependent decay of sound in a room quite inaccurately. Previous research suggests that a 5% error in the reverberation time (T60) can be audible. In this work, we propose to use an accurate graphic equalizer as the attenuation filter in a Feedback Delay Network re-verberator. We use a modified octave graphic equalizer with a cascade structure and insert a high-shelf filter to control the gain at the high end of the audio range. One such equalizer is placed at the end of each delay line of the Feedback Delay Network. The gains of the equalizer are optimized using a new weighting function that acknowledges nonlinear error propagation from filter magnitude response to reverberation time values. Our experiments show that in real-world cases, the target T60 curve can be reproduced in a perceptually accurate manner at standard octave center frequencies. However, for an extreme test case in which the T60 varies dramatically between neighboring octave bands, the error still exceeds the limit of the just noticeable difference but is smaller than that obtained with previous methods. This work leads to more realistic artificial reverberation.
... The dense set of late reflections are added by a feedback network. Matrix theory has been invoked for the design of "good reverberation" [5][6][7][8]. ...
... To preserve the stability of the system, the matrix U d has to be a unitary matrix, which restricts the design of U in the s-domain to skew-hermitian matrices. For practical applications it is suitable to design the matrix U d directly [5][6][7]. ...
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.
... Geometric-acoustic models have seen significant advancements, notably in the area of beamtracing [38], [47], enabling faster rendering for moving sources and receivers. Delay network-based models have been better characterised [48] and have begun to approach perceptual performance of geometric-acoustic models while maintaining low computational complexity [39], [49]- [51]. ...
The study of spatial audio and room acoustics aims to create immersive audio experiences by modeling the physics and psychoacoustics of how sound behaves in space. In the long history of this research area, various key technologies have been developed based both on theoretical advancements and practical innovations. We highlight historical achievements, initiative activities, recent advancements, and future outlooks in the research area of spatial audio recording and reproduction, and room acoustic simulation, modeling, analysis, and control.
... Designing an FDN often starts by creating a lossless prototype with an energy-preserving feedback loop [22,23]. This can be achieved by using an orthogonal feedback matrix, since it meets the condition for losslessness [24]. The advantage of initially designing a lossless FDN lies in the straightforward implementation of frequency-dependent decay that equally influences all system poles. ...
This paper seeks to improve the state-of-the-art in delay-network-based analysis-synthesis of measured room impulse responses (RIRs). We propose an informed method incorporating improved energy decay estimation and synthesis with an optimized feedback delay network. The performance of the presented method is compared against an end-to-end deep-learning approach. A formal listening test was conducted where participants assessed the similarity of reverberated material across seven distinct RIRs and three different sound sources. The results reveal that the performance of these methods is influenced by both the excitation sounds and the reverberation conditions. Nonetheless, the proposed method consistently demonstrates higher similarity ratings compared to the end-to-end approach across most conditions. However, achieving an indistinguishable synthesis of measured RIRs remains a persistent challenge, underscoring the complexity of this problem. Overall, this work helps improve the sound quality of analysis-based artificial reverberation.
... In recursive artificial reverberation architectures, using unilossless feedback mixing matrices such as in FDNs and IVNs [24], all feedback delay loops should approximate the same frequency-dependent decay rate [2]. However, each attenuation filter can be analyzed separately, as the amount of attenuation is determined by the target reverberation time, T 60 , and the delay-line length [4], [13], [14]. ...
Delay networks are a common parametric method to synthesize the late part of the room reverberation. A delay network consists of several feedback loops, each containing a delay line and an attenuation filter, which approximates the same decay rate by appropriately setting the frequency-dependent loop gain. A remaining challenge is the design of the attenuation filters on a wide frequency range based on a measured room impulse response. This letter proposes a novel two-stage attenuation filter structure, sharpening the design. The first stage is a low-order pre-filter approximating the overall shape and determining the decay at the two ends of the frequency range, namely at the dc and the Nyquist limit. The second filter, an equalizer, fine-tunes the gain at different frequencies, such as on one-third-octave bands. It is shown that the proposed design is more accurate and robust than previous methods. A design example applying the proposed method to an interleaved velvet-noise reverberator is also exhibited. The proposed two-stage attenuation filter is a step toward a realistic parametric simulation of measured room impulse responses.
... In particular, they realized that the method previously proposed by Stautner and Puckette could build up high time densities by using only few delay lines. In addition these algorithms are usually made to be lossless [15], meaning that no energy is lost during the delays; this allows to implement decay using absorbent filters, thus providing control over the frequency dependent reverberation time, something which was not possible with Schroeder's structures. The characteristics illustrated above, along with the lightweight and efficient design, make FDNs still widely appreciated and studied algorithms for modeling room impulse responses [16,17]. ...
The EMT 140 is a plate reverb that exploits the vibrations of a metallic plate to simulate reverberation. It was an industry standard in the 60s and 70s, and its sound is still widely appreciated nowadays. In this work, a physical model of the EMT 140 is developed, by adopting a modal approach. First, a modal decomposition of the plate vibration is performed; then, a physically-based model for damping is presented, based on previous works, including a simulation of the control-lable damper present on the real unit. The modes equations then are discretised with an exact numerical integrator. As physical models are quite computationally expensive, here an optimisation technique is also presented, based on linear least squares. Results show that the optimisation could reduce the computational time by 80%, while still maintaining a high sound quality.
... However, impulse response convolution with an audio signal doesn't provide any parametric control over the reverberation characteristics to the user. Also, the computing resources required to convolve an impulse response can be greater than the resources required for delay-based algorithmic reverberation [1]. ...
Full Text available here: http://www.aes.org/e-lib/browse.cfm?elib=21917
Recorded room impulse responses enable accurate and high-quality artificial reverberation. Used in combination with convolution, they can be computationally expensive and inflexible, providing little control to the user. On the other hand, reverberation algorithms are parametric which enable user control. However, they can lack realism and can be challenging to configure. To address these limitations, we introduce a multi-stage approach to optimize the coefficients of a Feedback Delay Network (FDN) reverberator to match a target room impulse response, thus enabling parametric control. In the first stage, we configure some FDN parameters by extracting features from the target impulse response. Then, we use a genetic algorithm to fit the remaining parameters to match the desired impulse response using a Mel-frequency cepstrum coefficients (MFCCs) cost function. We evaluate our approach across a dataset of impulse responses and conducted a subjective listening test. Our results indicate that the combination of the FDN with a short truncation of the target impulse response enables a better approximation, however, there are still differences with respect to the overall spectrum and the clarity factor in some more challenging cases.
... Such a criterion is, however, insufficient to obtain an adequately rapid increase in the echo density, especially when other unwanted effects, such as clustering and low-order dependencies, are not avoided [248,249]. The primary criterion for choosing the feedback matrix for an FDN is ensuring losslessness of the structure, i.e., the energy of the system should not decay when the attenuation filters are not in use [202,203,250]. Apart from this, the matrices are also used to enhance specific properties of the FDN, such as the increase in the echo density, computational efficiency, and spectral flatness [26,225,226,251,252,253]. ...
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.
... With the sets of incoming and outgoing signals being grouped into vectors, the scattering coefficients form a matrix, such that the matrix multiplication of the incoming signal vector by the scattering matrix gives the outgoing signal vector. The scattering matrices are designed to be unitary [16], which means that the scattering operation itself does not introduce or remove energy from the system [17]. Energy is only introduced by the input signal, and only removed by the attenuation filters, which guarantees stability. ...
Computer simulations of room acoustics suffer from an efficiency vs accuracy trade-off, with highly accurate wave-based models being highly computationally expensive, and delay-network-based models lacking in physical accuracy. The Scattering Delay Network (SDN) is a highly efficient recursive structure that renders first order reflections exactly while approximating higher order ones. With the purpose of improving the accuracy of SDNs, in this paper, several variations on SDNs are investigated, including appropriate node placement for exact modeling of higher order reflections , redesigned scattering matrices for physically-motivated scattering, and pruned network connections for reduced computational complexity. The results of these variations are compared to state-of-the-art geometric acoustic models for different shoebox room simulations. Objective measures (Normalized Echo Densities (NEDs) and Energy Decay Curves (EDCs)) showed a close match between the proposed methods and the references. A formal listening test was carried out to evaluate differences in perceived naturalness of the synthesized Room Impulse Responses. Results show that increasing SDNs' order and adding directional scattering in a fully-connected network improves perceived naturalness, and higher-order pruned networks give similar performance at a much lower computational cost.
... In the ball-within-the-box (BaBo) model [46], the matrix A can be interpreted as a scattering ball that redistributes the energy from incoming wavefronts into different directions, each corresponding to a planar wave loop, which produces a harmonic series. Indeed, the matrix A does not have to be unitary for the feedback structure to be lossless [52]. However, even staying within the class of unitary matrices, A can be chosen for its scattering properties, ranging from the identity matrix (no scattering at all) to maximally-diffusive structures [47,53]. ...
Humans have a privileged, embodied way to explore the world of sounds, through vocal imitation. The Quantum Vocal Theory of Sounds (QVTS) starts from the assumption that any sound can be expressed and described as the evolution of a superposition of vocal states, i.e., phonation, turbulence, and supraglottal myoelastic vibrations. The postulates of quantum mechanics, with the notions of observable, measurement, and time evolution of state, provide a model that can be used for sound processing, in both directions of analysis and synthesis. QVTS can give a quantum-theoretic explanation to some auditory streaming phenomena, eventually leading to practical solutions of relevant sound-processing problems, or it can be creatively exploited to manipulate superpositions of sonic elements. Perhaps more importantly, QVTS may be a fertile ground to host a dialogue between physicists, computer scientists, musicians, and sound designers, possibly giving us unheard manifestations of human creativity.
... where D i,j = x i − x j · F s /c is the delay between two scattering nodes, i and j, positioned at x i and x j , and S = 2 5 1 5×5 − I 5×5 is the scattering matrix. This specific scattering matrix can be shown to be unilossless, i.e. it results in lossless operation regardless of the length of the feedback delay lines [14]. ...
Artificial reverberators provide a computationally viable alternative to full-scale room acoustics simulation methods for deployment in interactive, immersive systems. Scattering delay network (SDN) is an artificial reverberator that allows direct parametric control over the geometry of a simulated cuboid enclosure as well as the directional characteristics of the simulated sound sources and microphones. This paper extends the concept of SDN reverberators to multiple enclosures coupled via an aperture. The extension allows independent control of the acoustical properties of the coupled enclosures and the size of the connecting aperture. The transfer function of the coupled-volume SDN system is derived. The effectiveness of the proposed method is evaluated in terms of rendered energy decay curves in comparison to full-scale ray-tracing models and scale model measurements.
... The lossless properties of the recirculating matrix was further developed in [184] with the introduction of circulant matrices, and [185] presented a general definition for the requirements of lossless matrices. Since these networks are designed as complex feedback filters, they do not have AP properties. ...
Available online with the related articles at: http://urn.fi/URN:ISBN:978-952-64-0472-1
In this dissertation, the reproduction of reverberant sound fields containing directional characteristics is investigated. A complete framework for the objective and subjective analysis of directional reverberation is introduced, along with reverberation methods capable of producing frequency- and direction-dependent decay properties. Novel uses of velvet noise are also proposed for the decorrelation of audio signals as well as artificial reverberation. The methods detailed in this dissertation offer the means for the auralization of reverberant sound fields in real-time, with applications in the context of Immersive sound reproduction such as virtual and augmented reality.
... where D i,j = x i − x j · F s /c is the delay between two scattering nodes, i and j, positioned at x i and x j , and S = 2 5 1 5×5 − I 5×5 is the scattering matrix. This specific scattering matrix can be shown to be unilossless, i.e. it results in lossless operation regardless of the length of the feedback delay lines [14]. ...
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ABSTRACT
Simulation of the acoustics of coupled rooms is an important problem not only in architectural acoustics but also in immersive audio applications that require acoustic simulation at interactive rates. Requirements for such applications are less demanding for accuracy but more demanding for computational cost. Scattering delay network (SDN) is a real-time, interactive room acoustics simulator for cuboid rooms. SDN affords an exact simulation of first-order early reflections, a gracefully degrading simulation of second and higher-order specular reflections and an accurate simulation of the statistical properties of the late reverberation. We propose coupled-volume SDN (CV-SDN) as an extension of the SDN model to simulate acoustics of coupled volumes. The proposed model retains the desirable characteristics of the original SDN model while allowing the simulation of double-slope decays with direct control over the simulated aperture size. The double-slope characteristics of room impulse responses simulated with CV-SDN agree well with those of measured impulse responses from a scale model and state-of-the-art room acoustics simulation software.
... In the ball-within-the-box (BaBo) model [46], the matrix A can be interpreted as a scattering ball that redistributes the energy from incoming wavefronts into different directions, each corresponding to a planar wave loop, which produces a harmonic series. Indeed, the matrix A does not have to be unitary for the feedback structure to be lossless [52]. However, even staying within the class of unitary matrices, A can be chosen for its scattering properties, ranging from the identity matrix (no scattering at all) to maximally-diffusive structures [47,53]. ...
Humans have a privileged, embodied way to explore the world of sounds, through vocal imitation. The Quantum Vocal Theory of Sounds (QVTS) starts from the assumption that any sound can be expressed and described as the evolution of a superposition of vocal states, i.e., phonation, turbulence, and supraglottal myoelastic vibrations. The postulates of quantum mechanics, with the notions of observable, measurement, and time evolution of state, provide a model that can be used for sound processing, in both directions of analysis and synthesis. QVTS can give a quantum-theoretic explanation to some auditory streaming phenomena, eventually leading to practical solutions of relevant sound-processing problems, or it can be creatively exploited to manipulate superpositions of sonic elements. Perhaps more importantly, QVTS may be a fertile ground to host a dialogue between physicists, computer scientists, musicians, and sound designers, possibly giving us unheard manifestations of human creativity.
... delay network (FDN) [3]- [5], can be converted into a pseudorandom noise generator by turning off the decay of sound. In practice, replacing all filters and attenuating coefficients in the algorithm with unity gains (i.e., no signal-processing operation) leaves only delay line and summation operations. ...
This paper proposes a novel algorithm for simulating the late part of room reverberation. A well-known fact is that a room impulse response sounds similar to exponentially decaying filtered noise some time after the beginning. The algorithm proposed here employs several velvet-noise sequences in parallel and combines them so that their non-zero samples never occur at the same time. Each velvet-noise sequence is driven by the same input signal but is filtered with its own feedback filter which has the same delay-line length as the velvet-noise sequence. The resulting response is sparse and consists of filtered noise that decays approximately exponentially with a given frequency-dependent reverberation time profile. We show via a formal listening test that four interleaved branches are sufficient to produce a smooth high-quality response. The outputs of the branches connected in different combinations produce decorrelated output signals for multichannel reproduction. The proposed method is compared with a state-of-the-art delay-based reverberation method and its advantages are pointed out. The computational load of the method is 60% smaller than that of a comparable existing method, the feedback delay network. The proposed method is well suited to the synthesis of diffuse late reverberation in audio and music production.
... When designing an FDN, a common practice is to first ensure that the energy of the system will not decay for any possible type of delay. Therefore, the matrix A should be unilossless [15]. To obtain a specific frequency-dependent RT, each of the delay lines must be cascaded with an attenuation filter, which approximates the target gain-per-sample expressed by ...
Reverberation is one of the most important effects used in audio production. Although nowadays numerous real-time implementations of artificial reverberation algorithms are available, many of them depend on a database of recorded or pre-synthesized room impulse responses, which are convolved with the input signal. Implementations that use an algorithmic approach are more flexible but do not let the users have full control over the produced sound, allowing only a few selected parameters to be altered. The real-time implementation of an artificial reverberation synthesizer presented in this study introduces an audio plugin based on a feedback delay network (FDN), which lets the user have full and detailed insight into the produced reverb. It allows for control of reverberation time in ten octave bands, simultaneously allowing adjusting the feedback matrix type and delay-line lengths. The proposed plugin explores various FDN setups, showing that the lowest useful order for high-quality sound is 16, and that in the case of a Householder matrix the implementation strongly affects the resulting reverberation. Experimenting with delay lengths and distribution demonstrates that choosing too wide or too narrow a length range is disadvantageous to the synthesized sound quality. The study also discusses CPU usage for different FDN orders and plugin states.
... Another decision concerning the design of the algorithm was the choice of the feedback matrix. In principle, the stability of an FDN is achieved, when the matrix is unilossless, i.e., it does not cause any loss of energy for any type of delay when no attenuation is introduced in the system [14]. To fulfill the above-mentioned requirement, in the present work a 16th-order Householder matrix is used. ...
Artificial reverberation algorithms aim at reproducing the frequency-dependent decay of sound in a room that is perceived as plausible for a particular space. In this study, we evaluate a feedback delay network reverberator with a modified cascaded graphic equalizer as an attenuation filter in terms of accurate reproduction of measured impulse responses of three rooms with different decay characteristics. First, the late reverb is synthesized by the proposed method and mixed with the early reflections separated from the original signal. The synthesized and measured signals are compared in terms of their decay characteristics and reverberation time values. The experiment shows that the proposed reverberator design reproduces real impulse responses well, although the decay-rate error exceeds the just noticeable difference of 5% in many cases. Additionally , perceptual qualities of the synthesized sounds were assessed through a listening test. Four qualities were tested for three room impulse responses and three kinds of stimuli. The results show that for the qualities reverberance, clarity, and distance, on average 75-79% of participants noticed only a slight or no difference between the measured and synthetic reverbs. Similar results were obtained for the speech and signing voice stimuli and the reverberation of lecture room and concert hall.
... An FFDN is lossless if A(z) is paraunitary, i.e., A(z −1 ) H A(z) = I, where I is the identity matrix and · H denotes the complex conjugate transpose [23]. Although also non-paraunitary FFMs may yield lossless FFDNs [15,24] deg A(z) = deg(det(A(z))). ...
Feedback delay networks (FDNs) are recursive filters, which are widely used for artificial reverberation and decorrelation. One central challenge in the design of FDNs is the generation of sufficient echo density in the impulse response without compromising the computational efficiency. In a previous contribution, we have demonstrated that the echo density of an FDN can be increased by introducing so-called delay feedback matrices where each matrix entry is a scalar gain and a delay. In this contribution, we generalize the feedback matrix to arbitrary lossless filter feedback matrices (FFMs). As a special case, we propose the velvet feedback matrix, which can create dense impulse responses at a minimal computational cost. Further, FFMs can be used to emulate the scattering effects of non-specular reflections. We demonstrate the effectiveness of FFMs in terms of echo density and modal distribution.
... Some important contributions in FDN research include [6,7] in which the authors propose a circulant feedback matrix for efficient implementation and maximum diffusion, and those by Schlecht [8,9] which deal with time-varying FDNs and their practical implementation. Schlecht also studied the properties of mixing matrices that produce lossless FDNs [10]. In a recent paper [11], he investigated the modal decomposition of feedback delay networks using the Ehrlich-Aberth iteration for finding poles and also studied the statistical distribution of mode frequencies and amplitudes. ...
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.
... They studied the use of absorptive filters in these systems to maximize the mode density of the response while controlling the decay rate of different frequencies and ensuring consistency within frequency bands. The central lossless property of FDNs has been extensively studied by Rocchesso and Smith [9] and by Schlecht and Habets [10]. ...
Artificial reverberation algorithms are used to enhance dry audio signals. Delay-based reverberators can produce a realistic effect at a reasonable computational cost. While the recent popularity of spatial audio algorithms is mainly related to the reproduction of the perceived direction of sound sources, there is also a need to spatialize the reverberant sound field. Usually, multichannel reverberation algorithms output a series of decorrelated signals yielding an isotropic energy decay. This means that the reverberation time is uniform in all directions. However, the acoustics of physical spaces can exhibit more complex direction-dependent characteristics. This paper proposes a new method to control the directional distribution of energy over time, within a delay-based reverberator, capable of producing a directional impulse response with anisotropic energy decay. We present a method using multichannel delay lines in conjunction with a direction-dependent transform in the spherical harmonic domain to control the direction-dependent decay of the late reverberation. The new reverberator extends the feedback delay network, retaining its time-frequency domain characteristics. The proposed directional feedback delay network reverberator can produce non-uniform direction-dependent decay time, suitable for anisotropic decay reproduction on a loudspeaker array or in binaural playback through the use of ambisonics.
... A lossless system then enables controlled reverberation time via the attenuation filters [20]. It is for the feedback matrix A A A sufficient to be unitary, or more generally, necessary and sufficient to be unilossless [21]. In this contribution, we limit the possible feedback matrices to the subgroup of orthogonal matrices, i.e., A A AA A A = I I I, where I I I is the identity matrix. ...
Feedback delay networks (FDNs) are an ef?cient tool for creating artificial reverberation. Recently, various designs for spatially extending the FDN were proposed. A central topic in the design of spatial FDNs is the choice of the feedback matrix that governs the interaction between spatially distributed elements and therefore the spatial impression. In the design prototype, the feedback matrix is chosen to be unilossless such that the reverberation time is infinite. However, in physics- and aesthetics-driven design of spatial FDNs, the target feedback matrix is not necessarily unilossless. This contribution proposes an optimization method for finding a close unilossless feedback matrix and improves the accuracy by relaxing the specification of the target matrix phase component and focussing on the sign-agnostic component.
... In this way, the application of both S W GW and S recombines the N (N − 1)(N − 1) channels present in the feedback loop back into the N (N − 1) inter-node wave variables required for reinsertion and further propagation through the system. Since the WGW makes use of essentially the same scattering operation as the SDN, it is similarly stable [10] regardless of the length of the delay lines connecting the nodes. As a result the addition of losses at the nodes will always result in a stable network. ...
Computer games and virtual reality require digital reverberation algorithms, which can simulate a broad range of acoustic spaces, including locations in the open air. Additionally, the detailed simulation of environmental sound is an area of significant interest due to the propagation of noise pollution over distances and its related impact on well-being, particularly in urban spaces. This paper introduces the waveguide web digital reverberator design for modeling the acoustics of sparsely reflecting outdoor environments; a design that is, in part, an extension of the scattering delay network reverberator. The design of the algorithm is based on a set of digital waveguides connected by scattering junctions at nodes that represent the reflection points of the environment under study. The structure of the proposed reverberator allows for accurate reproduction of reflections between discrete reflection points. Approximation errors are caused when the assumption of point-like nodes does not hold true. Three example cases are presented comparing waveguide web simulated impulse responses for a traditional shoebox room, a forest scenario and an urban courtyard, with impulse responses created using other simulation methods or from real world measurements. The waveguide web algorithm can better enable the acoustic simulation of outdoor spaces and so contribute towards sound design for virtual reality applications, gaming and auralisation, with a particular focus on acoustic design for the urban environment.
A common bane of artificial reverberation algorithms is spectral coloration in the synthesized sound, typically manifesting as metallic ringing, leading to a degradation in the perceived sound quality. In delay network methods, coloration is more pronounced when fewer delay lines are used. This paper presents an optimization framework in which a tiny differentiable feedback delay network, with as few as four delay lines, is used to learn a set of parameters to iteratively reduce coloration. The parameters under optimization include the feedback matrix, as well as the input and output gains. The optimization objective is twofold: to maximize spectral flatness through a spectral loss while maintaining temporal density by penalizing sparseness in the parameter values. A favorable narrow distribution of modal excitation is achieved while maintaining the desired impulse response density. In a subjective assessment, the new method proves effective in reducing perceptual coloration of late reverberation. Compared to the author’s previous work, which serves as the baseline and utilizes a sparsity loss in the time domain, the proposed method achieves computational savings while maintaining performance. The effectiveness of this work is demonstrated through two application scenarios where smooth-sounding synthetic room impulse responses are obtained via the introduction of attenuation filters and an optimizable scattering feedback matrix.
Room acoustic synthesis can be used in virtual reality (VR), augmented reality (AR) and gaming applications to enhance listeners' sense of immersion, realism and externalisation. A common approach is to use geometrical acoustics (GA) models to compute impulse responses at interactive speed, and fast convolution methods to apply said responses in real time. Alternatively, delay-network-based models are capable of modeling certain aspects of room acoustics, but with a significantly lower computational cost. In order to bridge the gap between these classes of models, recent work introduced delay network designs that approximate Acoustic Radiance Transfer (ART), a GA model that simulates the transfer of acoustic energy between discrete surface patches in an environment. This paper presents two key extensions of such designs. The first extension involves a new physically-based and stability-preserving design of the feedback matrices, enabling more accurate control of scattering and, more in general, of late reverberation properties. The second extension allows an arbitrary number of early reflections to be modeled with high accuracy, meaning the network can be scaled at will between computational cost and early reverberation precision. The proposed extensions are compared to the baseline ART approximating delay network as well as two reference GA models. The evaluation is based on objective measures of perceptually relevant features, including frequency-dependent reverberation times, echo density build-up, and early decay time. Results show how the proposed extensions result in a significant improvement over the baseline model, especially for the case of non-convex geometries or the case of unevenly distributed wall absorption, both scenarios of broad practical interest
In digital audio signal processing (DASP), there is a huge amount of literature about filtering systems designed for specific applications [1-88].
Artificial reverberation is an audio effect used to simulate the acoustics of a space while controlling its aesthetics, particularly on sounds recorded in a dry studio environment. Delay-based methods are a family of artificial reverberators using recirculating delay lines to create this effect. The feedback delay network is a popular delay-based reverberator providing a comprehensive framework for parametric reverberation by formalizing the recirculation of a set of interconnected delay lines. However, one known limitation of this algorithm is the initial slow build-up of echoes, which can sound unrealistic, and overcoming this problem often requires adding more delay lines to the network. In this paper, we study the effect of adding velvet-noise filters, which have random sparse coefficients, at the input and output branches of the reverberator. The goal is to increase the echo density while minimizing the spectral coloration. We compare different variations of velvet-noise filtering and show their benefits. We demonstrate that with velvet noise, the echo density of a conventional feedback delay network can be exceeded using half the number of delay lines and saving over 50% of computing operations in a practical configuration using low-order attenuation filters.
A computationally effcient digital filter structure for late reverberation modeling based on measured room acoustical data is presented. In our reverberator a dense response is obtained by inserting comb-allpass filters in comb filter loops which are connected in parallel. The sum of the comb filter outputs is fed back to their inputs. The advantage of the proposed structure is that a higher reflection density is obtained by a smaller computational burden than with former reverberators. To simulate the acoustics of an existing hall, the early reflections and the frequency dependent reverberation time are analyzed from measured room impulse responses. The data is used for deriving the parameters of the reverberator.
Recent research related to artificial reverberation is reviewed. The focus is on research published during the past few years after the writing of an overview article on the same topic by the same authors. Advances in delay networks, convolution-based techniques, physical room models, and virtual analog reverberation models are described. Many new developments are related to the feedback delay network, which is still the most popular parametric reverberation method. Additionally, three specific methods are discussed in detail: velvet-noise reverberation methods, scattering delay networks, and a modal architecture for artificial reverberation. It becomes evident that research on artificial reverberation and related topics continues to be as active as ever. The related conference paper is available from the AES E-Library at http://www.aes.org/e-lib/browse.cfm?elib=18061.
Increasing the size of a feedback delay network reverbera-tor produces more uniform frequency response and higher density of echoes, thus improving the perceptual quality of the output. Feedback delay networks mix delay line outputs back to the inputs using a matrix-vector multiplication that requires O(n 2) scalar multiplications in the general case or O(n log n) for some special matrices, where n is the number of delays in the network. Using the sparse matrix we present here, the mixing operaion can be done with O(n) multiplications , which permits us to use larger size networks without increasing computational cost. We show that this produces an impulse response with a flatter average FFT spectrum and discuss its effect on echo density.
An acoustic reverberator consisting of a network of delay lines connected via
scattering junctions is proposed. All parameters of the reverberator are
derived from physical properties of the enclosure it simulates. It allows for
simulation of unequal and frequency-dependent wall absorption, as well as
directional sources and microphones. The reverberator renders the first-order
reflections exactly, while making progressively coarser approximations of
higher-order reflections. The rate of energy decay is close to that obtained
with the image method (IM) and consistent with the predictions of Sabine and
Eyring equations. The time evolution of the normalized echo density, which was
previously shown to be correlated with the perceived texture of reverberation,
is also close to that of IM. However, its computational complexity is one to
two orders of magnitude lower, comparable to the computational complexity of a
feedback delay network (FDN), and its memory requirements are negligible.
This paper presents different methods for designing unitary mixing matrices for Jot reverberators with a particular emphasis on cases where no early reflections are to be modeled. Possible applications include diffuse sound reverberators and decorrelators. The trade-off between effective mixing between channels and the number of multiply operations per channel and output sample is investigated as well as the relationship between the sparseness of powers of the mixing matrix and the sparseness of the impulse response.
Digital waveguides and finite difference time domain schemes have been used in physical modeling of spatially distributed systems. Both of them are known to provide exact modeling of ideal one-dimensional (1D) band-limited wave propagation, and both of them can be composed to approximate two-dimensional (2D) and three-dimensional (3D) mesh structures. Their equal capabilities in physical modeling have been shown for special cases and have been assumed to cover generalized cases as well. The ability to form mixed models by joining substructures of both classes through converter elements has been proposed recently. In this paper, we formulate a general digital signal processing (DSP)-oriented framework where the functional equivalence of these two approaches is systematically elaborated and the conditions of building mixed models are studied. An example of mixed modeling of a 2D waveguide is presented.
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
Feedback delay networks (FDNs) are frequently used to generate artificial reverberation. This contribution discusses the temporal features of impulse responses produced by FDNs, i.e., the number of echoes per time unit and its evolution over time. This so-called echo density is related to known measures of mixing time and their psychoacoustic correlates such as auditive perception of the room size. It is shown that the echo density of FDNs follows a polynomial function, whereby the polynomial coefficients can be derived from the lengths of the delays for which an explicit method is given. The mixing time of impulse responses can be predicted from the echo density, and conversely, a desired mixing time can be achieved by a derived mean delay length. A Monte Carlo simulation confirms the accuracy of the derived relation of mixing time and delay lengths.
Digital waveguides and finite difference time domain schemes have been used in physical modeling of spatially distributed systems. Both of them are known to provide exact modeling of ideal one-dimensional (1D) band-limited wave propagation, and both of them can be composed to approximate two-dimensional (2D) and three-dimensional (3D) mesh structures. Their equal capabilities in physical modeling have been shown for special cases and have been assumed to cover generalized cases as well. The ability to form mixed models by joining substructures of both classes through converter elements has been proposed recently. In this paper, we formulate a general digital signal processing (DSP)-oriented framework where the functional equivalence of these two approaches is systematically elaborated and the conditions of building mixed models are studied. An example of mixed modeling of a 2D waveguide is presented.
The following paper analyzes the use of the Genetic Algorithm (GA) in conjunction with a length-4 feedback delay network for audio reverberation applications. While it is possible to manually assign coefficient values to the feedback network, our goal was to automate the generation of these coefficients to help produce a reverb with characteristics as similar to those of a real room reverberation as possible. We designed a GA to be used in a delay-based reverb that would be more desirable in the use of real-time applications than the more computationally expensive convolution reverb.
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 room acoustic modeling, Feedback Delay Networks (FDN) are known to efficiently model late reverberation due to their capacity to generate exponentially decaying dense impulses. However, this method relies on a careful tuning of the different synthesis parameters , either estimated from a pre-recorded impulse response from the real acoustic scene, or set manually from experience. In this paper we present a new method, which still inherits the efficiency of the FDN structure, but aims at linking the parameters of the FDN directly to the geometry setting. This relation is achieved by studying the sound energy exchange between each delay line using the acoustic Radiance Transfer Method (RTM). Experimental results show that the late reverberation modeled by this method is in good agreement with the virtual geometry setting.
Comb filters composed in a parallel or a serial way are a popular part of delay-line based artificial reverberators. Because the analysis of a complex comb filter structure can be tedious, there is a need for transforming such a structure into a compact and general representation. For this a transformation into the feedback delay network (FDN) filter structure is proposed as it is a general and well established framework to investigate the acoustic properties of the filter and therefore allows to compare different approaches. Index Terms ?? comb filter, feedback delay network, artificial reverberation
Let A, B be n × n matrices with entries in a field F. We say A and B satisfy property if B or Bt is diagonally similar to A. It is clear that if A and B satisfy property , then they have equal corresponding principal minors, of all orders. The question is to what extent the converse is true. There are examples which show the converse is not always true. We modify the problem slightly and give conditions on a matrix A which guarantee that if B is any matrix which has the same principal minors as A, then A and B will satisfy property . These conditions on A are formulated in terms of ranks of certain submatrices of A and the concept of irreducibility.
A simple algorithm is presented for testing the diagonal similarity of two square matrices with entries in a field. Extended forms of the algorithm decide various related problems such as the simultaneous di_gonal similarity of two families of matrices, the existence of a matrix in a subfield diagonally similar to a given matrix, the existence of a unitary matrix similar to a given complex matrix, and the corresponding problems for diagonal equivalence in place of diagonal similarity . The computational complexity of our principal algorithm is studied, programs and examples are given. The algorithms are based on the existence of a canonical form for diagonal similarity. In tpe first part of the paper theorems are proved which establish the existence of this form and which investigate its properties.
If R is an n × n matrix over the complex field which is the product of a diagonal matrix D and a permutation matrix P, then R is called a diagonally scaled permutation matrix. We present the eigenstructure of R by observing that R is permutation similar to the direct sum of diagonally scaled permutation matrices of the form DC where D is a diagonal matrix and C is the circulant permutation. The matrix DC is called a scaled circulant permutation matrix. We consider two cases for R = DC: when the scaling matrix D is nonsingular, and when D is singular. In the singular case R is nilpotent, and we are able to obtain upper and lower bounds on the index of nilpotency of R. We conclude with information about matrices that commute with a scaled permutation matrix. We are also able to represent an arbitrary n × n Toeplitz matrix as a sum of matrices of the form D(k,α,β)Ck for k = 1, …, n where D(k,α,β) is a diagonal matrix.
Let A, B be n × n matrices with entries in a field F. Our purpose is to show the following theorem: Suppose n⩾4, A is irreducible, and for every partition of {1,2,…,n} into subsets α, β with ¦α¦⩾2, ¦β¦⩾2 either rank A[α¦β]⩾2 or rank A[β¦α]⩾2. If A and B have equal corresponding principal minors, of all orders, then B or Bt is diagonally similar to A.
The paper is a tutorial intended to serve as a reference in the field of digital audio effects in the electronic music industry for those who are new to this specialization of digital signal processing. The effects presented are those that are demanded most often, hence they will serve as a good toolbox. The algorithms chosen are of such a fundamental nature that they will find application ubiquitously and often.
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.
This paper discusses efficient digital signal processing algorithms for real-time synthesis of dynamically controllable, natural-sounding artificial reverberation. A general modular framework is proposed for configuring a spatial sound processing and mixing system according to the reproduction format or setup and the listening conditions, over loudspeakers or headphones. In conclusion, the implementation and applications of a spatial sound processing software are described, and approaches to control interface design and effective distance effects are reviewed.
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.
We give a common, concise derivation of some important determinantal identities attributed to the mathematicians in the title. We also give a formal treatment of determinantal identities of the minors of a matrix.
A unitary n-input n-output linear network preserves the total energy of all input signals. Using the functional calculus of normal matrices, it is proved that feedback round a unitary circuit plus a direct path with suitable gain yields another unitary circuit. This has applications to the design of electronic reverberation units.
The digital waveguide mesh has been an active area of music acoustics research for over ten years. Although founded in 1-D digital waveguide modeling, the principles on which it is based are not new to researchers grounded in numerical simulation, FDTD methods, electromagnetic simulation, etc. This article has attempted to provide a considerable review of how the DWM has been applied to acoustic modeling and sound synthesis problems, including new 2-D object synthesis and an overview of recent research activities in articulatory vocal tract modeling, RIR synthesis, and reverberation simulation. The extensive, although not by any means exhaustive, list of references indicates that though the DWM may have parallels in other disciplines, it still offers something new in the field of acoustic simulation and sound synthesis
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