Scattering in Feedback Delay Networks

12/18/2019 ∙ by Sebastian J. Schlecht, et al. ∙ aalto 0

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.

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I Introduction

If a sound is emitted in a room, the sound waves travel through space and are repeatedly reflected at the room boundaries resulting in acoustic reverberation [1]. If the sound is reflected at a smooth boundary, the reflection is coherent (specular), while it is incoherent (scattered) when reflected by a rough surface. In small to large rooms, the first-order specular reflections arrive between 10 - 100 ms, whereas the time scale of incoherent reflections is a few milliseconds [2]. In geometric room acoustics, an incoherent reflection can be effectively generated from a set of closely spaced image sources [2], see Fig. 1. Consequently, each of the primary image sources (specular) is spread in time, and therefore the resulting echo density increases while the number of image sources still follows a polynomial of degree three [1]. In this work, we propose a method to introduce scattering-like effects into artificial reverberation filter structures effectively.

Fig. 1: Second-order reflections at two rough surfaces are represented with an equivalent set of image sources (inspired by [2]). The color saturation indicates the energy of the image sources.

Many artificial reverberators have been developed in recent years [3], among which the feedback delay network (FDN), initially proposed by Gerzon [4] and further developed in [5, 6], is one of the most popular. The FDN consists of delay lines combined with attenuation filters, which are fed back via a scalar feedback matrix . A significant challenge of FDN design is to achieve sufficient echo density because of an inherent trade-off between three aspects: computational complexity, mode density, and echo density. A higher number of delays increases both modal and echo density, but also the computational complexity. To reproduce a scattering-like effect, we require a set of long delays that are proportional to the mean free path [7, 8], and a set of filters to add the short-term density to each reflection (see Fig. 1). Feedforward-feedback allpass filters have been introduced with the delay lines to increase the short-term echo density [9, 10]. Alternatively, allpass filters may be placed after the delay lines [11, 12], which in turn doubles the effective size of the FDN [13]. Alternatively, scattering filters can be introduced in series to the FDN. The post-filtering is, however, not optimal because of the perceivable repetitions and an elaborate time-varying switching was proposed to overcome this problem [14].

Towards a possible solution, we introduced in [15] the delay feedback matrix (DFM), where each matrix entry is a scalar gain and a delay. In this contribution, we generalize the feedback matrix of the FDN to a filter feedback matrix (FFM), which then results in a filter feedback delay network (FFDN). As a special case of the FFM, we present the velvet feedback matrix (VFM), which can create ultra-dense impulse responses at a minimal computational cost. The VFM is inspired by the work of Karjalainen, Välimäki et al. [16, 17, 14, 3] in which velvet noise sequences, i.e., sparse time-domain responses with few

pulses, have been proposed to create perceptually white noise. Due to the general formulation, the proposed FFM is directly applicable for related techniques such as digital waveguides

[18], waveguide web [19], scattering delay networks [20] and directional FDNs [21].

The remainder of this work is organized as follows. In Section II, we review the general FDN structure, conditions for losslessness as well as prior techniques for introducing scattering-like effects. In Section III, we introduce a complete characterization of the proposed filter feedback matrices alongside various special cases, including the VFM. In Section IV, we study the effect the FFM has on the modal distribution. In Section V, we demonstrate the echo density of the proposed FFDN.

Fig. 2: Proposed filter feedback delay network (FFDN) with three delays, i.e., , and a filter feedback matrix (FFM) instead of the standard scalar feedback matrix .

Ii Filter Feedback Delay Network

In the following, we present the proposed FFM extension resulting in the filter feedback delay network (FFDN) and the corresponding conditions for losslessness and homogeneous loss.

Ii-a FFDN Structure

The proposed FFDN is similarly structured as the standard FDN, but employs a filter feedback matrix (FFM) instead of a scalar feedback matrix (see Fig. 2). The transfer function of the FFDN is

(1)

where the column vectors

, , and, scalar denote the input, output and direct gains, respectively. The lengths of the delays in samples are given by . is the diagonal delay matrix.

The FFM may consist of finite and infinite impulse response (FIR and IIR) filters. However, in this work, we largely focus on FIR FFMs. An FIR FFM may be expressed in terms of the scalar coefficient matrices , i.e.,

(2)

where is the maximum filter order of if . Every matrix element is an FIR filter with filter order at most . The degree of the FIR FFM, i.e., the effective number of required delay elements, is given by the polynomial degree of the matrix determinant [22, p.737]

(3)

An efficient FFT-based method for determining the polynomial matrix determinant is given in [23]. Thus, the minimal number of delay elements to realize the proposed FFDN in (1) is

(4)

To achieve a scattering-like effect as discussed in the introduction, we choose the FFM order relatively small compared to the main delays . In Sections IV and V, more details on on practical designs are provided.

Fig. 3: General structure of a FIR filter feedback matrix (FFM) as defined in (13) with and stages.

Ii-B Losslessness

FDNs are commonly designed as lossless systems, i.e., all system poles lie on the unit circle. The poles of the proposed FFDN are the eigenvalues of the polynomial matrix

, which in turn are the roots of the generalized characteristic polynomial (GCP) [24], i.e.,

(5)

The extra factor is necessary to turn into a proper polynomial. The lossless property of general unitary-networks, which in particular applies to the proposed FFDN, was described by Gerzon [25]. An FFDN is lossless if is paraunitary, i.e., , where

is the identity matrix and

denotes the complex conjugate transpose [25]. Although also non-paraunitary FFMs may yield lossless FFDNs [26, 15], here we focus on paraunitary FFMs only. Paraunitary matrices are particularly useful as they are closed under multiplication, i.e., if and are paraunitary, then is paraunitary as well [27].

Ii-C Lossy Feedback

Homogeneous loss is introduced into a lossless FFDN by replacing each delay element with a lossy delay filter , where is ideally zero-phase with a positive frequency-response. The frequency-dependent gain-per-sample relates to the resulting reverberation time by

(6)

where is the sampling frequency and is the angular frequency [5]. However, as substitution with lossy delays is impractical, the attenuation filters are lumped into a single filter per delay line. Thus, the lossy FFM is

(7)

where is paraunitary and a diagonal attenuation matrix. The attenuation matrix is chosen such that the frequency-dependent eigenvalue magnitudes are similar for

(8)

In standard FDNs with a scalar feedback matrix , (8) yields the well-known delay-proportional attenuation [5]

(9)

In Section IV, we give a general solution for FFMs. In the following section, we present various FFM designs.

Iii Filter Feedback Matrix

The filter feedback matrix (FFM) can be either realized by IIR or FIR filters. In the following, we discuss IIR FFMs and give a complete characterization of FIR FFMs. Then, we present particularly useful designs of FIR FFMs. We conclude this section by discussing the implementation and computational complexity.

Iii-a IIR FFMs

The IIR FFM, i.e., each matrix element is an IIR filter, is the most general FFM. IIR filters can produce long tails at little computational cost and therefore appear advantageous for increasing the density in artificial reverberation applications. In fact, allpass FDNs [11, 28, 12] are a special case of IIR FFMs with

(10)

where is a diagonal matrix of IIR allpass filters and

is a scalar unitary matrix.

The main drawback of IIR FFMs is that the additional IIR filter poles introduce an unrelated decay behavior, which can disturb the overall reverberation time [12]. For this reason, we do not consider further designs in this class and turn our attention to FIR FFMs, which can be parametrized and implemented more efficiently.

(a) Elemental block feedback matrix (EBFM) with in (12).
(b) Delay feedback matrix (DFM) with in (14).
(c) Random dense feedback matrix (RDFM) with in (16).
(d) Velvet feedback matrix (VFM) with and in (18).
Fig. 4: Paraunitary filter feedback matrices with . The subplots depict the filter coefficients of the matrix entries with . The pre- and post-delays and are zero except for the delay feedback matrix in Fig. 3(b).

Iii-B Characterizing FIR FFMs

Vaidyanathan gave a full characterization of paraunitary FIR FFMs in [22], which we reproduce in the following for completeness. The elemental building block is

(11)

where is a vector with unit norm, i.e., . The degree of the elemental building block is one, i.e., . Due to the closure under multiplication, the matrix product

(12)

with corresponding vectors yields a paraunitary matrix. It can be shown [22] that any FIR paraunitary matrix of degree can be factorized into the form (12).

A less rigorous, but more practical factorization of paraunitary FIR matrices is given by

(13)

where are scalar unitary matrices and are vectors of integer delays. It can be shown that for any , there are , , and, with , see [22, p. 733]. Thus, any paraunitary FIR FFM can be factored into form (13). In this formulation, the FFM mainly introduces delay and mixing stages within the main FDN loop (see Figure 3). A few examples of FIR FFMs are depicted in Fig. 4, and we explain their construction in the following.

Iii-C Elemental Block FIR FFMs

The characterization in (11) and (12) can be directly employed to create a paraunitary elemental block feedback matrix (EBFM) by choosing a set of vectors . Unfortunately, merely randomizing the vectors does not yield satisfactory results as the energy of the FIR tends to concentrate in time. Figure 3(a) shows an example matrix with , where all entries of the vectors

where picked from a Gaussian distribution and subsequently normalized. The concentration of energy is in direct opposition to the purpose of the FFM, namely to spread the energy of the reflection in time. We have tested various other strategies for choosing vectors

to no avail. In the following, we present alternative characterizations based on (13), which results in more intuitive designs.

Iii-D Delay Feedback Matrix

The delay feedback matrix (DFM) is a paraunitary FIR matrix, which was introduced by the present authors in [15]. It can be conveniently expressed in terms of (13) with a single stage, i.e, :

(14)

The implementation of the DFM matrix is similar to the scalar unitary matrix besides that the signal vector is read and written at dedicated points in the main delays, while all other arithmetic operations are unaltered. An example of the DFM is shown in Fig. 3(b).

Iii-E Dense FFMs and Paraunitary Hadamard

Through an iterative process, we compose dense paraunitary FFMs, i.e., no matrix entry of any coefficient matrix in (2) is zero. For this, we introduce the notion of a (strictly) column-distinct FFM, i.e., for any column index , there is at most one coefficient matrix with the column being non-zero.

We start the iteration with a dense unitary scalar matrix , i.e., all matrix entries are non-zero. The corresponding filter order . With delays , is column-distinct. For a dense unitary matrix , the FFM

(15)

is dense and the filter order is . The iteration is repeated analogously, however, with increased delays . Thus, is column-distinct and for a dense unitary matrix

(16)

is dense and the filter order is . One iteration step of (16) is depicted in Fig. 5. Fig. 3(c) gives an example of a random dense feedback matrix (RDFM) with random unitary matrices and iterations. Each matrix element is therefore a dense FIR filter with order . Please note that the presented formulation is not unique and many permutations and variations may exist.

(a)
(b)
(c)
Fig. 5: Random dense feedback matrices (RDFM) construction iteration from to for . The FFM order of is and delay is .

With a similar iterative procedure, we can construct a paraunitary Hadamard matrix , i.e., all entries are of equal magnitude111The classic Hadamard matrix has all entries either or and is orthogonal with . For notational convenience, we use nonetheless the orthonormal definition and scale such that . Hadamard matrices do not exist for all , but for and for most integer .. To this end, the unitary matrices in (16) are replaced with scalar Hadamard matrices , i.e., orthogonal matrices with all entries equal to . Thus, and

(17)

is column-distinct and

(18)

is dense. Now, we show that is also Hadamard. Because, (17) is column-distinct, any coefficient matrix has at most one non-zero column. In fact, due to the minimal delays, each such non-zero column has only entries with . Thus, the multiplication of such a coefficient matrix with the Hadamard matrix results in a dense matrix that has only entries .

Iii-F Sparse FIR FFM and Paraunitary Velvet

From perceptual studies of velvet noise sequences [29], it is known that reverberation tails do not necessarily require to be fully dense. In general, a few pulses per millisecond are considered to be sufficient [30, 31]. Inspired by velvet noise sequences, we propose FIR FFM for a given density of , i.e., the average number of pulses per sample (counted for each filter individually). In this sense, the dense FIR FFM correspond to a density of .

The construction in the preceding section can be readily adapted to incorporate density. Instead of using the smallest possible shift, we choose distinct values and set . Instead of choosing fully random values for , some regularity can be enforced to generate evenly distributed pulses [16]. To preserve the initial density, the subsequent delays are unaltered, i.e., with little variations to promote irregularity. Similar to (18), the sparse FFM can be based on Hadamard matrices such that all matrix entries are either of the same magnitude or zero. For this reason, we like to call this special sparse FFM, a velvet feedback matrix (VFM). In Fig. 3(d), a VFM for and is depicted. Each matrix entry is a velvet sequence with pulses.

Iii-G Implementation and Computational Complexity

In this section, we discuss the practical implementation of FFMs and the associated computational costs. For IIR FFM, the authors are not aware of a general way to accelerate implementation other than evaluating each IIR filter entry separately. Because of this reason, IIR FFM is considered rather computationally expensive for higher-order dense IIR FFMs. However, for FIR FFMs, we propose two approaches to implement the filtering efficiently: Firstly, by fast convolution and, secondly, cascaded multiplication. Further, any efficient unitary matrix multiplication like the Hadamard transform, Householder transform, and Circulant matrices are applicable similarly in the FFM context.

Iii-G1 Fast Convolution

As consists only of FIR filters, the path from every input to every output can be implemented by fast convolution. However, it is possible to reduce the necessary FFT transforms to

by processing the matrix mixing in the frequency domain. Every input signal is transformed into the frequency domain by an FFT, the complex vector is processed by a scalar complex matrix multiplication and then again back-transformed to the time domain by an inverse FFT. Compared to the scalar matrix

, the computational costs only increase by the forward and backward FFT, which depends on the lengths of the FIRs in . The computational complexity of the fast convolution for an FIR FFM with length and size is for the FFT and for the complex matrix multiplication.

Scalar
Addition & Subtraction 4 6 8 16 24 32 48 72 96 256
Multiplication 2 2 2 4 4 4 8 8 8 256
Delay Read & Write 12 16 20 24 32 40 48 64 80 0
Pulses per Filter 4 16 64 16 256 4096 64 4096 262144 1
TABLE I: Number of operations and filter order for paraunitary Hadamard matrix of size with stages. The number of operations is identical for the velvet feedback matrix (VFM). For comparison, a scalar matrix with is given.

Iii-G2 Cascade Multiplication

Alternatively, the FIR FFM can directly implement the cascaded form in (13) with alternating processing of delays and mixing . The diagonal delay matrix , implemented with ring buffers, require delay read and write operations plus circular pointer shifts. The computational complexity for stages in Fig. 3, is for the delay matrices and for the matrix multiplications. We want to mention the computational cost of the VFM specifically. The Hadamard transform, i.e., the multiplication with the fast Walsh-Hadamard transform, can be realized by addition and subtractions. In the VFM, the scaling factors of all Hadamard matrices can be summarized at the end of the VFM and realized by a total of multiplication. See Table I for examples of operations count both for the paraunitary Hadamard matrices and VFMs. For comparison, Table I also lists the operations count of a scalar matrix multiplication with . Besides that, the FFM of smaller matrix size has fewer operations in general, importantly also other operations such as attenuation filters get reduced with the smaller matrix sizes.

Iv Modal Distribution

In this section, we present the effect of FFMs on the modal distribution of the FFDN. First, we propose a method for lumped attenuation filters, which we subsequently evaluated by computing the modal decay distribution. In [24], we have developed a large-scale modal decomposition technique for scalar feedback matrices . In the following, we outline the general method and give an extension to FFMs .

Iv-a Attenuation Filters

In Section II-C, we introduced the lossy FFDN with the global gain-per-sample related to reverberation time by (6). For a frequency-independent gain , the criterium in (8) can be satisfied strictly

(19)

where

Thus, by pre-multiplying in the FFM stages, no additional filtering operations need to be performed. However for frequency-dependent gain , this approach adds a large number of extra filters. Instead, we propose a solution to (8) with a single lumped filter matrix . We extend the solution for the standard FDN (9) to

(20)

where the left and right group delay, and respectively, approximate the FFM group delay, i.e.,

(21)

where denotes the outer sum, i.e., and

(22)

is the matrix of group delays. For example, with a delay feedback matrix (DFM) in (14), the minimization in (21) would yield and with no approximation error. A general solution is outlined in Appendix A.

Any approximation error in (21) causes the modal decay to deviate from the specified reverberation time. In the following, we quantify the decay deviation by performing a modal decomposition.

Iv-B Modal Decomposition

The modal decomposition of FFDN computes the partial fraction decomposition of the transfer function in (14), i.e.,

(23)

where is the residue of the pole . The modal decomposition in (23) can be computed even for high orders with the polynomial Ehrlich-Aberth Method. The method starts with a set of initial poles

, which are typically placed along the unit circle. The Ehrlich-Aberth Iteration (EAI) provides the sequence of estimates

(24)

with the EAI step being

(25)

The Newton correction term is

(26)

and the deflation term is

(27)

and indicates the reversed GCP. 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 and the right side otherwise [32].

For the FFM, we give the explicit form of both Newton correction terms. The corresponding derivations can be found in Appendix B. The inverse Newton correction for a FFM is

(28)

The reversed GCP is

(29)

where

(30)

Similar to (26), we have

(31)

The matrix inverse is challenging to compute in general. However, given the cascaded representation in (13) allows to do a matrix inversion of each cascaded component by itself, i.e.,

(32)

In case of a paraunitary FFM, computing the inverse is trivial as .

Fig. 6: Modal decay distribution for VFMs with density and . Therefore, filter length is for . The VFM is similar to Fig. 3(d) with four main delays and a total system order of with reverberation time target of 5 seconds.

Iv-C Modal Decay Distribution

In general, the larger the group delay in the FFM , the larger is the possible error in (21) and therefore the deviation in modal decay. In the following, we give an example case to study the modal decay distribution. Consider a FFDN with and VFMs with density (see Section III-F). Therefore, the filter order of the VFMs is for . The main delays are such that the total system order (4) is . The target frequency-independent reverberation time is 5 seconds. Although, this can be solved accurately with (19), we attempt to solve it with the lumped formulation (20) to assess the frequency-dependent case.

Fig. 6

shows the probability distribution of modal decay for various sparse VFMs. Expectedly, the modal decay distribution becomes wider for longer filter length

. The mean absolute group delay error between and are about samples for , respectively. The maximum reverberation time error is below for , which is acceptable compared to the JND of for reverberation time [33, 34].

Fig. 7: Evolution of a reflection path compared VFM (blue) and a standard FDN (red). The dashed line indicates the boundaries between each reflection. The first pulse amplitudes for the standard FDN are and , but for better readability, the value range is limited to .
(a) Elemental block feedback matrix (EBFM) with in (12).
(b) Delay feedback matrix (DFM) with in (14).
(c) Random dense feedback matrix (RDFM) with in (16).
(d) Velvet feedback matrix (VFM) with and in (18).
(e) Scalar feedback matrix with (SFM-4).
(f) Scalar feedback matrix with (SFM-16).
Fig. 8: The impulse response of FFDNs with FFMs in Fig. 4. The red line indicates the echo density profile, while the mixing time predicted using [35] is shown in yellow.

V Temporal Density

In this section, we investigate the temporal density of the impulse response generated by the proposed FFDN.

V-a Echoes and Echo Paths

We first consider echo paths, which are paths through the FFDN. More formally, the echo path of length describes a sequence of delay lines, which is traversed by an impulse input. The resulting echo is the system response if considering only the corresponding echo path

(33)

In this sense, introducing FFMs into the FFDN does, in general, not change the number and overall temporal distribution of the echoes. However, most importantly, it changes the temporal spread of the echoes themselves. Fig. 7 shows the echoes through a VFM (depicted in Fig. 3(d)) with and . While the first pass-through the VFM results in a sparse echo response, the more often an echo traverses the VFM, it also becomes denser. The temporal extent of the spread is roughly the filter order of the VFM multiplied by the number of matrix pass-throughs. The cumulative spreading is also in accord with physical rooms where each reflection at a boundary introduces scattering (see Fig. 1). This is in stark contrast to the standard FDN with a scalar feedback matrix, where any echo response is an impulse, and no spreading in time occurs. This illustrates a key feature of the proposed method: while in standard FDN high echo density requires more delay lines [8], the FFDN can balance the number of echoes and their temporal extent. In particular, the echo density can be controlled by the number of mix and delay stages and the delays in the FFM.

V-B Echo Density Measure and Mixing Time

We employ a recently proposed echo density measure based on a smooth sorted density measure [35]. The echo density is computed using a two-pass procedure. The input impulse response is converted to an echogram. A local energy normalization factors out the energy decay envelope. The normalized echogram is analyzed with a rectangular sliding window centered at each sample. Then the sorted density is computed as a fraction of the window width. Processing for each sample and normalizing with the expected value for Gaussian noise yields echo density. This is fitted with a general power-law model motivated by theoretical considerations. Overall, it has been shown to provide a smoother and more reliable echo density curve compared to the model of Abel and Huang proposed in [8, 36, 37, 38].

In Fig. 8, the impulse responses of six FFDNs are plotted alongside their echo density profile. Four of the FFDNs are the same as in Fig. 4, and two additional standard FDNs with and for comparison. The four matrix types EBFM, RDFM, VFM, SFM-16 were parametrized such that the impulse responses mix at similar times, DFM and SFM do not mix within the first 0.5 seconds. In the following, the quantify the mixing time of FFDN via a Monte Carlo simulation.

We generate random variations of the six FFDN types as presented in Fig. 8: EBFM, DFM, RDFM, VFM, SFM-4, and SFM-16. The main delays are generated between and samples, and the FFM were generated with random seeds as described in Section III. We use the SFM-4 as the baseline, as it has the highest mixing time typically. Thus, the relative mixing time is the ratio between the evaluated mixing time and the baseline mixing time of SFM-4. Fig. 9 shows the probability distribution of the relative mixing time. The DFM reduces the mixing time by a factor of 2, while the EBDM reduces it by a factor of 5. The VFM and RDFM reduce the mixing time by a factor of 10 similar to the SFM-16. However, the computational cost of the VFM is only about 10% of the FDN-16 (see Table I).

Fig. 9: Probability distribution of the relative mixing time for FFM of type , i.e., .

Vi Conclusion

We proposed a novel artificial reverberation method, the filtered feedback delay network (FFDN), by introducing a scattering-like recursive filter to improve the echo density. Among the various designs for the filter feedback matrix (FFM), the velvet feedback matrix (VFM) is the most prominent innovation for its superior computational cost and induced echo density. A generalized design for attenuation filters in FFDNs was proposed. The effect of FFMs on the modal distribution was investigated by an extended modal decomposition method. The attenuation filter design keeps the modal decay deviation close to the JND of 5%. Similarly, the echo density was investigated for various FFM designs. We showed that highly efficient FFM designs can reduce the mixing time by a factor of 10. Thus, the FFDN solves the trade-off of standard FDN design between computational complexity, mode density, and echo density by introducing a computationally efficient scattering technique. The FFDNs render for the first time a low delay count viable for high-quality artificial reverberation. A possible future work is to connect the FFM design to a specified scattering effect, either derived from a perceptual or physical model.

Appendix A Group Delay Approximation

One method to solve the minimization problem in (21) is to convert it to a rank-1 approximation problem by applying an exponential function on both sides, i.e.,

(34)

where , and and the exponential function is applied element-wise. The rank-1 approximation in (34

) is then solved by taking the dominant vectors from a singular value decomposition of

. Applying a logarithm to and yields then a solution to the original problem. This is merely a practical solution can be possible further improved depending on the desired error norm in (21).

Appendix B Modal Decomposition for FFDN

Given a FFM and delays , the general characteristic polynomial is

(35)

and the corresponding derivative is

The quotient of the previous two terms yields than (28). The reversed GCP is (we write for convenience )

and as , it is

where

(36)

Finally, the derivative of the reversed GCP is

and the quotient of the previous equations yields the reversed Newton correction term in (31).

Acknowledgment

The authors would like to thank Nikunj Raghuvanshi and his colleagues for providing the MATLAB code of [35].

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