A llpass filters are unique as they preserve the signal’s energy and only alter the signal phase . Schroeder and Logan introduced delay-based allpass filters in the 1960s  to create “colorless” artificial reverberation. A decade later, Gerzon generalized the delay-based filters to feedback delay networks (FDNs)  and the single-input, single-output (SISO) allpass structure to multi-input, multi-output (MIMO) allpass networks .
An FDN essentially consists of a set of delay lines interconnected via a feedback matrix , see Fig. 1. 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 . Compared to general high-order allpass filters , FDNs are sparse filters, which are less flexible, but more computationally efficient. SISO allpass FDNs can be combined from simple allpass filters in series [2, 13] or by nesting  to create more complex structures while retaining the allpass characteristic. Rocchesso and Smith also suggested an almost allpass FDN with equal delays in [5, Th. 2]. MIMO allpass filters can be similarly generated from simple unitary building blocks [4, 15] or by generalizing the allpass lattice structure .
Both SISO and MIMO allpass FDNs were applied to a wide range of roles including: 1) increasing the echo density as preprocessing to an artificial reverberator [2, 17]; 2) increasing echo density of in the feedback loop of reverberators [18, 19, 20]; 3) decorrelation for widening the auditory image of a sound source [21, 22, 23]; 4) as reverberator in electro-acoustic reverberation enhancement systems [19, 16, 24, 25]; 5) linear dynamic range reduction [26, 27] ; and 6) dispersive system design [28, 29]. In the broader context of control theory, allpass FDNs are strongly related to Schur diagonal stability , e.g., stability properties of asynchronous networks.
In this work, we extend the theory of allpass FDNs for both SISO and MIMO. In particular, we study uniallpass111The term uniallpass is introduced here with similar motivation as unilossless feedback matrices in  which yields lossless FDNs regardless of delay lengths. FDNs, i.e., FDNs, which are allpass for arbitrary delay lengths. While not all allpass FDNs are uniallpass, the more straightforward design criterion significantly extends practical filter structures.
The feedback matrix determines many filter properties of the FDN. Thus, it is often desirable to first design the feedback matrix and subsequently choose the input, output, and direct gains such that the resulting FDN is allpass. We refer to this procedure as the completion problem. We call feedback matrices, which have a solution to the completion problem as being allpass admissible. A particularly useful class of feedback matrices are lossless mixing matrices in conjunction with diagonal delay-proportional absorption matrices. They result in homogeneous decay
of the impulse response, i.e., all system eigenvalues have the same magnitude. The main contributions of this work are
Necessary and sufficient conditions for SISO and MIMO FDNs to be uniallpass (Section III)
Characterization of admissible feedback matrices in uniallpass FDN (Section IV-B)
Completion algorithms for uniallpass SISO and MIMO FDNs (Section IV-D)
Characterization of uniallpass FDNs with homogeneous decay (Section V)
Embedding of previous designs in the proposed characterization (Section VI).
This work extends the design space of delay-based allpass filters from a handful of known structures to a freely parametrizable extensive class. In particular, the solution of the completion problem allows to combine feedback matrix design with the allpass property and potentially improves application designs mentioned above.
The remaining manuscript is structured as follows. Section II introduces FDN and allpass prior art and reviews a classic theorem on allpass state space systems. In Section III, we characterize uniallpass FDNs. Section IV presents a characterization of admissible feedback matrices and presents a completion algorithm. Section V derives a solution for uniallpass FDNs with homogeneous decay. In Section VI, we give examples of the proposed method and comparison to previous allpass FDN designs.
Ii Problem Statement and Prior Art
In the following, we state the problem formulation of this work and review the prior art.
Ii-a MIMO Feedback Delay Network
where and are the input and
output vectors at time sample, respectively. The FDN dimension is the number of delay lines. The FDN consists of the feedback matrix , the input gain matrix , the output gain matrix and the direct gain matrix . The lengths of the delay lines in samples are given by the vector . The vector denotes the delay-line outputs at time . The vector argument notation abbreviates the vector . Although, large parts of the derivations are general, we mainly focus with our results on FDNs with equal input and output channels, i.e., . We refer to an FDN where the number of delay lines is equal to the input and output channels as full MIMO, i.e., . A SISO FDN has , which is emphasized by notating vectors and scalars , and instead of matrices , and .
where the denominator is a scalar-valued polynomial
where denotes the determinant and the loop transfer function is
The numerator is a matrix-valued expression with
where denotes the adjugate of . The FDN system poles , where , are the roots of the generalized characteristic polynomial (GCP) in (4). Thus, the system poles are fully characterized by the delays and the feedback matrix .
Ii-B Allpass Property
A transfer function matrix with real coefficients is allpass if
denotes an identity matrix of appropriate size anddenotes the transpose operation. If , a MIMO system is allpass if is allpass [31, p. 772], i.e.,
In particular, is unitary for on the unit circle.
In the following, we present a classic result for allpass state space systems.
Ii-C Allpass State Space Systems
For a moment, we consider that all delays are single time steps, i.e.,, where denotes a vector or matrix of ones with appropriate size. The time-domain recursion in (1) reduces to the standard state space realization of a linear time-invariant (LTI) filter. We state a classic sufficient and necessary condition for state space systems to be allpass .
Assume that the transfer function has a realization .
There exists a solution of the equation
where , if and only if is an allpass function.
In the Section III, we present an extension of this theorem for allpass FDNs.
Ii-D Principal Minors and Diagonal Similarity
To demonstrate system properties of an FDN independent from delays , we have earlier developed a representation of based on the principal minors of [20, 6]. This representation is also useful to derive the uniallpass property of FDNs.
A principal minor of a matrix is the determinant of a submatrix with equal row and column indices . The set of all indices is denoted by and is the relative complement in , i.e., . indicates the cardinality of set .
For a given feedback matrix and delays , the generalized characteristic polynomial is given by
where . Note that for single sample delays, i.e., , is the standard characteristic polynomial of matrix . In contrast for , for and therefore each has a single summand in (11). Thus, principal minors of constitutes a powerful delay-invariant representation.
The principal minors of invertible matrices are related by Jacoby’s identity , i.e.,
Diagonally similar matrices and , i.e., there exists non-singular diagonal matrix with , have the same principal minors . The converse is not true in general , however, if and have the same principal minors, then they are diagonally similar [6, Th. 8].
In the following section, we derive the analogue of Theorem II-C for uniallpass FDNs with arbitrary delays .
Iii Uniallpass Feedback Delay Networks
The central question of the present work is which system parameters constitute an allpass transfer function in (2). In particular, we are interested in uniallpass FDNs, i.e., allpass FDNs with , , , and for arbitrary delays . The following theorem is our first main result, which is proofed at the end of this section.
Assume that the FDN transfer function has a realization .
There exists a solution of the equation
where is diagonal, if and only if is uniallpass, i.e., allpass for any . While in (13), is diagonal for uniallpass FDN, is not necessarily diagonal for specific delays . For instance, allpass FDNs with equal delays with , is only necessarily symmetric as in Th. II-C. For longer delays , it can become quickly impractical to determine the allpass property for specific such that the uniallpass property is more useful albeit slightly restrictive. However, based on observations of unilossless matrices, we conjecture for many that tends to be close to diagonal .
In the following subsections, we derive central aspects of Theorem III.
Iii-a System Matrix
First, we establish a convenient notation based on system matrices with , i.e.,
which is of size , where . The Schur complement of the invertible block in is a matrix defined by
and equivalently the Schur complement of the invertible block is
If , , , and are invertible, the block-wise inverse of the system matrix (14) is
Further, the inverse of the Schur complements are related by
Iii-B Delay-Independent Allpass Condition
The main challenge in Theorem III, is that we want the allpass property to be independent of the choice of the delays . Thus, we derive an allpass criterion which only depends on the system matrix .
For the special case , (21) holds if and only if
as for any such that each coefficient in (11) has a single summand. Thus, the principal minors of are directly related to the principal minors of . For arbitrary delays , (22) is sufficient for (21) to hold as the coefficients in (11) are merely summations for . In other words, an FDN is uniallpass if and only if (22) is satisfied.
Iii-C Proof of Theorem Iii
First, we assume is uniallpass with realization , , , and . As an uniallpass FDN is allpass for any , it is allpass also for and therefore satisfies (10) in Theorem II-C for some symmetric . Thus, the system matrix is similar to the inverse system matrix , i.e.,
Thus, and consequently
From Jacoby’s identity (12) with in ,
The lower right block in (23) yields
For the opposite direction let us assume, there exists diagonal matrix satisfying (23). Thus, is diagonally similar to . Therefore, (and also ) and have equal principal minors. Further, and have equal principal minors. Thus with (12) and (26), we have
for all as in (22). Therefore, is allpass.
Allpass FDNs are strongly related to unilossless matrices, i.e., feedback matrices such that all FDN poles are on the unit circle. In [6, Th. 1], an irreducible is unilossless if and only if there exists a non-singular diagonal matrix such that
However, compared to (23), the lower-right block of related to the input and output part is not necessarily .
In the following section, we present methods to design uniallpass FDNs based on a desired feedback matrix .
Iv Uniallpass FDN Completion
Uniallpass FDNs can be generated by a simple procedure for input and output channels and delay lines. First, generate an orthogonal system matrix of size with . Optionally, apply a similarity transform with a non-singular diagonal matrix . However, note that the similarity transform does not alter the transfer function, but may change computational properties. Lastly, divide the system matrix into the submatrices , , , and according to (14). However, this procedure does not allow to specify directly the feedback matrix and the resulting filter properties.
In this section, we present procedures related to the completion problem, i.e., determining , , and given such that is uniallpass. The following subsections are: IV-A) determining given uniallpass ; IV-B) characterize admissible feedback matrices ; IV-C) completion where ; and, IV-D) completion for any diagonal .
Iv-a Determining Diagonal Similarity
We give an alternative solution, which is helpful for the further development below. The system matrix satisfies (23), thus
is diagonally similar to an orthogonal matrix. We review here, key aspects of Engel and Schneider’s algorithm to determine the diagonal similarity.
A system matrix is diagonally similar to an orthogonal matrix if and only if is diagonally similar to a -matrix , i.e., . Operation denotes an element-wise division also called Hadamard quotient, i.e.,
Thus with (17), the similarity transform can be readily retrieved from
For fully connected matrices and , i.e., having only non-zero elements, contains only ones. Then, (32
) can be simply solved by a singular value decomposition. For non-fully connectedand , the computation is performed on the spanning tree of the adjacency graph of , for more details see .
Iv-B Admissible Feedback Matrix
In the following, we characterize the feedback matrix of uniallpass FDNs with system matrix . First, we assume that is orthogonal. The following theorem by Fiedler  gives sufficient and necessary conditions for such .
[Fiedler , Theorem 2.2] Every submatrix of an orthogonal matrix has at least singular values equal to one and singular values less than one.
Conversely, if is a matrix that has singular values equal to one and the remaining singular values less than one, then for every there exists an orthogonal matrix containing as a submatrix, and for no smaller than does such matrix exist.
In particular for the SISO case with , has exactly one singular value less than one and the other singular values are one. In the full MIMO case, i.e., , has all singular values less than one. Thus, any admissible feedback matrix of a uniallpass FDN is diagonally similar to a matrix with singular values as described above. There are various techniques to generate matrices with prescribed eigenvalues and singular values [38, 39]. Note, that for a stable FDN, the moduli of the eigenvalues of are less than one .
Iv-C Orthogonal Completion
The equations can be solved with a singular value decomposition, e.g., is the rank- decomposition of .
Particularly in the full MIMO case, any matrix with all singular values less than one can be completed to a uniallpass FDN. As demonstrated in the Section VI, this result is a large extension to prior designs.
Iv-D General Completion
Here, we complete a feedback matrix , which is part of any (not necessarily orthogonal) uniallpass FDN. The first part of the procedure is general, where as the latter part focuses on the SISO case. From (23) and (17), we have
Therefore, (18) is
Given the system matrix of a uniallpass FDN, thus, and are diagonally similar and the Hadamard quotient is diagonally similar to a -matrix. Thus,
is diagonally similar to a -matrix . In particular, the diagonal elements of are ones, and therefore
The remaining procedure is only for the SISO case, which is emphasized by notating vectors and scalars , and instead of matrices. From the uniallpass property, we have . We restate (40)
where , . We can also rewrite (41) for the SISO case, i.e.,
More concisely, we can write
where and denotes the element-wise product, also called Hadamard product. By inspecting the individual matrix entries for
we derive an important identity
Because is diagonally similar to a -matrix , we have
By Hadamard multiplying the equation with and substituting (46), we get
where denotes the element-wise square. Each matrix entry in (51) is a quadratic equation and can be solved independently. From the two possible solutions for each matrix entry, one is selected such that the solution matrix is of rank 1. From (37),
we can recover and therefore and from and . This concludes the completion algorithms for SISO uniallpass FDNs. In the following section, we study the completion of a special class of feedback matrices.
V Homogeneous Decay Allpass FDN
V-a Homogeneous Decay
A typical requirement in artificial reverberation and audio decorrelation is that all modes decay at the same rate, i.e., all system eigenvalues have the same magnitude, i.e., for . We refer to this property as homogeneous decay. In FDNs, this can be achieved by delay-proportional absorption in combination with a lossless matrix . Thus, the feedback matrix is
with unilossless matrix , diagonal matrix with 
For , the singular values of are then and the eigenvalues of have moduli less than 1. From Section IV-C, any such feedback matrix can be completed into a full MIMO uniallpass FDN. Note that this is a significant extension to Poletti’s design  as shown below in Section VI . In (54), can be a unilossless triangular matrix, i.e., with a diagonal of ones . In Section VI, we revisit this structure for series allpasses. In the following, we focus on the more intricate case of orthogonal .
V-B Siso Fdn
We right-multiply with and subsitute and such that
which is called a displacement equation . In the following, we denote the diagonal entries of a diagonal matrix with a single index, e.g., . The solution of the displacement equation (57) is the Cauchy-like matrix 
where the Cauchy matrix has elements
Then, the inverse of the Cauchy matrix is given by 
where the elements of vectors and are
Because , we have
need to be positive. And the unitary matrix is given by
V-C Admissible Parameters
Firstly, we give a sufficient condition for and to be admissible, i.e., and in (64) are positive. Secondly, for a given decay gains , we determine similarity matrix such that and are admissible. The choice of is effectively a parametrization of in (65) such that a uniallpass FDN exists with .
We show that following choice of and is admissible, i.e.,
Because of (62), we say that the zeros of and are strictly interlaced.
where denotes the sign operator. Similarly, because of (66), we have
Therefore, with (61), we have
Thus, for a given decay gain , we choose such that strictly interleaves . With (66), we have
and and are unconstraint. Note, that does not need to be sorted in any way. As we have not constraint the decay gains , we have shown that there exists SISO uniallpass FDNs with homogeneous decays for any delay and any decay rate . The similarity matrix acts as an additional design parameter within the constraints of (69).
In this section, we show that three well-known delay-based allpass structures are uniallpass FDNs: Schroeder’s series allpass , Gardner’s nested allpasses , and Poletti’s unitary reverberator . Reviewing these previous designs also reveals their limited design space and demonstrates the significant extension introduced by Theorem III. We conclude this section by presenting a complete numerical example of a SISO uniallpass FDN with homogeneous decay.
Vi-a SISO - Series Schroeder Allpass
The Schroeder series allpass of feedforward-feedback delay allpasses is
and the similarity transform in (13) is a diagonal matrix with diagonal elements
Fig. 1(b) depicts the system matrix of the Schroeder series allpass for . The feedback matrix is triangular with gains on the main diagonal. The remaining gains , , and are determined by the gains as well. Therefore, there exists with triangular unilossless and such that the Schroeder series allpass can have homogeneous decay, see (54). At the same time, the series allpass is a highly limited structure with a particular feedback matrix.
Vi-B SISO - Nested Allpass
The nested allpass as proposed by Gardner  is a recursive nesting of Schroeder allpasses, i.e.,
where and for
Figure 2(a) shows an instance of the nested allpass for . The corresponding state space realization is
where and for . The similarity transform in (13) is a diagonal matrix with diagonal elements
Fig. 2(b) depicts the system matrix of the nested allpasses for . The feedback matrix is Hessenberg and all gains including , , and are determined by the gains . Series allpasses are strongly related to nested allpasses as they share the same parameter space, however, differ in the structure. Interestingly, the feedback matrix of nested allpasses induce a much more complex decay pattern than the series allpass counterpart.
Vi-C MIMO - Poletti Reverberator
The MIMO reverberator proposed by Poletti is a direct multichannel generalization of the Schroeder allpass structure in lattice form, see Fig. 3(a). The loop gain controls the decay rate of the response tail such that
The state space realization is
and the similarity matrix in (13) is
Fig. 3(b) depicts the system matrix of Poletti’s allpass for and . While the direct and input gains, and , respectively, are scaled identity matrices, the feedback matrix and output gains are scaled versions of the unitary matrix . Interestingly, Poletti’s allpass has homogeneous decay only for equal delays, which is mostly an undesirable parameter choice.
Vi-D SISO Homogeneous Decay Uniallpass FDN
and from (69) we can choose
From (65), we can then compute
The feedback matrix results than from (54)
The remaining input, output and direct gains are determined by solving the completion problem in Section IV-D
Fig. 5 shows the system matrix for the numerical example. Interestingly, the feedback matrix exhibits a triangular-like shape which suggests that the homogeneous decay uniallpass FDN generalizes the triangular and Hessenberg shapes of the series and nested allpasses.
In this work, we developed a novel characterization for allpass feedback delay networks (FDNs). In particular, we presented a full characterization of uniallpass FDNs, which are allpass for any choice of delay lengths. Further, we introduced the uniallpass completion, i.e., completing a given feedback matrix to a uniallpass FDN. While the full MIMO case is relatively simple, also a solution to the SISO case was presented. Further, we solved the completion problem for a particular class of feedback matrices, which yields homogeneous decay of the impulse response. We reviewed three previous allpass FDN designs within this novel characterization and an additional numerical example for homogeneous decay uniallpass FDNs.
Future research questions should address application-specific designs of uniallpass FDNs, for instance, in audio signal processing, where additional constraints are required. Further research is also needed for the design of frequency-dependent FDN designs with the allpass property.
The author thanks Prof. Dario Fasino for his insights on orthogonal Cauchy-like matrices in Section V-B. Further thanks go to Dr. Maximilian Schäfer and Prof. Vesa Välimäki for proofreading and valuable comments.
-  P. A. Regalia, S. K. Mitra, and P. P. Vaidyanathan, “The digital all-pass filter: a versatile signal processing building block,” Proceedings of the IEEE, vol. 76, no. 1, pp. 19 – 37, 1988.
-  M. R. Schroeder and B. F. Logan, “”Colorless” artificial reverberation,” IRE Transactions on Audio, vol. AU-9, no. 6, pp. 209 – 214, 1961.
-  M. A. Gerzon, “Synthetic stereo reverberation: Part One,” vol. 13, pp. 632 – 635, 1971.
-  ——, “Unitary (energy-preserving) multichannel networks with feedback,” Electronics Letters, vol. 12, no. 11, pp. 278 – 279, 1976.
-  D. Rocchesso and J. Smith, “Circulant and elliptic feedback delay networks for artificial reverberation,” IEEE Transactions on Speech and Audio Processing, vol. 5, no. 1, pp. 51 – 63, 1997.
-  S. J. Schlecht and E. A. P. Habets, “On Lossless Feedback Delay Networks,” IEEE Transactions on Signal Processing, vol. 65, no. 6, pp. 1554 – 1564, 2016.
-  J. M. Jot and A. Chaigne, “Digital delay networks for designing artificial reverberators,” ser. Proc. Audio Eng. Soc. Conv., 1991, pp. 1 – 12.
-  K. Prawda, S. J. Schlecht, and V. Välimäki, “Improved Reverberation Time Control for Feedback Delay Networks,” ser. Proc. Int. Conf. Digital Audio Effects (DAFx), 2019, pp. 1 – 7.
-  E. D. Sena, H. Hacıhabiboğlu, Z. Cvetkovic, and J. O. Smith III, “Efficient synthesis of room acoustics via scattering delay networks,” IEEE/ACM Trans. Audio, Speech, Language Process., vol. 23, no. 9, pp. 1478 – 1492, 2015.
-  S. J. Schlecht and E. A. P. Habets, “Feedback delay networks: Echo density and mixing time,” IEEE/ACM Trans. Audio, Speech, Language Process., vol. 25, no. 2, pp. 374 – 383, 2017.
-  ——, “Modal Decomposition of Feedback Delay Networks,” IEEE Transactions on Signal Processing, vol. 67, no. 20, pp. 5340–5351, 2019.
-  J. S. Abel and J. O. Smith III, “Robust Design of Very High-Order Allpass Dispersion Filters,” ser. Proc. Int. Conf. Digital Audio Effects (DAFx), 2006, pp. 13 – 18.
-  S. J. Schlecht, “Frequency-Dependent Schroeder Allpass Filters,” Applied Sciences, vol. 10, no. 1, p. 187, 2019.
-  W. G. Gardner, “A real-time multichannel room simulator,” J. Acoust. Soc. Am., vol. 92, no. 4, pp. 1 – 23, 1992.
-  P. P. Vaidyanathan and Z. Doganata, “The role of lossless systems in modern digital signal processing: a tutorial,” IEEE Transactions on Education, vol. 32, no. 3, pp. 181–197, 1989.
-  M. A. Poletti, “A Unitary Reverberator For Reduced Colouration In Assisted Reverberation Systems,” ser. INTER-NOISE and NOISE-CON, vol. 5, 1995, pp. 1223 – 1232.
-  V. Välimäki, J. D. Parker, L. Savioja, and J. S. Smith III, Julius Oand Abel, “Fifty years of artificial reverberation,” IEEE/ACM Trans. Audio, Speech, Language Process., vol. 20, no. 5, pp. 1421 – 1448, 2012, readingList.
-  R. Väänänen, V. Välimäki, J. Huopaniemi, and M. Karjalainen, “Efficient and Parametric Reverberator for Room Acoustics Modeling,” ser. Proc. Int. Comput. Music Conf., 1997, pp. 200 – 203.
-  T. Lokki and J. Hiipakka, “A time-variant reverberation algorithm for reverberation enhancement systems,” ser. Proc. Int. Conf. Digital Audio Effects (DAFx), 2001, pp. 28 – 32.
-  S. J. Schlecht and E. A. P. Habets, “Time-varying feedback matrices in feedback delay networks and their application in artificial reverberation,” J. Acoust. Soc. Am., vol. 138, no. 3, pp. 1389 – 1398, 2015.
-  G. S. Kendall, “The Decorrelation of Audio Signals and Its Impact on Spatial Imagery,” Comput. Music J., vol. 19, no. 4, p. 71, 1995.
-  J. S. Abel and E. K. Canfield-Dafilou, “Dispersive Delay and Comb Filters Using a Modal Structure,” IEEE Signal Processing Letters, vol. 26, no. 12, pp. 1748–1752, 2019.
-  C. Gribben and H. Lee, “The Perception of Band-Limited Decorrelation Between Vertically Oriented Loudspeakers,” IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 28, pp. 876–888, 2020.
-  M. A. Poletti, “The Stability Of Multichannel Sound Systems With Frequency Shifting,” J. Acoust. Soc. Am., vol. 116, no. 2, pp. 853 – 871, 2004.
-  S. J. Schlecht and E. A. P. Habets, “The stability of multichannel sound systems with time-varying mixing matrices,” J. Acoust. Soc. Am., vol. 140, no. 1, pp. 601 – 609, 2016.
-  J. Parker and V. Välimäki, “Linear Dynamic Range Reduction of Musical Audio Using an Allpass Filter Chain,” IEEE Signal Processing Letters, vol. 20, no. 7, pp. 669 – 672, 2013.
-  J. A. Belloch, J. Parker, L. Savioja, A. Gonzalez, and V. Välimäki, “Dynamic range reduction of audio signals using multiple allpass filters on a GPU accelerator,” 2014, pp. 890–894.
-  V. Välimäki, J. D. Parker, and J. S. Abel, “Parametric Spring Reverberation Effect,” J. Audio Eng. Soc., vol. 58, no. 7/8, pp. 547 – 562, 2010.
-  J. Parker, “Efficient Dispersion Generation Structures for Spring Reverb Emulation,” EURASIP Journal on Advances in Signal Processing, vol. 2011, no. 1, pp. 547 – 8, 2011.
-  E. Kaszkurewicz and A. Bhaya, Matrix Diagonal Stability in Systems and Computation, 2000.
-  P. P. Vaidyanathan, Multirate Systems and Filter Banks, ser. Prentice Hall, 1993, this.
-  A. Ferrante and G. Picci, “Representation and Factorization of Discrete-Time Rational All-Pass Functions,” IEEE Transactions on Automatic Control, vol. 62, no. 7, pp. 3262–3276, 2016.
-  R. A. Brualdi and H. Schneider, “Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley,” Linear Algebra Appl., vol. 52-53, pp. 769 – 791, 1983.
-  R. Loewy, “Principal minors and diagonal similarity of matrices,” Linear Algebra Appl., vo