SUBMITTED FOR PUBLICATION IN IEEE TRANSACTIONS ON SIGNAL PROCESSING, JUNE 2016

On Lossless Feedback Delay Networks

arXiv:1606.07729v1 [] 24 Jun 2016

Sebastian J. Schlecht and Emanuël A.P. Habets, Senior Member, IEEE

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.

+

+
+

+

+

+

+

+

Index Terms—Feedback Delay Network, Lossless, Diagonal
Similarity, Artificial Reverberation.

Figure 1: Feedback Delay Network (FDN) structure for artificial reverberation as proposed by Stautner and Puckette [1]
and further developed by Jot and Chaigne [13].

I. I NTRODUCTION
EEDBACK Delay Networks (FDNs) consist of a set of
delays and a feedback matrix through which the delay
outputs are coupled to the delay inputs (see Fig. 1). FDNs
have been proposed by Stautner and Puckette [1] and gained
popularity as an efficient and flexible way to model and process artificial reverberation [2]. Many parametric reverberation
algorithms like Schroeder’s Cascaded Allpass and Moorer’s
extension [3], [4], Dattorro’s Allpass Feedback Network [5],
Dahl and Jot’s Absorbent Allpass FDN [6], can be expressed
by an FDN [7], [8]. Other physical modeling structures like
digital waveguide networks (DWN), digital waveguide mesh
(DWM), scattering delay networks (SDN) and finite difference
time domain (FDTD) simulation are in close relation to the
FDN structure [9]–[12].
It is common practice for FDNs to first design the system
to be lossless, i.e., all the system poles of the FDN lie on
the unit circle [9]. This later ensures a smooth frequencydependent pole magnitude by simply extending every delay
element with an attenuation [13]. From a practical point of
view it is desirable that the delays can be scaled without
changing the lossless property as this corresponds to scaling
the underlying physical model, e.g. the dimensions of a room
model [14] and the corresponding mixing time [15].
From the beginning, unitary and triangular feedback matrices have been found to constitute a lossless FDN independent
from the choice of the delays [1], [13]. In a more general
approach, Gerzon described unitary networks, i.e., frequencydependent unitary matrices of filters, in a feedback loop to
constitute a lossless FDN [16]. However, a non-unitary, nontriangular feedback matrix can constitute a lossless FDN as
well [8]. It has been shown that if an FDN is lossless then the
feedback matrix has only eigenvalues on the unit circle [9].

F

The paper is organized as follows. In Section II, we give motivating examples on the relation between feedback matrices
and the FDN system poles. Further, we show that the lossless
property is dependent on the delays as well. These examples
open the question about the definition of the lossless property
and its conditions. In Section III, we derive a new sufficient
and necessary condition for FDNs to be lossless independent
from the choice of delays. In Section IV, some well-known
reverb topologies and their representation as FDNs are given.
II. M OTIVATION & BACKGROUND
In this section, the FDN is defined and examples on lossless
FDNs are given. Then a definition of feedback matrices in
lossless FDNs is proposed and the full characterization of
lossless FDNs is formulated.
A. Feedback Delay Network
The standard FDN is given in time domain by the relation
[9]
y(n) =
si (n + mi ) =

N
X

i=1
N
X

ci si (n) + d x(n)

(1a)

aij sj (n) + bi x(n),

(1b)

j=1

where x(n) and y(n) are the input and output values respectively and si (n), 1 ≤ i ≤ N , are the delay outputs at time
sample n [9]. The double-indexed aij form a N × N feedback
matrix A. In a similar manner, we refer to the column vectors
of bi ’s and ci ’s as b and c respectively. The lengths of the N

S.J. Schlecht and E.A.P. Habets are with the International
Audio Laboratories Erlangen (a joint institution of the University
of Erlangen-Nuremberg and Fraunhofer IIS), Germany (e-mail:
{sebastian.schlecht,emanuel.habets}@audiolabs-erlangen.de).

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SUBMITTED FOR PUBLICATION IN IEEE TRANSACTIONS ON SIGNAL PROCESSING, JUNE 2016
delays in samples are given by m = [m1 , . . . , mN ] ∈ NN .
The transfer function of the FDN in the z-domain [9] is
H(z) =

Y (z)
= cT [Dm (z −1 ) − A]−1 b + d,
X(z)

A unitary
Lemma 1

(2)

A unilossless

where Dm (z) = diag ([z −m1 , z −m2 , . . . , z −mN ]) is the diagonal N × N delay matrix. If the notation is unambiguous, we
write D(z) instead of Dm (z). The order of the system given
in (2) is equal to the sum of the delays [9]
N

=

N
X

A triangular

mj .

(3)
Figure 2: Implication graph for unilossless matrices A in the
literature.

The zeros of the transfer function (2) are the roots of the
polynomial [9]


1 >
−1
qA,b,c,d,m (z) = det A − b c − Dm (z ) .
(4)
d

contradicts the claim presented in [9, Eq. (29)] that A having
unimodular eigenvalues and linearly independent eigenvectors
is sufficient for pA,m (z) to be lossless. In Appendix A, the
technical details of the flawed proof in [9] are given.
As an additional note, the stability of an FDN is not even
ensured if all eigenvalues of A lie within the unit circle. For


Λ
1.5
1
A = U −1 U =
(12)
−2 −1.5
2

The poles of the transfer function (2) are the roots of the
polynomial [9]
pA,m (z) = det[Dm (z −1 ) − A].

(5)

We call pA,m (z) the generalized characteristic polynomial of
A with delays m. Please note that pA,[1,...,1] (z) is the standard
characteristic polynomial found in literature [17]. Therefore,
the FDN is lossless if pA,m (z) has only unimodular roots,
i.e., all roots are of magnitude 1. We call such a polynomial
pA,m (z) to be lossless.

we have
pA,m (z) = z 3 + 1.5z 2 − 1.5z − 0.25.

Let us assume an FDN of dimension N = 2, with the delays
m1 = 1, m2 = 2 and feedback matrix


3
2
A=
.
(6)
−4 −3

The eigenvalue decomposition A = U −1 Λ U yields




2 1
1 0
U=
and Λ =
,
1 1
0 −1

(7)

such that U is an invertible matrix of eigenvectors, Λ is a
unimodular diagonal matrix of eigenvalues. The roots of
pA,m (z) = (z 2 − 3)(z 1 + 3) + 8
= z 3 − 3z 2 + 3z 1 − 1

z1,2,3 = [1, 1, 1]

(8)
C. Lossless FDN
The FDN’s losslessness should not depend on the choice
of the delays m, but only on the feedback matrix A. In
other words, we want to characterize all matrices A such that
pA,m (z) is lossless for any choice of m ∈ NN . We call such
a matrix to be unilossless.1
It has be shown in [1] that A being unitary, i.e., AAH = I,
is a sufficient condition for A to be unilossless. For completeness, we give an alternative proof.

(9)

and therefore the FDN is lossless. However on the contrary,
with m2 = 2, m1 = 1 the roots of
pA,m (z) = z 3 + 3z 2 − 3z − 1

= (z 2 + 4z + z)(z − 1)

z1,2,3 = [1, −2 −

√

3, −2 +

√

3]

(13)

The largest eigenvalue magnitude is then approximately 2.145
and therefore the FDN is unstable.
These examples illustrate that
• there are non-unitary, non-triangular A, for which the
FDN is lossless.
• there are A with only unimodular eigenvalues, for which
the FDN is not lossless.
• the losslessness of the FDN can be dependent on m.
• the eigenvalues of A being within the unit circle is not
a sufficient condition for stability.
In the following, the existing results on lossless FDNs are
summarized and a precise definition of the main result is
stated.

B. Motivating Examples

are

Eigenvalues
Λ unimodular

Lemma 2

j=1

are

Lemma 3

(10)

Lemma 1. Any unitary matrix A is unilossless.

(11)

1 Please note that matrices A with unimodular eigenvalues and linearly
independent eigenvectors have been named lossless in [9]. In this manuscript,
this terminology is avoided to reduce confusion and to emphasize the
dependency on the choice of the delays.

and therefore the FDN is not lossless. Consequently, A having
unimodular eigenvalues and linearly independent eigenvectors
is insufficient for the FDN to be lossless. This example
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Proof: Firstly, |pA,m (0)| = | det(A)| = 1, hence there is
no root of pA,m which is zero.
Secondly, if z 6= 0, such that pA,m (z) = 0 is equivalent to
the existence of v 6= 0√such that Av = D(z −1 )v. Writing
z −1 = ρeıt , where ı = −1, we obtain Av = D(ρ)D(eıt )v
. Using the fact that D(eıt ) is unitary, we obtain
−ıt

kvk2 = D(e

)Av

2

= kD(ρ)vk2 .

of independent matrix components. A matrix A is reducible
if there is a permutation matrix P with


E G
P −1 AP =
,
(16)
0 F
with E and F square. In other words, a matrix is reducible if
it is permutation conjugate to a block triangular matrix.

(14)

Theorem 1. A matrix A is unilossless if and only if either

As all delays m ≥ 1, two cases can be identified for all v and
ρ > 0:
kD(ρ)vk2 ≥ ρ kvk2 if ρ < 1
(15)
kD(ρ)vk2 ≤ ρ kvk2 if ρ > 1.

1) A is reducible and all irreducible components are
unilossless or
2) there exists a non-singular diagonal matrix E with
AEAH = E.

the result follows from (14).
A second class of unilossless matrices has been identified
by Jot in [13], the triangular matrices AT with unimodular
main diagonal, i.e., |tii | = 1 for all i and tij = 0 either for
i > j or i < j.

In this section, we explore the sufficient and necessary
conditions for the feedback matrix to be unilossless, and in
particular give a proof for Theorem 1. Figure 3 sketches the
dependencies of the following theorems and may guide the
reader towards the main result.

Lemma 2. Any triangular matrix with unimodular main
diagonal is unilossless.

A. Generalized Characteristic Polynomial

Proof: Given
QN triangular matrix AT with |tii | = 1, then
pAT ,m (z) = i=1 (z mi −tii ). The roots of (z mi −tii ) are unimodular and therefore all roots of pAT ,m (z) are unimodular
and AT is unilossless.
A necessary condition on unilossless matrices has been
given in [9].

To derive the conditions on the generalized characteristic
polynomial to have only unimodular roots, it is helpful to give
a closed form of pA,m (z). This closed form of the generalized
characteristic polynomial pA,m (z) in (5) was proven by the
authors in [8] in terms of principal minors. A principal minor
det A(I) of a matrix A is the determinant of a submatrix A(I)
with equal row and column indices I ⊂ hN i. |I| indicates
the cardinality of set I. The set of all indices is denoted by
hN i = {1, 2, . . . , N } and I c is the relative complement in
hN i, i.e., I c = hN i \ I.

Lemma 3. Any unilossless matrix A has only unimodular
eigenvalues.
Proof: As A is unilossless the roots of pA,m (z) are
unimodular for any m. Therefore, the roots of pA,[1,...,1] (z),
which are the eigenvalues of A, are unimodular.
The converse of Lemma 3 can be stated with an additional
condition that all delays are equal.

Theorem 2. For a given feedback matrix A and delays m,
the generalized characteristic polynomial pA,m is given by
pA,m (z) =

Lemma 4. Let A have only unimodular eigenvalues and all
delays are equal, i.e., m1 = · · · = mN , then pA,m (z) is
lossless.

N
X

ck z k

(17)

k=0

ck =

Proof: The roots of the characteristic polynomial ζI − A
with ζ ∈ C are the eigenvalues of A and therefore unimodular.
m
The substitution z1m = · · · = zN
= ζ maps the unit circle onto
itself. Consequently, the roots of pA,m (z) are unimodular.
For natural reverberation, the echo density usually increases
exponentially with time. When the delay lengths are equal, i.e.,
m1 = m2 = ... = mN , there is no increase of the echo density
across time such that the reverberation will be unnatural [18,
pp. 109]. The delays m employed in Lemma 4 are therefore a
special case. Figure 2 gives an overview of the known results
on unilossless matrices as described in the Lemmas 1, 2 and 3.
The next section is devoted to the derivation of the necessary
and sufficient conditions of unilossless matrices.

(P
0,

I∈Ik (−1)

N −|I|

det A(I c ), for Ik 6= ∅
otherwise
(18)

P
where Ik = {I ⊂ hN i| i∈I mi = k} is the set of index
sets I such that the delays with indices in I sum up to k. The
system order N is defined in (3).
There are 2N different principal minors, and for many
choices of m the coefficients ck are sums of multiple minors. Still, the principal minors can be assigned uniquely to
the coefficients, for example with exponential delays m =
[1, 2, 4, . . . , 2N −1 ].
However, for real matrices A there are no more than N 2
degrees of freedom defined by the square matrix entries,
which implies that there is a strong dependency between
the principal minors for larger matrices. In fact, it can be
shown that there are only N 2 − N + 1 degrees of freedom
[19] because of a similarity invariance which we discuss in
following Section III-B.

III. U NILOSSLESS M ATRICES
Before the main result of the present work on the characterization of unilossless matrices is given, we introduce the notion
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diagonal similarity. From (4), we have

Th. 3 & Lemma 1

∃F , B unitary: F −1 AF = B

qE −1 AE,b,c,d,m (z)


1
= det E −1 AE − b c> − D(z −1 )
d


(20)
1 > −1
−1
−1
= det E det A − Eb c E − D(z ) det E
d


1
>
0
0
−1
= det A − b c − D(z ) = qA,b0 ,c0 ,d,m (z)
d

A unilossless
Lemma 5

Th. 1

Th. 6 & 2

∃E : E −1 AH E = A−1

PM of A self-inverse
(29)
PM of A

−1

∗

= PM of A or A

with b0 = Eb and c0 = (E −1 )> c. Therefore, the diagonal
conjugation effects only the system feed and observation gains
b and c, respectively.
2) Rotation along the Unit Circle: A further invariance can
be derived from the fact that the generalized characteristic
polynomial can be rotated along the unit circle without changing the root magnitudes. Such a rotation can be realized by
the following theorem.

H

Th. 7
∃E : E −1 A∗ E = A−1
or E −1 AH E = A−1

Th. 8 & 4
+ A unilossless

Figure 3: Implication graph of theorems for unilossless and
irreducible A. All E and F are non-singular diagonal matrices.

Theorem 4. Given any feedback matrix A and any delay matrix D(z). The generalized characteristic polynomial
pD(γ)A,m (z) is lossless for any γ ∈ C with |γ| = 1 if and
only if pA,m (z) is lossless.
Proof: Substitute z = ζγ, where ζ ∈ C. Then

B. Extending the Set of Unilossless Matrices

pA,m (z) = pA,m (ζγ)

We will discuss three techniques to extend the known class
of unilossless matrices, which are described by the Lemmas
1, 2 and 3.

= det[D(ζ −1 γ −1 ) − A]

= det[D(γ −1 )D(ζ −1 ) − A]

1) Diagonal Similarity: We show that conjugating a
unilossless matrix with a diagonal matrix does not alter
the principal minors and therefore also not the unilossless
property.

= pD(γ)A,m (ζ)

Hence, r ∈ C is a root of pD(γ)A,m (z) if and only if rγ is a
root of pA,m (z).
This invariance does not translate to unilossless matrices A
as D(γ) is dependent on the choice of the delays m. Theorem
4 nevertheless plays an important role later in the proof the
main Theorem 1.
3) Reducible Matrices: We show here that the losslessness
of systems which can be decomposed into independent components can be determined from its parts. We show that the
lossless property of reducible matrices can be deduced from
the lossless property of the diagonal blocks.

Theorem 3. Given a non-singular diagonal matrix E and
any matrix A. The matrices A and E −1 AE have the same
principal minors.
Proof: We show that the generalized characteristic polynomials of A and E −1 AE are identical.


pE −1 AE,m (z) = det D(z −1 ) − E −1 AE


= det E −1 D(z −1 )E − E −1 AE


= det E −1 det D(z −1 ) − A det E

(21)

= det[D(ζ −1 ) − D(γ)A]

(19)

Theorem 5. Given a reducible matrix A and the permutation
conjugate like in (16). Then there is a partition m1 , m2 of
m with

= pA,m (z)

pA,m (z) = pE,m1 (z)pF ,m2 (z).
Consequently, all diagonal conjugations of a unilossless
matrix are unilossless again. As we have noted earlier, the
degree of freedom of the principal minors for real matrices is
N 2 minus the invariance of the diagonal conjugation of N − 1
degrees.

(22)

Proof: We show that the conjugate permutation of the
matrix is equivalent to a permutation of the delays:


pP −1 AP ,m (z) = det Dm (z −1 ) − P −1 AP


= det P −1 DmP −1 (z −1 )P − P −1 AP

It is interesting to note that the diagonal similarity is rather
a property of the system zeros than of the system poles. As
noted in Theorem 3, the principal minors A and E −1 AE are
equal and therefore also the system poles are invariant under

= pA,mP −1 (z),

(23)

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For A to be unilossless, pA,m has to be self-inversive for all
m, hence also for m = [1, 2, . . . , 2N −1 ] implying that Ik is
unique for every k in (18). Consequently as cN = det A(∅) =
1,
det A = (−1)N c0 = (−1)N  c∗N
(28)
=  (−1)N det A(∅) = (−1)N .

where the commutation in (23) follows from Lemma 2.2
in [20]. Further, we show that the generalized characteristic
polynomial can be decomposed as


pP −1 AP ,mP (z) = det DmP (z −1 ) − P −1 AP



E G
−1
(24)
= det DmP (z ) −
0 F

c
and because of Ik = IN−k
and |Ikc | + |Ik | = N

= pE,m1 (z)pF ,m2 (z),

c
det A(Ik ) = det A(IN−k
) = (−1)N −|IN−k | cN−k =

where [m1 , m2 ] = mP and follows from the determinant
formula for block matrices [21]. Together the proof concludes
with
pA,m (z) = pP −1 AP ,mP (z) = pE,m1 (z)pF ,m2 (z).

c

= (−1)N −|Ik |  c∗k
c

= (−1)2N −|Ik |−|Ik |  det A∗ (Ikc )
c

= (−1)3N −|Ik |−|Ik |  det A∗ det A−∗ (Ik )

(25)

(29)

= (−1)2N det A−∗ (Ik )
= det A−∗ (Ik )

Consequently, the matrix A is unilossless if and only if
E and F are unilossless. This also gives a more general
argument for the unilossless triangular matrices characterized
in Lemma 2. For the intuition, we remark that the diagonal
blocks correspond to strongly connected components of the
signal graph, i.e., a subset of delays where every delay is
connected to each other by a chain of non-zero gains in
the feedback matrix [22]. The off-diagonal blocks correspond
to connections between the strongly connected components.
Because of the block triangular form, once a signal leaves a
strongly connected component it never returns, which explains
the independent stability property of these components.

In other words, pA,m is self-inversive for all m only if
the principal minors of A are the complex conjugate principal
minors of A−1 . This holds for A with
A−1 = A∗ or A−1 = AH .

A matrix A−1 = A∗ is called conjugate-involutory. In the
following, we show that these two options are essentially
exhaustive if we additionally allow diagonal similarity.
3) Diagonal Similarity and Principal Minors: In the following, we characterize the matrices with property (29). The
following theorem is build upon results by Hartfiel and Loewy
[24], [25].

C. Characterizing the Set of Unilossless Matrices

Theorem 7. Let A be irreducible and invertible. If A∗ and
A−1 have equal principal minors then there exists a nonsingular diagonal matrix E with either E −1 A∗ E = A−1 or
E −1 AH E = A−1 .

In this section, we develop the necessary conditions for
unilossless matrices, which eventually results in a full characterization.
1) Roots on the Unit Circle: The starting point is a classical
theorem on polynomials which have all roots on the unit circle.
In the following, p0 denotes the derivative of the polynomial
p.

Proof: The proof can be found in Appendix B.
Theorem 7 gives a characterization of (30) with additional
diagonal similarity. This result can be expressed also in terms
of diagonal similarity of the matrices themselves.

Theorem 6 (A. Cohn, [23]). All zeros of a polynomial p lie
on the unit circle if and only if
1) p is self-inversive
2) all zeros of p0 lie in or on the unit circle.
PN
k
A polynomial p(z) =
of order N is selfk=0 ck z
inversive if
∃ ∈ C, ∀j : cN−j =

 c∗j .

Lemma 5. If E −1 A∗ E = A−1 then there exists a diagonal
non-singular matrix F with B = F −1 AF such that B −1 =
B ∗ . Also, if E −1 AH E = A−1 then there exists a diagonal
non-singular matrix F with B = F −1 AF such that B −1 =
BH .
E

(26)

det A

det A(I c )
.
(I) =
det A

−1

Proof: Let B = F −1 AF with F ∗ F −1 = E. Given
A∗ E = A−1 , then equivalently A∗ EA = E and
B ∗ B = F −∗ A∗ F ∗ F −1 AF = F −∗ A∗ EAF

Criterion 2 of Theorem 6 is similarly difficult to verify as the
original problem, however Criterion 1 gives a simple starting
point to create a necessary condition on unilossless matrices.
The main advantage of Criterion 1 is that it is independent
from the choice of the delays.
2) Self-Inversive Principal Minors: We derive the conditions on A such that pA,m is self-inversive. For this, we recall
Jacoby’s identity for invertible matrices A [21]: Given index
set I ⊂ hN i
−1

(30)

= F −∗ EF = I.

(31)

Let B = F −1 AF with F −∗ F −1 = E. Given
E AH E = A−1 , then equivalently AH EA = E and
−1

B H B = F ∗ AH F −∗ F −1 AF = F ∗ AH EAF
= F ∗ EF = I.

(32)

Because of Theorem 7 and Lemma 5, any irreducible
and invertible matrices having self-inversive principal minors
are either diagonally similar unitary or conjugate-involutory

(27)

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SUBMITTED FOR PUBLICATION IN IEEE TRANSACTIONS ON SIGNAL PROCESSING, JUNE 2016
matrices. More precisely, if B is a unilossless and irreducible
matrix, then there is a non-singular matrix E with A =
E −1 BE such that A satisfies (30).
4) Irreducible Unilossless is Necessarily Diagonally Similar Unitary: Theorem 7 shows that any irreducible unilossless matrix is either diagonally similar to a unitary or
conjugate-involutory matrix. The following theorem shows
that conjugate-involutory, but non-unitary matrices cannot be
unilossless.

D. Proof of the Main Result
Figure 3 sketches the dependencies of the theorems we have
proven so far for the irreducible matrices A and the relation
to the main result Theorem 1. We conclude this section with
the proof of Theorem 1.
Proof of Theorem 1: Both cases can be treated independently.
1) If A is reducible, the result follows from Theorem 5.
2) Let A be irreducible. If A is unilossless then there is
a non-singular diagonal matrix E with AEAH = E
(see Theorem 8). On the contrary, any unitary matrix
B is unilossless (see Lemma 1). For any non-singular
diagonal matrix F , A = F −1 BF is unilossless (see
Theorem 3). Let E = F −1 F −∗ and finally

Theorem 8. Let A be an irreducible unilossless matrix then
there exists a non-singular diagonal matrix E with AEAH =
E.
Proof: Let A be a irreducible and unilossless matrix.
From Theorem 7 follows that there exists a non-singular
diagonal matrix E with AEA∗ = E or AEAH = E.
1) Assume AEAH = E: X
2) Assume AEA∗ = E:
Given a unimodular γ and m with mi 6= mj for all
i 6= j. With Theorem 4 it follows pA,m (z) is lossless
if and only if pAD(γ),m (z) is lossless. We write B =
AD(γ). If pB,m (z) is lossless, then BEB ∗ = E or
BEB H = E.
a) Assume BEB H = E:

AEAH = F −1 BF EF ∗ B H F −∗ = E.

Theorem 1 fully characterizes the set of unilossless matrices.
The following section discusses briefly some aspects of nonunilossless matrices and their connection to lossless FDNs.
E. Lossless, but not Unilossless
This section discusses some aspects of lossless FDNs which
are not based on unilossless feedback matrices. For the FDN
to be lossless, following Theorem 6, pA,m (z) needs to be selfinversive and therefore A is characterized by Theorem 7. In
the following, a parameterization of the unitary and conjugateinvolutory matrices are given.
1) Characterizing the Unitary and Conjugate-Involutory
Matrices: Here we would like to give a characterization for
all matrices A with AAH = I or AA∗ = I. Both classes
can be identified with the exponential map. It is well known
that for any matrix with AAH = I, there is a skew-symmetric
matrix M such that A = exp(M ) [17]. A similar map can
be shown for any matrix with AA∗ = I. Namely, there is a
pure imaginary matrix M such that A = exp(M ). A proof
is given in Appendix C.
It can be seen that for any eigenvalue λA of A there is an
eigenvalue λM of M with λA = exp(λM ). Therefore, A has
only unimodular eigenvalues if and only if M has only pure
imaginary eigenvalues. Interestingly, skew-symmetric matrices
have only pure imaginary eigenvalues, whereas pure imaginary
matrices clearly do not need to have only pure imaginary
eigenvalues. The question when a pure imaginary matrix has
only pure imaginary eigenvalues is a rather difficult question
and to the best of our knowledge no simple criterion exists.
2) On Lossless 2 × 2 FDNs: We give some illustrations on
matrices A which can constitute a lossless FDN for certain
values of delays m. Given the delays m = [m1 , m2 ] and
a 2 × 2 matrix A, following Theorem 2, the generalized
characteristic polynomial is:

E = BEB H = AD(γ)ED H (γ)AH = AEAH
(33)
b) Assume BEB ∗ = E:
BEB ∗ = E
E −1 BE = B −∗
−1

(D(γ)E)

AD(γ)E = A

(34)

−∗

We write A−∗ = [a0ij ]N ×N . Because of Assumption 2,
E −1 AE = A−∗

(35)

ei
aij = a0ij .
ej

(36)

we write

From (34), we have
γ m i ei
aij = a0ij .
γ m j ej
And together, this gives
 mi

γ
ei
−
1
aij = 0.
γ mj
ej

(39)

(37)

(38)

As γ mi 6= γ mj and ei 6= 0, we have aij = 0 for
all i 6= j. Then A is a diagonal matrix, such that
A∗ = AH , and therefore AEAH = E.

pA,m (z) = z m1 +m2 − a22 z m1 − a11 z m2 + det A.

Given any matrix B, it is possible to test algorithmically
whether B is diagonally similar to a unitary matrix and to
determine the diagonal similarity matrix. More details can be
found in [26].

(40)

Since pA,m (z) is necessarily self-inversive, we have det A =
 with || = 1 and a∗22 = ∗ a11 . The off-diagonal elements
a12 and a21 have no direct influence on the generalized
characteristic polynomial except the determinant.
6

SUBMITTED FOR PUBLICATION IN IEEE TRANSACTIONS ON SIGNAL PROCESSING, JUNE 2016
A. Schroeder’s Parallel Comb and Serial Allpass
In [3], Schroeder presented an early topology for a delay
based network. It follows a two stage design: four parallel
comb filters to create the reverberation tail and then two serial
allpass filters to increase the reflection density.
We define the main building block: the feedbackfeedforward comb filter
z −m − gf
.
(43)
Cm,gb ,gf (z) =
1 − gb z −m
where m is the delay in samples, gb and gf are the feedback
and feedforward gains, respectively. Comb allpass filters are
feedback-feedforward comb filters with identical feedback and
feedforward gains, i.e., Cm,g (z) = Cm,g,g (z). The filter
Cm,g (z) has a unit magnitude response and therefore called
allpass. The Schroeder structure is then given by
! 6
4
X
Y
HS (z) =
Cmi ,gi ,0 (z)
Cmj ,gj (z).
(44)

3
2
k
k
k
k
k
k

=(a11 )

1
0

−1

=
=
=
=
=
=

2
3
4
5
6
7

−2
−3
−3 −2 −1

0

1

2

3

4

<(a11 )

Figure 4: Stability region of a11 with det A = −1 and
m = [1, k] for which pA,m (z) is lossless. The depicted lines
indicate the outer border of the valid set of values.

i=1

The corresponding feedback matrix is

g1
0
0
0
0
 0
g
0
0
0
2

 0
0
g
0
0
3
AS = 
 0
0
0
g
0
4

 1
1
1
1
g5
−g5 −g5 −g5 −g5 1 − g52

If A is unitary, then |a11 | ≤ 1 and pA,m (z) is lossless for
any m. Nonetheless there are certain choices of delays m such
that pA,m (z) is lossless even with |a11 | > 1. For example, for
m1 = 1, m2 = 2:
pA,m (z) =
p0A,m (z)

=

z − a11 z − a∗11 z +
3z 2 − 2a11 z − a∗11 .
3

The roots of p0A,m (z) are

2



(41)

a11 ±


0
0

0

0

0
g6

cS = [g5 g6 , g5 g6 , g5 g6 , g5 g6 , g6 (g52 − 1), 1 − g62 ]

p

(45)

(46)

with m = [m1 , . . . , m6 ]. The triangular feedback matrix AS
has the block matrix form


g1
0>

g2
0> 



g3
0> 


AS = 
(47)
g4
0> 



g5 0> 

a211 + 3a∗11
.
(42)
3
According to Theorem 6, pA,m (z) is lossless if and only if
all roots of p0A,m (z) are on or within the unit circle, i.e.,
|r1/2 | ≤ 1. For  = −1 and a11 = 3, the roots r1/2 = 1
and the corresponding FDN is lossless. This coincides with
the example (8) given in the beginning. Figure 4 depicts the
stability region of a11 , i.e., the set of values a11 for which
pA,m (z) is lossless under fixed det A and m. This plot shows
the outer border of the stability region for m = [1, k] and
det A = −1. It can be observed that for higher k, the stability
regions approximate the unit circle, which is guaranteed by
the unilossless condition. Also, this analysis illustrates that
the practice of employing non-unilossless matrices A for the
FDN design results in rather subtle and difficult determination
of the lossless property. This can become especially difficult
for long delays as the determination of the system poles suffers
from numerical intractability.
r1/2 =

j=5

g10

g20

g30

g40

g50

g6

such that according to (16), AS is reducible. According to
Condition 1 of Theorem 1 is a reducible matrix unilossless if
and only if the irreducible components gi are unilossless, i.e.,
|gi | = 1 for all i. However for real gi , the output from the 5th
and 6th delays is zero, which is consistent to the fact that a
lossless allpass filter cannot have a decaying tail.
B. Dahl and Jot’s Absorbent Allpass FDN
The absorbent allpass FDN suggested in [6] extends every
delay of the FDN by a feedback-feedforward allpass filter.
Similar structures have been discussed also earlier [5], [27].
The open feedback loop is described by

IV. U NILOSSLESS R EVERB T OPOLOGIES
Various topologies for delay based reverberation algorithms
have been proposed over the last 50 years. As the delay lines
can be reordered and the feedback paths can be summarized
in an adjacency matrix, any of these topologies can be represented by an FDN structure. Whereas the computational
efficiency of the original structure may be better than a
general matrix computation, the system analysis benefits from
a standardized representation. In the following, we discuss
some of the classic examples and their FDN versions.

Dm (z −1 )Cm0 ,g (z −1 ) − A,

(48)

where Cm0 ,g (z) = diag [Cm01 ,g1 (z), . . . , Cm0N ,gN (z)] is a
diagonal matrix of allpass filters. In [8], the authors showed
that any allpass FDN can be represented as a standard FDN
with twice the delay lines and the feedback matrix


−A G A
AAP =
,
(49)
I − G2 G
7

SUBMITTED FOR PUBLICATION IN IEEE TRANSACTIONS ON SIGNAL PROCESSING, JUNE 2016
a

where 1 = [1, . . . , 1]> , y = [y1 , . . . , yN ]> and h·, ·i denotes
the scalar product. Matrix AS2 is a Householder reflection,
which is unitary and therefore a unilossless matrix (see Lemma
1). Matrix AS1 satisfies

g

x(n)

z −m1

z −m2

y(n)

H

AS1 Y AS1 = Y ,

−g

where Y = diag (y). Consequently, AS1 is diagonally similar
to a unitary matrix and according to the Condition 2 of
Theorem. 1, AS1 is a unilossless feedback matrix. The matrix
AS1 is therefore an example of an explicitly non-unitary, but
unilossless matrix in application.

(a) Allpass delay Line with feedback



−ag
1 − g2

a
g

(53)



V. C ONCLUSION
x(n)

z −m1

The present work proposed a new characterization of lossless FDNs based on unilossless feedback matrices, which
are independent from the choice of the delays. Further, a
sufficient and necessary condition for unilossless feedback
matrices was given, which sets the complete framework for
any FDN design based on lossless prototypes. Additionally,
some aspects of lossless FDNs with non-unilossless feedback
matrices were discussed. The proposed framework was applied
to three existing artificial reverberation topologies suggesting
a unified analysis approach of delay based physical modeling
topologies based on FDNs.

y(n)

z −m2

(b) Equivalent standard FDN

Figure 5: An allpass filter in series with a delay filter in
a feedback loop with feedback gain a and the equivalent
representation as a standard FDN.

where G = diag ([g1 , . . . , gN ]) and mAP = [m, m0 ]. Figure
5 depicts a single delay line allpass FDN and an equivalent
two delay line standard FDN. It has been shown in [6] that the
allpass FDN is lossless if all |gi | ≤ 1. This can be confirmed
by demonstrating that AAP is diagonally similar to a unitary
matrix:


−A G A
AAP =
I − G2 G




(50)
I 0
−A G AG0 I
0
=
,
0 G0
G0
G
0 G0−1
√
where G0 = I − G2 and

 


−A G AG0
A 0 −G G0
=
(51)
G0
G
0 I
G0 G

A PPENDIX A
F LAWED P ROOF IN [9]
By assuming that each delay line is longer than two samples,
(1) can be restated in the state-space description as follows [9]:
y(n) = cT s(n) + d x(n)
s(n + 1) =

d =
bT
c

T

=
=

d

(55)

[0, ..., 0, bT ]

(56)

T

[0, ..., 0, c , 0, ..., 0].
| {z }

(57)

N

The state-transition matrix is

U1 0
0
 0 U2 0

 ..
..
..

.
.
A =  .
 0
0
0

 P1 P2 P3
0
0
0

C. De Sena’s Scattering Delay Network

···
···
..
.

0
0
..
.

···
···
···

UN
PN
0

where

The scattering delay network (SDN) proposed in [11], [28]
extends the core structure of the FDN and draws the parameterization of the filter coefficients from a room geometry based
analysis. Two different feedback matrix types are employed:
AS1

(54)

where

is unitary. Consequently, AAP is diagonally similar to a unitary
matrix and irreducible such that according to the Condition
2 of Theorem 1 AAP is unilossless. Whereas the implementation complexity might be beneficial for the allpass FDN,
this analysis suggests that in terms of the sonic design of
reverberation, unilossless FDNs offers the full potential of
delay based systems.

2
2
>
=
1y > − I and AS2 =
2 yy − I
h1, yi
kyk2

A s(n) + b x(n)

Uj

(52)

8


0 1 0
0 0 1


=  ... ... ...

0 0 0
0 0 0

···
···
..
.
···
···


R1
R2 

.. 
. 
 (58)
0 RN 

0
0 
A
0
0
0
..
.


0
0

.. 
.

1
0

(59)

SUBMITTED FOR PUBLICATION IN IEEE TRANSACTIONS ON SIGNAL PROCESSING, JUNE 2016

Rj


0···0
0 · · · 0


=  ...

0 · · · 0
| {z }

···
···
..
.

0
0
..
.

···
···
..
.

1

0

···

j−1

and

Pj

=



 0

j − 1  ...


0

1

0

.
 ..
0

Let us define
Uj
Rj
Pj

0
..
.
0
0
0
..
.
0

···
..
.
···
···
···
..
.
···


0
0

.. 
.

0


0
.. 
.

0

0

0

.. 
.
0

>

(60)

(61)

∈ C(mj −2)×(mj −2)

(62)

∈ CN ×(mj −2) .

(64)

∈ C(mj −2)×N

is orthogonal and for this A and A− are diagonally similar despite the fact that the rank property is violated by
rankA(12|34) = 1. The following shows that the rank
property can be dropped in Theorem 10 for the special case
of B = A−∗ . Please refer to the original proof in [24] for
additional technical information on the given proof. Similarly
to the proof of Theorem 10, the following proof is by induction
on the matrix dimension N , where as the base case is N = 4.
The following lemma, which is a combination of Lemma 1
and 4 in [24], provides the inductive step:
Lemma 6. Let N ≥ 4 and suppose A, B ∈ CN ×N are
irreducible. Let
S1 = {i : 1 ≤ i ≤ N : B(ic ) is diagonally similar to A(ic )}
n
o
S2 = i : 1 ≤ i ≤ N : B(ic )> is diagonally similar to A(ic ) .
Then,
1) if |S1 | ≥ 3, then B is diagonally similar to A;
2) if |S2 |S
≥ 3, then B is diagonally similar to A;
3) if |S1 S2 | ≥ 5, then either B or B > is diagonally
similar to A.

(63)

Given the so-called generalized admittance


e = IN−N 0 ,
Γ
0
Γ

Let N ≥ 5. The inductive assumption provides that for
irreducible A, B ∈ CN −1×N −1 , either B or B > is diagonally
similar
S to A. Hence, for sets S1 and S2 as defined in Lemma 6,
|S1 S2 | ≥ 5. Therefore, the induction is valid by Lemma 6.
It remains to proof the base case N = 4.
Let N = 4. The proof of Theorem 4 in [24] applies
Theorem 9 to the 3×3 principal submatrices and arrives at the
following representation based solely on A being irreducible
and A and B having equal corresponding principal minors
and without exploiting the additional rank property:

(65)

where Γ ∈ CN ×N is a Hermitian, positive-definite matrix, we
have
e = diag ([Im −2 , . . . , Im −2 , B]) .
A> ΓA
(66)
1

N

The proof fails in (29) of [9] as
 T


I
A ΓA 0
B=
instead of B =
0
I
0


0
.
AT ΓA

(67)



A PPENDIX B
P ROOF OF T HEOREM 7
We first recall two main results on which the proof of
Theorem 7 is based upon.
Theorem 9 (Hartfiel, [24]). Let N = 2 or N = 3 and A
is irreducible. Then if A and B have equal principal minors
then there is a non-singular diagonal matrix E with either
E −1 AE = B or E −1 AE = B > .

and

a11
 a21
A=
a13 f3
a14 e4


a12
a22

a23 f3
f2
a24 e4
f2

a13
a23
a33
a43

a21
a11
a13
f2
a12 f2 a22 a23
B=
a13 f3 a23 f3 a33
f2
e4
a14 e4 a24
a43
f2

a11 a12 f2 a13 f3
a23 f3
 a21
a22
>
f2
f2
B =
a13
a23
a33
a14
a24
a34

Consequently for N < 4, all matrices A−1 with property
(29) are diagonally similar to B = A∗ or B > = AH . For
higher N , an additional condition has to be posed on A for
the general case.
Theorem 10 (Loewy, [25]). Let N ≥ 4 and A is irreducible,
and for every partition of {1, 2, ..., N } into subsets α, β with
|α| ≥ 2, |β| ≥ 2 either rankA(α|β) ≥ 2 or rankA(β|α) ≥ 2.
Then if A and B have equal principal minors then there is
a non-singular diagonal matrix E with either E −1 AE = B
or E −1 AE = B > .


a14
a24 

a34 
a44

(69)


a14
a24 
,
a34 
a44

a14 e4
a24 e4
f2

a43
a44



(70)


,


where f2 , f3 , e3 , e4 6= 0 are coefficients from diagonal similarity transformations. Because B is by construction diagonally
similar to A−∗ , Jacoby’s identity (27) yields
det A(13|24) = det B(24|13)




a12 a14
a f a23
= det 12 2
det a23 f3
a34
a14 e4 a43
f2
a23 f3
= a12 f2 a43 − a23 a14 e4
a12 a34 − a14
f2


f3
a12 (a34 − f2 a43 ) = a14 a23
− e4
f2

The following example shows that the additional rank
property in Theorem 10 is not necessary. The matrix


−1 4 −2 −2
1 −4 1
2
2

A= 
(68)
2 −1 4 
5 2
−2 −2 −4 1
9

(71)

SUBMITTED FOR PUBLICATION IN IEEE TRANSACTIONS ON SIGNAL PROCESSING, JUNE 2016
and

det A(24|13) = det B(13|24)
#
"


a21
a14
a21
a23
f2
det
= det a23 f3
a14 e4 a43
a34
f2

a21
a23 f3
a34 − a14
f2
f2


a34
f3
a21 (a43 −
) = −a14 a23
− e4 .
f2
f2

[8] ——, “Time-varying Feedback Matrices in Feedback Delay Networks
and Their Application in Artificial Reverberation,” The Journal of the
Acoustical Society of America, vol. 138, no. 3, pp. 1389–1398, Sep.
2015.
[9] D. Rocchesso and J. O. Smith III, “Circulant and elliptic feedback delay
networks for artificial reverberation,” Speech and Audio Processing,
IEEE Transactions on, vol. 5, no. 1, pp. 51–63, 1997.
[10] M. Karjalainen and C. Erkut, “Digital Waveguides versus Finite Difference Structures: Equivalence and Mixed Modeling,” EURASIP Journal
on Advances in Signal Processing, vol. 2004, no. 7, pp. 978–989, 2004.
[11] E. De Sena, H. Hacihabiboglu, Z. Cvetkovic, and J. O. Smith III,
“Efficient Synthesis of Room Acoustics via Scattering Delay Networks,”
Audio, Speech, and Language Processing, IEEE Transactions on, vol. 23,
no. 9, pp. 1478–1492, 2015.
[12] D. Murphy, A. Kelloniemi, J. Mullen, and S. Shelley, “Acoustic Modeling Using the Digital Waveguide Mesh,” Signal Processing Magazine,
IEEE, vol. 24, no. 2, pp. 55–66, Mar. 2007.
[13] J. M. Jot and A. Chaigne, “Digital delay networks for designing
artificial reverberators,” in Audio Engineering Society Convention 90,
Paris, France, Feb. 1991, pp. 1–12.
[14] D. Rocchesso, “The Ball within the Box: A Sound-Processing
Metaphor,” Computer Music Journal, vol. 19, no. 4, pp. 47–57, 1995.
[15] S. J. Schlecht and E. A. P. Habets, “Feedback Delay Networks: Echo
Density and Mixing Time,” submitted for publication.
[16] M. A. Gerzon, “Unitary (energy-preserving) multichannel networks with
feedback,” Electronics Letters, vol. 12, no. 11, p. 278, 1976.
[17] G. H. Golub and C. F. Van Loan, Matrix computations. Baltimore :
Johns Hopkins University Press, 1996.
[18] H. Kuttruff, Room Acoustics, Fifth Edition. CRC Press, Jun. 2009.
[19] E. B. Stouffer, “On the Independence of Principal Minors of Determinants,” American Mathematical Society, Transactions of the, vol. 26,
no. 3, p. 356, Jul. 1924.
[20] J. L. Stuart and J. R. Weaver, “Diagonally scaled permutations and
circulant matrices,” Linear Algebra And Its Applications, vol. 212-213,
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Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley,”
Linear Algebra And Its Applications, vol. 52-53, pp. 769–791, Jul. 1983.
[22] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction
To Algorithms. MIT Press, 2001.
[23] A. Cohn, “Über die Anzahl der Wurzeln einer algebraischen Gleichung
in einem Kreise,” Mathematische Zeitschrift, vol. 14, no. 1, pp. 110–148,
Dec. 1922.
[24] D. J. Hartfiel and R. Loewy, “On matrices having equal corresponding
principal minors,” Linear Algebra And Its Applications, vol. 58, pp.
147–167, Apr. 1984.
[25] R. Loewy, “Principal minors and diagonal similarity of matrices,” Linear
Algebra And Its Applications, vol. 78, pp. 23–64, Jun. 1986.
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Similarity of Matrices and Related Problems,” Algebraic Discrete Methods, SIAM Journal on, vol. 3, no. 4, pp. 429–438, Jul. 2006.
[27] R. Väänänen, V. Välimäki, J. Huopaniemi, and M. Karjalainen, “Efficient and Parametric Reverberator for Room Acoustics Modeling,” in
International Computer Music Conference. The International Computer
Music Association, 1997, pp. 200–203.
[28] E. De Sena, H. Hacihabiboglu, and Z. Cvetkovic, “Scattering Delay
Network: An Interactive Reverberator for Computer Games,” in 41th
International Conference of the Audio Engineering Society. Audio
Engineering Society, Feb. 2011.

(72)

a21 a43 − a23 a14 e4 =

The combination of both identities is then
a34
)
f2
a12 f2 (a34 − f2 a43 ) = a21 (a34 − f2 a43 )
a12 (a34 − f2 a43 ) = −a21 (a43 −

(73)

(a12 f2 − a21 )(a34 − f2 a43 ) = 0.

If the first term (a12 f2 − a21 ) = 0, then A = B.
If the second term (a34 − f2 a43 ) = 0, then B > =
−1
diag (1, f2 , f3 , e4 ) Adiag (1, f2 , f3 , e4 ). Thus, the matrices
A and B or B > are diagonally similar.
A PPENDIX C
PARAMETERIZATION OF C ONJUGATE -I NVOLUTORY
M ATRICES
Lemma 7. For any A with AA∗ =√I, there is a real matrix
M with A = exp(ıM ), where ı = −1.

Proof: Let M = U +ıV , where U , V are real. AA∗ = I
is equivalent to U 2 +V 2 = I and U V = V U . Consequently,
there is a real matrix M with U = cos(M ) and V = sin(M )
such that A = exp(ıM ).
ACKNOWLEDGMENT

The authors would like to thank a number of researchers for
their helpful comments and discussions: G. H. E. Duchamp for
the alternative proof of Lemma 1, V. Protsak for pointing out
the diagonal similarity invariance, R. Bryant and L. Blanc for
the proof of Lemma 7, R. Loewy and D. J. Hartfiel for helpful
comments on their work and D. Rocchesso for his fundamental
contribution to FDN design and supportive discussion of the
present idea.
R EFERENCES
[1] J. Stautner and M. Puckette, “Designing Multi-Channel Reverberators,”
Computer Music Journal, vol. 6, no. 1, p. 52, 1982.
[2] V. Välimäki, J. D. Parker, L. Savioja, J. O. Smith III, and J. S. Abel,
“Fifty Years of Artificial Reverberation,” Audio, Speech, and Language
Processing, IEEE Transactions on, vol. 20, no. 5, pp. 1421–1448, Jul.
2012.
[3] M. R. Schroeder, “Natural Sounding Artificial Reverberation,” Journal
of Audio Engineering Society, vol. 10, no. 3, pp. 219–223, 1962.
[4] J. A. Moorer, “About This Reverberation Business,” Computer Music
Journal, vol. 3, no. 2, pp. 13–17, Jun. 1979.
[5] J. Dattorro, “Effect Design, Part 1: Reverberator and Other Filters,”
Journal of Audio Engineering Society, vol. 45, no. 9, pp. 660–684, 1997.
[6] L. Dahl and J. M. Jot, “A Reverberator based on Absorbent All-pass
Filters,” in Digital Audio Effects, Workshop on, Verona, Italy, Dec. 2000,
pp. 1–6.
[7] S. J. Schlecht and E. A. P. Habets, “Connections Between Parallel and
Serial Combinations of Comb Filters and Feedback Delay Networks,”
in Acoustic Signal Enhancement, International Workshop on. VDE,
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