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IPSJ SIG Technical Reportexactly ensured, one should at least look for outputs which are “as consistent as
possible”. This is in particular the case when trying to resynthesize a signal froma modified magnitude STFT spectrogram, for which there is in general no phasesuch that their combination is consistent. Carefully studying the structure ofcomplex STFT spectrograms and quantifying the consistency of a set of complex
numbers is thus a crucial issue.
In this article, we derive explicit consistency constraints for STFT spectrogramsas the kernel of a simple linear operator in the complex time-frequency domain
with coefficients depending on the window length, the frame shift and the analysis
and synthesis windows used to build the spectrogram or which the spectrogram isassumed to have been obtained from. The norm of the image of a set of complex
numbers by this linear operator defines a numerical consistency criterion, which
can for example be used as a prior distribution on complex spectrograms when
performing separation tasks in the complex time-frequency domain, or, as we shallinvestigate here, as an objective function on the phase when trying to recover
the most coherent phase for a given magnitude spectrogram.
We will first derive consistency constraints for STFT spectrograms, then explain how to define a numerical consistency criterion based on them, and finallyintroduce an algorithm for phase reconstruction based on the optimization of an
objective function derived from the consistency criterion and show how it can beused to develop a flexible real-time time-scale modification algorithm.

Spectrogram consistency and its application
to phase reconstruction
Jonathan Le Roux,†1 Hirokazu Kameoka,†1
Nobutaka Ono†2 and Shigeki Sagayama†2
In this article, we derive the constraints which a set of complex numbers mustverify to be a consistent STFT spectrogram, i.e., to be the STFT spectrogramof an actual real-valued signal, and describe how they lead to an objective function measuring the consistency of a set of complex numbers as a spectrogram.
We then present a flexible phase reconstruction algorithm based on a local approximation of the consistency constraints and derive a real-time time-scale
modification algorithm.

1. Introduction
Many acoustical signal processing techniques, developed for a wide range ofapplications such as source separation [7, 11–13], noise canceling [2], time-scale
and pitch-scale modifications or more generally audio modification [6], involvea processing of the time-frequency representation of a signal obtained by theshort-time Fourier transform (STFT).
However, as the STFT representation is usually obtained from overlapping
frames of a waveform, it is redundant and characterized by a particular structure. Starting from a set of complex numbers in the complex time-frequency
domain, it is thus not guaranteed whether there exists a signal in the time domain whose STFT is equal to that set of complex numbers. Therefore, to avoid
irrealistic solutions, any processing performed on STFT spectrograms should ensure that its outputs are “consistent spectrograms”, i.e., that they all are the
STFT of some time-domain signal, or, when dealing with magnitude or power
spectrograms, that there exist phases which can be coupled with the ouput magnitude spectrograms to give consistent spectrograms. If consistency cannot be

2. Characterization of consistent STFT spectrograms
2.1 Perfect reconstruction constraints on the window functions
Let (x(t))t∈Z be a digital signal. We review here the conditions for perfectreconstruction of the signal through STFT and inverse STFT [1, 4]. Let N be
the window length, R the window shift, w the analysis window function and s thesynthesis window function. We suppose that w and s are zero outside the interval
0 ≤ t ≤ N − 1. We assume that the window length N is an integer multiple ofthe shift R, and we note Q = N/R. We shall denote by θ = (w, s, N, R) the setof all parameters related to analysis and synthesis.
It can be shown [4] that there is perfect reconstruction through the inverse
STFT under the following necessary conditions:Q−1∑
1=w(t − qR)s(t − qR), ∀t.
(1)†1 NTT コミュニケーション科学基礎研究所
NTT Communication Science Laboratories†2 東京大学大学院情報理工学系研究科
Graduate School of Information Science and Technology, The University of Tokyoq=0

1

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By introducing the coefficients
k+qR
1 ∑
(θ)
αq,p
=
w(k)s(k + qR)e−j2πp N − δp δq ,N

2.2 Consistency operator
Let (Hm,n )0≤m≤M −1,0≤n≤N −1 be a set of complex numbers, where m will correspond to the frame index and n to the frequency band index, and w and s beanalysis and synthesis windows verifying the perfect reconstruction conditions (1)for a frame shift R. STFTw will denote the STFT with analysis window w, andiSTFTs the inverse STFT with synthesis window s. When not indicated, we willassume that we use w and s respectively.
As we are interested here in spectrograms corresponding to real-valued signals,
we shall assume in the following that N = 2P and that all the sets of complex
numbers considered are “conjugate symmetric”, i.e., ∀n ∈ [[1; P − 1]], H(m, P +n) = H(m, P − n) and the elements at frequency band indices 0 and P are realvalued. These sets form an R-vector subspace of CM ×N , denoted by S . STFT
spectrograms obtained from real-valued signals are examples of such sets, but
the point to be made in this paper is that not all such sets can be obtained
as STFT spectrograms of real-valued signals. For the set H to be a consistent
STFT spectrogram, it needs to be the STFT spectrogram of a signal x(t), whichby perfect reconstruction can be none other than the result of the inverse STFTof the set (Hm,n ). A necessary and sufficient condition for H to be a consistent
spectrogram is thus for it to be equal to the STFT of its inverse STFT. The point
here is that, for a given window length N and a given frame shift R, the operation
iSTFTs ◦STFTw from the space of real-valued signals of length T = (M −1)R+N
to itself is the identity, while STFTw ◦ iSTFTs from S to itself is not.
Altogether, the set of consistent spectrograms can be described as the kernelof the R-linear operator from S to itself defined by
Fw,s (H) = (STFTw ◦ iSTFTs − I M N )(H),(2)
where I L denotes the L×L identity matrix. We shall refer to this operator as the“consistency operator”, and write F instead of Fw,s when there is no ambiguity.
2.3 Derivation of explicit consistency constraints
We can derive consistency constraints for STFT spectrograms in the timefrequency domain by explicitly stating that a consistent spectrogram H must bein the kernel of F. A simple computation leads to the following expression for
the image of H by F:
Q−1
N
−1
∑
∑
′ k+qRn
1 ∑F(H)m,n =w(k)e−j2πk N
s(k + qR)
Hm−q,n′ ej2πn N − Hm,n .
N′kq=−(Q−1)(4)kwhere −(N − 1) ≤ p ≤ N − 1 and δi is the Kronecker delta (δ0 = 1 and δi = 0
for i ̸= 0), and stating that F(H)m,n must be equal to 0 for all m and n, we
can obtain the consistency constraints we are looking for, summarized in thefollowing
Proposition. For analysis and synthesis windows w and s verifying the perfectreconstruction conditions (1) for a frame shift R, a set of complex numbers H ∈S is a consistent spectrogram if and only if, ∀m ∈ [[0, M − 1]], ∀n ∈ [[0, N − 1]],
Q−1
N
−1
∑
∑
qR
(θ)
F(H)m,n =ej2π N n αq,n−n′ Hm−q,n′ = 0,(5)n′ =0 q=−(Q−1)(θ)

where the coefficients α are defined by (4).
The interesting point in the above relation is that we characterized the consistency of a set of complex numbers directly in the time-frequency domain, without
having to make apparent the connection with a certain time domain signal.
3. Numerical consistency criterion
3.1 Relaxing the constraints
Equation (5) represents the relation between a set of complex numbers andthe STFT of its inverse STFT, as its left member is F(H). Instead of enforcing
consistency through the “hard” constraints (5), which may be difficult to handle,
these can be relaxed by introducing any vector norm of F(H): this indeed leads
to a numerical criterion which can be used to quantify how far a set of complex
numbers is from being consistent. We can for example consider in particular theL2 norm of F(H), as we will show later on that this special choice leads to acriterion which is related to that used by Griffin and Lim to derive their iterative
STFT algorithm [4]. This defines a consistency criterion I(H) = ||F(H)||2 asfollows:∑
2
I(H) =
F(H)m,n .
(6)m,n3.2 Consistency as a penalty function
We can think of using numeral consistency criterions such as (6) as penalty
functions, for example when working on source separation or spectrogram modification algorithms in the complex time-frequency domain. In such situations

n =0

(3)

2

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where one attempts to estimate complex spectrograms following certain properties from an input signal, ensuring that the estimated sets of complex numbers
are consistent spectrograms is both likely to lead to better, because more consistent and thus more meaningful, results, and to ease the optimization process by
reducing the dimension of the problem, as it will tend to discard solutions which
are not coherent.
We shall however leave these issues to forthcoming papers and investigate here
the use of the consistency criterion introduced above to define an objective function on phase when the magnitude is fixed, as described in the next section.

d(x(k+1) , A)Ĩ(φ(k) )x̂(k+1)x̂(k)jφ(k+1)

Ae

Aejφ

(k)

4. Phase reconstruction for a modified STFT spectrogram
We consider here the application of the consistency criterion (6) to develop an
algorithm for reconstructing the most consistent phase given a modified magnitude STFT spectrogram. The iterative STFT algorithm introduced by Griffinand Lim [4] is the reference for such algorithms. Its principle is to find the consistent STFT spectrogram with magnitude closest to a given modified magnitude
spectrogram. On the other hand, the algorithm we propose looks for a phasesuch that the spectrogram obtained by associating it with that magnitude spectrogram is as consistent as possible. Although conceptually close to the iterativealgorithm, a crucial difference is that it operates directly in the time-frequency
domain.
4.1 Objective function for phase reconstruction problems
In the problem of phase reconstruction, we are given a set of real non-negative
numbers Am,n which are supposedly the amplitude part of an STFT spectrogram,for example obtained through modifications of the power spectrum of a sound.
The goal is to estimate the phase ϕm,n to adjoin to A such that Am,n ejϕm,nis as close as possible to be a consistent STFT spectrogram. This amounts tominimizing the consistency criterion I w.r.t. the phase ϕ, with the amplitude A
given, defining the following objective function:
∑
∑
2
qR
(θ)
ej2π N n αq,pĨ(ϕ) =
Am−q,n−p ejϕm−q,n−p ,(7)|H| = A
Fig. 1 Illustration of the iterative STFT algorithm and the relation between the objectivefunction Ĩ and the squared distance d(x, A).
(k+1)

(k+1)

x̂(k+1) the STFT of x(k+1) , and ϕm,n the phase of x̂(k+1) , ϕm,n = ∠x̂(k+1) .
They showed that this procedure estimates a real-valued signal x which minimizes (at least locally) the squared distance∑2d(x, A) =|x̂|m,n − Am,n ,(8)m,ni.e., the squared error between the magnitude of the STFT x̂ of x, and themagnitude spectrogram A. As can be seen in Fig. 1, we shall note that theobjective function Ĩ measures a slightly different quantity from the squared distance (8), but that the iterative STFT algorithm also converges to a minimumof (7). Indeed, both quantities become equivalent near convergence, as one canshow that d(x(k+1) , A) ≤ Ĩ(ϕ(k) ) ≤ d(x(k) , A) [4]. However, the objective function Ĩ we introduced has the advantages to be explicit and defined directly in thetime-frequency domain, and in its general version (6) not to be limited to phasereconstruction problems with fixed magnitude.
4.2 Direct optimization of Ĩ
The iterative STFT algorithm, as mentioned above, can be used to minimize
Ĩ. However, this can be considered as an indirect minimization, and it is worth
looking at a direct minimization of Ĩ through classical optimization methods.

m,n p,q s.t. |n−p|∈[[0,N −1]]

In [4], Griffin and Lim presented the iterative STFT algorithm, which consists(k)in iteratively updating the phase ϕm,n at step k by replacing it with the phaseof the STFT of its inverse STFT while keeping the magnitude A. The algorithm(k)is illustrated in Fig. 1, where x(k+1) denotes the inverse STFT of Am,n ejϕm,n ,3

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Vol.2009-MUS-81 No.8
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IPSJ SIG Technical ReportpThis will indeed provide us with the freedom to modify/approximate the objectivefunction on one hand, and to select how each bin will be dealt with on the other.
One could for example decide not to update the phase for certain bins which
are considered reliable. Further in this direction, one could try to reconstructbins which have been discarded by a binary mask from the bins which have been
determined as reliable by minimizing the consistency criterion (6) with respect toboth the magnitude and phase of those bins while keeping others fixed. Finally,
one could also imagine introducing weights depending on frequency to emphasize
stronger consistency in certain frequency regions based on perceptual criteria, oron magnitude to give more importance to the reduction of inconsistency aroundbins with large magnitude. The bin selection presented in Section 4.4 is a discrete
version of this last idea.
4.3 Approximate objective function and phase coherence
Here, we will make the following two approximations. We will first neglect
the influence of ϕm,n in all the terms F(H)m′ ,n′ other than F(H)m,n , which(θ)
is the one where it is multiplied by α0,0 . The motivation behind this first ap(θ)
proximation is that the coefficient α0,0 dominates over the other coefficients. By
assuming the other phase terms fixed, we will then update each bin’s phase ϕm,n(θ)
so that α0,0 Am,n ejϕm,n is in opposite direction with the terms coming from theneighboring bins, while keeping its amplitude Am,n fixed. This corresponds to(θ)
performing a coordinate descent [14]. More precisely, as α0,0 = 1/Q − 1 < 0, theupdate for bin (m, n) is
(
)
∑
qR
(θ)
ej2π N n αq,pϕm,n ← ∠Hm−q,n−p .
(9)

8
4
0
−4
−8
−1

0
q

1

(θ)

Fig. 2 Magnitude of the central coefficients αq,p for N = 512, R = 256, a Hanning analysis
window and a rectangular synthesis window.

would be involved if using
terms. The update becomes: )( all the∑qR(θ)
Hm−q,n−p ,ϕm,n ← ∠ej2π N n αq,p(10)(p,q)̸=(0,0) s.t. |p|≤lwhere frequency indices are understood modulo N . For l = 2 and a 50 % overlap,
for example, we only consider 5 × 3 coefficients.
4.4 Taking advantage of sparsenessAs evoked above, using a direct optimization of the objective function Ĩ enables
us to select which bins to update. This can be the key to deal with problems
where only a part of the spectrogram has to have its phase reconstructed, but
it can also in general be used to lower the computational cost. Indeed, we canuse the sparseness of the acoustic signal to limit the updates to bins with asignificant amplitude, or progressively decrease the amplitude threshold above
which the bins are updated, starting with the most significant bins and refining
afterwards. This idea can be related to the peak picking techniques in [5, 6].
4.5 Exact optimization through an auxiliary function method
We note that the objective function (7) can be minimized based on an auxiliary
function method. Let (Gm,nm′ ,n′ ) be the matrix representation of F. We then have∑m,nF(H)m,n = m′ ,n′ Gm,nm′ ,n′ Hm′ ,n′ . If we introduce auxiliary variables Z m′ ,n′ such
∑
m,n
m,n
that ∀m, n, m′ ,n∑′ Z m′ ,n′ = 0, then for any such Z and for any δm′ ,n′ > 0 suchm,nthat for all m, n, m′ ,n′ δm′ ,n′ = 1, we can show that
∑
2
1
m,n
m,n
jϕm′ ,n′||F(H)||2 ≤.
(11)m,n Z m′ ,n′ − Gm′ ,n′ Am′ ,n′ e
δm′ ,n′′′(p,q)̸=(0,0) s.t. |n−p|∈[[0,N −1]](θ)

Second, noticing that most of the weight in the coefficients αq,p is actuallyconcentrated near (0, 0), as can be seen in Fig. 2 for a window length N = 512
and a frame shift R = 256, with a Hanning analysis window and a rectangularsynthesis window, we can further approximate the update equations (9) by using
only (2l + 1) × (2Q − 1) central coefficients instead of the total N × (2Q − 1),
where l ≪ N . This approximation is motivated by the importance of local phasecoherences, in particular the so-called “horizontal” and “vertical” coherences, toobtain a perceptually good reconstructed signal, and can be considered close tophase-locking techniques [5, 6, 10].
This approximation enables us to compute directly the update of each bin
through the summation of a few terms, instead of the whole convolution whichm,n,m ,nMinimizing the auxiliary function in the right-hand term first w.r.t. Z then w.r.t.
ϕ leads to the following update(equation for ϕ:)
ϕm,n ← Arg am,n Hm,n − (F H F(H))m,n ,(12)4

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IPSJ SIG Technical Report∑m′ ,n′ 2 m′ ,n′where am,n =depends on the choice of δ. As δ is ofm′ ,n′ |Gm,n | /δm,nvery large dimension, it is preferable to set it in such a way that a can bem,ncomputed ∑directly without explicitely computing δ. If we consider δm′ ,n′ =
m,n q
m,n q|Gm′ ,n′ | / m′ ,n′ |Gm′ ,n′ | , where q > 0 is a tunable exponent, we get
∑ (θ)
∑ (θ)
am,n =|αm′ ,n′ |2−q|αm′ ,n′ |q = a.
(13)
m′ ,n′and introduce artifacts of their own [6]. These artifacts have been shown to bemainly connected to phase incoherences, and special care must thus be taken
when estimating the phases in the modified signal’s STFT spectrogram. The
iterative STFT algorithm of Griffin and Lim has been proposed as a way to correctsuch phase incoherences, although the computational cost and the slow speedof convergence have been obstacles to its adoption in commercial applications.
The algorithm we introduced is a flexible alternative to iterative STFT, and byan active use of sparseness and the reduction of the number of multiplications
involved at each step, should lead to a lower computational cost.
4.6.2 Sliding-block analysis for real-time processing
Inspired by an idea in [9], we derive a real-time optimization scheme for theobjective function introduced above based on a sliding-block analysis. The spectrogram is not processed all at once, but progressively from left to right, making
it possible to change the parameters while sound is being played. The waveform
to be time-scaled is read N samples at a time and with an average frame shift off R where f is the time-scale multiplication factor. The phase updates describedabove are performed only on a sliding block of b frames. Frames exiting thesliding block are then used to resynthesize the output signal, with a frame shift
R, leading to a signal whose length is 1/f times the length of the original signal.
This way of building intermediate frames of the spectrogram is arguably morereliable than interpolating neighboring spectrogram frames and leads to an initial
estimate of the phase which can be used as a starting point for both the iterative
STFT and the proposed method.
4.6.3 Experimental evaluation
We implemented the proposed method and the iterative STFT algorithm andcompared their convergence speed on the time-scale modification of the first 23 sof Chopin’s Nocturne No. 2. The time-scale modification factor was set to 0.7, theframe length to 1024 and the frame shift to 512, for a final length of approximately(θ)
32 s. We used a 5 × 3 approximation of the coefficients αq,p for our algorithm.
We ran our algorithm in two different experimental conditions, one where all thebins are updated at each iteration, and the other relying on sparseness, where atiteration k, only bins whose amplitude is larger than Ae−Bk are updated. Here,
we used A = 1 and B = 0.005, which were determined experimentally. The
objective function I(H) is used as a measure of convergence, and represented indecibels, with the initial value as a reference.
A comparison of the speed of convergence in terms of the decrease of I(H) w.r.t.

m′ ,n′

Intuitively, a acts as a learning weight in Eq. (12): the larger a, the slower ϕ will
move from its current value. As convergence is guaranteed anyway, we should
thus look∑ for a as small as possible, which is the case for q = 1, where we havea = ( |α|)2 . We then have()ϕ ← Arg aH − F H F(H) .
(14)
It is interesting to note the link with Griffin and Lim’s update, which can be
written with our notations as()
ϕ ← Arg H + F(H) ,(15)
In the special case where the windows w and s are equal (for example with thesine window), then F H = F and one can see that F H F(H) = −F(H). The only
difference is then the a factor. We haven’t been able so far to find a setting forδ which would lead to a equal to or close to 1, and noticed through experiments
that, due to that factor, Griffin and Lim’s update was faster than the auxiliary
approach one in terms of the decrease of inconsistency per iteration. We plan toinvestigate this issue in the future as well.
4.6 Time-scale modification
4.6.1 Need for an efficient frequency-domain algorithm
Many methods for time-scale and pitch-scale modification of acoustic signals
have been proposed, and the interest on this subject intensified in recent years
with the increase in the commercial application of such techniques. So far, mostcommercial implementations rely on time-domain methods, usually variations
on Synchronous Overlap and Add (SOLA) or Pitch Synchronous Overlap andAdd (PSOLA) techniques [8]. Their advantages are a low computational costand good quality results for small modification factors (smaller than ±20 %or ±30 %) and monophonic sounds. For larger factors, polyphonic sounds or
non-pitched signals, however, the quality of the results drops severely. On theother hand, frequency-domain methods, such as the phase vocoder [3], are notlimited to such constraints, but they involve a much higher computational cost5

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STFT spectrograms, and explained how it could be used to introduce penalty
functions in the complex time-frequency domain and to define an objective function on phase when working in the time-frequency magnitude or power domain.
We developed a flexible phase reconstruction algorithm which can take advantageof the sparseness of the input signal and can take into account regions where phaseis reliable. We applied this algorithm for time-scale modification, and explained
how to perform a real-time processing based on a sliding-block analysis.

Inconsistency (dB)0
Iterative STFT
Proposed method
Proposed method (sparse)

−5
−10
−15
0

50

100
Nb. of iterations

150

200

Acknowledgments

Fig. 3 Comparison of the evolution of the inconsistency measure I(H) w.r.t. the number of
iterations for the iterative STFT algorithm and the proposed method.

The authors would like to thank Dr. Emmanuel Vincent for fruitful discussions.
References

Table 1 Computation time (s) required to reach a certain decrease of the inconsistency
measure I
Time to reach
Iterative STFT (G&L)
Proposed method
Proposed method (sparse)

-10dB
3.9s
0.6s
0.2s

-13dB
10.9s
2.4s
0.8s

1) Allen, J.B. and Rabiner, L.R.: A unified approach to short-time Fourier analysis,
Proc. of the IEEE, Vol.65, No.11, pp.1558–1564 (1977).
2) Boll, S.: Suppression of Acoustic Noise in Speech Using Spectral Subtraction, IEEE
Trans. ASSP, Vol.27, pp.113–120 (1979).
3) Dolson, M.: The phase vocoder: A tutorial, Computer Music Journal, Vol.10, No.4,
pp.14–27 (1986).
4) Griffin, D.W. and Lim, J.S.: Signal Estimation from Modified Short-Time Fourier
Transform, IEEE Trans. ASSP, Vol.32, No.2, pp.236–243 (1984).
5) Karrer, T., Lee, E. and Borchers, J.: PhaVoRIT: A Phase Vocoder for Real-TimeInteractive Time-Stretching, Proc. ICMC, pp.708–715 (2006).
6) Laroche, J. and Dolson, M.: Improved Phase Vocoder Time-Scale Modification of
Audio, IEEE Trans. SAP, Vol.7, No.3, pp.323–332 (1999).
7) Le Roux, J., Kameoka, H., Ono, N., de Cheveigné, A. and Sagayama, S.: Single
Channel Speech and Background Segregation through Harmonic-Temporal Clustering, Proc. WASPAA, pp.279–282 (2007).
8) Moulines, E. and Laroche, J.: Non-Parametric Techniques for Pitch-Scale andTime-Scale Modification of Speech, Speech Comm., Vol.16, pp.175–206 (1995).
9) Ono, N., Miyamoto, K., Kameoka, H. and Sagayama, S.: A Real-time Equalizer ofHarmonic and Percussive Components in Music Signals, Proc. ISMIR, pp.139–144(2008).
10) Puckette, M.S.: Phase-locked vocoder, Proc. WASPAA (1995).
11) Schmidt, M.N. and Mørup, M.: Nonnegative Matrix Factor 2-D Deconvolution forblind Single Channel Source Separation, Proc. ICA, pp.700–707 (2006).
12) Smaragdis, P.: Discovering Auditory Objects Through Non-Negativity Constraints, Proc. SAPA (2004).
13) Virtanen, T.: Monaural Sound Source Separation by Nonnegative Matrix Factorization With Temporal Continuity and Sparseness Criteria, IEEE Trans. ASLP,
Vol.15, No.3, pp.1066–1074 (2007).
14) Zangwill, W.I.: Nonlinear Programming: A Unified Approach, Prentice Hall (1969).

-15dB
23.8s
7.6s
1.6s

the number of iterations (or, equivalently, the block size) is shown in Fig. 3. One
can see that, although our algorithm is based on an approximation of the original
objective function I(H), it outperforms the iterative STFT algorithm in termsof speed of convergence as the number of iterations required to reach a given
decrease of inconsistency. We also compared the computation times of the threemethods, measuring only the time required by the phase reconstruction part of
the algorithm as the other parts are identical. With our implementations, for
200 iterations, the iterative STFT algorithm took 29.1 s, our method with full
updates 24.2 s, and our method using sparseness 2.1 s. Looking at the amountof time required to reach a certain decrease in inconsistency illustrates how ourmethod, by combining speed of convergence and lower computational cost, leads
to a much shorter computational time than iterative STFT, as illustrated inTable 1 for the experimental conditions described above.
In terms of flexibility, speed of convergence and computation time, our algorithm thus outperforms the iterative STFT algorithm.
5. Conclusion
We introduced a general framework to assess for the consistency of complex

6

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