3-10-3
Phase initialization schemes for faster spectrogram-consistencybased signal reconstruction ∗◎ Jonathan Le Roux, Hirokazu Kameoka (NTT CS Labs),
Nobutaka Ono and Shigeki Sagayama (The University of Tokyo)

1Introduction

In previous contributions [1, 2], we presented a fast algorithm for the reconstruction of a time-domain signal from amagnitude short-time Fourier transform (STFT) spectrogram,and showed experimentally that it could lead to substantial reduction in computation time compared to previous work byGriffin and Lim [3] when both algorithms are initialized using zero phase. We study here other strategies for phase initialization, compare the performance of our method with thestate-of-the-art RTISI-LA algorithm [4], and consider several
extensions and combinations of the two approaches.

2Signal reconstruction from magnitude
spectrograms

Let (Am,n )0≤m≤M −1,0≤n≤N −1 ∈ CM N denote an arrayof real non-negative numbers, where m will correspond to theframe index and n to the frequency band index. This array,for example obtained through modifications of the magnitude
spectrogram of a sound, is assumed to be the magnitude partof a hypothetical STFT spectrogram. The problem of signal
reconstruction is to find the “most relevant” phase for A.
Griffin and Lim [3] formulated the problem as that of estimating a real-valued time-domain signal x such that the magnitude of its STFT is closest to the magnitude spectrogram A
in a least-squares sense. To solve it, they proposed the iterative STFT algorithm, which consists in iteratively updating
the phase ϕ(k) at step k by replacing it with the phase of theSTFT of its inverse STFT while keeping A fixed. Alternatively, we showed in [1] that this problem can also be formulated as that of estimating a phase ϕ such that H = Aejϕ is“as consistent as possible”, where we call “consistent spectrograms” the elements of CM N which can be obtained as theSTFT spectrogram of a time-domain signal. The lack of consistency, or inconsistency, of any array H is numerically characterized by the L2 norm of I(H) = ∥G(H) − H∥2 , whereG(H) = STFT(iSTFT(W )) denotes the STFT of the inverse
STFT of H. The two formulations can be shown to be equivalent near convergence, and we can actually interpret the iterative STFT as the application of the auxiliary function method tothe minimization of Ĩ. Our consistency-based formulation has
the advantage to be defined directly in the time-frequency (T-F)
domain, to allow for fast computation of G(H) thanks to local
approximations, and to justify the use of partial updates limited
to a subset of T-F bins. Based on it, we developed a signal reconstruction algorithm which can achieve the same reductionof inconsistency 10 to 60 times faster than the iterative STFT
algorithm. We shall refer to [2] for more details, and only give
here the update equations: at iteration k, we update the phase
ϕm,n of all bins whose magnitude is larger than ck , where ck
is exponentially decaying with k, through)( ∑qR
ϕm,n ← ∠ej2π N n αq,p Hm−q,n−p , (1)(p,q)̸=(0,0)|p|≤lwhere the αq,p are weights depending on the analysis windowof the STFT, N the window length, R the window shift, l a∗truncation order typically set to 5, and H is the current spectrogram estimate.

3 RTISI-LA
In our previous papers, we performed all comparisons bysimply initializing the algorithms with zero phase. Our intent was to compare the performances of Griffin and Lim’smethod (“G&L”) and our method under the same conditions,leaving the question of using better initial phase estimates tofuture works. In order to investigate further the applicability ofour approach, we consider here its comparison with a state-ofthe-art method relying on better initial phase estimates, Zhu etal.’s real-time iterative spectrogram inversion with look-ahead(RTISI-LA) [4], as well as several extensions and combinationsof the two approaches.
RTISI-LA is based on a sliding-block approach: some processing is performed on the block constituted of frames m tom+k; when it is over, frame m exits the block and is no longer
updated, while a new frame with index m + k + 1 is inserted in
the block. The processing performed by RTISI-LA consists in:1) computing the spectrum of the short-term signal obtained byoverlap-adding the contributions at frame m + k of the previous frames m to m + k − 1 overlapping with frame m + k,using the best available phase estimates for these frames. The
initial phase of frame m + k is determined as the phase part ofthis spectrum; 2) performing p iterative STFT updates on thecurrent block, taking into account all past frames but discarding the contribution of future frames with index larger than orequal to m + k + 1. The rationale for discarding future framesis that the absence of a reliable phase estimate for them is likely
to make their contribution more of a disturbance than a usefulclue. In order to partially compensate for the above asymmetry
between previous and future frames, Zhu et al. propose to use
asymmetric analysis windows with a reverse effect when analyzing frame m + k. Although the above procedure relies onheuristic considerations, the authors show that it leads to muchbetter performance than G&L for a given number of iterations
per block p (note that, as RTISI-LA is an online algorithm discarding future frames, the result of the algorithm with a given
number of iterations p + 1 cannot be deduced from the resultfor p).

4

Further developments

Other than just comparing our previously proposed methodwith RTISI-LA, we shall also consider new developments
stemming from insights gained from both approaches: combined usage of initial phase estimation through RTISI-LA, followed by our method (“RTISI-LA Then Proposed”); introduction of a consistency-based RTISI-LA-like initial phase estimation in our method (“No future initialization”); design of aconsistency-based version of RTISI-LA directly in the timefrequency domain (“TF-RTISI-LA”). The first development isself-explanatory, and we shall only explain the two others.
In RTISI-LA, the first phase estimate for a new frame is obtained by discarding the contributions of that frame and of future ones. This amounts to considering that the phase values forthese frames are uniformly distributed, and that the expectation

スペクトログラム無矛盾性に基づく位相付加アルゴリズムにおける位相初期値選択による高速化の検討、
ルルー・ジョナトン、亀岡弘和 (NTT)、小野順貴、嵯峨山茂樹 (東大情報理工)日本音響学会講演論文集- 601 -2010年9月

0
Griffin−Lim−5

RTISI−LA−10Prop.
RTISI−LA then Prop.

Inconsistency (dB)

Inconsistency (dB)0−15−20−25

0510152025

Griffin−Lim−10

RTISI−LA−20Prop.
RTISI−LA then Prop.

−30−4030051015Time (s)Fig. 1 Evolution of C (music, f = 0.7).

35

RTISI−LAProp.
q>0q>0 + asym.
q>0 or q=0 & p>0
q>0 or q=0 & p>0 + asym.

Griffin−Lim−5RTISI−LA−10Prop.
RTISI−LA then Prop.

Inconsistency (dB)

Inconsistency (dB)30

Evolution of C (speech, f = 1).

Fig. 3−5−15−202500051015202530

354045

Evolution of C (speech, f = 0.7).

of the complex value of the corresponding T-F bins is thus zero.
This can be incorporated in our framework by performing afirst swipe from left to right over the spectrogram, and limiting
the sum in (1) to q > 0, thus considering only the bins which
are in the frames to the left. We can also consider using the already initialized bins in the same frame, i.e., include the termswith q = 0 and p > 0. Once all bins have been initialized, weuse the classical consistency-based updates.
As our approach consists in replacing computations involving round trips between the T-F domain and the time domain
by local computations in the T-F domain, one can easily design
a T-F domain equivalent of RTISI-LA. The computation of Gas STFT ◦ iSTFT is simply replaced by its corresponding local computations similar to (1). The asymmetric windows caneven be introduced by computing specific sets of weight parameters for the updates of the newest frame. These weights
are computed in the same way as the ones used in (1), only replacing the analysis window by the asymmetric one. We shallhowever leave the evaluation of this method to future works.

Experimental evaluation

We investigate the performances of the various proposed
methods and compare them with that of RTISI-LA and G&L
in terms of the decrease of a normalized inconsistency meaI(H)sure defined as C(H) = 10 log10 ∥H∥
2 (i.e., the objectivefunction I(H) represented in decibels with the total energyof the original spectrogram as a reference) w.r.t. the computation time. The task we consider is that of the estimation of
the most consistent phase for a given magnitude spectrogram
for the purpose of time-scale modification. All methods wereimplemented in C. As the data, we used speech from male andfemale speakers divided in 10 files of average length 33 s, and 8
excerpts of music pieces [5], each 23 s long. The sampling rate
was 16 kHz. The sine window was used as analysis and synthesis window. The time-scaled spectrograms are built through asliding-block analysis technique described in [1]. The timescale modification factors f used were 0.7 (slow down by30 %), 1 (no time-scale modification), and 1.3 (pace up by30 %). We used a frame length of 1024 samples and two settings for the frame overlap, namely 50 % and 75 %, although
we shall show here only the results for 75% overlap. G&L
and the previously proposed method use the magnitude of thetime-scaled spectrograms as initial point, and are stopped after
200 iterations.
The sparseness threshold at iteration k is set toc
ae−bk E, where E denotes the mean magnitude of the given
magnitude spectrogram. The parameters were set to a = 100,b = 0.1 and c = 1 based on a preliminary experiment.

日本音響学会講演論文集−15−20−30024681012Time (s)“No future” initialization (music, f = 0.7).

Fig. 4
0RTISI−LAProp.
q>0q>0 + asym.
q>0 or q=0 & p>0q>0 or q=0 & p>0 + asym.

−5Inconsistency (dB)

Fig. 2−10−25Time (s)520Time (s)−10−15−20−25−30−35

024681012141618Time (s)Fig. 5 “No future” initialization (speech, f = 0.7).
We can see in Fig. 1 that our previously proposed method isoutperformed by RTISI-LA on the music data for 75 % overlap.
This is however not the case on the speech data, as shown inFigs. 2 and 3, as well as for 50 % overlap (omitted). In all cases,
using RTISI-LA as an initialization step for our method leads
to the most reliable results. We think that the difference of behavior on speech and music data is due to the greater stabilityof sounds in music data, as starting from incoherent neighboring bins is then more likely to lead to local minima.
Figures 4 and 5 show the evolution of C for various kindsof “No future” initialization steps on music and speech data
respectively, for f = 0.7. The best results are obtained for
an initialization involving only the bins for q > 0 and using
weights α computed using an asymmetric window. Similar
results were obtained for 50 % overlap and other values of f .

6Conclusion

We compared the consistency-based method for signal reconstruction that we previously proposed to the RTISI-LA
method of Zhu et al., and considered several combinations of
those methods, leading to yet faster reconstruction algorithms
outperforming the state-of-the-art methods.

References[1] Le Roux et al., Proc. SAPA, 23–28, 2008.
[2] Le Roux et al., Proc. DAFx-10, 2010.
[3] Griffin and Lim, IEEE Trans. ASSP, 32(2), 236–243, 1984.
[4] Zhu et al., IEEE Trans. ASLP, 15(5), 1645–1653, 2007.
[5] Goto, Proc. ICA, 553–556, 2004.

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