Designing nearly tight window for improving time-frequency masking
Tsubasa KUSANO(1) ; Yoshiki MASUYAMA(2) ; Kohei YATABE(3) ; Yasuhiro OIKAWA(4)
(1)

(2)

Waseda University, Japan, tsubasa.k@suou.waseda.jp
Waseda University, Japan, mas-03151102@akane.waseda.jp(3)
Waseda University, Japan, k.yatabe@asagi.waseda.jp(4)
Waseda University, Japan, yoikawa@waseda.jp

Abstract
Many audio signal processing methods are formulated in the time-frequency (T-F) domain which is obtained by
the short-time Fourier transform (STFT). The properties of the STFT are fully characterized by window function,
number of frequency channels, and time-shift. Thus, designing a better window is important for improving theperformance of the processing especially when a less redundant T-F representation is desirable. While many
window functions have been proposed in the literature, they are designed to have a good frequency response foranalysis, which may not perform well in terms of signal processing. The window design must take the effectof the reconstruction (from the T-F domain into the time domain) into account for improving the performance.
In this paper, an optimization-based design method of a nearly tight window is proposed to obtain a window
performing well for the T-F domain signal processing.
Keywords: Discrete Gabor transform (DGT), Short-time Fourier transform (STFT), Window design, Speechenhancement, Non-convex optimization.

1

INTRODUCTION

Many audio signal processing methods are formulated as modifications of the signal in the time-frequency (T-F)
domain, which is often called T-F masking. For converting the signal into the T-F domain, the short-time Fouriertransform (STFT) [1]1 is usually utilized owing to its simplicity and easily understandable structure [2–9]. Whilemost of the research has concentrated on the method of modification in the T-F domain (the way how toconstruct a T-F mask), the method of converting a signal into the T-F domain is also important for improving
the performance of processing.
When STFT is considered as the conversion to the T-F domain, its property is fully characterized by thewindow function since STFT is a highly structured transform. Aiming to obtain a better T-F representation,
many window functions have been proposed to improve their frequency responses [10–16]. For example, theHann window is one popular window which has a good sidelobe decay. The Nuttall window was proposedto achieve a better sidelobe decay, while the Kaiser window was proposed so that its frequency response wasadjustable by a tuning parameter. Although such research on window functions has provided a better T-F
representation, most research only has considered the analysis side. That is, there is little research on windowfunctions considering the reconstruction point of view.
To realize processing in the T-F domain, the signal must be reconstructed back into the time domain after the
T-F-domain processing. The reconstruction side of STFT is achieved by the (pseudo-) inverse STFT which also
involves a window function. Therefore, for the T-F domain signal processing, a window function must be chosenin accordance with not only STFT but also the inverse STFT. Indeed, incorrect choice of the pair of windowfunctions (for STFT and the inverse STFT) makes reconstruction impossible. To allow a reasonable window for
the reconstruction, the T-F representation is usually chosen to be redundant, and the error-minimizing windowcalled the canonical dual window is often used for the reconstruction (see Section 2).
For some applications favoring less redundant T-F representation, choice of the window function is morecritical for the reconstruction (and thus, critical for processing). One example is T-F masking in low-powerdevices which allow a little computation [17, 18]. In such cases, redundancy should be lowered because higher
redundancy directly results in higher computational cost. Another very important example is speech enhancement based on deep learning. Recent study has shown that non-redundant T-F representation can improve
1 STFT is also often called the Gabor transform based on [1]. Note that some literature strictly distinguishes STFT from the Gabor transformby their mapping properties [2], while others do not. In this paper, we may utilize the term “STFT” in the sense of the discrete Gabor transform(DGT), which is a common habit especially in the acoustical signal processing community.

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the performance of enhancement using deep neural networks (DNN) [19]. This is because less redundant TF representation reduces the number of parameters to be learned, which makes the training easier. For those
applications, redundancy of STFT should be lowered by increasing the window shifting width. However, theinverse transform becomes more sensitive to the error of signal processing when the redundancy is reduced (seeSection 2.3), which also degrades the performance. Although there exists a type of window insensitive to suchprocessing error, called tight window, it has a drawback that its frequency response is often poor (sidelobe levelis high). Therefore, a window function which is less sensitive to processing error and, at the same time, has agood frequency response is desired for realizing a better processing in a less redundant situation.
In this paper, we propose a window design method to simultaneously meet both requirements. It aims to
make a window function closer to a tight window, while its frequency response is constrained to be better. Since
the designed window is not strictly tight, we call it nearly tight window. The proposed method is formulated
as an optimization problem so that it can easily control the trade-off between the two requirements, and it issolved by the linearized alternating direction method of multipliers (ADMM).

2PRELIMINARIES

While the discrete and downsampled T-F transform is called “STFT” in acoustical signal processing, the literature of T-F analysis calls it the discrete Gabor transform (DGT) [2]. Hereafter, we will utilize the language
used in DGT to express the T-F representation because it will be easier for explaining the proposed method.
2.1 Gabor system and discrete Gabor transform (DGT)
Let a window be denoted by g = [g[0], g[1], . . . , g[L−1]]T ∈ RL . DGT is a T-F transform based on a collectionof windowed sinusoids,G(g, a, M ) = {gm,n }m=0,...,M −1, n=0,...,N −1 ,(1)which is called the Gabor system, where a ∈ N is the time-shifting width, M ∈ N is the number of frequency
channels,
2πml(2)gm,n [l] = ei M g[l − an],√L
is a windowed complex sinusoid, and i = −1. DGT of a discrete signal f ∈ R is defined by the following
inner product:
L−1
X(Gg f )[m + nM ] = hf , gm,n i =f [l] gm,n [l],(3)l=0where x is the complex conjugate of x, and Gg ∈ CM N ×L is the matrix consisting of all the elements in
the Gabor system in Eq. (1). That is, multiplying Gg to a signal obtains the vectorized version of its T-F
representation which is often called “spectrogram.”
2.2 Reconstruction of time-domain signal from T-F domain
A system G(g, a, M ) is said to be a frame [20, 21] if there exist 0 < A, B < ∞ such thatX
2
2
A kf k2 ≤|hf , gm,n i|2 ≤ B kf k2 ,

(4)

m,n

for all f ∈ RL , where k · kp is the `p norm. A and B are called the lower and upper frame bound, respectively.
If the Gabor system is a frame, a time-domain signal can be reconstructed from its T-F domain representation.
The inverse DGT, reconstructing a signal from its coefficients c ∈ CM N , with respect to G(g, a, M ) is definedbyX
fsyn =
c[m + nM ] gm,n = G∗g c,(5)n,mwhere G∗g denotes the complex-conjugate transpose of Gg . If a Gabor system G(g, a, M ) is a frame, then there
exists the corresponding dual Gabor frame G(h, a, M ) = {hm,n } which satisfiesXf=hf , gm,n ihm,n ,
(6)
n,m

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2πml

where hm,n [l] = ei M h[l − an], and h is a dual window of g. That is, a time-domain signal can be reconstructed if (1) G(g, a, M ) is a frame, and (2) h is a dual window of g. These conditions are decided by thewindow pair g, h, the time-shifting width a, and the number of frequency channels M .
When a Gabor system G(g, a, M ) is redundant, the corresponding dual window h is not unique, and infinitely many variation of h can satisfy the reconstruction formula, Eq. (6). One standard choice among all
possible dual windows is the canonical dual windowg̃ = S−1
g g,(7)

where Sg = G∗g Gg is the so-called frame operator defined as
Sg f =X

hf , gm,n igm,n = (G∗g Gg ) f .

(8)

m,n

The canonical dual window is optimal in the sense that its synthesis operator corresponds to the Moore–Penrose
pseudo-inverse:
X
X
∗
−1 ∗
c[m + nM ] g̃m,n =c[m + nM ] S−1
Gg c.
(9)
g gm,n = (Gg Gg )
n,m

n,m

In this paper, the canonical dual window is considered for inverse DGT as it is the standard choice in acousticalsignal processing. One reason for such popularity should be because of the optimality to the following leastsquares signal reconstruction problem:2
minimize kGg x − ĉk2 ,(10)x∗
whose solution is Gg̃ ĉ as can be confirmed from the fact in Eq. (9).

2.3 Influence of window functions on signal processing
A signal processing framework in T-F domain is illustrated in Figure 1. In words, some processing is performed in the T-F domain to modify the Gabor coefficient c to ĉ, and then the inverse DGT is applied toobtain the processed result f̂ . While the quality of the processing is important for obtaining a good result, thetransformation pair, DGT and inverse DGT, is also important since it decides the coefficient c.
To see the effect of the window pair in terms of T-F domain signal processing, a preliminary experiment
was performed. 200 speech signals [22] from TIMIT database [23] were degraded by adding the Gaussian noisein the time domain so that the signal-to-noise ratio (SNR) became 0 dB. They were enhanced by the Wiener
filter (T-F masking based on the power ratio of noisy and clean signals) with a minimum mean-square error(MMSE) estimator of noise power [8] and the decision-directed approach [6]. The redundancy was changed bychanging a while fixing the window length to 256 and M = 256. Its performance was compared with the ideal
Wiener filter and the condition number of Gg ,p
(11)
κ(Gg ) = σmax (Gg ) / σmin (Gg ) = B /A ,which is the standard measure of numerical stability of Eq. (9), where σmax (Gg ) and σmin (Gg ) denote themaximum and minimum singular value of Gg , respectively.
Three types of window functions were utilized for comparison: the Hann window, the Kaiser window (α =10), and the canonical tight window of the Kaiser window. A window gT is said to be tight if its canonicaldual window is itself (i.e., self-dual) [24]. Then,
S = G∗gT GgT = A I(12)holds, where I is the identity. Thus, the condition number of a tight window is always 1. Particularly, a tightwindow with A = 1 is called the Parseval tight window. The canonical tight window of a window g can beobtained by inverting square root of the frame operator:−1gT = Sg 2 g,

(13)

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Gabor coefficients(Spectrogram)

Input signal

Modified coefficients

Modification ofGabor coefficients
DGT

Inverse DGT

Modified signal

Figure 1. Framework of the signal processing in the T-F domain.

Figure 2. (left) Condition number of the DGT matrix κ(Gg ). (center) Denoising result using the ideal Wiener
filter. (right) Denoising result using the Wiener filter with MMSE noise power estimation.
which corresponds to the solution of the following problem [25]:minimize kg − xk2 ,
x∈T

(14)

where T is the set of all Parseval tight windows. Thus, the canonical tight window is the closest Parseval tightwindow from the window g.
Results of the experiment are shown in Figure 2, where SNR is the average among all speech signals. For
both ideal and realistic Wiener filters (center and right), the processing performances of the Hann and Kaiser
windows were degraded as the redundancy decreased (horizontal axes are related to the redundancy). In contrast,
the performance of the tight Kaiser window was not degraded much. These results can be predicted from thecondition numbers (left). Based on this experiment, a window g should be designed so that the condition
number κ(Gg ) becomes lower. A tight window is the best window in this sense because its condition number
is the lowest. However, as in the figure, a tight window is not always the best in terms of processing, which
is because the frequency response of a tight window is usually not better than one of a non-tight window (seeFigure 3). Therefore, in this paper, a design method of a nearly tight window is proposed so that the condition
number is lowered while its frequency response is kept well.
2.4 Related works on Gabor window design
For designing a low-condition-numbered window, design methods of tight windows have been proposed [26,27].
These methods aim to find a tight window with better frequency responses. However, since the constraint to
the tight window greatly limits the set of variables, desired characteristics may not be obtained.
On the other hand, some methods of nearly-tight window design have been proposed [28–30]. One approach of this research is to minimize the difference between the frame operator and identity operator using thegradient-based optimization [28, 29]. These methods minimize the distance to the set of tight windows by thegradient method, whereby they have a possibility of falling into the local minima. Another approach is to replace the non-convex cost of measuring the distance to the tight window with convex functions [30]. Since thatmethod is formulated as convex optimization, it is guaranteed that globally optimal solutions can be obtained,
so a trade-off between the condition number and the frequency response can be easily considered. However,2888

as a result of approximating the cost function, the obtained solutions may not be close to the original solution
which is tight. The cost should be reduced strictly without approximation, while the trade-off should be easilyadjusted.

3

PROPOSED METHOD

In this section, we propose a design method of nearly tight window that can easily control the trade-off between
the desired frequency response and the condition number. At first, we formulate the nearly tight Gabor window
design as a constrained minimization problem. Then, an algorithm solving this problem through the proximal
operators is introduced. Since a window whose support is shorter than the signal length is used in most signal
processing, the formulation considers g[l] = 0 for l = K, . . . , L − 1, i.e., only g[l] (l = 0, . . . , K − 1) are treatedas the variables in this paper.
3.1 Problem formulation for designing nearly tight window
To propose an easily adjustable window function design, the desired frequency response is considered as aconstraint, and the window is made closer to a tight window as possible. Its direct formulation isminimizeg∈C

1 2
d (g),
2 T(15)

where dT (g) is the distance to the set of Parseval tight windows T ,dT (g) = min kg − xk2 ,
x∈T

(16)

and C is the set of windows satisfying the desired frequency response. Since the magnitude response should beconsidered in decibels for audio applications, a popular choice for the set C to constrain the frequency response
into desired one, in filter design [31], isC = {g ∈ RK | k log10 |F̃g| − log10 dk∞ ≤ log10 β},

(17)

K̃×K
is the zero-padded
where d ∈ RK̃
+ (K̃ ≥ K) is magnitude of the desired frequency response, F̃ ∈ C
discrete Fourier transform,
2πmn
1F̃[m, n] = √ e−i K̃ ,(18)
K̃and β ≥ 1 is a parameter for controlling the amount of error. However, directly treating this constraint is noteasy because taking difference after absolute value results in the non-convex set.
Since the requirement in window design (in contrast to filter design) is to lower the sidelobe level towards
zero (i.e., increasing the magnitude of sidelobe is usually not desired), it should be sufficient to constrain only
the upper bound. Based on this observation,C˜ = {g ∈ RK | |(F̃g)[n]| ≤ βd[n] for n = 0, . . . , N − 1},(19)is considered as the constraint set instead of Eq. (17). Consequently, our formulation becomes a minimization
problem on the convex set:
1 2
d (g).
(20)
minimize
2 T
g∈C̃This model directly handles the distance function dT instead of approximation as in [30], while the desired
frequency response is strictly imposed by the constraint C˜ as opposed to [28, 29].
3.2 Algorithm for solving problem using linearized ADMM
To solve Eq. (20), linearized ADMM [32–34] is utilized in this paper. It is an algorithm solving problems
written in the following form:
minimize F (x) + G (Ax),
(21)
x

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Figure 3. Designed nearly tight windows by the proposed method. Each column shows (from left to right)
the obtained window shapes, their frequency responses, their condition numbers, denoising results for the ideal
Wiener filter, and those for the Wiener filter with MMSE noise power estimation. Each row shows (from top
to bottom) the results of the Hann-based windows for a = 128, 192, and those of the Kaiser-based windows(α = 10) for a = 128, 192. The transition of colors from blue to yellow represents a change in parameter βof the proposed method, where the blue represents go and brighter color (larger β) means closer to tight. Redlines indicate the canonical tight window of go .
where F (x) and G (x) are proper lower semi-continuous functions, and A is a linear operator. By using theproximity operator [34],12proxρF (x) = argmin F (y) + ky − xk2 ,
(22)
2ρ
y
the linearized ADMM algorithm is given as the following procedure:µx[k+1] = proxµF x[k] − A∗ (Ax[k] − z[k] + u[k] ) ,
λ
[k+1]
[k+1]
z
= proxλG (Ax+ u[k] ),
u

[k+1]

=u

[k]

+ Ax

[k+1]

−z

[k+1]

,

(23)
(24)
(25)

where λ and µ are real numbers satisfying 0 < µ ≤ λ/kAk2op , and k · kop is the operator norm.
For applying this linearized ADMM algorithm to Eq. (20), it is rewritten as the equivalent problem having
the form of Eq. (21):1minimize d2T (g) + ι(F̃g),(26)
g2
where ι(z) is the indicator function corresponding to Eq. (19),
0 (|z[n]| ≤ βd[n] for n = 0, . . . , N − 1)
ι(z) =.
∞ (otherwise)

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(27)

Then, Eq. (26) is solved by iterating the following procedure:µg[k+1] = prox µ2 d2 g[k] − F̃∗ (F̃g[k] − z[k] + u[k] ) ,
T
λ
[k+1]
z
= proxι (F̃g[k+1] + u[k] ),
u

[k+1]

=u

[k]

+ F̃g

[k+1]

−z

[k+1]

,

(28)
(29)
(30)

where prox µ2 d2 (·) and proxι (·) in Eqs. (28) and (29) are given by
T

1
µ
−1
g+
Sg 2 g,
T
1+µ
1+µ


d[n]
, 1 z[n].
proxι (z)[n] = min β|z[n]|

prox µ2 d2 (g) =

(31)
(32)

Thanks to the property of the canonical tight window in Eq. (14), Eq. (31) can be expected to give an appropriate descent direction even though the cost function d2T is non-convex. Therefore, this algorithm should be
able to effectively manage the difficulty associated with the non-convexity of d2T .

4

NUMERICAL EXPERIMENTS

The shapes, frequency responses and condition numbers of the windows designed by the proposed method werecompared with the denoising performance provided by the same experiment in Section 2.3 using ideal and
MMSE Wiener filters. For the initial window inputted to the algorithm, the Hann and Kaiser windows, whoseenergies were normalized to a/M , were chosen in accordance with Section 2.3. By iterating the algorithm fromthese windows denoted by go , the designed windows are expected to have characteristics similar to go with abetter condition number. The frequency responses d for the constraint set C˜ were constructed by interpolating
the maxima of log10 |F̃go | by the cubic C 2 -splines.
The obtained nearly tight windows by the proposed method and the denoising results for a = 128, 192 aresummarized in Figure 3. When the parameter β was set to a higher value (brighter color), then the obtained
windows got closer to a tight window, which can be confirmed by the condition numbers. Note that the canonical tight window has the highest level of the first side lobe which may prevent a denoising method to be work
correctly. It can be seen that some windows obtained by the proposed method outperformed both the original
window (blue) and the canonical tight window (red) in terms of the denoising results. These results indicatethat the proposed method can design a window having better characteristics for T-F domain signal processing
than the original and the canonical tight window. The performance was adjustable by the single parameter β,which enables to look for a better window by a simple line search.

5

CONCLUSION

In this paper, the nearly tight window designing method for signal processing in the T-F domain is proposed.
The proposed method can obtain nearly tight windows having desired frequency responses, which can result ina better performance of T-F masking than those of original and canonical tight windows. Future work includes
the automatic adjustment of β as well as the generalization of the method.

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