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A STUDY OF INVERSE SHORT-TIME FOURIER TRANSFORM Bin Yang Chair of System Theory and Signal Processing, University of Stuttgart, Germany
ABSTRACT In this paper, we study the inverse short-time Fourier transform (STFT). We propose a new vector formulation of STFT. We derive a family of inverse STFT estimators and a least squares one. We discuss their relationship and compare their performance with respect to both additive and multiplicative modiﬁcations to STFT. The inﬂuence of window, overlap, an

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A STUDY OF INVERSE SHORT-TIME FOURIER TRANSFORM
Bin Yang
Chair of System Theory and Signal Processing, University of Stuttgart, Germany
ABSTRACT
In this paper, we study the inverse short-time Fourier trans-form (STFT). We propose a new vector formulation of STFT.We derive a family of inverse STFT estimators and a leastsquares one. We discuss their relationship and compare theirperformance with respect to both additive and multiplicativemodiﬁcations to STFT. The inﬂuence of window, overlap, andzero-padding are investigated as well.
IndexTerms
—
inverseshort-timeFouriertransform, time-frequency analysis, least squares methods
1. INTRODUCTION
The short-time Fourier transform (STFT) is a useful tool toanalyze nonstationary signals and time-varying systems. Ithas been successively applied to a large number of signalprocessing applications like time-frequency analysis, speechenhancement, echo cancelation, and blind source separation.The problem of the inverse STFT (ISTFT) is to constructthe srcinal time domain sequence from a modiﬁed STFT.The modiﬁcation could be additive or multiplicative. Sur-prisingly, ISTFT has not been studied systematically in theliterature. Almost all papers up to now including many recentpublications use a heuristic overlap-add method for comput-ing ISTFT [1, 2, 3, 4]. It combines the results of the inverseFourier transform of different sections by overlap-add. In[5], a weighted overlap-add procedure has been proposed, butwithout a guideline about the optimum weighting.STFT is a linear operation. Due to section overlappingandzero-padding, theresultofSTFTcontainsalargernumberof samples than the srcinal time domain sequence. Clearly,ISTFT is a linear overdetermined problem for which the leastsquares (LS) approach is well applicable [6].In this paper, we take a detailed look at ISTFT. In com-parison to [6], our contributions are: 1) We derive a novelcompact vector formulation of STFT. It facilitates the work-ing with STFT and is also useful for other purposes. 2) Wederive a family of heuristic ISTFT estimators including theclassical overlap-add method. 3) We also derive a LS esti-mator. While [6] assumed a continuous-frequency represen-tation, we focus on the discrete-frequency Fourier transform.In addition, we allow zero-padding and non-equally spacedfrequencies. 4) We clarify the relationship between differ-ent ISTFT estimators. 5) We compare their performance withrespect to both additive and multiplicative modiﬁcations toSTFT.6)Finally, wealsostudytheinﬂuenceofwindow, over-lap, and zero-padding on ISTFT.The following notations are used in the paper. Matricesand column vectors are represented by boldface and under-lined characters. The superscript
T
and
H
denote transposeand Hermitian transpose, respectively.
·
is the Euclideanvector norm.
diag(
·
)
describes a diagonal matrix.
2. VECTOR FORMULATION OF STFT
We ﬁrst derive a new vector formulation of STFT. This willbe the basis for further investigations.A time domain sequence
x
(
n
) (
n
≥
0)
is divided into
M
overlapping sections. Each section has the length
N
. Theshift length from section to section is
1
≤
S
≤
N
. Theoverlap length between two adjacent sections is
N
−
S
, seeFig. 1. Let
x
m
= [
x
(
mS
)
,x
(
mS
+1)
,...,x
(
mS
+
N
−
1)]
T
∈
C
N
(1)denote the
m
-th section (
0
≤
m
≤
M
−
1
) of
x
(
n
)
.
M
such overlapping sections with a shift length
S
contain a totalnumber of
J
= (
M
−
1)
S
+
N
samples
x
(0)
,...,x
(
J
−
1)
with
J
≤
MN
. Let
x
= [
x
(0)
,x
(1)
,...,x
(
J
−
1)]
T
∈
C
J
(2)be the vector of all involved samples. The relationship be-tween
x
and
x
m
in (1) is described by
[
x
T
0
,x
T
1
,...,x
T M
−
1
]
T
=
O
x
(3)where
O
=
⎡⎢⎢⎢⎢⎢⎣
I
N S
I
N
...
I
N
⎤⎥⎥⎥⎥⎥⎦
∈
R
MN
×
J
(4)is a so called overlap matrix. It consists of
M
identity ma-trices
I
N
along the main diagonal. Each identity matrix isshifted by
S
columns to the right with respect to the aboveone. The left-multiplication of
x
by
O
corresponds to its seg-mentation into
M
overlapping sections as shown in Fig. 1.Each section
x
m
is now weighted by a real valued win-dow
w
(
n
)
of the same length. We assume that
w
(
n
)
is non-zero and thus invertible. Then this windowed sequence is ap-pended by
K
−
N
zeros before the
K
-point Fourier trans-form
X
m
(
k
)
is calculated at
K
discrete frequencies
ω
k
(0
≤
k
≤
K
−
1)
with
K
≥
N
. In general,
ω
k
do not need to
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x
⇓
N N N
x
0
x
1
x
2
SS
Fig. 1
.
O
x
: Divide a sequence
x
into overlapping sectionsbe equally spaced like in discrete Fourier transform (DFT). Inmatrix-vector-notation, the Fourier transform is described by
X
m
= [
X
m
(0)
,...,X
m
(
K
−
1)]
T
=
FPW
x
m
∈
C
K
(5)with
W
= diag(
w
(0)
,...,w
(
N
−
1))
∈
R
N
×
N
,
P
=
I
N
0
(
K
−
N
)
×
N
∈
R
K
×
N
,
(6)
F
= [
e
−
jω
k
n
]
0
≤
k,n
≤
K
−
1
∈
C
K
×
K
.
W
is a diagonal window matrix.
P
consists of an
N
×
N
identity and a
(
K
−
N
)
×
N
zero matrix and describes thezero-padding.
F
is the square Fourier transform matrix withthe element
e
−
jω
k
n
at the
k
-th row and
n
-th column, bothcounted starting from zero.We stack the Fourier transforms of all
M
sections into asingle column vector
X
= [
X
T
0
,X
T
1
,...,X
T M
−
1
]
T
∈
C
MK
.
(7)By using (3), (5), and the notation
A
⊗
B
= [
a
ij
B
]
i,j
forthe Kronecker tensor product, we ﬁnally obtain the linear re-lationship between the complete time domain sequence
x
andits windowed zero-padded STFT
X X
=
⎡⎢⎣
FPW
...
FPW
⎤⎥⎦⎡⎢⎣
x
0
...
x
M
−
1
⎤⎥⎦
=
H
x,
H
= (
I
M
⊗
(
FPW
))
O
.
(8)The meaning of the different matrices in (8) is self-explained:
ã
O
: overlapped segmentation
ã
W
: windowing
ã
P
: zero-padding
ã
F
: discrete-frequency Fourier transform
ã
I
M
⊗
: section-by-section processing
3. A FAMILY OF HEURISTIC ISTFT ESTIMATORS
Starting from (8), we now study how to compute the corre-sponding ISTFT.If
X
denotes the exact STFT of a given time domain se-quence
x
, we can uniquely determine
x
. In practical applica-tions, however, the STFT is often modiﬁed before it is trans-formed back into the time domain. In this case, there is ingeneral no time domain sequence
x
whose STFT matches ex-actly
X
because
X
contains more elements than
x
if
S < N
(true overlapping) or
K > N
(true zero-padding). The prob-lem is overdetermined. The question is how to compute areasonable estimate
ˆ
x
for
x
from
X
?A simple but heuristic idea is based on the following ob-servation: If we multiply both sides of (8) by the
J
×
MK
matrix
O
H
(
I
M
⊗
A
p
)
from left with
A
p
=
W
p
−
1
P
H
F
−
1
∈
C
N
×
K
, weobtainaccordingto
(
A
⊗
B
)(
C
⊗
D
) =
AC
⊗
BD
[7] the following result
O
H
(
I
M
⊗
A
p
)
X
=
O
H
(
I
M
⊗
A
p
)(
I
M
⊗
(
FPW
))
O
x
=
O
H
(
I
M
⊗
(
A
p
FPW
))
O
x
=
O
H
(
I
M
⊗
W
p
)
O
x,
(
p
= 0
,
1
,
2
...
)
.
(9)
W
p
is a diagonal matrix containing the diagonal elements
w
p
(
n
)
. This motivates the following family of estimates
ˆ
x
p
=
D
−
1
p
O
H
(
I
M
⊗
A
p
)
X,
D
p
=
O
H
(
I
M
⊗
W
p
)
O
.
(10)We call it the
p
-ISTFT estimate.Step Meaning1)
F
−
1
inverse Fourier transform of
X
m
2)
P
H
keep only the ﬁrst
N
samples of
F
−
1
X
m
3)
W
p
−
1
weight these
N
samples by
w
p
−
1
(
n
)
4)
I
M
⊗
do the above computations for all
M
sections5)
O
H
overlap-add of the results of all sections6)
D
−
1
p
ﬁnal normalization
Table 1
. Steps of
p
-ISTFTNote that Eq. (10) has a simple interpretation. Table1 summarizes all steps of
p
-ISTFT. The ﬁrst four steps areeasy to understand. According to the deﬁnition of the over-lap matrix
O
in (4), the operation
O
H
z
in the 5-th step with
z
= [
z
T
0
,...,z
T M
−
1
]
T
∈
C
MN
describes the well knownoverlap-add of the
M
sections
z
m
. The shadowed areas inFig. 2 represent the overlap-add regions. Fig. 3 illustratesthe matrix overlap-add operation
O
H
(
I
M
⊗
U
)
O
where
U
is any
N
×
N
matrix. The result is a
J
×
J
matrix in whichadjacent matrices
U
along the main diagonal overlap and addin an
(
N
−
S
)
×
(
N
−
S
)
area. Clearly, if
U
=
W
p
=diag(
w
p
(0)
,...,w
p
(
N
−
1))
isdiagonal, thenthematrix
D
p
=
O
H
(
I
M
⊗
W
p
)
O
is diagonal as well
D
p
= diag(
d
p
(0)
,...,d
p
(
J
−
1))
∈
R
J
×
J
.
(11)The diagonal elements
d
p
(
n
)
are obtained by overlap-adding
M
sections of
w
p
(0)
,...,w
p
(
N
−
1)
as in Fig. 2. The lastnormalization step in Table 1 involves thus only
J
scalar di-visions.
3.1. LS inverse STFT estimator
Since ISTFT is a linear overdetermined problem, it is naturalto apply the least squares (LS) approach to the signal model
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N N N
z
0
z
1
z
2
++
SS
O
H
z
⇓
Fig. 2
.
O
H
z
: Overlap-add of vectors
UUU
N N SS
...
Fig. 3
.
O
H
(
I
M
⊗
U
)
O
: Overlap-add of matrices(8). After some calculations, we obtain the LS estimate
ˆ
x
LS
= (
H
H
H
)
−
1
H
H
X
=
D
−
1LS
O
H
(
I
M
⊗
(
WP
H
F
H
))
X,
D
LS
=
H
H
H
=
O
H
(
I
M
⊗
U
)
O
,
U
=
WP
H
F
H
FPW
.
(12)It is referred to as LS-ISTFT.
4. DISCUSSIONS
Below we discuss the relationship between different ISTFT:
ã
Theclassicaloverlap-add method[1,2,3,4]isaspecialcase of our
p
-ISTFT with
p
= 1
.
ã
The LS solution from [6] is identical to
p
-ISTFT with
p
= 2
. It assumed, however, a continuous-frequencyFourier transform.
ã
OurLS-ISTFTisderivedforthediscrete-frequencycase.In general, i.e. for arbitrary discrete frequencies
ω
k
,
U
and thus
D
LS
in (12) are not diagonal. In this case, thematrixinversion
D
−
1LS
isexpansive and
ˆ
x
LS
differsfrom
ˆ
x
p
and the LS solution from [6].
ã
In the special case of DFT with equally spaced discretefrequencies
ω
k
=
2
πkK
(0
≤
k
≤
K
−
1)
,
F
H
F
=
K
I
K
,
U
=
K
W
2
,
F
H
=
K
F
−
1
, and
ˆ
x
LS
simpliﬁesto
ˆ
x
2
.Besides the choice of the estimator, the inverse STFT alsodepends on a number of other factors like the modiﬁcation of the STFT, the choice of the window, the shift length
S
, andthe number of appended zeros
K
−
N
. Below we study theinﬂuence of these factors on ISTFT.
ã
If
X
denotestheexactSTFTofatimedomainsequence
x
, then both
ˆ
x
p
and
ˆ
x
LS
return the same
x
. This can beeasily shown be combining (8) and (10) as well as (12).
ã
If we use the rectangular window
w
(
n
) = 1
, then allestimators
ˆ
x
p
return the same result. In this case,
W
=
I
N
and
ˆ
x
p
in (10) does not depend on
p
.
ã
If there is no overlap between adjacent sections (
S
=
N
), thenallestimators
ˆ
x
p
returnthesameresultaswell.In this case,
O
=
I
MN
and
ˆ
x
p
simpliﬁes to
(
I
M
⊗
(
W
−
1
P
H
F
−
1
))
X
.We expect a small difference between various
p
-ISTFT esti-mators if the modiﬁcation to STFT or the deviation of
w
(
n
)
from the rectangular window or the overlap length is small.Concerning the computational complexity, all
p
-ISTFTestimators and the DFT-based version of LS-ISTFT are com-parable to the classical overlap-add method. For each section,one inverse Fourier transform has to be computed. The onlydifference is the use of different windows
w
p
−
1
(
n
)
for thescaling of the inverse Fourier transform and the computationof the ﬁnal normalization sequence
d
p
(
n
)
in (11).
5. EXPERIMENTS
Inthissection, wecomparetheperformanceofdifferentISTFTestimators. We use clean speech signals of roughly 6 sec-ond duration sampled at 16 kHz in our experiments. For eachspeech signal
x
, we compute its STFT and modify it by ad-ditive noise or multiplicative masking. Then we compute thesignal estimate
ˆ
x
for different ISTFT estimators. Since DFT(FFT) is used,
ˆ
x
LS
is identical to
ˆ
x
2
and will not be consid-ered separately. The performance measure is the signal-to-distortion ratio (SDR) in the time domain after the signal re-construction
SDR
p
= 10log
10
(
x
2
/
x
−
ˆ
x
p
2
) dB
.
(13)In particular, we focus on
SDR
2
of the LS estimator and theperformance loss of other estimators
ΔSDR
p
= SDR
2
−
SDR
p
with respect to that. For statistical averaging, we use8 different utterances from a male and a female speaker andcalculate the average values of
SDR
2
and
ΔSDR
p
over these8 speech signals. The default parameter set for the Fouriertransform is hamming window, window length
N
= 512
,shift length
S
=
N/
2
, and FFT length
K
=
N
unless other-wise stated.
Additive distortion
First we consider additive distortion.
X
is modeled as
H
x
+
N
.
H
x
is the exact STFT of
x
and
N
contains realizationsof zero-mean uncorrelated random variables with equal vari-ance
σ
2
. The variance is chosen to achieve a certain signal-to-noise ratio (SNR) in the time-frequency domain
SNR =10log
10
(
H
x
2
/
N
2
)
. For this particular signal model, itis well known that the LS estimator
ˆ
x
LS
= ˆ
x
2
achieves thesmallest variance among all linear unbiased estimators like
ˆ
x
p
. In addition, the variance of
ˆ
x
p
increases linearly with
σ
2
.The default value of SNR is 10 dB. In Table 2, we usedifferent invertible windows. The window names are takenfrom MATLAB. In Table 3, we vary the overlap length
N
−
S
.In Table 4, we change the FFT length
K
.
Multiplicative distortion
Inasecondseriesofexperiments, wemultiplytheexactSTFT
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with a mask, a typical operation in underdetermined blindsource separation based on the spareness of speech signalsin the time-frequency domain [4]. Due to limited space, weonly consider a binary mask. The mask is determined suchthat
p
mask
percentage of the time-frequency points with thehighest amplitude of
H
x
pass the mask. In all other time-frequency points, the binary mask is zero.In Table 5, we choose different values for
p
mask
. Theother parameters are identical to the default choice in the pre-vious subsection. In Table 6 to 8, we keep
p
mask
= 30%
andrepeat the same performance study with respect to window,overlap, and zero-padding as previously.window
SDR
2
ΔSDR
0
ΔSDR
1
ΔSDR
3
rectwin 13.00 0 0 0kaiser (
β
=0) 13.00 0.01 0 0hamming 12.47 9.26 1.08 0.10triang 7.22 20.30 0.33 0.03
Table 2
. Additive distortion: Varying window
N
−
S
SDR
2
ΔSDR
0
ΔSDR
1
ΔSDR
3
0 0.23 0 0 0
N/
8
5.00 0.82 0.17 0.07
N/
4
8.50 3.97 0.55 0.09
N/
2
12.47 9.26 1.08 0.10
3
N/
4
15.27 9.13 1.15 0.20
Table 3
. Additive distortion: Varying overlap length
K
SDR
2
ΔSDR
0
ΔSDR
1
ΔSDR
3
N
12.47 9.28 1.08 0.10
1
.
5
N
14.25 9.30 1.10 0.10
2
N
15.50 9.34 1.10 0.10
Table 4
. Additive distortion: Varying zero-padding
Observations
We draw the following conclusions from the above experi-ments:
ã
In additive distortion,
ˆ
x
LS
= ˆ
x
2
is the best one as ex-pected. In binary masking,
ˆ
x
LS
is almost the best oneamong the tested linear estimators, but not always be-cause of
ΔSDR
3
<
0
in Table 6 and 8.
ã
ˆ
x
0
has a poor performance. The weighting of the in-verse Fourier transform with
w
−
1
(
n
)
ampliﬁes the dis-tortion if
w
(
n
)
is close to zero. This happens at bothend of each section, particularly for the window “tri-ang”.
ã
The classical overlap-add method
ˆ
x
1
is always worsethan the LS one with a SDR loss of up to 1 dB. Inter-estingly, it is also worse than
ˆ
x
3
.
ã
There is almost no performance difference between
ˆ
x
2
and
ˆ
x
3
.
ã
As expected, the larger the overlap length is, the largerthe SDR improvement of
ˆ
x
2
is.
ã
Zero-padding has a fairly small impact to the perfor-mance difference.
ã
In order to achieve a good absolute performance
SDR
2
,a large overlap is highly desirable resulting in a largernumber of noisy samples. Also zero-padding is advan-
p
mask
SDR
2
ΔSDR
0
ΔSDR
1
ΔSDR
3
10% 21.32 5.48 0.29 0.0720% 28.16 6.31 0.43 0.0630% 33.60 7.41 0.60 0.0640% 38.74 8.75 0.82 0.06
Table 5
. Multiplicative distortion: Varying mask window
SDR
2
ΔSDR
0
ΔSDR
1
ΔSDR
3
rectwin 29.18 0 0 0kaiser (
β
=0) 29.40 0.18 0.09
−
0
.
09
hamming 33.60 7.41 0.60 0.06triang 33.52 21.74 0.36 0.06
Table 6
. Multiplicative distortion: Varying window
N
−
S
SDR
2
ΔSDR
0
ΔSDR
1
ΔSDR
3
0 23.51 0 0 0
N/
8
28.85 1.59 0.45
−
0
.
03
N/
4
31.33 3.88 0.43 0.09
N/
2
33.60 7.41 0.60 0.06
3
N/
4
34.49 5.87 0.51 0.02
Table 7
. Multiplicative distortion: Varying overlap length
K
SDR
2
ΔSDR
0
ΔSDR
1
ΔSDR
3
N
33.60 7.41 0.60 0.06
1
.
5
N
33.90 5.95 0.38 0.06
2
N
34.02 5.72 0.35 0.06
Table 8
. Multiplicative distortion: Varying zero-paddingtageous though the improvement is much smaller.
ã
Foradditivedistortion, ﬂatwindowslike“rectwin, kaiser,hamming”arebetter. Formultiplicativedistortion, “ham-ming, triang” windows are preferred. Hence the ham-ming window seems to be a good compromise.
6. REFERENCES
[1] J. B. Allen and L. R. Rabiner, “A uniﬁed approach toshort-time Fourier analysis and synthesis,”
Proc. IEEE
,vol. 65, pp. 1558–1564, 1977.[2] L. R. Rabiner and R. W. Schafer,
Digital processing of speech signals
, Prentice-Hall, 1978.[3] “Inverse short-time FFT,” MATLAB Signal ProcessingBlockset Documentation.[4] S. Araki, H. Sawada, et al., “Underdetermined blindsparse source separation for arbitrarily arranged multi-ple sensors,”
Signal Processing
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