Fast Gabor Wavelet Transform Based on Synthesis
of Gabor Spectrum using Convolution of Gaussian
Takanobu Ishikawa
Ryosuke Takayama
Shuichi Arai
Tokyo City University, Japan
ishikawa12@ipl.cs.tcu.ac.jp
Tokyo City University, Japan
takayama11@ipl.cs.tcu.ac.jp
Tokyo City University, Japan
arai.s@cs.tcu.ac.jp
Abstract—Gabor wavelet transform is often used in timefrequency analysis for non-stationary signals. However, the
calculation of continuous wavelet transform including Gabor
wavelet transform is very complex. Mallat algorithm in discrete
wavelet transform is representative of a speeding-up method
of wavelet transform, but we can’t use this algorithm for the
Gabor wavelet since the Gabor wavelet is a non-orthogonal
wavelet. Some methods, which approximate the basis function
to orthonormal function, have been proposed to solve this issue.
However, approximate accuracy of each algorithm is low.
In this paper, we focus on mathematical characteristics of
a Gaussian which is used as a basis function, and propose a
synthetic method of wavelet coefficients using Gabor spectra
which are calculated using FFT. In addition, we discussed the
synthetic accuracy and calculation complexity, then made it clear
that we can reduce the calculation complexity to about 1/20 in
regard to general CWT maintaining the desired accuracy.
Index Terms—Gabor Transform, Gabor Wavelet Transform,
complexity reduction, spectrum synthesis
I. I NTRODUCTION
Wavelet transform [1] [2] is often used in time-frequency
analysis for non-stationary signals. Wavelet transform uses
little wavelike functions known as mother wavelets. Since
mother wavelets are scaled, the analytic length is different
for each frequency. With this method, time-frequency resolution can be controlled properly as compared to Short Time
Fourier Transform (STFT) [3] because the analytic length
of STFT is constant regardless of frequency [4]. The timefrequency resolution is determined by a mother wavelet in
wavelet transform. Hence, it is important to decide what kind
of basis function to employ. Gabor function [5] is wellknown for minimizing the product of time and frequency
resolution. Gabor wavelet transform, which employs Gabor
function as the mother wavelet, has been used in a variety
of fields [6]–[8]. However, the calculation of Continuous
Wavelet Transform (CWT) is very complex. Moreover, in
recent years, the calculation complexity has tended to increase
along with the high sampling rate of signals. Therefore, a
speeding-up method such as Fast Fourier Transform (FFT)
in STFT is being sought.Mallat algorithm [9] [10] in Discrete
Wavelet Transform (DWT) is representative of the speeding-up
method of wavelet transform, but the basis function must have
orthonormality in DWT. Thus, we cannot use this algorithm for
Gabor wavelet since Gabor wavelet is non-orthogonal wavelet
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
[11]. Some methods, which approximate the basis function to
orthonormal function, have been proposed to solve this issue
[12] [13]. These methods show that Mallat algorithm can be
used in Gabor wavelet transform. However, the approximate
accuracy of each algorithm is low. A method to improve this
approximate accuracy have been proposed [14]. However, a
high approximate accuracy cannot be achieved.
The objective of this study is fast and high accuracy CWT.
To achieve this aim, we employ Gabor transform, which is
STFT with a Gaussian window. We have proposed a method to
calculate high-frequency resolution Gabor spectra using lowfrequency resolution Gabor spectra [15]. Taking advantage
from this method, we propose a method to synthesize hightime resolution Gabor spectra using low-time resolution Gabor
spectra. This method make it possible to obtain spectra in any
frequency position and change resolution for each frequency.
Therefore we can calculate wavelet coefficients using Gabor
spectra. In this paper, first of all, we show that a wavelet
coefficient can be synthesized from some Gabor spectra using
the mathematical characteristics of a Gaussian. Next, we
discuss the calculation accuracy, and show a design method of
parameters based on calculation accuracy. Finally, we discuss
the calculation complexity, and show that we can reduce the
calculation complexity of CWT.
II. G ABOR T RANSFORM
We perform fast Gabor wavelet transform using Gabor
transform in the procedure of Fig. 1. In Gabor Transform,
Fig. 1. Outline of proposed method
Fourier transform of the Gaussian window wσ (t) and the
Gaussian gσ (t) are
Z
2
− ω1 2
−jωt
2( )
σ
Wσ (ω) =
wσ (t) · e
dt = e
(4)
Z
2
√
− ω1 2
Gσ (ω) =
gσ (t) · e−jωt dt = 2πσ · e 2( σ ) . (5)
Fig. 2. Difference of Gabor spectrum and wavelet coefficient
we cut out a part of signal using the Gaussian window, and
perform Fourier Transform. Gabor Transform of input signal
x(t) is defined by
Z
Xσ (t0 , ω) =
wσ (t − t0 ) · x(t) · e−jωt dt
(1)
where the Gaussian window wσ (t) is defined as
wσ (t) = √
t2
1
1
gσ (t) = √
e− 2σ2
2πσ
2πσ
(2)
using the Gaussian gσ (t).
A. Determine Frame length using Standard deviation and
Quantization error
It is necessary to decide the frame length for cutting out a
input signal in Gabor transform. Since a Gaussian has infinite
length, we must cut off the Gaussian if we use it as the window
function. As might be expected, an error occurs if we cut off
the Gaussian at finite length. However, an actual signal also
has the error in consequence of quantization. If we cut off a
Gaussian in consideration of dynamic range of the signal by
quantization, the calculation accuracy do not become worse.
Then, we can decide cutting off the window at the range
[− L2 , L2 ] that an amplitude is smaller than the acceptable error
ε. Thus, the frame length L[sec] can be written as
√
(3)
L = 2 −2 ln ε · σ.
B. Difference of Gabor spectrum and Wavelet coefficient
In Gabor transform, a standard deviation σ of the Gaussian
window determines time-space resolution that is constant as
shown in Fig. 2. On the other hand, wavelet coefficients, which
are synthesis target, have different resolution for each frequency. In addition, frequency bins are linear interval in STFT,
but analysis frequencies of wavelet transform are non-linear
interval. Hence, we must be corresponded the resolution and
analysis frequency of Gabor spectrum to wavelet coefficient.
III. S YNTHESIS M ETHOD
In general, in order to obtain a different resolution spectrum,
it is necessary to perform Gabor transform using different
window length again. If we can control σ, we can change
the resolution for each frequency without reanalysis. Thus, we
focused on two mathematical characteristics of the Gaussian.
First, a frequency response of a Gaussian is also a Gaussian.
Second characteristic, convolution of two Gaussians become
a Gaussian with different standard deviation. In frequency
domain, this characteristics is
Z
As
Wασ (ω) =
Wσ (ω − γ)Gσs (γ)dγ
(6)
2π
where σs and As are
σs
=
√
α
σ,
1 − α2
As =
1
α
(α < 1).
(7)
This formula means the frequency response of Gaussian window with any scale standard deviation ασ can be obtained by
convolving the frequency response of the Gaussian window
with σ and the Gaussian with σs . Substituting Eq.(6) into
Gabor transform, synthetic formula is obtained undergoing a
process of follow.
Z
Xασ (t, ω0 ) = wασ (τ − t) · x(τ ) · e−jω0 τ dτ
Z
= Wασ (ω) · X(ω + ω0 ) · ejωt dω
Z
Z
As
Wσ (ω − γ) · Gσs (γ)dγ
=
2π
·X( ω + ω0 ) · ejωt dω
Z Z
As
=
Wσ (ω − γ) · X(ω + ω0 ) · ej(ω−γ)t dω
2π
·Gσs (γ) · ejγt dγ
Putting ω − γ = ω 0 , then dω is dω 0
Z Z
0
As
=
Wσ (ω 0 ) · X(ω 0 + ω0 + γ) · ejω t dω 0
2π
·Gσs (γ) · ejγt dγ
The underline of Eq.(8) means Gabor transform using Gaussian window with standard deviation σ. Thus, the synthetic
formula can be written as
Z
As
Xασ (t, ω0 ) =
ejγt · Xσ (t, ω0 + γ) · Gσs (γ)dγ. (9)
2π
This formula means that a spectrum of different resolution can
be calculated by convolving a Gaussian with spectra in respect
to frequency as shown in Fig. 3.
Then, we can get any resolution spectra by changing σs of
the Gaussian Gσs (γ) for each frequency. And moreover, we
can get spectrum in any frequency position by changing the
center of convolution. Therefore, we can control the resolution
and analysis frequency conformed to wavelet coefficients. Up
to this point, we showed the synthetic method of wavelet
coefficients from Gabor spectra, but the calculation complexity
remain high since Eq.(9) is in continuum region. Therefore, we
(8)
Fig. 3. Spectrum synthesis method
consider discretization of convolution. Discretization of Eq.(9)
is
eασ (t, ω0 )
X
Mω
As X
=
{Xσ (t, ω0 + mΩ)·Gσs (mΩ)·ej(mΩ)t }·Ω. (10)
2π
Fig. 4. Characteristics of the syntheric Gaussian window in time domain
m=−Mω
Normally, the number of synthetic term Mω is infinite, but
we must determine Mω as finite term. Additionally, we must
determine discretization interval Ω of the Gaussian. In short,
we must determine two discretization parameters.
IV. D ECISION
OF
D ISCRETIZATION PARAMETERS
In Eq.(10), the wider Ω becomes, the more we can reduce
the complexity cost. And, the fewer Mω becomes, the more
we can reduce the complexity cost. However the synthetic
error which is caused by discretization becomes higher in this
condition. There, we set an acceptable error ε, and determine
Ω and Mω which satisfy the condition that the synthetic error
is smaller than the ε In this paper, we treat the quantization
error as the acceptable error.
First, Eq.(10) can be written that convolution of frequency
response of synthetic Gaussian window Wασ (ω − ω0 ) with
Fourier transform X(ω) of input signal.
eασ (t, ω0 )
X
Mω
As X
{Xσ (t, ω + mΩ) · Gσs (mΩ) · ej(mΩ)t } · Ω
=
2π
m=−Mω
Z
Mω
As X
=
{ Wσ (ω − ω0 − mΩ) · X(ω) · ejωt dω
2π
m=−Mω
·Gσs (mΩ) · ej(mΩ)t } · Ω
Z
Mω
As X
{Wσ (ω − ω0 − mΩ) · Gσs (mΩ)} · Ω
= X(ω) ·
2π
m=−Mω
=
Z
·e
jωt
dω
fασ (ω − ω0 ) · ejωt dω
X(ω) · W
(11)
In other words, the synthetic error is controlled by the frequency response of synthetic Gaussian window. Thus, we
quantify the error of the frequency response of synthetic
Gaussian window, and determine Ω and Mω which satisfy
the condition that the synthetic error is smaller than the ε.
Since the frequency response of synthetic Gaussian window
is obtained by convolving the frequency response of the original Gaussian window with the discrete Gaussian as shown in
underline of Eq.(11), the synthetic Gaussian window becomes
product in time domain.
w
eασ (t)
= As · wσ (t) ·
∞
X
k=−∞
gσs (t −
2πk
)
Ω
(12)
The Gaussian is a monotonically decreasing function, but the
time domain signal becomes periodic signal like the blue line
shown in Fig. 4 since the Gaussian discretized in frequency
domain. As a result, the synthetic Gaussian window w
eασ (t)
has error components. The error component is caused multiple.
However, An amplitude of high order error components are
quite small in comparison to the first error component, because
the Gaussian window wσ (t) is also a monotonically decreasing
function. Hence, the peak level of the first error component
dominates the accuracy of the synthesis. When we calculate
the first error component in positive time, range of k is 0 or
1. Thereby Eq.(12) becomes
2π
w
eασ (t) = As · wσ (t) · gσs (t) + gσs (t −
) . (13)
Ω
The error E(t) between the synthetic window and a target
window is
E(t)
= |wασ (t) − w
eασ (t)|
2π
= As · wσ (t) · gσs t −
Ω
n
2π 2 o
t−
Ω
− 12 ( σt )2 +
1
σs
. (14)
= As · √
·e
2πσ
Furthermore, the derivative the time tp , which the error component is peak, is zero. Setting derivative of E(tp ) with respect
to tp equal to zero, we can obtain the formula of tp .
2π
(1 − α2 )
(15)
Ω
Substituting Eq.(15) into Eq.(14), the peak of the error component is
tp
=
2
1 1 2 2π 2
1
· e− 2 ( σ ) ( Ω ) (1−α ) .
(16)
2πασ
As shown in Eq.(16), the peak of the error components
becomes smaller when Ω becomes smaller. On the other
hand, the calculation complexity is lower when Ω becomes
larger. Therefore, we determine the longest Ω which satisfy the
condition that the synthetic error is smaller than the acceptable
error ε. As with Ep.(3), if the peak of the error components fall
within dynamic range of the signal by quantization, we can
synthesize the spectrum without degradation of the calculation
accuracy. Sinse
√ the maximum amplitude of the Gaussian
window is 1/( 2πασ), the condition for obtaining Ω is
E(tp )
=
√
1
· ε.
(17)
2πασ
Solving this formula for Ω, the condition of Ω can be written
as
√
p
2π
· 1 − α2 .
(18)
Ω ≤ √
− ln ε · σ
E(tp ) ≤
√
We can determine the longest Ω without degradation of the
calculation accuracy when we satisfy this condition.
However, actual spectra have been calculated at intervals
of frequency resolution ∆f [Hz]. Substituting Eq.(3) and Ω =
2πF into Eq.(18), the condition of descretization interval is
p
1
F ≤ 2 1 − α2 · .
(19)
L
√
If 2 1 − α2 ≥ 1 isn’t satisfied, we can’t synthesize in practice
beause 1/L means frequency resolution ∆f . Thus, α is limited
by a ceiling.
√
3
(20)
α ≤
2
Next, we discuss the relationship between the number of
synthetic term Mω and the synthetic error. The synthetic error
is lower when Mω is larger, but the calculation complexity is
lower when Mω is smaller. Hence, we determine the smallest
Mω based on the acceptable error ε as with Ω.
The synthetic Gaussian window w
eασ is product of the
original Gaussian window wσ and the discrete Gaussian gσs
as shown in Eq.(12). Thereby we determine the range of
convolution based on the range which an amplitude of the
discrete Gaussian is adequately attenuated. If the damping ratio
as against the maximum amplitude of the frequency response
of the discrete Gaussian Gσs is smaller than the ε, we can
synthesize the spectrum without degradation of the calculation
accuracy. The condition for obtaining Mω is
√
Gσs (ω) ≤
2πσs ε.
(21)
Since ω = Mω Ω， the condition of Mω can be written as
p
2(1 − α2 ) √
· − ln ε
Mω Ω ≥
(22)
ασ
substituting Eq.(18) into this formula,
1
(− ln ε).
(23)
πα
We can determine the smallest Mω without degradation of the
calculation accuracy when we satisfy this condition.
As above, we made clear the condition of discretization
parameters based on the desired accuracy.
Mω
≥
V. C OMPUTATIONAL C OST
In this section, we compare the calculation complexity of
our method with the general CWT about number of times
of multiplication. We calculate wavelet coefficients by scaling
mother wavelet from 1 to S in CWT. However, scaling factor
s is discretized on a computer. When we perform the scaling
at an interval of every ∆s, scaling factor s can be written as
s = a · ∆s + 1 (a : 0 ∼ (S − 1)/∆s)
(24)
where a is integer and S is maximum of scaling.
The calculation complexity of Gabor wavelet transform have
been already obtained in [16]. The calculation complexity of
the general CWT Ow is
(S−1)/∆s
Ow
=
X
(25)
Ow (a)
a=0
Ow (a)
√
= 2(2 −2 ln εfs σ(a∆s + 1) + 1)
(26)
where fs is sampling frequency. We determine standard deviation σ in consideration of the density of spectrum. We design
σ based on a condition that spectra on highest frequency and
adjacent frequency bin are crossed at half energy.
√
∆s + 2
ln 2
σ =
·
(27)
∆s
2πf0
On the other hand, the calculation complexity of proposed
method is a sum of FFT and synthesis. The calculation
complexity of synthesis is 2Mω +1 from Eq.(10). Substituting
Eq.(18) into Mω , the calculation complexity of proposed
method Op is
(S−1)/∆s
Op
Op (a)
= 4 · N log2 N + N +
X
Op (a)
(28)
a=0
= 4 · (2Mω + 1)
1
(− ln ε) + 1
= 4· 2
πα
(29)
where N [sample] is frame length. The first term in Eq.(28) is
the cost of FFT. The second term is the cost of windowing.
And the third term is the cost of synthesis. In this regard, we
For example, when we set fs = 44.1[kHz] and S = 256
(analysis frequency is 55∼14080[Hz]), we can reduce the
calculation complexity to about 1/20. In this way, the proposed
method is effective in calculation complexity reduction of
CWT.
TABLE I
E XPERIMENTAL CONDITION
ω0
σ
fs
ε
∆s
S
2π · f0 [rad/sec] (f0 = 14080[Hz])
1.98 · 10−4
44.1, 96, 192 [kHz]
3.05 · 10−5 (16bit quantization error)
0.1
32, 64, 128, 256
TABLE II
R ATE OF COMPLEXITY REDUCTION
PP
fs
PP S
P
P
44.1
96
192
32
64
128
256
0.3005
0.2695
0.2755
0.1658
0.1465
0.1483
0.0905
0.0791
0.0794
0.0491
0.0425
0.0423
must consider the calculation of complex number. Thereby
FFT and synthesis are needed quadruple cost. We scale up
σ depending on frequency using scaling factor s in general
CWT, but we can only scale down the standard deviation σ in
proposed method. Hence, we scale down standard deviation
σ, which is largest. So, α is
α =
(S − a · ∆s)
.
S
(30)
However, α has a condition shown in Eq.(20). Therefore, we
determine standard deviation σw used in Gabor transform as
shown in Eq.(31).
√
2 3
(31)
σw = Sσ ·
2
Also, we determine the frame length N depending on σw .
Using Eq.(3), we define the frame length N as
√
N ≥ 2 −2 ln ε · fs · σw .
(32)
In this regard, FFT exert high-speed performance when the
frame length is power-of-two. Thus, we define N as the
shortest length of power-of-two which satisfy Eq.(32).
We calculated the rate of complexity reduction r = Op /Ow
using parameters shown in table I. And show the result in table
II. The calculation complexity ratio remain unchanged when
sampling frequency is changed. The calculation complexity of
the general CWT increases when sampling frequency becomes
higher. However, FFT cost also increases because necessary
frame length becomes longer depending on sampling frequency. Since FFT cost dominate the calculation complexity
of the proposed method, the calculation complexity ratio is
scarcely affected by sampling frequency. On another front,
the calculation complexity ratio becomes decreases when S
becomes larger. For the above mentioned reasons, the calculation complexity of proposed method is not change largely
by changing S. Thus, the proposed method have an advantage
over the general CWT if the number of analysis frequency
increases.
VI. C ONCLUSION
In this paper, we proposed a synthesis method of wavelet
coefficient using FFT in order to the calculation complexity
reduction of CWT. We focused on the mathematical characteristics of the Gaussian that convolution of two Gaussians
become a Gaussian with different standard deviation. Using
this characteristics, we showed the synthetic method of wavelet
coefficients can be calculated by convolving a Gaussian with
some spectra obtained by Gabor transform in frequency domain. Next, we discussed the synthetic accuracy, and showed
the condition of discretization parameters which satisfy the
desired accuracy. Furthermore, we discussed the calculation
complexity, and showed that we can reduce the calculation
complexity to about 1/20 in regard to general CWT.
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