2020 13th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics (CISP-BMEI)
Spectrum Representation Based on STFT
Xuebao Wang, Tao Ying, Wei Tian
Department of Electronic Engineering
Naval University of Engineering
Wuhan, China
Abstract—A new spectrum representation method based on short
time Fourier transform (STFT) is proposed. The formula of new
spectrum representation is derived base on energy cumulant of
short time Fourier transform (EC-STFT), which indicates that
EC-STFT has the spectrum characteristics. Simulations on the
linear frequency modulation (LFM) signal show that the ECSTFT spectrum is closer to the ideal spectrum curve than FT
spectrum. During calculating EC-STFT, a time frequency
domain iterative mean threshold (TFD-IMT) denoising method is
presented to remove the addictive white Gaussian noise (AWGN),
by which the EC-STFT spectrum has better anti-noise capacity
than FT spectrum. Theoretical analyses and simulations verify
the advantages of the EC-STFT over FT in conditions of low
SNR.
Keywords-spectrum representation;
iterative mean threshold
I.
STFT;
TFD
II.
DERIVATION OF EC-STFT
STFT is a widely used tool to analyse and process the nonstationary signals [1, 6-8]. The STFT of a non-stationary signal
x(t) is defined as
STFTx (t , f ) = 
+∞
x(τ )g * (τ − t )e− j 2π f τ dτ
−∞
(1)
where g(t) is the window function and * donates complex
conjugate. Based on [5], we redefine EC-STFT as
ECSTFTx ( f ) =
1 T
STFTx (t , f ) dt
T 0
(2)
where | ⋅ | is the absolution operation and T is the time length
of x(t).
denosing;
In equation (2), EC-STFT calculates the marginal
distribution of STFT absolute energy form at the frequency
axis. According to equation (1), when g (t ) = 1 (∀t ) , STFT
INTRODUCTION
Fourier transform (FT) and short time Fourier transform
(STFT) are proposed to show the spectrum characteristic of
signals from different aspects [1, 2]. Commonly, it is difficult
to remove the addictive white Gaussian noise (AWGN) in the
time or frequency domain, and the FT spectrum easily
deteriorates in low-SNR conditions. Studies about STFT
mainly concentrate on the resolution improvement and local
feature extraction [3, 4], while the concept of STFT global
feature has not been concerned. In [5], the concept of STFT
global feature is given, the energy cumulant of short time
Fourier transform (EC-STFT), which is used as a new spectrum
feature of radar intra-pulse signals. During calculating the ECSTFT, it is more advantageous to remove the AWGN in two
dimensional time frequency domain (TFD) than it in single
time or frequency domain. However, these viewpoints are
confined to experiences, and the theoretical derivation is not
completed.
turns into FT. Let y(τ , t ) = x(τ ) g (τ − t ) , substitute it into (1),
and equation (1) can be expressed as
STFTx (t , f ) = 
+∞
−∞
y (τ , t )e − j 2π f τ dτ = FTy (t , f )
(3)
where FTy (t , f ) is the FT of y(τ , t ) . By the FT convolution
property, equation (3) can be written as
STFTx (t , f ) = FTy (t , f ) =
1
FTx ( f ) ⊗ [ FTg ( f )e j 2π ft ] (4)
2π
It leads to
ECSTFTx ( f ) =
1
2π T

T
0
FTx ( f ) ⊗ [ FTg ( f )e j 2π ft ] dt
(5)
Substitute e j 2π ft = cos 2π ft + j sin 2π ft into equation (5),
then we have
T
1
ECSTFTx ( f ) =
∆12 +∆ 2 2 dt
(6)
2π T 0
In this paper, the global feature of STFT is discussed and
the EC-STFT is redefined. The marginal distribution of STFT
absolute energy form is calculated as the new generated
spectrum curve. To decrease the effect of noise on the
spectrum, the feasibility of reduce noise in TFD is analyzed.
Multiple iterative mean thresholds are conducted to suppress
the AWGN in TFD, which has better denosing performance
than other methods. Above all, the main work in this paper
includes: (i) a new spectrum representation method is
proposed, which has more ideal spectrum parameters; (ii) a
TFD denoising method is presents during the process of
calculating EC-STFT, which increases the anti-noise capacity
of new spectrum.
where ∆1 = FTx ( f ) ⊗ FTg1 ( f , t ) , ∆ 2 = FTx ( f ) ⊗ FTg2 ( f , t ) ,
FTg1 ( f , t ) = FTg ( f )cos2π ft , FTg2 ( f , t ) = FTg ( f )sin 2π ft
and ⊗ refers to convolution operation. In equation (6),
FTgi ( f , t ) can be regarded as a new window function’s FT,
which is respectively modulated by cos 2π ft or sin 2π ft with
Corresponding author: yingtao_scholar@163.com
This work was supported by Natural Science Foundation of China
(61803379) and China postdoctoral Science Foundation (2017M613370,
2018T111129).
978-0-7381-0545-1/20/$31.00 ©2020 IEEE
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every time sample t. The length of t can influence the time
resolution and frequency resolution in STFT.
According
to
equation
(6),
FTx ( f )
traverses
all
i
g
FT ( f , t ) when t gets different values, and convolution
operation relates the whole spectrum sample of FTx ( f ) .
Therefore, the integration adds these spectrums processed by
different window functions [9-16].
Different from the time domain and common image denosing
[7, 8], TFD denoising concentrates on a specific area.
Suppose that the energy of x(t) and noise are Ex and En,
which are modulus of their STFT. In frequency domain, Ex
mainly distributes in the band LB, and En distributes in the
whole spectrum LF randomly. We assume that the energy is
equally distributed in the area, and have
Ex = Ex / LB
(9)
Suppose there exists an equivalent window function v(t)
satisfying
ECSTFTx ( f ) = FTx ( f ) ⊗ FTv ( f )
(7)
Ideally, the noise can be eliminated by comparing Ex and
where FTv ( f ) is the FT spectrum of v(t). In equation (7),
En with setting a threshold. However, En is a hypothetical
i
FTx ( f ) and FTg ( f , t ) are known and we have the time
average as y, FTx ( f ) as u, and FTv ( f ) as v. When we
provide a solution to y = u ⊗ α , it indicates that EC-STFT
relates to FT.
Set y = [ y (1),..., y (2 M − 1)]T , u = [u (1), u (2),..., u ( M )]T and
α = [α (1), α (2),..., α ( M )]T , in which u and α are FT results
with the same length M. The convolution operation is
expanded as
y (1) = u (1)α (1)
y (2) = u (1)α (2) + u (2)α (1)
...
(8)
y ( M ) = u (1)α ( M ) + u (2)α ( M − 1) + ... + u ( M )α (1)
...
y (2M − 1) = u (n)α (n)
Hence, the equation (8) can be expressed as y = Uα , and
the convolution matrix U is defined as
L
0
0 
 u (1)
 u (2)
L
u (1)
0 

 M
M
O
M 


U = u ( M ) u ( M − 1) L u (1)  .
 0
u (M )
L u (2) 


M
O
M 
 M
 0
0
L u ( M ) 

En = En / LF
value and Ex might be covered in En at low SNR, which
makes the denoising difficult. Same as above mentioned, the
energy distribution in TFD can be expressed as
Ex′ = Ex / S B
(10)
En′ = En / ( LF LT )
where S B is the energy distributed area of signals in TFD
and LT is the length of time samples. Given LB 《 LF , we
have S B ≈ LB LT 《 LF LT . Therefore, the relative mean noise
decreases as
Ex′ / Ex 》 En′ / En
which indicates that the threshold margin to process the
AWGN in TFD has been improved than it in single time or
frequency domain. Aiming to this problem, we propose the
TFD iterative mean threshold (IMT) denoising method. The
method is summarized in alrithm “TFD-IMT denoising”.
Algorithm: TFD-IMT denoising
Input: TFD A, initial MSE: ε (0) = ∞ .
Process:
1. Generate the vector of energy form TFD,
A M × N → β = [ β1 , β1 ,..., β M × N ] .
We choose M adjacent equations from equation (8) to form U1
and y1 to attain y 1 = U 1α . Commonly, we ensure
2. for i=1,2,…,L do
3. Remove the mean and keep positive values,
β%i = [ β − mean( β )]+ .
rank ( U1 ) = M , and the solution is α = U 1-1 y 1 .
As a result, the equivalent window function v(t) is
established: v (t ) = IFT[ U 1-1 y 1 ] , by which it is proved that ECSTFT has the spectrum characteristic [17-22].
III.
(11)
4. Compute the MSE, ε (i ) =
1
M
 β% − β
2
.
5. Compare ε (i) with ε (i − 1)
TFD ITERATIVE MEAN THRESHOLD
6. if ε (i ) < ε (i − 1)
The FT spectrum deteriorates severely in noisy environment.
The AWGN is difficult to be removed in single time or
frequency domain, especially in conditions of low SNR.
7. β = β%i .
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Table I. Parameter comparison on EC-STFT and FT
curves
8. else
% , and break.
9. Get matrix energy form, β%i −1 → A
10. end if
Curve
B (Hz)
IBF
ARS
EC-STFT
47.3
0.646
72.7
FT
47.7
6.34
66.6
11. end for
% .
Output: Denoised TFD A
SIMULATIONS AND RESULTS
The LFM signal is used to verify the proposed methods.
The FT spectrum of LFM signal is represented by complex
Fresnel integral, so the spectrum presents waving [23-28]. As
shown in Fig.1, the curve of EC-STFT is flatter than FT’s, and
the former is more closed to ideal rectangle. Parameters in
TABLE I are given to compare the EC-STFT and FT: 3 dB
bandwidth (B), in-band flatness (IBF) and average rising slope
(ARS), which are defined by
B=
As shown in Fig.2, the correlation coefficient and MSE of
denoised and non-denoised TFD images present crosscurrents
with the iteration L increasing: the correlation coefficient
shows a convex trend and MSE shows a concave trend.
Furthermore, the correlation decreases and MSE increases
when SNR decreasing, which indicates that the denoising
performance degrades. According to the Algorithm, the
iterative number is determined by the least MSE between
deniosed and ideal TFD images. Weighting these factors, we
choose L=6 for IMT deniosing to attain a better denoising
performance.
( x2 − x1 ) N s
T
1
N pp
IBF =  yk − yk
k =1
0.9
y j − yi
x j − xi
where Ns is the sample number of x(t), T is the time length of
x(t), and Npp is sample number between start-peak and endpeak in the spectrum.
1
EC-STFT
10dB(corr)
0dB(corr)
-5dB(corr)
0.7
1
10dB(MSE)
5dB(MSE)
0.6
0.5
0dB(MSE)
-5dB(MSE)
0.5
0.6
(x i, y i)
0.2
x1
0
0
5
L
10
0
15
Fig. 2 Denoising result of TFD images with different
iterations (L) and SNR
(x j, y j)
0.4
0
1.5
5dB(corr)
0.8
FT
0.8
normalized amplitude
c orrelation c oefficient
ARS =
2
(12)
MSE
IV.
0.1
0.2
x2
0.3
0.4
0.5
normalized frequency
Fig. 1 Spectrum curves of EC-STFT and FT
(Ns = 2048, T = 5s, fstart = 100Hz, fend = 150Hz, Gaussian
window function, Ng =255.)
Comparing parameters of EC-STFT with FT, we find that
both curves have the similar bandwidth, while EC-STFT has
better in-band flatness and larger ARS. The result indicates
that EC-STFT spectrum has better performance parameters
than FT spectrum.
Table II gives the denosing results with methods in TFD
and time domain (TD). Correlation coefficient is used to
evaluate the denoising performance between the denoised and
ideal spectrum curves. We use the classical adaptive wavelet
soft threshold (AWST) [7] for TD denoising, mean filtering
(MF) and non-local mean filtering (NLMF) [8] in TFD
denoising for comparison. According to the result in Table II,
the TD-AWST does not work well for FT spectrum curve,
comparing with the non-denoising one. The same occurs to the
MF and NLMF methods in TFD for EC-STFT spectrum curve.
However, the proposed IMT denoising method has better
performance than the former methods. Besides, NLMF is
combined with IMT together for denosing in TFD, which
further improves the result. By contrast, the IMT proposed in
this paper contributes more to noise reduction.
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TABLE II. Correlation coefficient between denoised and
ideal spectrum curves with different SNR
SNR(dB)
10
5
0
-5
TD-Non
0.944
0.813
0.517
0.224
TD-AWST
0.944
0.813
0.518
0.226
TFD-Non
0.988
0.958
0.850
0.600
TFD-MF
0.987
0.954
0.873
0.570
TFD-NLMF
0.989
0.960
0.889
0.585
TFD-IMT
0.998
0.994
0.968
0.796
NLMF-IMT
0.997
0.995
0.967
0.882
V.
CONCLUSION
In this paper, we proposed a new spectrum representation
method based on STFT, the EC-STFT spectrum. The new
representation has better in-band flatness and is closer to the
ideal form than FT spectrum with similar bandwidths.
Moreover, a TFD iterative mean threshold denoising method
was presented, which increases the new spectrum’s anti-noise
capability into EC-STFT computing process.
Moreover, the EC-STFT curve can be used as a
characteristic for the radar signal in the radar location and
identification field for its good performance against niose.
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