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8/3/2019 2 Rajeev Aggrawal Research Article April 2011
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Available ONLINE ww w.visualsoftindia.com/ journal.html
VSRD-IJEECE, Vol. 1 (2), 2011, 62-71
____________________________
1Research Scholar, 2Professor, 12Department of Electronics & Communication, Sri Satya Sai Institute of Science &
Technology, Sehore, Madhya Pradesh, INDIA. 3Assistant Professor, Department of Electronics & Communication, IMS
Engineering College, Ghaziabad, Uttar Pradesh, INDIA.4Professor, Department of Electronics, UIT, RGPV, Bhopal,
Madhya Pradesh, INDIA.*Correspondence : [email protected]
RRR EEE SSS EEE AAA RRR CCC HHH AAA RRR TTT III CCC LLL EEE
Elimination of White Noise from Speech Signal
Using Wavelet TransformBy Soft and Hard Thresholding
1Rajeev Aggarwal*,
2Jay Karan Singh,
3Vijay Kr. Gupta and
4Dr. Anubhuti Khare
ABSTRACT
In this paper, Discrete-wavelet transforms based algorithms are used for speech signal denoising. Analysis is
done on noisy speech signal corrupted by white noise at 0dB, 5dB, 10dB and 15dB signal to noise ratio levels.
Here both hard thresholding and soft thresholding are used for denoising. Simulation & results are performed in
MATLAB 7.10.0 (R2010a). Output signal to noise ratio and mean square error is calculated & compared using
both types of thresholding methods. Soft thresholding method performs better than hard thresholding at all input
signal to noise ratio levels. Hard thresholding shows a maximum of 23.4494 dB improvement whereas soft
thresholding shows a maximum of 43.9606 dB improvement in output signal to noise ratio.
Keywords : Hard Thresholding, Soft Thresholding, Multi-Resolution Analysis, Signal To Noise Ratio.
INTRODUCTION
Wavelets have become a popular tool for speech processing, such as speech analysis, pitch detection and speechrecognition. Wavelets are successful front end processors for speech recognition, this by exploiting the time
resolution of the wavelet transform. For the speech recognition, the mother wavelet is based on the Hanning
window. The recognition performance depends on the coverage of the frequency domain. The goal for good
speech recognition is to increase the bandwidth of a wavelet without significantly affecting the time resolution.
This can be done by compounding wavelets [8].
White noise is the most difficult to detect and to remove. The harmonic signal content is closely correlated,
resulting in larger coefficients than the uncorrelated noise. The noise can thus be removed by discarding small
coefficients. White noise can be handled either by hard and soft thresholding. Hard thresholding sets all the
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Page 63 of 71
wavelet coefficients below the given threshold value equal to zero and exhibits artifacts. Soft thresholding
smoothen the signal by reducing the wavelet coefficients by a quantity equal to the threshold value and modifies
the signal energy [7]. For the denoising of the signal it is assumed that the noise can be approximated by a
Gaussian distribution. The speech components will have large values compared to the noise. The computation of
the coefficients is done using a multi-resolution wavelet filter bank. The filter choice depends on the noise level
and other parameters. For a good denoising result, a good threshold level has to be estimated. The wavelet
function and the decomposition level also play an important role in the quality of the denoise signal [7].
Recently, various wavelet based methods have been proposed for the purpose of speech denoising. The wavelet
split coefficient method is a speech denoising procedure to remove noise by shrinking the wavelet coefficients
in the wavelet domain. The method is based on thresholding in the signal that each wavelet coefficient of the
signal is compared to a given threshold. Using wavelets to remove noise from a signal requires identifying
which components contain the noise, and then reconstructing the signal without those components. The
application areas of wavelet transform are wavelet modulation in communication channels, producing &
analyzing irregular signals etc.
The paper is organized as follows: Section 2 Wavelet Transforms, Section 3 Multi-resolution analysis, Section 4
Noise removal algorithm using discrete wavelet transform, Section 5 Results and discussion, Section 6
conclusion.
WAVELET TRANSFORM
A wavelet is a small wave, which has its energy concentrated in time to give a tool for the analysis of non-
stationary, or time varying phenomena [3]. Wavelet transforms convert a signal into a series of wavelets and
provide a way for analyzing waveforms, bounded in both frequency and duration. By using wavelet Transform,
we can get the frequency information which is not possible by working in time-domain.
There are many different wavelet systems that can be used effectively. A wavelet system is a set of building
blocks to represent a signal. One advantage of wavelet transform analysis is the ability to perform local analysis
and the calculation of the coefficients from the signal can be done efficiently. Wavelet analysis is able to express
signal appearance that other analysis techniques miss such as breakdown points, discontinuities etc.
The continuous wavelet transform performs a multi-resolution analysis by contraction and dilatation of the
wavelet functions and discrete wavelet transform (DWT) uses filter banks for the construction of the multi-
resolution time-frequency plane [7].
DWT provides sufficient information both for analysis and synthesis and reduce the computation time
sufficiently.
MULTIRESOLUTION ANALYSIS
The time-frequency resolution problem is caused by the Heisenberg uncertainty principle and exists regardless
of the used analysis technique. By using an approach called multi-resolution analysis (MRA) it is possible to
analyze a signal at different frequencies with different resolutions [7]. It is more suitable for short duration of higher frequency and longer duration of lower frequency components. It
is assumed that low frequencies appear for the entire duration of the signal, whereas high frequencies appear
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from time to time as short interv
calculates the correlation between
between the signal and the analy
resulting in a two dimensional re
mother wavelet.
Wavelet decomposition involves fil
content of the signal in half. It the
redundancy. Next, Wavelet thresh
These coefficients are again combi
to remove the noise completely).
Process after decomposition or an
coefficients. Wavelet reconstruc
components can be assembled back
Figure 1 shows the three-level wave
Fig.
NOISE REMOVAL ALGORI
We assume noisy speech signal (
=
+
Where represents original signal,[5]
. Now, we want to find threshold
International Journal of Electrical, Electronics & Com
al. This is often the case in practical applications.
the signal under consideration and a wavelet funct
ing wavelet function is computed separately for di
resentation. The analyzing wavelet function (t) is
tering and down-sampling. The low and high-pass fil
efore seems logical to perform a downsampling wit
lding is applied to the approximation coefficient
ed to reconstruct the noise free signal (where as prac
lysis is called synthesis where we reconstruct the si
tion process consists of up-sampling and filteri
into the original signal without loss of information.
let decomposition and reconstruction structure.
1. Wavelet decomposition and reconstruction structure
HM USING DISCRETE WAVELET TRAN
):
i = 1, 2, 3 -----N (1)
is standard deviation of noise and represents arvalue that will use to remove noise from noisy signal,
. Engg. Vol. 1 (2), 2011
Page 64 of 71
The wavelet analysis
ion (t). The similarity
ifferent time intervals,
also referred to as the
ters split the frequency
a factor two to avoid
nd detail coefficients.
tically it’s not possible
gnal from the wavelet
ng. In reconstruction,
SFORM
ray of random numbersbut also recover the
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Page 65 of 71
original signal efficiently. If the threshold value is too high, it will also remove the contents of original signal
and if the threshold value is too low, denoising will not work properly.
One of the first methods for selection of threshold was developed by Donoho and Johnstome [4] and it is called
as universal threshold. Different universal threshold was proposed in [2]:
ℎ = 2 2 () (2)
Where N denotes number of samples of noise and is standard deviation of noise. Later found that threshold
obtained by Eq. (2) is too high.
Again Universal threshold was proposed in [5] and modified with factor ‘k’ in order to obtain higher quality
output signal:
ℎ = . 2 2 () (3)
During our research it is noticed that if we use two factors i.e. k & m, threshold value obtained by Eq. (3) gives
better results. We will discuss this in next section 5.
The soft and hard thresholding methods are used to estimate wavelet coefficients in wavelet threshold denoising
[7]. Hard thresholding zeros out small coefficients, resulting in an efficient representation. Soft thresholding
softens the coefficients exceeding the threshold by lowering them by the threshold value. When thresholding is
applied, no perfect reconstruction of the original signal is possible.
Hard thresholding can be described as the usual process of setting to zero the elements whose absolute values
are lower than the threshold. The hard threshold signal is x if x thr and is 0 if x thr, where ‘thr’ is a
threshold value. Soft thresholding is an extension of hard thresholding, first setting to zero the elements whose
absolute values are lower than the threshold, and then shrinking the nonzero coefficients towards 0. If x thr,
soft threshold signal is (sign (x). (x - thr)) . Figure 2 shows the comparision between hard thresholding and soft
thresholding (threshold value ‘thr = 0.5’).
() = || ≥ ℎ0 || < ℎ (4)
() = ().( − ℎ) ≥ ℎ0 −ℎ ≤ < ℎ
(
) .(
+
ℎ)
<
−ℎ
(5)
Original signal Soft threshold Hard threshold
Fig. 2. Original signal, hard thresholding and soft thresholding
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Denoising algorithm scheme is showed in Figure 3. Inverse DWT is applied to get denoise time domain signal.
Fig. 3. Denoise Algorithm
RESULTS AND DISCUSSION
There are several software packages available to study, experiment with, and apply wavelet signal analysis.
Math Works, Inc. has a Wavelet Toolbox; Donoho’s group at Stanford has Wave Tool, Taswell at Stanford has
Wave Box and many more software’s are available[3]
. We implemented white noise removal algorithm in
Matlab 7.10.0 (R2010a). Wavelet toolbox in Matlab has large collection of functions for wavelet analysis. Input
of our simulation is original signal in Wave format and Noisy signal is created by adding random generated
numbers to the original signal i.e. see Eq. (1).
It is assumed that high amplitude DWT coefficients represent signal, and low amplitude coefficients represent
noise. It is considered that some samples of noisy signal contain only noise, so we choose that samples for noise
calculation.
It’s very important to select the proper level in multi-resolution analysis. In multi-resolution analysis, the
approximation signals (output of Low-pass filter & then decimation) are splits upto the certain level.
Original signal Xi
White noise ni
Noisy signal Yi
DWT
Thresholding
IDWT
De-noised signal
+
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Figure 4 shows the Original speech
Figure 5 shows the noisy signal.
We choose 5-level DWT and Da
threshold value is obtained by replaℎ =
Where, 0 < k < 1 and 0 < m < 1. ‘N’
We use two factors ‘k’ and ‘m’. I
different range of threshold value w
We apply this threshold value to
thresholding separately. Finally u
thresholding results are more efficie
Figure 6 shows the reconstructed (d
International Journal of Electrical, Electronics & Com
signal.
Fig. 4. Original signal
Fig. 5. Noisy signal
ubechies filters with 5 coefficients are used i.e. d
ing threshold ‘thr’ value of Eq. (3) with.. 2 2 ()(6)
denotes number of noise samples and is standard
t is found that if we fix one factor & vary other f
hich gives improved results especially for low level n
approximation coefficient & all detail coefficients.
se these new coefficients to reconstruct the signa
nt than hard thresholding.
enoise) signal.
Fig. 6. Reconstructed signal for white noise
of 15 db using soft threshold
. Engg. Vol. 1 (2), 2011
Page 67 of 71
b5 wavelet. Improved
deviation of noise.
ctor, than we can get
oise.
We use soft & hard
l. We found that soft
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Figure 7 shows the noise in noisy si
Figure 8 shows the noise in denoise
In example 1 as shown in table 1,
the value of Post SNR also increase
Table 1. Results of n
Std.
Dev.
σ
k
0.0061 0.1
0.0061 0.2
0.0061 0.3
0.0061 0.4
0.0061 0,5
0.0061 0.6
0.0061 0.7
0.0061 0.8
0.0061 0.9
International Journal of Electrical, Electronics & Com
gnal.
Fig. 7. Noise in noisy signal
signal.
Fig. 8. Noise in de-noise signal
hen fix the value of factor m= 0.9 & varies the value
s. Graph 1 shows the comparison between factor ‘k’
oise removal algorithm for modified soft thresholding method (exa
m thr
Input
MSE
(10-4)
Output
MSE
(10-4)
SNR
Pre
(db)
S
Po
(d
0.9 0.0029 0.3696 0.1578 15 18.
0.9 0.0059 0.3696 0.0592 15 22.9
0.9 0.0087 0.3696 0.0203 15 27.6
0.9 0.0118 0.3696 0.0043 15 34.3
0.9 0.0147 0.3696 0.0004 15 43.9
0.9 0.0178 0.3696 0.00043 15 44.3
0.9 0.0207 0.3696 0.0002 15 47.0
0.9 0.3779 0.3696 0.000089 15 51.1
0.9 0.0255 0.3696 0.000048 15 53.7
. Engg. Vol. 1 (2), 2011
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of ‘k’ from 0.1 to 0.9,
Post SNR.
ple 1)
R
st
)
02
623
097
095
606
655
607
848
823
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Input MSE is defined as [2]:
Where, is original signal and iOutput MSE is defined as:
Where,
is reconstructed signal.
In example 2 as shown in table 2, w
SNR level of noisy signal. Mean sq
Table 2. Results of n
Std.
Dev.
σ
k
0.0343 0.5
0.0189 0.5
0.0108 0.5
0.0061 0.5
Denoising is successful if output M
two signals.
Graph 2 shows comparison between
Graph 3 shows comparison between
thresholding method.
International Journal of Electrical, Electronics & Com
Graph 1 . K v/s SNR ( Post ) db
∑ ( − ) (7)
noisy signal.
∑ ( − ) (8)
e fix the value of both factor ‘k’ & ‘m’ and analyse th
are error (MSE) is mostly used for estimating signal
oise removal algorithm for modified soft thresholding method (exa
m thr
Input
MSE
(10-4)
Output
MSE
(10-4)
SNR
Pre
(db) (
0.9 0.083 11.7601 0.0253 0 26
0.9 0.0457 3.5838 0.0093 5 31
0.9 0.0262 1.1837 0.0024 10 36
0.9 0.0147 0.3696 0.0004 15 43
SE is lower than input MSE. Lower MSE means th
input MSE with Output MSE at different SNR leve
SNR (Pre) db with SNR (Post) db at different SNR l
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
K SNR (Post) db
. Engg. Vol. 1 (2), 2011
Page 69 of 71
e post SNR at different
uality.
ple 2)
NR
ost
db)
.6606
.0089
.8071
.9606
closer match between
l.
vel by using soft
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Graph 2 . Input MSE v/s O
Another method to analyse the deno
much a signal has been corrupted b
corrupting the signal. Denoising is s
Signal to Noise Ratio is defined as:
= 10 , ,In example 3 as shown in table 3,
compare to analysis done by using s
Graph 4 shows comparison betwe
SNR (pre) db with SNR (Post) db at
Table 3. Results of n
Std.
Dev.
σ
k
0.0343 0.5
0.0189 0.5
0.0108 0.5
0.0061 0.5
Graph 4 . Input MSE v/
11.7601
3.5838 1.1837
0.0253 0.0093 0.002
Input MSE Output
11.7601
3.5838
1.183
1.0415 0.3341
Input MSE O
International Journal of Electrical, Electronics & Com
utput MSE Graph 3 . SNR ( Pre ) db v/s S
ise signal is by Signal to Noise Ratio (SNR). SNR is
noise. It is defined as the ratio of signal power to the
uccessful if Post SNR is higher than Pre SNR.
= , − ,
(9)
hard thresolding method is used and found that res
oft thresholding method.
n input MSE with Output MSE and Graph 5 show
different SNR level by using hard thresholding meth
ise removal algorithm for modified hard thresholding method (exa
m thr
Input
MSE
(10-4)
Output
MSE
(10-4)
SNR
Pre
(db)
SN
Po
(d
0.9 0.083 11.7601 1.0415 0 10.5
0.9 0.0457 3.5838 0.3341 5 15.4
0.9 0.0262 1.1837 0.1305 10 19.5
0.9 0.0147 0.3696 0.0529 15 23.4
s Output MSE Graph 5 . SNR ( Pre ) db v/s
0.3696
0.0004
MSE
0 5 10
26.660631.0089
36.8071
SNR
Pre
(db)
S
P
(d
70.3696
.1305 0.0529
utput MSE
0 5 10
10.5073
15.4447
19.526
SNR(Pre)db SN
. Engg. Vol. 1 (2), 2011
Page 70 of 71
NR ( Post ) db
used to quantify how
noise power
ults are not so good as
s comparison between
od.
mple 3)
R
st
)
73
47
265
94
SNR ( Post ) db
15
43.9606
R
st
b)
15
5
23.4494
R (Post)db
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Page 71 of 71
CONCLUSION
In this paper we used wavelet transform for denoising speech signal corrupted with white noise. Speech
denoising is performed in wavelet domain by thresholding wavelet coefficients. We found that by using
modified universal threshold, We can get the better results of de-noising, especially for low level noise. During
different analysis we found that soft thresholding is better than hard thresholding because soft thresholding gives
better results than hard thresholding. Higher threshold removes noise well, but the part of original signal is also
removed with the noise. It is generally not possible to filter out all the noise without affecting the original signal.
We can analyse the denoise signal by signal to noise ratio (SNR) and mean square error (MSE) analysis.
REFERENCES
[1] Adhemar Bultheel, September 22 (2003), “Wavelets with applications in signal and image processing”.
[2] Alexandru Isar, Dorina Isar , May (2003): “Adaptive denoising of low SNR signals”, Third International
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Transforms”
[4] D.L. Donoho, May (1992), Stanford University: "De-noising by Soft Thresholding”, IEEE
TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 3.
[5] MATKO SARIC, LUKI BILICIC, HRVOJE DUJMIC (2005), “White Noise Reduction of Audio Signal
using Wavelets Transform with Modified Universal Threshold”, University of Split, R. Boskovica b. b HR
21000 Split, CROATIA.
[6] Michel Misiti, Yves Misiti, Georges Oppenheim, Jean Michel Poggi (2007), “Wavelet and their
applications”
[7] Prof. Dr. Ir. M. Steinbuch, Dr. Ir. M.J.G. van de Molengraft, June 7 (2005), Eindhoven University of
Technology , Control Systems Technology Group Eindhoven, “Wavelet Theory and Applications”, a
literature study, R.J.E. Merry, DCT 2005.53
[8] R.F. Favero (1994), Compound wavelets: “wavelets for speech recognition”, IEEE, pages 600–603.
[9] Yuan Yan Tang (2001), “Wavelet analysis and its applications”: second international conference, Springer-
Verlag, (U.K.)