2 rajeev aggrawal research article april 2011

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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



Rajeev Aggarwal et. al / VSRD International Journal of Electrical, Electronics & Comm. Engg. Vol. 1 (2), 2011

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



Rajeev Aggarwal et. al / VSRD

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




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




Page 66 of 71

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

+




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


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

7 of 71

b5 wavelet. Improved

deviation of noise.

ctor, than we can get

oise.

We use soft & hard

l. We found that soft




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


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

Page 68 of 71

of ‘k’ from 0.1 to 0.9,

Post SNR.

ple 1)

R

st

)

02

623

097

095

606

655

607

848

823




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


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.


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




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


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


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




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

Conference on WAA 2003, Chongqing, P. R. China, 29-31, p.p. 821-826[3] C. Sidney Burrus, Ramesh A. Gopinath, Haitao Guo (1998), “Introduction to Wavelets and Wavelet

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.)

2 rajeev aggrawal research article april 2011

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