Circuits and Systems, 2013, 4, 459-465
Published Online November 2013 (http://www.scirp.org/journal/cs)
http://dx.doi.org/10.4236/cs.2013.47060
Open Access CS
A Robust Denoising Algorithm for Sounds of Musical
Instruments Using Wavelet Packet Transform
Raghavendra Sharma, Vuppuluri Prem Pyara
Department of Electrical Engineering, Dayalbagh Educational Institute, Agra, India
Received September 17, 2013; revised October 17, 2013; accepted October 24, 2013
Copyright © 2013 Raghavendra Sharma, Vuppuluri Prem Pyara. This is an open access article distributed under the Creative Com-
mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited.
ABSTRACT
In this paper, a robust DWPT based adaptive bock algorithm with modified threshold for denoising the sounds of musi-
cal instruments shehnai, dafli and flute is proposed. The signal is first segmented into multiple blocks depending upon
the minimum mean square criteria in each block, and then thresholding methods are used for each block. All the blocks
obtained after denoising the individual block are concatenated to get the final denoised signal. The discrete wavelet
packet transform provides more coefficients than the conventional discrete wavelet transform (DWT), representing ad-
ditional subtle detail of the signal but decision of optimal decomposition level is very important. When the sound signal
corrupted with additive white Gaussian noise is passed through this algorithm, the obtained peak signal to noise ratio
(PSNR) depends upon the level of decomposition along with shape of the wavelet. Hence, th e optimal wav elet and lev el
of decomposition may be different for each signal. The obtained denoised signal with this algorithm is close to the
original signal.
Keywords: DWPT; Adaptive Blo ck Denoising; Peak Signal to Noise Ratio; Wavelet Thresho lding
1. Introduction
In the field of denoising the sounds of musical instru-
ments, time frequency based transforms play an impor-
tant role. They allow u s to work with a sound sign al fro m
both time and frequency perspectives simultaneously.
Such transforms have traditionally been useful in study-
ing the nature of the sound signal, noise, and in facilitat-
ing the application of aesthetically interesting and novel
modification to specific sound signals [1]. We are inter-
ested in a transform that is useful in working with musi-
cal instrument sound signals, and we look at the applica-
tion of the discrete wavelet packet transform (DWPT) to
remove the additive white Gaussian noise. There are
several reasons for choosing the DWPT, it is inherently
multi-resolution, making it more suited to human psyc-
hoacoustics than fixed resolution transforms as short time
Fourier transform (STFT) [2]. It is easily reconfigured to
allocate time frequency resolution in different ways
through various basis selection approaches. Furthermore,
efficient discrete time algorithms are available, and the
transform basis function is inherently time localized
without the introduction of a separate window function.
Signals may be transformed, modified and re-synthesized
using DWPT without affecting the quality of the signal
[3].
Noise has been a major problem for all signal proc-
essing applications. An unwanted signal gets superim-
posed over clean undistu rbed signal. Noise exists in high
frequency, but the sound signal is primarily low fre-
quency. Since the wavelet transform decomposes the sig-
nal into approximation (low frequency) and detail (high
frequency) coefficients [4,5], much of the noise is con-
centrated in detail coefficients. This suggests a method to
denoise the signal, simply reducing the size of the detail
coefficients before using them to reconstruct the signal,
which is called thresholding or shrinkage rule [6]. We
cannot eliminate the detail coefficients entirely, because
they contain some important information of the signal.
Various kinds of thresholding have been proposed in
literature [7], but the choice depends upon the application
at hand. The two important types of thresholing, hard and
soft have been used to denoise the signal. In hard thresh-
olding the wavelet coefficients below the given threshold
are set to zero, but in soft thresholding the wavelet coef-
ficients are reduced by a quantity equal to the threshold

R. SHARMA, V. P. PYARA
460
value. The extension of discrete wavelet transform is
discrete wavelet packet transform in which we split both
low pass and high pass filters at all scales in filter bank
implementation to obtain flexible and detail analysis
transform for denoising the sound signals [8]. In [9],
wavelet packet approach which deals with heterogeneous
noise for preprocessing of mass spectrometry data is
discussed which incorporate a variance change point de-
tection method in thresholding. Wavelet packet method
has been used to reduce the Additive White Gaussian
Noise from the speech signal which shows significant
SNR improvement [10]. The rest of the article is organ-
ized as follows: In Section 2, brief theory of discrete
wavelet packet transform (DWPT) is given. Wavelet
packet adaptive block denoising scheme is discussed in
Section 3, which is preceded by block denoising algo-
rithm based on DWPT in Section 4. The various experi-
mental results are discussed in Section 5. Section 6 gives
the concluding remarks based on the experimental re-
sults.
2. Discrete Wavelet Packet T r a nsform
(DWPT)
Discrete wavelet packet transforms are used to get the
advantage of better frequency resolution representation.
When the wavelet transform is generalized to wavelet
packet transform, not only the low pass filter output is
decomposed through further filtering, but the high pass
filter output decomposed as well. The ability to decom-
pose the high pass filter outputs means that the wavelet
packet allows for more than one basis function at a given
scale, versus the wavelet transform which has one basis
function at each scale other than the deepest level, where
it has two.
The set of wavelet packets collectively make up the
complete family of possible basis, and many potential
basis can be constructed from them. If only the low pass
filter is decomposed, the result is wavelet basis. If all low
pass and high pass filters are decomposed, the complete
tree basis results. This basis has the time frequency parti-
tioning like STFT. Between these two extremes lie a
large number of possible basis and their associated sub
trees. Nodes can be merged or split based on the re-
quirement of application. In all cases, the leaves of each
connected sub tree of the complete wavelet packet tree
from the basis of initial space; they span the space in
linearly independent fashion. The tree diagram of a
depth-3 complete tree basis is shown in the Figure 1.
As with the wavelet transform tree diagram in [11],
denotes the depth within the transform and the posi-
tion of each node k


,
j
k, but now the position index
conveys more information, specifically which wavelet
packet it corresponds to a given scale. We refer to the
associate wavelet packet as ,,
j
kp
w analogus to the
Figure 1. Depth-3 discrete wavelet packet transform tree.
wavelet ,kp. The tree diagram does not convey time
domain information, so the index
w
p
is not used in node
naming. Hence in wavelet packet, if all the packets are at
the same scale, we may simply refer to them as as
shown in the Figure 1. k
w
Furthermore, ,
j
k is either the scaling function, or
derived from the scaling function. DWPT does not re-
quire the explicit definition of wavelet, only filter d efini-
tions are enough. To see the wavelet packet at given level
of decomposition, we can do a recursion of them at
each node moving down the tree, to get the wavelet at
next level. Specifically, if we split a wavelet packet node
at level and position into two nodes at level
w
jk
1j

and locations and , we get the follow-
ing two packets: 2k2k1


m
  
1
1,20 ,
0
2
M
jk jk
m
wn hmwnm



 (1)
and
  
1
1,211,
0
2
M
jk jk
m
wnhmwn

 

 (2)
Then the wavelet packet transform coefficients ,,
j
kp
c
are given by:


1
,, ,,
0
M
jk pjk p
m
csmw


m (3)
And the original signal can be expressed in terms of
these coefficients and the corresponding wavelet packets
as:




,, ,,
,,
,
jk pjk p
jkp
s
mcwm (4)


,
j
k

all leaf nodes of basis.
where p ranges over all time offsets at scale j for which
signal s is defined.
3. Wavelet Packet Adaptive Block Denoising
The wavelet packet based denoising technique employs
the decomposition concept in adaptive base of wavelets.
This technique is efficient in denoising the musical sound
signal corrupted with additive white Gaussian noise
(AWGN), which is evenly distributed over the entire
signal, and removal of AWGN from noisy signal is dif-
ficult task. Donoho and Johnstone pioneered the work of
Open Access CS

R. SHARMA, V. P. PYARA
Open Access CS
461
filtering the additive white Gaussian noise using wavelet
thresholding [12]. The block denoising is explained in
the following sub sections:

, if
0, if
, if
s
xx
fx x
xx













(7)
3.1. Thresholding Based Denoising In [13], we see that the soft thresholding gives lesser
mean square error. Due to this reason soft thresholding is
preferred over hard thresholding, but in case of some si-
gnals, we could see that hard thresholding results in les-
ser amount of mean square error.
A noise reduction technique developed by donoho, uses
the wavelet coefficients contraction and its principle
consists of three steps;
1) Apply discrete wavelet transform to noisy signal:
Wy WsWz  (5) 3.2. Block Selection
where ,,
y
sz
and W are the noisy musical instru-
ment sound, original clean sound signal, noise signal
and the matrix associated to the discrete wavelet trans-
form respectively.
Most of the musical instrument sound signals are far too
long to be processed in their entirety; for example a 10
second sarangi sound signal sampled at 44.1 KHz will
contain 441,000 samples. Thus, as with spectral methods
of noise reduction, it is necessary to divide the time do-
main signal in multiple blocks and process the each block
individually. The block formation of the signal is shown
in the Figure 2. The important task is to choose the block
length. Berger et al. [14] shows that, blocks which are
too shorts fail to pick important time structures of the
signal. Conversely, blocks which are too long miss cause
the algorithm to miss the important transient details in
the musical instrument sound signal. Due to the binary
splitting nature of the tree bases in wavelet analysis to
decompose the signal, it is better to choose the length of
each block with a number of samples to a power of two.
2) Threshold the obtained wavelet coefficients.
3) Reconstruct the desired signal by applying the inverse
wavelet transform to the thresholded wavelet coeffi-
cients.
The thresholding function which is also known as
wavelet shrinkage function is categorized as hard thresh-
olding and soft thresholding function. The hard thresh-
oldingfunction retains the wavelet coefficients which are
greater than the threshold λ and sets all other to zero. The
hard thresholding is defined as:

, if
0, otherwise
h
xx
fx





 (6)
As discussed previously, the block size chosen must
strike a balance between being able to pick up important
transient detail in the sound signal, as well as recognizing
longer duration, sustained events. Tables 1 and 2 shows
the PSNR values which are quality measures, obtained
for various block sizes and fo r different signals.
The threshold

is chosen according to the signal
energy and the standard deviation

of the noise. If the
wavelet coefficient is greater than

, then it is assumed
that it is significant and contributes to the original signal.
Otherwise it is due to the noise and discarded. The soft
thresholding function shrinks the wavelet coefficients by

towards zero. Hence this function is also called as Tables 1 and 2 show that the PSNR values for differ-
ent wavelets are varying with the block size. Hence the
optimum block is that for which we have maximum
PSNR or minimum mean square error. The optimal block
shrinkage function. The soft thresholding function is de-
fined as:
Table 1. PSNR values obtained for different block length on shehnai sound with different wavelets.
Samples/block length ( ms) haar db10 sym3 coif5 dmey
1024/23 23.06 33.99 30.99 36.42 36.41
2048/46 23.37 34.82 30.27 36.07 36.57
4096/92 23.96 36.45 31.73 36.9 39.62
8192/185 22.50 34.06 30.73 36.81 38.37
16,384/371 23.12 35.85 30.86 34.59 36.31
32,768/743 23.60 34.64 31.28 35.38 35.76
65,536/1486 22.50 33.21 30.95 35.25 36.25

R. SHARMA, V. P. PYARA
462
Table 2. PSNR values obtained for different block length on dafli sound with different wavelets.
Samples/block length ( ms) haar db10 sym3 coif5 dmey
1024/23 07.39 13.20 09.22 08.91 08.84
2048/46 32.74 34.84 23.43 34.15 37.22
4096/92 35.78 35.42 36.96 37.66 37.28
8192/185 37.17 44.95 43.17 46.76 42.57
16,384/371 43.77 50.17 45.63 49.68 47.20
32,768/743 41.89 44.79 42.88 42.58 39.49
65,536/1486 40.84 45.29 44.32 45.37 38.54
Block 1Block 2Blo ck N. . . . . . . . .
Total length of signal
Figure 2. Block formation of signal
size for shehnai sound is 4096 samples and for dafli
16,384 samples. The informal listening test agree with
this statement in a general sense, hence the block size is
variable for musical instrument sound signals.
3.3. Threshold Selection
Donoho and Johnstone derived a general optimal univer-
sal threshold for the Gaussian white noise under a mean
square error (MSE) criterion described in [12]. However
this threshold is not ideal for musical instrument sound
signals due to poor correlation between the MSE and
subjective quality and the more realistic presence of cor-
related noise. Here we use a new time frequency de-
pendent threshold estimation method. In this method first
of all the standard deviation of the noise,

is calcu-
lated for each block. For given

, we calculate the
threshold for each block. Noise component removal by
thresholding the wavelet coefficients is based on the ob-
servation that in musical instrument sound signal, energy
is mostly concentrated in small number of wavelet di-
mensions. The coefficients of these dimensions are rela-
tively very large compared to other dimensions or to any
other signal like noise that has its energy spread over a
large number of coefficients. Hence by setting smaller
coefficients to be zero, we can optimally eliminate noise
while preserving important information of the signal. In
wavelet domain noise is characterized by smaller coeffi-
cients, while signal energy is concentrated in larger coef-
ficients. This feature is useful for eliminating noise from
signal by choosing the appropriate threshold. Generally
the selected threshold is multiplied by the median value
of the detail coefficients at some specified level which is
called threshold processing.
At each level of decomposition, the standard deviation
of the noisy signal is calculated. The standard deviation
is calculated by Equation (8):


medi an
0.6745
j
j
c
  (8)
where
j
c are high frequency wavelet coefficients at jth
level of decomposition, which are used to identify the
noise components and
j

is Median Absolute Devia-
tion (MAD) at this level. This standard deviation can be
further used to set the threshold value based on the noise
energy at that level. The modified threshold value [15]
can be obtained by the equation (9):

2
2log log
hj j
TkL L   j (9)
where h is threshold value, T
j
L is the length of each
block of noisy signal and k is the constant whose value is
varying between 0 - 1. For determining the optimum
threshold, value of k should be estimated.
4. Denoising Algorithm
The proposed wavelet packet based block denoising al-
gorithm for reduction of white Gaussian noise is ex-
plained in the following steps:
1) Take a musical instrument sound signal of suitable
length.
2) Add White Gaussian Noise to the original signal de-
pending upon the standard deviation

.
3) Divide the noisy signal into blocks of different length
depending upon the length of the signal in time do-
main, and the number of samples should be to a po-
wer of two.
4) Determine the optimal block size based on minimum
mean square error criteria.
5) Compute the discrete wavelet packet transform (DWPT)
of one block of the noisy signal at level 1.
6) Estimate the standard deviation of the noise using
Equation (8) and determine the threshold value using
Equation (9), then apply the different thresholding
techniques for time and level dependent wavelet co-
Open Access CS

R. SHARMA, V. P. PYARA 463
efficients using Equations (6) and (7).
7) Take inverse discrete wavelet packet transform (IDWPT)
of the coefficients obtained through step 6, which has
reduced noise.
8) Calculate mean square error (MSE), peak signal to
noise ratio (PSNR) fo r de nois e d signal.
9) Repeat steps 4 to step 7 for other level of decomposi-
tion 2 - 5.
10) Concatenate all the blocks of the denoised signals
obtained through step 8 and do averaging operation
for MSE and PSNR of the musical instrument sound
signal.
The complete DWPT based denoising algorithm is
shown graphically in Figure 3.
5. Results and Discussions
The denoising algorithm developed in the previous sec-
tion is applied to the sound samples of the various Indian
musical instruments sampled at 44.1 K samples per sec-
ond. For experimental purpose the sounds of three musi-
cal instruments shehnai, dafli and flute are taken. For
comparing the performance of the various wavelets for
musical instrument sound signals, six wavelets haar,
db10, sym3, coif5, dmey and bior 2.2 are taken. Besides
observing the performance of the wavelets, the effect of
decomposition is also discussed.
For comparing the performance and measurement of
quality of denoising, the peak signal to noise ratio (PSNR)
is determined between the original signal and the
signal denoised by our algorithm. i
S
,
d
S
2
max
10
PSNR 10logMSE
S




(10)
where max is the maximum value of the signal and is
given by,
S






max max max,max i
SSd S (11)
And MSE is mean square error, given by:
original
signal
noisy
signal
gaussian
noise
dividenoisy
signalinto
blocks
DWPT
ofblock
level
dependent
modified
thresholding
IDWPTof
modified
coefficients
Denoised
block
concatenate
alldenoised
blocks
denoisedsignal
select
optimal
block
Figure 3. DWPT based block denoising algorithm with
modified threshold.
 
2
1
1N
di
l
M
SESlSl
N




(12)
where N is the length of the signal. The PSNR vaues
obtained for different wavelets applied on shehnai, dafli
and flute signals at different level of decomposition are
shown in Tables 3-5. The additive white Gaussian noise
is taken at 0.1


, which is approximately 50% of the
signal value.
It is observed from Tables 3-5 that the PSNR values
are dependent upon the shape of the wavelet, type of
thresholding and the level of decomposition. Hard
thresholds are better than soft thresholds for denoising
the musical instrument sound signals. The selection of
level of decomposition plays a significant role, and
should be optimal for best denoising results. Hence, the
shehnai sound will give best results when denoised with
db 10 wavelet at level 5, dafli sound with dmey at level 5
and flute sound with db10 at level 4, respectively. The
different signals denoised with optimal wavelet and level
of decomposition are shown in the Figures 4-6.
6. Conclusion
Adaptive wavelet packet transform has been widely used
in denoising the sounds of musical instruments and
2 4 6 810 12 14
x 10
4
-1
0
1
Samples
Amplit ude
Or
i
g
i
na
l
s
i
gna
l
2 4 6 810 12 14
x 10
4
-1
0
1
Samples
Amplit ude
Nois y s ignal
2 4 6 810 12 14
x 10
4
-1
0
1
Samples
Amplitude
De-noised s ignal
Figure 4. Original, noisy and denoised shehnai signal with
db 10 at level 5.
1 2 34 5 6
x 10
4
-1
0
1
Samples
Amplitude
Noisy s ignal
1 2 34 5 6
x 10
4
-1
0
1
Samples
Amplitude
De-noised signal
1 2 34 5 6
x 10
4
-1
0
1
Samples
Amplitude
Original signal
Figure 5. Original, noisy and denoised dafli signal with
mey at level 5. d
Open Access CS

R. SHARMA, V. P. PYARA
Open Access CS
464
Table 3. PSNR values of shehnai sound after decompostion at different levels.
Level 2 Level 3 Level 4 Level 5
Wavelet soft hard soft hard soft hard soft hard
haar 26.02 31.33 18.57 18.72 23.09 27.65 16.89 25.32
db10 23.95 23.43 18.53 18.55 20.14 27.23 16.52 31.62
sym3 26.44 25.68 18.56 18.56 19.65 27.21 14.85 30.39
coif5 24.02 23.71 18.68 19.65 21.05 27.22 18.13 29.65
dmey 23.92 23.78 18.57 18.76 23.34 23.48 22.76 23.43
bior2.2 31.06 26.27 18.44 18.54 25.26 25.75 20.61 26.93
Table 4. PSNR values of dafli sound after decomposition at different levels.
Level 2 Level 3 Level 4 Level 5
Wavelet soft hard soft hard soft hard soft hard
haar 20.07 19.88 17.56 17.72 27.76 27.58 26.08 27.48
db10 20.14 19.56 17.66 17.75 24.30 24.56 26.05 27.67
sym3 20.12 19.86 17.46 17.56 26.12 25.24 26.92 26.93
coif5 19.65 20.05 17.50 17.59 25.88 24.88 27.05 26.79
dmey 20.05 19.78 17.44 17.65 24.47 25.18 27.05 27.97
bior2.2 20.00 19.77 17.35 17.53 23.58 21.73 24.21 23.11
Table 5. PSNR values of flute sound after decomposition at different levels.
Level 2 Level 3 Level 4 Level 5
Wavelet soft hard soft hard soft hard soft hard
haar 21.83 22.99 19.55 19.21 16.16 18.82 9.07 18.97
db10 23.73 24.02 19.09 19.40 13.65 35.71 11.31 33.00
sym3 24.50 24.64 19.36 19.52 16.49 28.64 11.48 30.47
coif5 23.98 24.10 19.40 19.63 13.75 33.98 11.42 29.68
dmey 23.75 24.37 19.25 19.53 13.74 34.53 11.76 30.26
bior2.2 28.80 27.72 19.41 19.48 13.46 18.94 10.87 19.70
Providing better performance in terms of PSNR values
than the other denoising techniques. In this paper, dis-
crete wavelet packet transform is used for denoising-
shehnai, dafli and flute sound signal corrupted with addi-
tive white Gaussian noise, 50% of the signal strength.
First, sound signal is divided into multiple blocks de-
pending upon the optimal block size for each signal. De-
noising of signal is performed with these optimal block
sizes in wavelet packet domain by thresholding the
wavelet coefficients at different level of decomposition.
It is observed that hard thresholding gives better PSNR
than soft thresholding at all the decomposition levels.
The choice of the optimal level of decomposition is im-
portant, and different for each sound signal. If the level
of decomposition is not optimal then the PSNR value
2 4 6810 12 14 16
x 104
-1
0
1
Samples
Amplitude
Noisy s ignal
2 4 6810 12 14 16
x 104
-1
0
1
Samples
Amplitude
De-nois ed s ignal
2 4 6810 12 14 16
x 104
-1
0
1
Samples
Amplitude
Original signal
Figure 6. Original, noisy and denoised flute signal with
db10 at level 4.

R. SHARMA, V. P. PYARA 465
will not be maximum, hence denoising will not be the
best. Maximum PSNR value for shehnai sound is at level
5 with db10 wavelet, dafli at level 5 with dmey and flute
at level 4 with db10 respectively. When each block is
denoised, all the blocks are concatenated to form the fi-
nal denoised signal. It is also observed that when modi-
fied threshold with is used, the PSNR values are in-
creased. Higher thresholds remove the noise well but
some parts of the original signal are also removed be-
cause it is not possible to remove the noise without af-
fecting the original signal.
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