Digital Signal Processing 20 (2010) 1352–1364

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Digital Signal Processing

www.elsevier.com/locate/dsp

Convolution wavelet packet transform and its applications to signalprocessing

Xuezhi Zhao ∗, Bangyan Ye

School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Available online 7 January 2010

Keywords:Convolution wavelet packet transformDecomposition and reconstructionLength invarianceSignal processingNoise reduction

The length of decomposition results of traditional wavelet packet transform (WPT) willdecrease by half in the next level for downsampling, then the length of sequences inthe last level will become very short, and this is very inconvenient for further analysis ofthese sequences. One kind of WPT based on convolution definition is put forward, its fastdecomposition and reconstruction algorithms are given, and the outstanding characteristicof this convolution WPT is that no matter how many levels a signal is decomposed, thelength of sequences got in every level will never decrease and can always keep the sameas that of the original signal, so the defect of traditional WPT is overcome. For traditionalWPT, to achieve the same effect of direct decomposition of convolution WPT, reconstructionoperation must be done and the calculation will greatly increase. Based on the lengthinvariance property of convolution WPT, a noise reduction algorithm is proposed, andsignal processing example shows that its denoising performance is better than that oftraditional WPT, and also much better than that of wavelet transform.

© 2010 Elsevier Inc. All rights reserved.

1. Introduction

As is well known, wavelet packet transform (WPT) is a further decomposition for wavelet transform, and both theapproximation and detail signals obtained by wavelet transform at each level will be further decomposed by WPT, so resultswith higher resolution in time and frequency domain can be got. According to the traditional wavelet packet theory, theWPT of signal x(t) is defined as follows

xn, jp = 2− j/2

∫R

x(t)μn(2− jt − p

)dt, 0 � j � S, 0 � n � 2S − 1, (1)

where μn(t) is wavelet packet function, j is the number of decomposition level, or so-called scale parameter, p is theposition parameter, n is the channel number, S is the maximum decomposition level. For signal x(t), after it is decomposedby WPT, 2S sequences can be got in the Sth level.

The corresponding fast decomposition algorithm for this kind of WPT is [1]⎧⎪⎪⎪⎨⎪⎪⎪⎩

x2n, j+1k =

∑p∈Z

h(p − 2k)xn, jp ,

x2n+1, j+1k =

∑p∈Z

g(p − 2k)xn, jp ,

(2)

* Corresponding author. Fax: +86 20 87111038.E-mail address: mezhaoxz@scut.edu.cn (X. Zhao).

1051-2004/$ – see front matter © 2010 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2010.01.007

X. Zhao, B. Ye / Digital Signal Processing 20 (2010) 1352–1364 1353

where h(i) and g(i) are wavelet quadrature mirror filter (QMF) coefficients. Eq. (2) is the classical decomposition formula ofWPT, and in the wide engineering applications of WPT, such as feature extraction for heart rate variability (HRV) signal [2]and electroencephalogram (EEG) signals [3], the analysis for the impulsive energy of the vibration signal of water hydraulicmotor [4], gearbox fault detection [5], online tracking of bearing wear [6], fault diagnosis of rotating machinery and moni-toring of machining processes [7–9], texture classification [10] and so on, they are all based on this decomposition formula.The outstanding characteristic of this formula is the downsampling, which means that when computing the sequences inthe ( j + 1)th level, for the data of sequences in the jth level, only those ones in the even points can be sampled to calculatethe sequences in the ( j + 1)th level. So compared with the length of sequences in the jth level, the one in the ( j + 1)thlevel will decrease by half. For example, if the length of original signal is 1024, then after decomposition of WPT, the lengthof the sequences got in the first level will decrease to 512, in the second level will decrease to 256, etc., like this way thelength of sequences in the next level will always decrease by half, so for the sequences in the last level, their length willbecome very short. This length degression of WPT may be very useful in data compression domain [10,11], but it’s verydiscommodious in fault diagnosis domain [4–9]. Because for getting the reliable diagnosis results, those sequences in thelast level will always be further analyzed, but their short length will make against the further analysis. For example, inorder to get the accurate fast Fourier transform (FFT) result of a sequence with high frequency resolution, in general thelength of this sequence is required to be more than 64 at least. Supposing the length of original signal is 1024, and themaximum decomposition level is S = 5, then after decomposition of WPT, for those sequences in the fifth level, their lengthwill all decrease to 32, and their lengths are so short that the FFT results with high frequency resolution can’t be got. Inaddition, it’s also very difficult to observe the waveform feature of these sequences for their too short length. In order tofurther analyze these sequences of the last level or observe their waveform feature, the only way is to use them to doreconstruction calculation. The reconstruction formula of WPT is

xn, jp =

∑k∈Z

[h(p − 2k)x2n, j+1

k + g(p − 2k)x2n+1, j+1k

]. (3)

Corresponding to the downsampling in the decomposition formula, the characteristic of this reconstruction one is upsam-pling, i.e. before computing xn, j

p , for sequences x2n, j+1k and x2n+1, j+1

k , one zero will be inserted between every two adjacentdata of these two sequences. By virtue of this reconstruction operation, the length of the sequences in the last level willrecover to that of the original signal, then the further analysis for these sequences can go on. For example, in Refs. [4–9],to reveal the feature in the decomposition results of WPT, the reconstruction operation of WPT had to be made so that thedecomposition results can recover to the same length of original signal and then the features can be clear shown. However,this reconstruction process is a troublesome matter, moreover its calculation will also greatly increase. Whether there is aWPT by which a signal is decomposed, the length of sequences, no matter which level these sequences are located in, willnever decrease and can always keep the same as that of the original signal? To solve this problem, one kind of convolutionWPT is put forward in this paper, and the fast decomposition and reconstruction algorithm for this WPT are given. The out-standing characteristic of this convolution WPT is that no matter how many levels a signal is decomposed, the sequencesgot in the every level can always keep the same length as that of the original signal, so the defect of traditional WPT iscompletely overcome and the further analysis for these sequences can conveniently go on without reconstruction operation.The signal processing example shows that the direct decomposition results of this convolution WPT are completely in linewith the reconstruction ones of traditional WPT, while its calculation quantity is much less than that of traditional WPT. Bymeans of the length invariance property of convolution WPT, a noise reduction algorithm is proposed, and the denoisingexample shows that its noise reduction performance is better than that of the traditional WPT, and also better than that ofwavelet transform.

2. Definition of convolution WPT and its fast algorithm

As is well known, for wavelet transform, besides the inner product definition, there is another definition, i.e. the convo-lution definition, which is proposed to detect the local singularity of signal by Mallat [12], but for WPT, there is only theinner product definition, as shown in Eq. (1). Whether the convolution definition is suitable for WPT? In this paper thisproblem is studied and it’s proved that the convolution definition for WPT is also feasible. For this purpose, let’s define theconvolution WPT as follows:

For signal x(t) ∈ L2(R), supposing that function series {2− j/2μn(2− jt − k) | k ∈ Z} make up of the orthonormal bases ofwavelet packet subspace Un

j , then the convolution WPT can be defined as follows

xn, jp = 1

2 j

∫R

x(t) · μn

(p − t

2 j

)dt = 1

2 j· x(t) ∗ μn

(t

2 j

), 0 � j � S, 0 � n < 2S , (4)

where j is the number of decomposition level, or so-called scale parameter, p is the position parameter, n is the channelnumber, S is the maximum decomposition level.

1354 X. Zhao, B. Ye / Digital Signal Processing 20 (2010) 1352–1364

For inner product WPT, its fast decomposition formula is shown in Eq. (2). Similarly, for the practicality of convolutionWPT, its fast decomposition formula should also be found. The definition of wavelet packet is⎧⎪⎪⎨

⎪⎪⎩μ2n(t) = √

2∑k∈Z

h(k)μn(2t − k),

μ2n+1(t) = √2∑k∈Z

g(k)μn(2t − k),(5)

where h(i) and g(i) are the wavelet QMF coefficients. The function series {μn(t) | n ∈ Z} shown in Eq. (5) are called theorthogonal wavelet packet.

In Eq. (5), let t = 2− j x, and do Fourier transform in both sides, then we obtain⎧⎪⎪⎪⎨⎪⎪⎪⎩

μ̂2n(2 jω

) = 1√2

∑k∈Z

h(k)e−i2 j−1ωkμ̂n(2 j−1ω

),

μ̂2n+1(2 jω

) = 1√2

∑k∈Z

g(k)e−i2 j−1ωkμ̂n(2 j−1ω

).

(6)

Define⎧⎪⎪⎪⎨⎪⎪⎪⎩

H(ω) = 1√2

∑k∈Z

h(k)e−iωk,

G(ω) = 1√2

∑k∈Z

g(k)e−iωk.

(7)

Then Eq. (6) can be written as{μ̂2n

(2 jω

) = H(2 j−1ω

)μ̂n

(2 j−1ω

),

μ̂2n+1(2 jω

) = G(2 j−1ω

)μ̂n

(2 j−1ω

).

(8)

According to the convolution theorem, converting Eq. (4) into frequency domain, we can get

x̂n, jω = x̂(ω)μ̂n

(2 jω

). (9)

In Eq. (9), replace ‘n’ by ‘2n’ and ‘ j’ by ‘ j + 1’, then this equation can be written as

x̂2n, j+1ω = x̂(ω)μ̂2n

(2 j+1ω

). (10)

While according to Eqs. (8) and (9), the right part of Eq. (10) can be expressed

x̂2n, j+1ω = x̂(ω)H

(2 jω

)μ̂n

(2 jω

) = H(2 jω

)x̂n, jω .

Considering the definition of H(ω), we obtain

x̂2n, j+1ω = 1√

2

∑k∈Z

h(k)e−i2 jωkx̂n, jω . (11)

As is well known, if the Fourier transform of x(t) is x̂(ω), then the Fourier one of x(t − b) is e−iωbx̂(ω), and this is thetranslation property of Fourier transform. According to this property, converting Eq. (11) into time domain, then we obtain

x2n, j+1p = 1√

2

∑k∈Z

h(k) · xn, jp−2 jk

. (12)

Correspondingly, the computation formula for x2n+1, j+1p can also be obtained by the similar deduction, and the only differ-

ence is that h(k) is replaced by g(k), so the fast decomposition formula for convolution WPT can be obtained as follows⎧⎪⎪⎪⎨⎪⎪⎪⎩

x2n, j+1p = 1√

2

∑k∈Z

h(k) · xn, jp−2 jk

,

x2n+1, j+1p = 1√

2

∑k∈Z

g(k) · xn, jp−2 jk

.

(13)

It can be seen that the concrete form of wavelet packet function μn(t) is not involved in this fast decomposition for-mula, and the decomposition of convolution WPT for any signal can be conveniently realized just by dint of wavelet QMFcoefficients h(k) and g(k), and this is just like Eq. (2). However, compared with Eq. (2), the fast decomposition formula of

X. Zhao, B. Ye / Digital Signal Processing 20 (2010) 1352–1364 1355

Fig. 1. Vibration acceleration signal of a ball bearing.

Fig. 2. The sequences of 8 channels in the 3rd level got by the traditional WPT.

traditional WPT, the obvious characteristic of Eq. (13) is that there is no downsampling, and to compute the sequences inthe ( j + 1)th level, just a translation of 2 jk is done for the sequences in the jth level, so the length of sequences in the( j + 1)th level will not decrease and can always keep the same as that of the original signal.

Although without the reconstruction operation, the direct decomposition results of convolution WPT can also keep thesame length as that of the original signal, for the completeness of convolution WPT, its reconstruction algorithm is alsogiven here. It can be known from the wavelet theory that for wavelet QMF h(k) and g(k), their relationship in the frequencydomain is∣∣H(ω)

∣∣2 + ∣∣G(ω)∣∣2 = 1. (14)

According to Eqs. (9) and (14), we have

x̂n, jω = x̂(ω)μ̂n

(2 jω

)= x̂(ω)μ̂n

(2 jω

)[∣∣H(2 jω

)∣∣2 + ∣∣G(2 jω

)∣∣2]= x̂(ω)μ̂n

(2 jω

)H

(2 jω

)H

(2 jω

) + x̂(ω)μ̂n(2 jω

)G(2 jω

)G(2 jω

).

Considering Eqs. (8) and (10), we obtain

x̂n, jω = x̂(ω)μ̂2n

(2 j+1ω

)H

(2 jω

) + x̂(ω)μ̂2n+1(2 j+1ω

)G(2 jω

)= x̂2n, j+1

ω H(2 jω

) + x̂2n+1, j+1ω G

(2 jω

).

The definition of H(ω) and G(ω) is taken into consideration, then we can get

x̂n, jω = 1√

2

∑k∈Z

h(k)ei2 jωkx̂2n, j+1ω + 1√

2

∑k∈Z

g(k)ei2 jωkx̂2n+1, j+1ω . (15)

According to the translation property of Fourier transform, converting Eq. (15) into time domain, then we can obtain thereconstruction formula for convolution WPT

xn, jp = 1√

2

∑k∈Z

h(k)x2n, j+1p+2 jk

+ 1√2

∑k∈Z

g(k)x2n+1, j+1p+2 jk

. (16)

Next a signal processing example will be given to test the validity of this convolution WPT, in addition, the comparison withthe traditional WPT will also be made.

3. Signal processing example and the comparison with the traditional WPT

A vibration acceleration signal of a ball bearing is shown in Fig. 1, the length of this signal is 1024 and the samplefrequency for this signal is 15 000 Hz. In this bearing some injuries exist in the ball track and the shocks will be causedwhen the balls pass these injuring points, as a result, some impulses will take place in the vibration acceleration signal.However, because the injuries are slight and also there is noise interference, these impulses are difficult to be identified and

1356 X. Zhao, B. Ye / Digital Signal Processing 20 (2010) 1352–1364

Fig. 3. The comparison between the reconstruction results of traditional WPT and direct decomposition ones of convolution WPT. (a) Reconstruction resultsof traditional WPT; (b) decomposition results of convolution WPT.

confirmed in the original signal. Here WPT method is used to reveal these impulses and locate their positions. The signalis firstly processed by traditional WPT, in which the wavelet QMF coefficients are selected as ‘db3’ Daubechies waveletones and the maximum decomposition level is set as 3, then 8 sequences can be got in the third level. However, for thedownsampling, the lengths of these 8 sequences all decrease to 128. These 8 sequences are placed in the same rank andtheir total length is 1024, as shown in Fig. 2. Because the length of each sequence is so short that no useful impulse featurecan be identified, to say the least, even though the impulses can be revealed in these sequences, the accurate positions ofimpulses in original signal can’t be located by them. Because the lengths of these sequences are much shorter than that ofthe original signal, it’s impossible to find out that the impulses in these sequences correspond to which positions in theoriginal signal. In order to display the impulses and locate their positions, the only way is to utilize these 8 sequences to doreconstruction operation of WPT, so that they can recover the same length of the original signal and then the impulses canbe revealed. The reconstruction results of these 8 sequences are shown in Fig. 3(a), here the impulses and their positionsare clear revealed, and these impulses mean the shocks caused by the injuring points in the ball track. However, thisfeature extraction process is very troublesome: firstly the decomposition operation of WPT should be done, and then thereconstruction operation of WPT for each sequence in the third level should be done too.

X. Zhao, B. Ye / Digital Signal Processing 20 (2010) 1352–1364 1357

Fig. 4. Spectrums comparison. (a) Spectrums of reconstruction results of traditional WPT; (b) spectrums of decomposition results of convolution WPT.

Now let’s see the decomposition results of convolution WPT. The decomposition formula (13) is used to decomposethe original signal into 3 levels, and 8 sequences got in the third level are illustrated in Fig. 3(b). It can be seen that theimportant difference from the ones of traditional WPT in Fig. 2 is that the lengths of these 8 sequences are all 1024, i.e.the length of original signal. While compared Fig. 3(b) with Fig. 3(a), one can easily see that whether the impulses or theirpositions in the direct decomposition results of convolution WPT are all in agreement with the ones in the reconstructionresults of traditional WPT. Further comparison is made in frequency domain, and the spectrums of all sequences in Fig. 3are illustrated in Fig. 4. It can be seen that whether for the frequency range or for the amplitude, the spectrums of thedirect decomposition results of convolution WPT are all in line with those of the reconstruction results of traditional WPT.These comparisons in time and frequency domain demonstrate the validity of convolution WPT, but its procedure is muchsimpler than that of the traditional WPT because reconstruction operation is completely avoided.

The algorithm complexity of these two kinds of WPT can be analyzed here. Supposing that the length of original signalis N , the maximum decomposition level is S , and the length of wavelet QMF coefficients is L, then for traditional WPT, tomake all the sequences in the last level recover their length to N , the total needed number of multiplication operation in theprocess of decomposition and reconstruction is SN[(2S+1 + 1)L + 1], and the number of addition operation is SN[2S (4L +1) + 2L]. While for convolution WPT, to obtain the same effect of traditional WPT, the reconstruction operation can beavoided, and in its decomposition process, the total needed number of multiplication operation is 2N(2S − 1)(2L + 1),

1358 X. Zhao, B. Ye / Digital Signal Processing 20 (2010) 1352–1364

Fig. 5. The comparison of reconstruction error. (a) Boundary error of traditional WPT; (b) boundary error of convolution WPT; (c) left boundary errorcomparison; (d) right boundary error comparison.

and the number of addition operation is 4N L(2S − 1). So compared with the convolution WPT, the additional number ofmultiplication operation needed in the traditional WPT is

2S+1[(S − 2)L − 1]N + [

(S + 4)L + S + 2]N.

While the additional number of addition operation needed in the traditional WPT is

2S[4(S − 1)L + S]N + 2(S + 2)N L.

According to these two formulas, if S = 1, for traditional WPT, the additional number of multiplication and addition op-eration is N(L − 1)and N(6L + 2) respectively. With the increasing of S , compared with the convolution WPT, the additionalnumber of multiplication and addition operation in the traditional WPT will increase with exponential speed, and this in-crease is very considerable. In this example, S = 3, N = 1024, L = 6, so compared with the convolution WPT, the additionalnumber of multiplication and addition operation in the traditional WPT is 130 048 and 479 232 respectively. In a computerwhose CPU frequency is 2000 MHz, the calculation time of traditional WPT for this example is 27.6066 ms, while the oneof convolution WPT is only 10.3409 ms.

Besides the less calculation quantity, the advantage of avoiding reconstruction is also reflected in the less boundaryerror. Since data extension is necessary in the calculation process of both kinds of WPT, the boundary error will surelyexist in their processing results. However, for traditional WPT, to recover the decomposition results to the same length oforiginal signal, reconstruction operation must be made, so there are two kinds of errors, i.e. decomposition error and re-construction one. Furthermore, since the decomposition results in which boundary error exist are used to do reconstructioncomputation, the error will be accumulated to make the boundary error of reconstruction results further increase, while theconvolution WPT needs no reconstruction, so there is only one kinds of error, i.e. decomposition error. In this respect thedirect decomposition results of convolution WPT are more reliable than the reconstruction ones of traditional WPT.

However, it is very difficult for us to obtain the decomposition error, for that to compute the boundary error, two signalsmust be offered, one is the ideal signal, and the other is the real signal, while we never know what the ideal decompositionresult is and can never get it, so here it is very difficult for us to compare the boundary error of the results shown inFig. 3. Nevertheless, we can easily get the reconstruction error, because the ideal reconstruction signal is just the originalsignal, and the difference of real reconstruction result and original signal is the reconstruction error. For this example, thereconstruction errors of two kinds of WPT are illustrated in Fig. 5. To reveal the errors more clearly, the front 30 data ofboth errors are plotted in the same figure, as shown in Fig. 5(c), and the last 30 data of both errors are also plotted inthe same figure, as shown in Fig. 5(d). The standard deviation of error and the average of absolute value of error are alsocalculated, as listed in Table 1. From these error curves and the data in Table 1 it can be easily seen that the reconstructionerror of convolution WPT is also less than that of the traditional WPT. In fact this result can also demonstrate that thedecomposition error of convolution WPT is less than that of the traditional WPT, because that the reconstruction signalis obtained from the decomposition results, and the decomposition results with less error will produce the reconstructionsignal with less error. The less the decomposition error is, the less reconstruction error is.

4. Noise reduction algorithm based on the convolution WPT

In the decomposition results of wavelet transform, maxima produced by white noise can’t be transmitted along thescales, and they will gradually disappear in the biggish scale. This characteristic is the foundation of noise reduction al-gorithms based on wavelet transform. Many noise reduction algorithms based on wavelet transform have been developed

X. Zhao, B. Ye / Digital Signal Processing 20 (2010) 1352–1364 1359

Table 1The statistical parameters of reconstruction error.

Standard deviation of error Average of absolute value of error

Convolution WPT 0.0105 0.0012Traditional WPT 0.0188 0.0017

Fig. 6. The comparison between the decomposition structure of WPT and wavelet transform.

[13–17], in which one kind of algorithm can be summarized as follows: supposing that decomposition level is 3, firstlythe original signal is decomposed by wavelet transform and approximation signal V 3, detail signals W3, W2 and W1 canbe got. Secondly, the maxima in W3 being used as criterion, the maxima in W2 and W1 are checked and those ones thatcan’t be transmitted to W3 are all set to zero, so new W2 and W1 can be obtained. Thirdly V 3, W3, and new W2 and W1are used to do reconstruction operation of wavelet transform, and the reconstruction result is just the denoised signal. Ormaxima in W3 are not used as criterion, while different threshold methods are used, and in all detail signals W3, W2 andW1, those wavelet transform coefficients who are less than the threshold will be all set to zero, so new W3, W2 and W1can be obtained, then reconstruction operation is made by the V 3, new W3, W2 and W1, and the reconstruction result isthe denoised signal. About detailed schemes of wavelet threshold, please see Refs. [14–17].

WPT is the further decomposition for the results of wavelet transform, and certainly it can be used to reduce noise.In wavelet transform, assuming that the width of frequency window of wavelet is �ψ̂ , then what detail signal W j ( j � 1)

reflects is the local information of original signal in a frequency window with center 3×2− j�ψ̂ and width H j = (2− j+1�ψ̂,

2− j+2�ψ̂), with the decreasing of j, the frequency center of W j will move to high frequency direction and the width H jwill become broad, and this means that frequency resolution of wavelet transform in high frequency area will become bad,in such a situation many high frequency noises can’t be isolated, and this will influence the denoising effect. While after thefurther decomposition of WPT, and in the kth level, detail signal W j will be further resolved into 2k− j subsequences, andfrequency band H j will also be further divided into 2k− j sub-bands, so frequency resolution will be greatly improved andthose high frequency noises that can’t be isolated by wavelet transform will be completely isolated. For example, if a signalis decomposed by wavelet transform into 3 levels, approximation signal V 3, detail signals W3, W2 and W1 can be got;while this signal is also decomposed by WPT into 3 levels, and in the third level 8 sequences U 0

3, U 13, . . . , U 7

3 can be got, inwhich U 0

3 is the approximation signal V 3, and U 13 is the detail signal W3; while U 2

3 and U 33 are the further decomposition

results of W2; U 43 , U 5

3 , U 63 and U 7

3 are the further decomposition results of W1, and this relationship is shown in Fig. 6.After this further decomposition, some high frequency noises that can’t be isolated in W1 will be completely isolated intothe sequences U 4

3 , U 53 , U 6

3 and U 73 ; some high frequency noises that can’t be isolated in W2 will be completely isolated

into the sequences U 23 and U 3

3 , and what these high frequency noises exhibit in these sequences are quite a number ofserried maxima. In such a situation if maxima in U 1

3 , i.e. W3 is used as criterion, check the maxima in U 23, U 3

3, . . . , U 63 and

U 73 , and eliminate those maxima that can’t be transmitted to U 1

3 , because high frequency noises are completely isolatedinto the back sequences especially U 3

3, . . . , U 63 and U 7

3 by WPT, surely much more noise maxima will be eliminated, then allsequences are used to do reconstruction operation of WPT, and it’s sure that the better denoising effect than that of wavelettransform can be achieved.

However, in the traditional WPT, the lengths of sequences will decrease by half in the next level as a consequenceof downsampling, then for the sequences in the last level, their lengths, compared with that of the original signal, willbecome very short, and many maxima, no matter they are produced by normal signal or noise, will be lost in the processof downsampling, in this situation the foundation of maxima-based noise reduction method will not exist any more. Whilefor convolution WPT, there is no downsampling, the lengths of sequences in every level will never decrease and they canalways keep the same as that of the original signal, then in every sequence all its maxima can be reserved, so maxima-basednoise reduction method can be realized.

Based on above analysis, a simple maxima-based noise reduction algorithm using the convolution WPT is proposed asfollows:

1360 X. Zhao, B. Ye / Digital Signal Processing 20 (2010) 1352–1364

Fig. 7. The comparison of denoising effect among the different methods. (a) Original signal; (b) denoised result of convolution WPT; (c) denoised one ofwavelet transform; (d) denoised one of traditional WPT. (For interpretation of the references to color, the reader is referred to the web version of thisarticle.)

Step (1): Decompose the original noisy signal f (t) by the convolution WPT, and supposing that the maximum decomposi-

tion level is S , then in the Sth level 2S sequences can be got, i.e. U 0S , U 1

S , . . . , U 2S −1S , and they will keep the same length

as that of the original signal.Step (2): Extract the maxima in the sequence U 1

S , and record the position coordinates of these maxima as (x1, x2, . . . , xM),where ‘M ’ is the total number of maxima in U 1

S .

Step (3): The position coordinates (x1, x2, . . . , xM) are used as criterion, for sequence U jS (2 � j � 2S − 1), search for the

maxima in interval (xi, xi+1), i = 1,2, . . . , M −1, and set these maxima to zero; search for the maxima in interval (0, x1)

and interval (xM , N − 1), where N is the length of original signal, and set these maxima to zero, then let j = j + 1,repeat the operation of step (3) till j = 2S − 1.

Step (4): After the processing of step (3), noise maxima in sequences U jS (2 � j � 2S − 1) have already been eliminated,

now all the sequences in the Sth level are used to make the reconstruction of convolution WPT, and the reconstructionresult is just the denoised signal.

In this algorithm, because the denoised result is obtained by the reconstruction operation of WPT, its phase will keep thesame as that of the original signal, so this is a zero phase shift denoising algorithm. In general when decomposition level isS = 4, the good denoising effect can be achieved. Furthermore, in this noise reduction algorithm no special conditions arerequired for the noised signals because that signal types, signal-to-noise ratio (SNR) and threshold are no involved in thesteps of algorithm.

5. Noise reduction example

For signal f (t) = sin(2t) + sin(6t) + sin(18t), 1024 data are sampled in interval [0,2π ], white noise chosen from normaldistribution N(0,1) is added to this signal and signal-to-noise ratio (SNR) of the combined signal is 1.6775. The noise is sostrong that the waveform of f (t) can’t be distinguished, as shown in Fig. 7(a). Convolution WPT with the QMF coefficientsh[ ] = {0.125,0.375,0.375,0.125} and g[ ] = {−2.0,2.0} is used to eliminate the noise in this signal, and the decompositionlevel is set as 4, then 16 sequences can be got in the fourth level. By the processing of above noise reduction algorithmfor these 16 sequences, the denoised result can be got, which is illustrated in Fig. 7(b) and the red dashed line is the ideal

X. Zhao, B. Ye / Digital Signal Processing 20 (2010) 1352–1364 1361

Fig. 8. The noise reduction error curves. (a) The error curve of convolution WPT; (b) the one of wavelet transform; (c) the one of traditional WPT.

Table 2The statistical parameters of three noise reduction errors.

Standard deviation of error Average of absolute value of error

Convolution WPT 0.2676 0.2185Wavelet transform 0.4146 0.3240Traditional WPT 0.3237 0.2510

f (t), now SNR is improved to 20.4116. It can be seen that the principal part of noise has been eliminated, the waveformof denoised result is very smooth, furthermore there is no phase deviation, and signal waveform is very close to the idealone. As a comparison, the denoised result got by the corresponding wavelet transform is shown in Fig. 7(c), in which thered dashed line is the ideal f (t). It can be easily seen that this denoising effect is not so good as that of convolution WPT,and many noises especially some impulses still exist in this denoised result, which make the signal waveform look not verycontinuous, and SNR of this result is 8.6270.

For traditional WPT, many maxima in the sequences of the last level are lost as a result of downsampling, so this WPT isnot suitable to be used to realize the above maxima-based noise reduction algorithm. However, the soft threshold denoisingmethod [16] is still suitable for this WPT. For a data x, supposing that t is the threshold, then the soft threshold adjustmentregulation can be described as follows [16]

x̂ ={

0 |x| � t,

sgn(x)(|x| − t) |x| > t.(17)

For the sequences in the last level got by the traditional WPT, except for the first sequence, every data in the other se-quences is adjusted by this soft threshold regulation, then all these sequences are used to make WPT reconstruction, andthe reconstruction result is just the denoised signal. For a sequence xi , i = 1,2, . . . ,n, in order to achieve the good denoisingeffect, the threshold can be decided by the following formula [16]

t = √2 log(n)

n∑i=1

|xi |/(0.6745 · n). (18)

By dint of this processing, the denoised result of traditional WPT can be got, as shown in Fig. 7(d), in which the reddashed line is the ideal f (t), and SNR of this result is 14.1014. One can see that its denoising performance is better thanthat of wavelet transform, and much more noise is eliminated, but this denoising effect is not yet so good as that of theconvolution WPT, because some waveforms of this denoised signal become very sharp, while the ones of convolution WPTare much more smooth and there is no this waveform distortion phenomenon.

To further analyze the denoising performance of these three methods with more accuracy, their noise reduction errorsare calculated, which are illustrated in Fig. 8, and the standard deviation of these errors, the average of absolute value ofthese errors are also calculated, as listed in Table 2. From these error curves and especially the data in Table 2, it can be

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Fig. 9. The comparison between the original sequences of last 5 channels got by convolution WPT and the processed ones in which noise maxima areeliminated. (a) Original sequences; (b) sequences after noise maxima are eliminated.

Fig. 10. The comparison between the original detail signals of 4 scales got by wavelet transform and the processed ones in which noise maxima areeliminated. (a) Original detail signals; (b) detail signals after noise maxima are eliminated.

seen that whether judged by standard deviation or by the average, the denoising effect of convolution WPT is always thebest, and the traditional WPT is the second best, while wavelet transform is always in the third class.

In order to explain the differences of denoising effect, we might as well see what happened in the intermediate denoisingprocess. The last 5 sequences in the fourth level got by convolution WPT are shown in Fig. 9(a), one can see that the

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Fig. 11. The comparison between the original sequences of traditional WPT and the ones processed by soft threshold adjustment regulation. (a) Originalsequences; (b) sequences after soft threshold adjustment.

isolated maxima are very dense, which are produced by noise but can’t be isolated by wavelet transform, while after furtherdecomposition of WPT, they are completely isolated, especially isolated into these 5 back sequences, then by the processingof noise maxima elimination algorithm, large numbers of noise maxima are eliminated and this effect is clearly shown inFig. 9(b). As a comparison, the four detail signals got by wavelet transform are shown in Fig. 10(a). It can be easily seenthat the isolated maxima are much less than those of convolution WPT, and the elimination effect of noise maxima is alsonot so good as that of convolution WPT, as shown in Fig. 10(b). This comparison confirms the preceding analysis about thedenoising principle of convolution WPT and directly reveals the inherent reason why denoising performance of convolutionWPT is better than that of wavelet transform.

The intermediate process of traditional WPT can also be illustrated. The 16 sequences in the fourth level got by thetraditional WPT are shown in Fig. 11(a), as a result of downsampling, the length of every sequence has decreased to 64 andtheir total length is 1024, while after the soft threshold adjustment, the new 16 sequences are illustrated in Fig. 11(b). Onecan see that nearly all the data in the last 15 sequences are adjusted to new values by soft threshold adjustment regulation;however, this adjustment may be excessive, which leads to the waveform distortion of reconstruction result.

6. Conclusions

In the traditional WPT, the length of sequences will decrease by half in the next level, which is inconvenient for furtheranalysis of these sequences. To solve this problem, the concept of convolution WPT is put forward, and its fast decompo-sition and reconstruction algorithms are deduced. The signal processing examples testify the validity of this kind of WPT.Summarizing the total contents of this paper, we can draw the following conclusions:

(1) For convolution WPT, no matter how many levels a signal is decomposed, the lengths of sequences in every level willnever decrease and can always keep the same as that of the original signal, so the defect of traditional WPT is overcome,and this characteristic will also make convolution WPT have the advantage in fault diagnosis.

(2) Signal processing example shows that the direct decomposition results of convolution WPT are completely in line withthe reconstruction ones of traditional WPT, while for traditional WPT, to achieve the same effect of feature extractionobtained by the direct decomposition results of convolution WPT, the reconstruction operation must be done, so signalprocessing procedure of convolution WPT is much simpler, and furthermore, convolution WPT needs less calculationand has less boundary error.

(3) By virtue of the length invariance of sequences, a noise reduction algorithm based on convolution WPT is proposed. Thedenoising example shows that convolution WPT can achieve the excellent denoising effect, which is better than that ofthe traditional WPT, and also, much better than that of wavelet transform.

Acknowledgments

The support from National Natural Science Foundation of China (NSFC, Grant Nos. 50305005 and 50875086) for thisresearch is gratefully acknowledged. The authors also thank the two anonymous reviewers for their valuable suggestions.

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Xuezhi Zhao received his MS and Ph.D. degrees from South China University of Technology, in 1998 and 2001, respectively. Currentlyhe is an Associate Professor at the School of Mechanical and Automotive Engineering in South China University of Technology. His researchinterests include information fusion, pattern recognition, signal processing and scientific computation.

Bangyan Ye is a Professor at the School of Mechanical and Automotive Engineering in South China University of Technology; he isalso the senior member of Chinese Mechanical Engineering Society. He obtained Ph.D. degree in 1989 from South China University ofTechnology. He has worked on variety of topics including signal processing, fault diagnosis, neural networks and artificial intelligence.