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1 Digital Signal Processing 0 010) Contents lists available at ScienceDirect Digital Signal Processing Convolution wavelet packet transform and its applications to signal processing Xuezhi Zhao, Bangyan Ye School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou , People s Republic of China article info abstract Article history: Available online 7 January 010 Keywords: Convolution wavelet packet transform Decomposition and reconstruction Length invariance Signal processing Noise reduction The length of decomposition results of traditional wavelet packet transform WPT) will decrease by half in the next level for downsampling, then the length of sequences in the last level will become very short, and this is very inconvenient for further analysis of these sequences. One kind of WPT based on convolution definition is put forward, its fast decomposition and reconstruction algorithms are given, and the outstanding characteristic of this convolution WPT is that no matter how many levels a signal is decomposed, the length of sequences got in every level will never decrease and can always keep the same as that of the original signal, so the defect of traditional WPT is overcome. For traditional WPT, to achieve the same effect of direct decomposition of convolution WPT, reconstruction operation must be done and the calculation will greatly increase. Based on the length invariance property of convolution WPT, a noise reduction algorithm is proposed, and signal processing example shows that its denoising performance is better than that of traditional WPT, and also much better than that of wavelet transform. 010 Elsevier Inc. All rights reserved. 1. Introduction As is well known, wavelet packet transform WPT) is a further decomposition for wavelet transform, and both the approximation and detail signals obtained by wavelet transform at each level will be further decomposed by WPT, so results with higher resolution in time and frequency domain can be got. According to the traditional wavelet packet theory, the WPT of signal xt) is defined as follows x n, j p = j/ xt)μ n j t ) p dt, 0 j S, 0 n S 1, 1) R where μ n t) is wavelet packet function, j is the number of decomposition level, or so-called scale parameter, p is the position parameter, n is the channel number, S is the maximum decomposition level. For signal xt), after it is decomposed by WPT, S sequences can be got in the Sth level. The corresponding fast decomposition algorithm for this kind of WPT is [1] x x n, j+1 k n+1, j+1 k = p Z = p Z hp k)x n, j p, gp k)x n, j p, ) * Corresponding author. Fax: address: mezhaoxz@scut.edu.cn X. Zhao) /$ see front matter 010 Elsevier Inc. All rights reserved. doi: /j.dsp

2 X. Zhao, B. Ye / Digital Signal Processing 0 010) where hi) and gi) are wavelet quadrature mirror filter QMF) coefficients. Eq. ) is the classical decomposition formula of WPT, and in the wide engineering applications of WPT, such as feature extraction for heart rate variability HRV) signal [] and electroencephalogram EEG) signals [3], the analysis for the impulsive energy of the vibration signal of water hydraulic motor [4], gearbox fault detection [5], online tracking of bearing wear [6], fault diagnosis of rotating machinery and monitoring 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 in the j + 1)th level, for the data of sequences in the jth level, only those ones in the even points can be sampled to calculate the sequences in the j + 1)th level. So compared with the length of sequences in the jth level, the one in the j + 1)th level will decrease by half. For example, if the length of original signal is 104, then after decomposition of WPT, the length of the sequences got in the first level will decrease to 51, in the second level will decrease to 56, etc., like this way the length of sequences in the next level will always decrease by half, so for the sequences in the last level, their length will become very short. This length degression of WPT may be very useful in data compression domain [10,11], but it s very discommodious in fault diagnosis domain [4 9]. Because for getting the reliable diagnosis results, those sequences in the last level will always be further analyzed, but their short length will make against the further analysis. For example, in order to get the accurate fast Fourier transform FFT) result of a sequence with high frequency resolution, in general the length of this sequence is required to be more than 64 at least. Supposing the length of original signal is 104, and the maximum decomposition level is S = 5, then after decomposition of WPT, for those sequences in the fifth level, their length will all decrease to 3, and their lengths are so short that the FFT results with high frequency resolution can t be got. In addition, it s also very difficult to observe the waveform feature of these sequences for their too short length. In order to further analyze these sequences of the last level or observe their waveform feature, the only way is to use them to do reconstruction calculation. The reconstruction formula of WPT is x n, j p = [ n, j+1 n+1, j+1] hp k)x + gp k)x k k. 3) Corresponding to the downsampling in the decomposition formula, the characteristic of this reconstruction one is upsampling, i.e. before computing x n, j n, j+1 n+1, j+1 p, for sequences x and x, one zero will be inserted between every two adjacent k k data of these two sequences. By virtue of this reconstruction operation, the length of the sequences in the last level will recover 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 the decomposition 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 a WPT by which a signal is decomposed, the length of sequences, no matter which level these sequences are located in, will never decrease and can always keep the same as that of the original signal? To solve this problem, one kind of convolution WPT is put forward in this paper, and the fast decomposition and reconstruction algorithm for this WPT are given. The outstanding characteristic of this convolution WPT is that no matter how many levels a signal is decomposed, the sequences got in the every level can always keep the same length as that of the original signal, so the defect of traditional WPT is completely 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 line with the reconstruction ones of traditional WPT, while its calculation quantity is much less than that of traditional WPT. By means of the length invariance property of convolution WPT, a noise reduction algorithm is proposed, and the denoising example shows that its noise reduction performance is better than that of the traditional WPT, and also better than that of wavelet transform.. 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 convolution definition, which is proposed to detect the local singularity of signal by Mallat [1], but for WPT, there is only the inner product definition, as shown in Eq. 1). Whether the convolution definition is suitable for WPT? In this paper this problem is studied and it s proved that the convolution definition for WPT is also feasible. For this purpose, let s define the convolution WPT as follows: For signal xt) L R), supposing that function series { j/ μ n j t k) k Z} make up of the orthonormal bases of wavelet packet subspace U n, then the convolution WPT can be defined as follows j x n, j p = 1 j R ) p t xt) μ n j dt = 1 ) t xt) μ j n, j 0 j S, 0 n < S, 4) where j is the number of decomposition level, or so-called scale parameter, p is the position parameter, n is the channel number, S is the maximum decomposition level.

3 1354 X. Zhao, B. Ye / Digital Signal Processing 0 010) For inner product WPT, its fast decomposition formula is shown in Eq. ). Similarly, for the practicality of convolution WPT, its fast decomposition formula should also be found. The definition of wavelet packet is μ n t) = hk)μ n t k), μ n+1 t) = 5) gk)μ n t k), where hi) and gi) are the wavelet QMF coefficients. The function series {μ n t) n Z} shown in Eq. 5) are called the orthogonal wavelet packet. In Eq. 5), let t = j x, and do Fourier transform in both sides, then we obtain ˆμ n j ω ) = 1 hk)e i j 1 ωk ˆμ n j 1 ω ), ˆμ n+1 j ω ) = 1 gk)e i j 1 ωk ˆμ n j 1 ω ) 6). Define Hω) = Gω) = 1 hk)e iωk, 1 gk)e iωk. Then Eq. 6) can be written as { ˆμn j ω ) = H j 1 ω ) ˆμ n j 1 ω ), ˆμ n+1 j ω ) = G j 1 ω ) ˆμ n j 1 ω ). 7) 8) According to the convolution theorem, converting Eq. 4) into frequency domain, we can get ˆx n, j ω = ˆxω) ˆμ n j ω ). In Eq. 9), replace n by n and j by j + 1, then this equation can be written as n, j+1 ˆx ω = ˆxω) ˆμ n j+1 ω ). While according to Eqs. 8) and 9), the right part of Eq. 10) can be expressed n, j+1 ˆx ω = ˆxω)H j ω ) ˆμ n j ω ) = H j n, j ω )ˆx ω. Considering the definition of Hω), weobtain n, j+1 ˆx ω = 1 hk)e i j ωk ˆx n, ω j. 9) 10) 11) As is well known, if the Fourier transform of xt) is ˆxω), then the Fourier one of xt b) is e iωb ˆxω), and this is the translation property of Fourier transform. According to this property, converting Eq. 11) into time domain, then we obtain n, j+1 xp = 1 hk) x n, j. p j k 1) n+1, j+1 Correspondingly, the computation formula for xp can also be obtained by the similar deduction, and the only difference is that hk) is replaced by gk), so the fast decomposition formula for convolution WPT can be obtained as follows n, j+1 xp = 1 hk) x n, j, p j k n+1, j+1 xp = 1 gk) x n, j p j k. It can be seen that the concrete form of wavelet packet function μ n t) is not involved in this fast decomposition formula, and the decomposition of convolution WPT for any signal can be conveniently realized just by dint of wavelet QMF coefficients hk) and gk), and this is just like Eq. ). However, compared with Eq. ), the fast decomposition formula of 13)

4 X. Zhao, B. Ye / Digital Signal Processing 0 010) Fig. 1. Vibration acceleration signal of a ball bearing. Fig.. 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 in the j + 1)th level, just a translation of j k 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 the same length as that of the original signal, for the completeness of convolution WPT, its reconstruction algorithm is also given here. It can be known from the wavelet theory that for wavelet QMF hk) and gk), their relationship in the frequency domain is Hω) + Gω) = 1. 14) According to Eqs. 9) and 14), we have ˆx n, ω j = ˆxω) ˆμ n j ω ) = ˆxω) ˆμ n j ω )[ H j ω ) + G j ω ) ] = ˆxω) ˆμ n j ω ) H j ω ) H j ω ) + ˆxω) ˆμ n j ω ) G j ω ) G j ω ). Considering Eqs. 8) and 10), we obtain ˆx n, ω j = ˆxω) ˆμ n j+1 ω ) H j ω ) + ˆxω) ˆμ n+1 j+1 ω ) G j ω ) n, j+1 = ˆx ω H j ω ) n+1, j+1 + ˆx ω G j ω ). The definition of Hω) and Gω) is taken into consideration, then we can get ˆx n, ω j = 1 hk)e i j ωk n, j+1 ˆx ω + 1 gk)e i j ωk n+1, j+1 ˆx ω. 15) According to the translation property of Fourier transform, converting Eq. 15) into time domain, then we can obtain the reconstruction formula for convolution WPT x n, j p = 1 n, j+1 hk)x + 1 n+1, j+1 gk)x. 16) p+ j k p+ j k Next a signal processing example will be given to test the validity of this convolution WPT, in addition, the comparison with the 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 104 and the sample frequency for this signal is Hz. In this bearing some injuries exist in the ball track and the shocks will be caused when 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

5 1356 X. Zhao, B. Ye / Digital Signal Processing 0 010) Fig. 3. The comparison between the reconstruction results of traditional WPT and direct decomposition ones of convolution WPT. a) Reconstruction results of 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 signal is firstly processed by traditional WPT, in which the wavelet QMF coefficients are selected as db3 Daubechies wavelet ones and the maximum decomposition level is set as 3, then 8 sequences can be got in the third level. However, for the downsampling, the lengths of these 8 sequences all decrease to 18. These 8 sequences are placed in the same rank and their total length is 104, as shown in Fig.. Because the length of each sequence is so short that no useful impulse feature can be identified, to say the least, even though the impulses can be revealed in these sequences, the accurate positions of impulses in original signal can t be located by them. Because the lengths of these sequences are much shorter than that of the original signal, it s impossible to find out that the impulses in these sequences correspond to which positions in the original signal. In order to display the impulses and locate their positions, the only way is to utilize these 8 sequences to do reconstruction operation of WPT, so that they can recover the same length of the original signal and then the impulses can be revealed. The reconstruction results of these 8 sequences are shown in Fig. 3a), here the impulses and their positions are clear revealed, and these impulses mean the shocks caused by the injuring points in the ball track. However, this feature extraction process is very troublesome: firstly the decomposition operation of WPT should be done, and then the reconstruction operation of WPT for each sequence in the third level should be done too.

6 X. Zhao, B. Ye / Digital Signal Processing 0 010) 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 decompose the original signal into 3 levels, and 8 sequences got in the third level are illustrated in Fig. 3b). It can be seen that the important difference from the ones of traditional WPT in Fig. is that the lengths of these 8 sequences are all 104, i.e. the length of original signal. While compared Fig. 3b) with Fig. 3a), one can easily see that whether the impulses or their positions in the direct decomposition results of convolution WPT are all in agreement with the ones in the reconstruction results of traditional WPT. Further comparison is made in frequency domain, and the spectrums of all sequences in Fig. 3 are illustrated in Fig. 4. It can be seen that whether for the frequency range or for the amplitude, the spectrums of the direct 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 much simpler 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 signal is N, the maximum decomposition level is S, and the length of wavelet QMF coefficients is L, then for traditional WPT, to make all the sequences in the last level recover their length to N, the total needed number of multiplication operation in the process of decomposition and reconstruction is SN[ S+1 + 1)L + 1], and the number of addition operation is SN[ S 4L + 1) + L]. While for convolution WPT, to obtain the same effect of traditional WPT, the reconstruction operation can be avoided, and in its decomposition process, the total needed number of multiplication operation is N S 1)L + 1),

7 1358 X. Zhao, B. Ye / Digital Signal Processing 0 010) Fig. 5. The comparison of reconstruction error. a) Boundary error of traditional WPT; b) boundary error of convolution WPT; c) left boundary error comparison; d) right boundary error comparison. and the number of addition operation is 4NL S 1). So compared with the convolution WPT, the additional number of multiplication operation needed in the traditional WPT is S+1[ S )L ] 1 N + [ S + 4)L + S + ] N. While the additional number of addition operation needed in the traditional WPT is S[ 4S 1)L + S ] N + S + )NL. According to these two formulas, if S = 1, for traditional WPT, the additional number of multiplication and addition operation is NL 1)and N6L + ) respectively. With the increasing of S, compared with the convolution WPT, the additional number of multiplication and addition operation in the traditional WPT will increase with exponential speed, and this increase is very considerable. In this example, S = 3, N = 104, L = 6, so compared with the convolution WPT, the additional number of multiplication and addition operation in the traditional WPT is and respectively. In a computer whose CPU frequency is 000 MHz, the calculation time of traditional WPT for this example is ms, while the one of convolution WPT is only ms. Besides the less calculation quantity, the advantage of avoiding reconstruction is also reflected in the less boundary error. Since data extension is necessary in the calculation process of both kinds of WPT, the boundary error will surely exist in their processing results. However, for traditional WPT, to recover the decomposition results to the same length of original signal, reconstruction operation must be made, so there are two kinds of errors, i.e. decomposition error and reconstruction one. Furthermore, since the decomposition results in which boundary error exist are used to do reconstruction computation, the error will be accumulated to make the boundary error of reconstruction results further increase, while the convolution WPT needs no reconstruction, so there is only one kinds of error, i.e. decomposition error. In this respect the direct 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 signals must be offered, one is the ideal signal, and the other is the real signal, while we never know what the ideal decomposition result is and can never get it, so here it is very difficult for us to compare the boundary error of the results shown in Fig. 3. Nevertheless, we can easily get the reconstruction error, because the ideal reconstruction signal is just the original signal, and the difference of real reconstruction result and original signal is the reconstruction error. For this example, the reconstruction errors of two kinds of WPT are illustrated in Fig. 5. To reveal the errors more clearly, the front 30 data of both errors are plotted in the same figure, as shown in Fig. 5c), and the last 30 data of both errors are also plotted in the same figure, as shown in Fig. 5d). The standard deviation of error and the average of absolute value of error are also calculated, as listed in Table 1. From these error curves and the data in Table 1 it can be easily seen that the reconstruction error of convolution WPT is also less than that of the traditional WPT. In fact this result can also demonstrate that the decomposition error of convolution WPT is less than that of the traditional WPT, because that the reconstruction signal is obtained from the decomposition results, and the decomposition results with less error will produce the reconstruction signal 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 the scales, and they will gradually disappear in the biggish scale. This characteristic is the foundation of noise reduction algorithms based on wavelet transform. Many noise reduction algorithms based on wavelet transform have been developed

8 X. Zhao, B. Ye / Digital Signal Processing 0 010) Table 1 The statistical parameters of reconstruction error. Standard deviation of error Average of absolute value of error Convolution WPT Traditional WPT 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, firstly the original signal is decomposed by wavelet transform and approximation signal V 3, detail signals W 3, W and W 1 can be got. Secondly, the maxima in W 3 being used as criterion, the maxima in W and W 1 are checked and those ones that can t be transmitted to W 3 are all set to zero, so new W and W 1 can be obtained. Thirdly V 3, W 3, and new W and W 1 are used to do reconstruction operation of wavelet transform, and the reconstruction result is just the denoised signal. Or maxima in W 3 are not used as criterion, while different threshold methods are used, and in all detail signals W 3, W and W 1, those wavelet transform coefficients who are less than the threshold will be all set to zero, so new W 3, W and W 1 can be obtained, then reconstruction operation is made by the V 3,newW 3, W and W 1, and the reconstruction result is the 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 ˆψ, then what detail signal W j j 1) reflects is the local information of original signal in a frequency window with center 3 j ˆψ and width H j = j+1 ˆψ, j+ ˆψ), with the decreasing of j, the frequency center of W j will move to high frequency direction and the width H j will 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 the further decomposition of WPT, and in the kth level, detail signal W j will be further resolved into k j subsequences, and frequency band H j will also be further divided into k j sub-bands, so frequency resolution will be greatly improved and those high frequency noises that can t be isolated by wavelet transform will be completely isolated. For example, if a signal is decomposed by wavelet transform into 3 levels, approximation signal V 3, detail signals W 3, W and W 1 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 1 3,...,U 7 3 can be got, in which U 0 3 is the approximation signal V 3, and U 1 3 is the detail signal W 3; while U 3 and U 3 3 are the further decomposition results of W ; U 4 3, U 5 3, U 6 3 and U 7 3 are the further decomposition results of W 1, and this relationship is shown in Fig. 6. After this further decomposition, some high frequency noises that can t be isolated in W 1 will be completely isolated into the sequences U 4 3, U 5 3, U 6 3 and U 7 3 ; some high frequency noises that can t be isolated in W will be completely isolated into the sequences U 3 and U 3 3, and what these high frequency noises exhibit in these sequences are quite a number of serried maxima. In such a situation if maxima in U 1 3,i.e.W 3 is used as criterion, check the maxima in U 3, U 3 3,...,U 6 3 and U 7 3, and eliminate those maxima that can t be transmitted to U 1 3, because high frequency noises are completely isolated into the back sequences especially U 3 3,...,U 6 3 and U 7 3 by WPT, surely much more noise maxima will be eliminated, then all sequences are used to do reconstruction operation of WPT, and it s sure that the better denoising effect than that of wavelet transform can be achieved. However, in the traditional WPT, the lengths of sequences will decrease by half in the next level as a consequence of downsampling, then for the sequences in the last level, their lengths, compared with that of the original signal, will become very short, and many maxima, no matter they are produced by normal signal or noise, will be lost in the process of downsampling, in this situation the foundation of maxima-based noise reduction method will not exist any more. While for convolution WPT, there is no downsampling, the lengths of sequences in every level will never decrease and they can always keep the same as that of the original signal, then in every sequence all its maxima can be reserved, so maxima-based noise reduction method can be realized. Based on above analysis, a simple maxima-based noise reduction algorithm using the convolution WPT is proposed as follows:

9 1360 X. Zhao, B. Ye / Digital Signal Processing 0 010) Fig. 7. The comparison of denoising effect among the different methods. a) Original signal; b) denoised result of convolution WPT; c) denoised one of wavelet transform; d) denoised one of traditional WPT. For interpretation of the references to color, the reader is referred to the web version of this article.) Step 1): Decompose the original noisy signal f t) by the convolution WPT, and supposing that the maximum decomposition level is S, theninthesth level S sequences can be got, i.e. U 0 S, U 1 S,...,U S 1 S, and they will keep the same length as that of the original signal. Step ): Extract the maxima in the sequence U 1 S, and record the position coordinates of these maxima as x 1, x,...,x M ), where M is the total number of maxima in U 1 S. Step 3): The position coordinates x 1, x,...,x M ) are used as criterion, for sequence U j S j S 1), searchforthe maxima in interval x i, x i+1 ), i = 1,,...,M 1, and set these maxima to zero; search for the maxima in interval 0, x 1 ) and interval x M, 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 = S 1. Step 4): After the processing of step 3), noise maxima in sequences U j S j S 1) have already been eliminated, now all the sequences in the Sth level are used to make the reconstruction of convolution WPT, and the reconstruction result is just the denoised signal. In this algorithm, because the denoised result is obtained by the reconstruction operation of WPT, its phase will keep the same as that of the original signal, so this is a zero phase shift denoising algorithm. In general when decomposition level is S = 4, the good denoising effect can be achieved. Furthermore, in this noise reduction algorithm no special conditions are required for the noised signals because that signal types, signal-to-noise ratio SNR) and threshold are no involved in the steps of algorithm. 5. Noise reduction example For signal f t) = sint) + sin6t) + sin18t), 104 data are sampled in interval [0, π], white noise chosen from normal distribution N0, 1) is added to this signal and signal-to-noise ratio SNR) of the combined signal is The noise is so strong that the waveform of f t) can t be distinguished, as shown in Fig. 7a). Convolution WPT with the QMF coefficients h[] ={0.15, 0.375, 0.375, 0.15} and g[] ={.0,.0} is used to eliminate the noise in this signal, and the decomposition level is set as 4, then 16 sequences can be got in the fourth level. By the processing of above noise reduction algorithm for these 16 sequences, the denoised result can be got, which is illustrated in Fig. 7b) and the red dashed line is the ideal

10 X. Zhao, B. Ye / Digital Signal Processing 0 010) 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 The statistical parameters of three noise reduction errors. Standard deviation of error Average of absolute value of error Convolution WPT Wavelet transform Traditional WPT f t), now SNR is improved to It can be seen that the principal part of noise has been eliminated, the waveform of denoised result is very smooth, furthermore there is no phase deviation, and signal waveform is very close to the ideal one. As a comparison, the denoised result got by the corresponding wavelet transform is shown in Fig. 7c), in which the red 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 very continuous, and SNR of this result is For traditional WPT, many maxima in the sequences of the last level are lost as a result of downsampling, so this WPT is not suitable to be used to realize the above maxima-based noise reduction algorithm. However, the soft threshold denoising method [16] is still suitable for this WPT. For a data x, supposing that t is the threshold, then the soft threshold adjustment regulation can be described as follows [16] { 0 x t, ˆx = 17) sgnx) x t) x > t. For the sequences in the last level got by the traditional WPT, except for the first sequence, every data in the other sequences is adjusted by this soft threshold regulation, then all these sequences are used to make WPT reconstruction, and the reconstruction result is just the denoised signal. For a sequence x i, i = 1,,...,n, in order to achieve the good denoising effect, the threshold can be decided by the following formula [16] t = logn) n x i / n). i=1 By dint of this processing, the denoised result of traditional WPT can be got, as shown in Fig. 7d), in which the red dashed line is the ideal f t), and SNR of this result is One can see that its denoising performance is better than that of wavelet transform, and much more noise is eliminated, but this denoising effect is not yet so good as that of the convolution WPT, because some waveforms of this denoised signal become very sharp, while the ones of convolution WPT are 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 errors are calculated, which are illustrated in Fig. 8, and the standard deviation of these errors, the average of absolute value of these errors are also calculated, as listed in Table. From these error curves and especially the data in Table, it can be 18)

11 136 X. Zhao, B. Ye / Digital Signal Processing 0 010) Fig. 9. The comparison between the original sequences of last 5 channels got by convolution WPT and the processed ones in which noise maxima are eliminated. 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 are eliminated. 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 the best, 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 denoising process. The last 5 sequences in the fourth level got by convolution WPT are shown in Fig. 9a), one can see that the

12 X. Zhao, B. Ye / Digital Signal Processing 0 010) Fig. 11. The comparison between the original sequences of traditional WPT and the ones processed by soft threshold adjustment regulation. a) Original sequences; 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 further decomposition of WPT, they are completely isolated, especially isolated into these 5 back sequences, then by the processing of noise maxima elimination algorithm, large numbers of noise maxima are eliminated and this effect is clearly shown in Fig. 9b). As a comparison, the four detail signals got by wavelet transform are shown in Fig. 10a). It can be easily seen that the isolated maxima are much less than those of convolution WPT, and the elimination effect of noise maxima is also not so good as that of convolution WPT, as shown in Fig. 10b). This comparison confirms the preceding analysis about the denoising principle of convolution WPT and directly reveals the inherent reason why denoising performance of convolution WPT 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 the traditional WPT are shown in Fig. 11a), as a result of downsampling, the length of every sequence has decreased to 64 and their total length is 104, while after the soft threshold adjustment, the new 16 sequences are illustrated in Fig. 11b). One can 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 further analysis of these sequences. To solve this problem, the concept of convolution WPT is put forward, and its fast decomposition 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 will never 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. ) Signal processing example shows that the direct decomposition results of convolution WPT are completely in line with the reconstruction ones of traditional WPT, while for traditional WPT, to achieve the same effect of feature extraction obtained by the direct decomposition results of convolution WPT, the reconstruction operation must be done, so signal processing procedure of convolution WPT is much simpler, and furthermore, convolution WPT needs less calculation and has less boundary error. 3) By virtue of the length invariance of sequences, a noise reduction algorithm based on convolution WPT is proposed. The denoising example shows that convolution WPT can achieve the excellent denoising effect, which is better than that of the traditional WPT, and also, much better than that of wavelet transform. Acknowledgments The support from National Natural Science Foundation of China NSFC, Grant Nos and ) for this research is gratefully acknowledged. The authors also thank the two anonymous reviewers for their valuable suggestions.

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