Overview of wavelet packet transform methods

Wavelet transform only decomposes the low-frequency band of the signal . Wavelet packet transform inherits the time-frequency analysis characteristics of wavelet transform , Further decompose the high-frequency band signal that is not decomposed in the wavelet transform , Make different resolution choices for various frequencies at different levels , On all scales , It provides a series of time-domain waveforms of subbands in the full band range . Wavelet packet analysis is to further subdivide the frequency band of wavelet subspace in binary way , In order to improve the frequency resolution . The relationship between wavelet transform and wavelet packet transform is shown in the figure below .

2、 The principle of construction

(1)、 The second generation wavelet packet transform is also composed of decomposition and reconstruction . Set the data sequence as S={s(k),k∈Z} , Divide the data sequence into odd sample sequence So(k) And even sample sequence Se(k) Two parts , among S(k) by S No k Samples .

(2)、 Then calculate the wavelet packet number by the following formula L Each band signal of layer decomposition . among P,U It is the predictor and updater of the second generation wavelet transform .

(3)、 Reconstruct the signal according to the following formula .

3、 The example analysis

Next, we use a simple example to illustrate the difference between wavelet analysis and wavelet packet analysis . Suppose the sampling frequency is 1000Hz, Then the analysis frequency is 500Hz. The following figure is a simple signal , contain 100Hz,200Hz,300Hz,400Hz Four frequency components . We do one-level and two-level wavelet decomposition and wavelet packet decomposition for the signal , Then perform single branch reconstruction . One layer decomposition results in 2 Frequency band , We use it (1,1),(1,2) Express , Two layer decomposition results in 4 We use (2,1),(2,2),(2,3),(2,4) Express .

Wavelet analysis can be realized (1,1),(1,2),(2,1),(2,2)4 Single branch reconstruction of frequency bands , Each single reconstruction retains the frequency component of the corresponding frequency band .

Wavelet packets in addition to the above 4 Single branch reconstruction of frequency bands , It can also be realized (2,3),(2,4)2 Frequency band Single branch reconstruction of .

4、 Conclusion

Wavelet packet analysis is actually an extension of wavelet analysis , Wavelet analysis only realizes the reconstruction of low-frequency signal , Wavelet packet complements the reconstruction of the high-frequency part , In this way, we realize the reconstruction of the full frequency band of the signal . In complex signal processing , We often want to filter out most of the invalid frequency components , Keep only the active ingredients . Wavelet packet analysis can undoubtedly meet this demand , We can calculate the frequency band of useful signals in advance , Using a single branch reconstruction of a certain frequency band to recover the waveform signal of that frequency band . The computational complexity of wavelet packet algorithm is also o(n) Level