Visualizing Wavelet Packet Terminal Nodes in Wavelet Signal Analyzer

A wavelet packet decomposition of level $$L$$ consists of $$2^L$$ terminal nodes. By design, the Wavelet Signal Analyzer app lists and plots at most 32 terminal nodes. This example demonstrates how the app selects the nodes. The example also shows how to plot a subset of the nodes, based on the frequency range.

Import Data

Load the Kobe earthquake data set.

load kobe

Obtain Decimated Wavelet Packet Decomposition

Open Wavelet Signal Analyzer and import the Kobe data. By default, the app obtains the nondecimated discrete wavelet decomposition of the data. Obtain the decimated wavelet packet decomposition of the signal down to level 6 using the fk18 wavelet. The decomposition consists of $$2^6=64$$ nodes.

1. On the Analyzer tab, click Add and choose Decimated Wavelet Packet. The kobe2 scenario appears in Scenarios.

2. On the Wavelet tab, set Wavelet to fk, Number to 18, and Level to 6.

3. Click Analyze to update the decomposition.

4. Switch back to the Analyzer tab.

Node Selection Algorithm

Recall that in the wavelet packet transform, each node (wavelet packet) corresponds to an equal-width subband filtering of the input signal. Furthermore, when you use an orthogonal wavelet to obtain the wavelet packet decomposition, you end up with a partitioning of the signal energy among the equal-width subbands. Wavelet Signal Analyzer uses the following procedure to select which nodes of a wavelet packet decomposition to list and display.

1. For the current frequency range, [0,Fs/2), where Fs is the sample rate, the app determines the relative energies of the nodes in that range.

2. The app sort the nodes by relative energy in descending order.

3. In that same order, Wavelet Signal Analyzer lists the nodes in Levels, and plots those coefficients in Decomposition Coefficients. If there are more than 32 nodes in the decomposition, the app displays the first 32 nodes.

Reproduce Order in Workspace

Reproduce the ordered list of nodes in your workspace. Click Export, and then under Decomposition Coefficients, select Generate MATLAB® Script. After a few moments, an untitled script appears in your editor. Modify the output of the dwpt function to include the center frequencies of the approximate subbands. Save the script and execute.

% Perform the decomposition using DWPT
% wpt       - Terminal nodes
% l         - Bookkeeping vector necessary for inverse DWPT
% frq       - Center frequencies of approximate subbands
% relEnergy - Energy by node as a fraction of the total energy
numberOfLevels = 6;
[wpt,l,~,frq,relEnergy] = dwpt(kobe,"fk18",Level=numberOfLevels,Boundary="periodic");

% Energy by node as a percentage of the total energy
relativeEnergyByNode = 100*cell2mat(relEnergy);

Confirm that there are 64 terminal nodes in the decomposition and the sum of the nodal energies equals 100 percent.

length(wpt)

ans = 
64

sum(relativeEnergyByNode)

ans = 
100.0000

Sort the nodes by relative energy in descending order. Each node corresponds to an approximate subband. List the center frequencies of the sorted subbands. Confirm that Levels lists the nodes in the same order. Keep in mind that indexing is zero-based.

[sortedRelativeEnergies,nodeIndex] = sort(relativeEnergyByNode,"descend");
reorderedCenterFrequencies = frq(nodeIndex);
nodeIndex = nodeIndex-1;
nodeList = table(nodeIndex,sortedRelativeEnergies,reorderedCenterFrequencies)

nodeList=64×3 table
    nodeIndex    sortedRelativeEnergies    reorderedCenterFrequencies
    _________    ______________________    __________________________

        5                28.799                     0.042969         
        6                12.707                     0.050781         
        0                 10.72                    0.0039062         
        7                10.001                     0.058594         
        4                9.3465                     0.035156         
        3                7.1272                     0.027344         
        8                4.5254                     0.066406         
       22                2.1989                      0.17578         
       24                1.8151                      0.19141         
       21                1.3272                      0.16797         
       23                1.2751                      0.18359         
        9                1.1619                     0.074219         
       26               0.89599                      0.20703         
       25                 0.891                      0.19922         
        1               0.86739                     0.011719         
       20               0.86029                      0.16016         
      ⋮

Plot the energy by level. In the Statistics section, click the arrow to open the gallery. Under For All Levels, click Energy by Level. Confirm the energies are plotted in descending order.

Display Nodes in Specific Frequency Range

By default, to determine which 32 nodes to display, Wavelet Signal Analyzer sorts the terminal nodes based on the frequency range $$[0,\frac{F_s}{2})$$, where $$F_s$$ is the sample rate. You can observe the terminal nodes from the following four frequency ranges:

• $$[0,\frac{F_s}{8})$$

• $$[\frac{F_s}{8},\frac{F_s}{4})$$

• $$[\frac{F_s}{4},\frac{3F_s}{8})$$

• $$[\frac{3F_s}{8},\frac{Fs}{2})$$

To plot the nodes with the most energy in the frequency range $$\frac{F_s}{8}\le F<\frac{F_s}{4}$$, click Coefficients > Terminal Nodes Selected. Under By Frequency Range, select Specify the frequency range for packet coefficients > [Fs/8,Fs/4).

The Levels table and plots update to reflect the range. The app lists and plots all 16 nodes in the range in descending order of relative energy.

Reproduce the order in your workspace.

subFreqInd = find((nodeList.reorderedCenterFrequencies >= 1/8) & ...
    (nodeList.reorderedCenterFrequencies < 1/4));
nodeList(subFreqInd,:)

ans=16×3 table
    nodeIndex    sortedRelativeEnergies    reorderedCenterFrequencies
    _________    ______________________    __________________________

       22                 2.1989                    0.17578          
       24                 1.8151                    0.19141          
       21                 1.3272                    0.16797          
       23                 1.2751                    0.18359          
       26                0.89599                    0.20703          
       25                  0.891                    0.19922          
       20                0.86029                    0.16016          
       19                0.73645                    0.15234          
       27                0.49414                    0.21484          
       28                0.48409                    0.22266          
       29                0.27693                    0.23047          
       18                0.22348                    0.14453          
       30                0.17528                    0.23828          
       31                0.17136                    0.24609          
       17                0.16958                    0.13672          
       16               0.064556                    0.12891