Wavelet Power Spectrum

Background

The wavelet coefficients yield information as to the correlation between the wavelet (at a certain scale) and the data array (at a particular location). A larger positive amplitude implies a higher positive correlation, while a large negative amplitude implies a high negative correlation.

A useful way to determine the distribution of energy within the data array is to plot the wavelet power, equivalent to the amplitude-squared. By looking for regions within the Wavelet Power Spectrum (WPS) of large power, you can determine which features of your signal are important and which can be ignored.

Method

Given the wavelet transform Wi of a multi-dimensional data array, Ai, where i=0...N–1 is the index and N is the number of points, then the Wavelet Power Spectrum is defined as the absolute-value squared of the wavelet coefficients, |Wi|2.

One-dimensional Vector

For a vector (such as a time series) the coefficients of wavelet power can be rearranged to yield a two-dimensional picture, where the first dimension is the independent variable (e.g. time) and the second dimension is the wavelet scale (e.g. 1/frequency).

Two-dimensional Array

The wavelet transform of a 2D array is also two-dimensional, and is arranged so that the smallest scales are in the upper-right quadrant (assuming that index [0, 0] is in the lower-left).

Example

Use the "Chirp" dataset that is included in the Wavelet sample file. This dataset contains a time series with a sine wave that has an exponentially-increasing frequency. You can use the Multiresolution Analysis viewer to examine the time series.

Try the following steps:

  1. From the main window, select the Chirp dataset and start the Wavelet Power Spectrum viewer using either the Visualize Menu or the Toolbar button. The WPS can be seen under Wavelet Power Spectrum.
  2. Select the Morlet wavelet function from the Family dropdown box. You should be able to see the exponential increase in frequency as a band of high power extending from left to right, and ranging from about Scale=256 sec. near the beginning to Scale=16 sec. near the end of the time series.
  3. To bring out the features more clearly, change the Energy Scaling dropdown item from Power to Magnitude.
  4. Notice the large peak near Scale=256 sec. This is primarily due to the discontinuity that occurs when the dataset is wrapped around from the end back to the beginning. Move the Order slider bar from 6 to 4 to make the peak more narrow.

    5. Tip
    You can use your mouse to rotate, zoom in or out, or move the plot.

  5. To find the chirp peaks, select the Zero Phase Lines check box.
  6. Now deselect the 3D check box to view the surface from above.

Figure 3-1: The Wavelet Power Spectrum of the Chirp Signal

wavewps.gif