2.3 Power Spectrum Estimation

1. Periodograms
2. The Maximum Entropy method

The power spectrum function activates an interactive session allowing the user to display the one-dimensional power spectral density (PSD) for the main image window. The plot shows normalized frequency values on the X-axis and power in decibels (dB) on the Y-axis. The power spectrum function makes use of a viewport (just as the main image window) for any portion of the plot which falls outside the power spectrum window. This viewport can be enlarged by resizing the power spectrum window.

The periodogram method of spectral estimation is found by calculating the magnitude squared of the FFT of the input sequence, divided by the number of samples in the sequence. Since it is based on the FFT, the number of points in the plot is a power of two (the next higher power of two if the input sequence is not a power of two). The periodogram can be greatly affected by the presence of noise in a sequence.

The maximum entropy method is a good estimator of the power spectrum if one can properly select the order parameter. A low order estimation can result in an overly smooth spectrum whereas a high order estimation can be greatly affected by noise in the data. In general, one must experiment with the order parameter for different types of data.

The peaks of the power spectrum can rise sharply, often falling between points being evaluated, giving a false impression of the amplitude of the spectrum. Using the maximum entropy method, however, one can evaluate a portion of the spectrum at different granularities. This is done by changing the number of points, start frequency, and end frequency parameters--the spectrum is estimated at the specified number of points starting at the specified starting frequency and ending at the specified ending frequency. All frequencies are normalized, that is, a normalized frequency of 0.5 corresponds to PI/2. Both forms of spectral estimation yield real-valued sequences which are symmetric about the Y-axis.

More information on these methods may be found in:

1. Press, Flannery, Teukolsky, and Vetterling. Numerical Recipes--The Art of Scientific Computing. Cambridge University Press; 1986.
2. Haddad and Parsons. Digital Signal Processing--Theory, Applications, and Hardware. Computer Science Press; 1991.
3. Blackman and Tukey. The Measurement of Power Spectra. Dover Publications; 1958.

   

   Maximum Entropy:

   Calculate and display a power spectral density plot using the maximum entropy method. The plot is drawn using the color indicated when the toggle is selected.

   Periodogram:

   Calculate and display a power spectral density plot using a periodogram. The plot is drawn using the color indicated when the toggle is selected.

   Start Frequency:

   Initial output frequency is plotted on the X-axis of the power spectrum plot. This value is always 0 for the periodogram plot but may be modified by the user for the maximum entropy calculation. If both the maximum entropy and the periodogram plot are displayed, start frequency will be set to 0 and it cannot be modified.

   End Frequency:

   Ending output frequency is plotted on the X-axis of the power spectrum plot. This value is always .5 for the periodogram plot but may be modified by the user for the maximum entropy calculation. If both the maximum entropy and the periodogram plot are displayed, end frequency will be set to .5 and it cannot be modified.

   Number of Points:

   The number of points evalulated. The default number of points depends on the dimension selected.

   If the X dimension is selected, the default number of points for the maximum entropy method is the number of displayed samples divided by two (number_samples / 2.0). The default number of points for the periodogram is the number of displayed samples rounded to the next greatest power of 2 divided by 2.0 (nearest_power_two(number_samples) / 2.0).

   If the Y dimension is selected, the default number of points for the maximum entropy method is the number of displayed lines divided by two (number_lines / 2.0). The default number of points for the periodogram is the number of displayed lines rounded to the next greatest power of 2 divided by 2.0 (nearest_power_two(number_lines) / 2.0).

   If the Z dimension is selected, the default number of points for the maximum entropy method is the number of displayed bands divided by two (number_bands / 2.0). The default number of points for the periodogram is the number of displayed bands rounded to the next greatest power of 2 divided by 2.0 (nearest_power_two(number_bands) / 2.0).

   If both the maximum entropy and periodogram plots are displayed, the number of points will be set using the periodogram method and cannot be modified.

   Order:

   Enter the order of the maximum entropy calculation. As this number is increased, the maximum entropy plot will more closely resemble the periodogram plot. The default is 25.

   The X, Y, and Z radio buttons allow the user to select the dimension the power spectrum is to be calculated in.

   X:

   The input data for the power spectral density plot is a single line of image data.

   Y:

   The input data for the power spectral density plot is a single column of image data.

   Z:

   The input data for the power spectral density plot is an array of data made up of a single pixel from each of the displayed bands of image data.

   The automatic and manual radio buttons allow the user to select the update mode for the power spectrum plot.

   Automatic:

   Update the power spectrum plot each time the user moves the mouse within the main image window.

   Manual:

   Update the power spectrum plot only when the user clicks the left mouse button in the main image window.

   The continouous and discrete radio buttons allow the user to select the type of line to be drawn.

   Continuous:

   Draw the plot line as a continuous line between points.

   Discrete:

   Draw the plot line as discrete points.

   Dismiss:

   Close the power spectrum panel.