Time-Frequency Spectral Analysis Object and Template – Continuous Wavelet Transform (CWT) Spectrum
The Continuous Wavelet Transform (CWT) spectral option presents the Continuous Wavelet Transform (CWT) in a limited 3D spectral format. The CWT is a multiresolution time-frequency technique that is useful for exploring data known to be non-stationary. It is also of value for determining whether data can be safely judged wide-sense stationary as required by many of the spectral procedures.
Spectrum Type
The frequency domain information can be output in an variety of formats. In the following table, Re is the real component of the FFT at a given frequency, Im is the imaginary component, n is the data set size, δX is the sampling interval, and σ² is the data set variance.
Spectrum Type |
Formula/Description |
|---|---|
dB |
10 * log10(Re² + Im²) |
dB normalized |
10 * log10(Re² + Im²) - dBmax Decibels, normalized to 0 for time-frequency node with maximum power. |
Integral=TISA |
δX * (Re² + Im²) / 2 Surface integral is Time-Integral Squared Amplitude power. |
Integral=MSA |
(Re² + Im²) / n / 2 Surface integral is Mean Squared Amplitude power. |
Integral=SSA |
(Re² + Im²) / 2 Surface integral is Sum Squared Amplitude power. |
Variance |
(Re² + Im²) / σ² / 2 |
Magnitude² |
Re²+Im² |
Magnitude |
sqrt(Re² + Im²) |
In wavelet spectra, because of multiresolution analysis, the contour magnitudes (spectral peak heights) are not linearly proportional to power. If you need this property, you should use the Short-Time Fourier Transform. The most useful options for CWT spectra are the dB formats for visualization.
In a CWT graph, spectral information within the Cone of Influence is not displayed. The corresponding values in the Y data matrix of the result are set to void. The cone of influence defines the spectral region where edge effects can be present.
Wavelet
The Morlet, Paul, and Gaussian Derivative wavelets are available for CWT spectral analysis. The adjustable parameter (Adjustment) for the Morlet wavelet it is its wavenumber (from 6 to 200). For the Paul wavelet it is an order that can vary from 4 to 40. For the Derivative of Gaussian wavelet, it is the order of the derivative (from 2 to 80). The wavelets are complex.
Parameters - Convolution Zero Pad
The Zero Pad field specifies the amount of zero padding during fast convolution. The CWT uses an FFT-based fast convolution procedure that requires zero padding in order to be free of wraparound effects (aliasing). It is often possible to zero pad to the next power of two and avoid any perceptible wraparound and at the same time achieve the fastest possible convolutions. The number of time values in the spectrum is always the data count, irrespective of zero padding.
Parameters - Frequency
The CWT offers the ability to generate a wavelet spectrum using any set of frequencies desired. The default sets the frequency range from the lowest unit frequency to the Nyquist frequency. Any Start Frequency and End Frequency within this range can be specified. If your data set does not contain time values, then you have to enter the Nyquist-normalized frequencies in the range from 0 to 0.5.
The Log Spacing option specifies that the frequencies should use a logarithmic spacing. This is useful when most of a signal's energy is at lower frequencies. When this option is not checked, the frequency spacing will be linear.
The Number of Frequencies field specifies the count of frequencies in the wavelet spectrum. The default of 40 usually gives a respectable coverage, although it may be insufficient to catch closely spaced high frequency components when a log spacing is used, or closely spaced low frequency components when a linear spacing is used. Bear in mind that each frequency requires a separate FFT, so computation times and memory requirements for large data sets will go up appreciably when high frequency counts are specified.
Options - Maximum dB Range and Time Values
The Maximum dB range field is enabled only for the dB and dB normalized formats. A dB limit is used to furnish limits for the automatic scaling of the Y range of the graph as well as to specify the exact Y gradient that will be rendered. Please enter 0 in this field to specify an unlimited dB range.
The value Maximum CWT time values is used to set the threshold for decimation by averaging. This decimation is done to maintain a manageable size for the 3D rendering engine. Decimated spectra retain power, although there will be some attenuation of the spectral peaks. The maximum of 10,000 time values, allows for a 10,000 data length with no decimation imposed. Please enter 0 in this field to do not want to reduce the number of time values. The unlimited size of the CWT grid is the number of frequencies x number of data values.
Options - Set/Clear Reference (Analysis Wizard Only)
This function lets you compare various spectral procedures and settings. You can view a copy of the currently displayed spectrum in the lower pane by pressing Set Reference. Next, you can adjust additional settings that affect the display in the upper pane. With Clear Reference you can remove the copy and the time signal will appear again.
Result
Traditionally, STFT contour plots use frequency as the horizontal variable. In order to make the STFT and CWT comparable, both plot orientations are offered.
Particularly for evaluating run-ups, you can specify a synchronically measured Speed signal to replace the time in the result with the speed. In this case, FlexPro assigns the applicable rotational speed at the time it is captured to each single spectrum. The spectra are automatically sorted in the result according to increasing speed.
High Frequency Resolution Morlet Wavelets
High Morlet wave numbers can offer dramatically improved frequency resolution when used with large data sets (i.e., 16K or greater in size). These high wave numbers must be used with caution, however, since narrowband features can completely vanish between the discrete frequencies computed in the CWT. To insure an accurate analysis, first identify the primary spectral features using a lower Morlet wave number, one that produces significant "fuzziness" in frequency. Then, at the high wave number, insure that the same key spectral features still appear by using up to the maximum count of 500 frequencies, logarithmic scaling, or limit the frequencies to a specified band of interest. To prevent washing out spectral information, it is also necessary to be cautious of using too high a wave number with data that have too few oscillations or too short a sampling length. You must not overspecify the wavelet; the data sequence being analyzed should have appreciably more oscillations than the wavelet.
Higher Morlet wave numbers increase the fuzziness within time. A data set with one million total samples will have perhaps one thousand samples in time for each pixel in a rendering system. A wavelet with ultra-fine time resolution can produce a spectrum where a feature appears and disappears entirely within the one thousand time units. In such a case, the averaging decimation that is used for surface rendering may result in such local features in time being lost. A high wave number Morlet wavelet produces sufficient fuzziness in time to insure proper mapping within the surface decimation.
Memory Issues
Separate FFTs are made for each scale or frequency in the CWT. For memory reasons, the number of evaluated CWT frequencies is limited to a maximum of 500. In the CWT, zero padding is only used to prevent wraparound effects in the convolution.
In the CWT, the memory relationship is linear. Doubling the frequency count doubles the amount of physical memory needed. Typically, there is little to gain beyond 50 to 60 CWT frequencies.