Wigner-Ville Distribution (Advanced Signal Processing Toolkit)

LabVIEW 2014 Advanced Signal Processing Toolkit Help

Edition Date: June 2014

Part Number: 372656C-01

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With the Wigner-Ville Distribution (WVD) quadratic time-frequency analysis method, you do not need to specify a window type like you do with the STFT spectrogram method. The WVD returns many useful signal properties for signal analysis, such as marginal properties, the mean instantaneous frequency, and the group delay. The WVD also has time and frequency shift invariance, which means that the components of two signals that are the time-shifted versions of each other look the same regardless of location in the time-frequency plane. The WVD is a Cohen's class method.

You can use the WVD on signals that have simple, widely separated signal components for which you require a fine time-frequency resolution for the corresponding time-frequency representation. The WVD also is a good choice when you want to extract signal features from a signal that contains only a single component.

Use the TFA Wigner-Ville Distribution VI to compute the WVD.

Cross-Term Interference

One serious disadvantage of the WVD is cross-term interference. Cross-terms are artifacts that appear in the WVD representation between auto-terms, which correspond to physically existing signal components. These cross-terms falsely indicate the existence of signal components between auto-terms.

The following figure shows the WVD of the example frequency hopper signal. This signal has four auto-terms. Each component has a different time center and a different frequency center.

1 Auto-terms

Compared to the ideal time-frequency representation of the hopper signal, the previous figure includes many signal components that do not correspond to the four auto-terms. These artifacts are the cross-terms. Notice that the cross-terms are strongest at the midpoints between the auto-terms and that the cross-terms have a higher peak magnitude than the auto-terms. The cross-terms also oscillate, or form bands in the time-frequency domain, with the band spacing proportional to the distance between the auto-terms. In general, as the number of auto-terms increases, the auto-terms and the cross-terms overlap. Consequently, distinguishing the auto-terms from cross-terms can be challenging.

The time-frequency plane includes positive frequencies and negative frequencies. Signal components present at positive frequencies in real-valued signals, such as the example frequency hopper signal, have mirrored, symmetric components at negative frequencies. The example frequency hopper signal contains four signal components at positive frequencies and four corresponding signal components at negative frequencies, which are not shown. The cross-terms appear between auto-terms at positive frequencies, between auto-terms at negative frequencies, and between auto-terms at positive and negative frequencies.

In the previous example frequency hopper signal, if you convert this real-valued signal into a complex-valued analytic signal by removing the auto-terms at negative frequencies before you apply the WVD, you can reduce the number of cross-terms in the WVD, as shown in the following figure.

1 Auto-terms

Notice that in contrast to the figure of the WVD of the hopper signal, no cross-terms appear near the horizontal axis in the previous figure. The analytic frequency hopper signal example has the same spectral content at positive frequencies as the original, real-valued signal but has no spectral content at negative frequencies. By converting the real-valued signal to an analytic signal, you remove the cross-terms between auto-terms at negative frequencies and the cross-terms between auto-terms at positive frequencies and negative frequencies.

In addition to converting real-valued signals to analytic signals to reduce cross-terms in the WVD, you can use other Cohen's class methods and the Gabor expansion-based spectrogram, also called the Gabor spectrogram, to reduce cross-term interference. The Gabor spectrogram is a method unique to the LabVIEW Time Frequency Analysis Tools.

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