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If the periodic components of a time series vary over time, you cannot use traditional cepstrum estimation methods to identify echoes and periodic components of that time series. In this situation, you can identify time-varying periodic components of a time series by observing the time-cepstrum of the time series.
A time-cepstrum is a function of time and quefrency that indicates how the cepstral content of a signal evolves over time. A time-cepstrum uses a sliding window to estimate each real cepstrum of a signal. Sliding windows, also called window functions, are functions in which the amplitude tapers gradually and smoothly toward zero at the edges. The time-cepstrum first partitions the time-domain input signal into several disjointed or overlapped blocks by multiplying the signal with a window function. Then, the time-cepstrum applies the real cepstrum to each block. Because each block occupies different time periods, the resulting time-cepstrum indicates the cepstral content of the signal at each corresponding time period.
You can observe the cepstral changes of a nonstationary bearing vibration signal in the Cepstrogram graph in the following figure.
You can display the Cepstrogram on an intensity graph and observe how the cepstral content of the signal evolves over time. The intensity legend represents the time-cepstrum values in decibels. In the previous figure, the peaks in the time-cepstrum appear as intersecting lines. These peaks do not appear in a real cepstrum, because the periodic components vary over the length of the signal in the time domain.
Use the TSA Time-Cepstrum VI to compute the time-cepstrum of a time series. As with the real cepstrum estimation method, you can estimate the time-cepstrum by using the fast Fourier transform (FFT) or the AR model of the time series. Use the TSA Configure Cepstrogram Indicator VI to display the time-cepstrum of a time series on an intensity graph.
Refer to the Bearing Time-Cepstrum Analysis VI in the labview\examples\Time Series Analysis\TSAApplications directory for an example that demonstrates how to compute the time-cepstrum of a univariate time series by using the TSA Time-Cepstrum VI.