Frequency-time representations

5.6 Frequency-time representations

It is often useful to think of signal representations in a frequency-time phase space, shown schematically in Figure 15

. There are many ways to tile frequency-time, e.g., Fourier transforms, wavelets, chirplets, Gabor-transforms (and variants, depending on the temporal windowing used); there is a correspondingly large literature. Depending on the situation each tiling can have special merit (e.g., if additional information suggests a specific candidate signal is likely to project preferentially onto a small fraction of a particular mathematical basis while the noise does not).

Figure 15:

Schematic diagram of signals in a frequency-time space. Sinusoids are “on” for all time and have horizontal tracks, linear chirps are straight lines with non-zero slope, bursts are time-localized, etc. These localizations in frequency-time suggest different detection approaches for different classes of signal. This space can be tiled in many ways. Particular tilings can have special merit for particular waveforms, e.g., if a candidate signal projects preferentially onto a small fraction of a particular mathematical basis while the noise does not.

As an example, Figure 16 shows normalized Fourier power as a function of frequency-time for the Cassini two-way Ka-band track on 2001 DOY 350 (time series shown in Figure 8 ). This plot was constructed by taking the unwindowed power spectrum of sequential 102.4 s data segments (in this case 75% overlapped in time). The heavy white line indicates the two-way light time at the beginning of the data set. The normalized power – power at a given (frequency, time) point divided by the estimated local continuum power near that point – is plotted. This is a nondimensional measure of the contrast (and potential statistical significance) of the Fourier power at that point relative to a local background. The color code runs from black (low values) through green to red (very high values). Points with estimated contrast ratio $> 10$ are marked with white circles. If two high-contrast features are at the same frequency and separated by a two-way light time, they are connected with a thin white line. The FTS glitch of Figure 8 is clearly evident in both the time series and in T₂-separated bands of high-contrast Fourier power in the lower plot. Additional features not evident in the time series but paired at T₂ are detected near the beginning of the data set and at low-frequency. See also Figures 17 and 18 .

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Figure 16:

Frequency-time representation of Cassini two-way Ka-band data on 2001 DOY 350. Upper panel: time series of two-way Doppler data with $≃$ 1 s time constant, sampled at 0.2 s/sample. At this resolution the visual impression of the plot is set by relatively high frequency noise. Lower right panel: low frequency resolution power spectrum of the full data set shown in upper panel. Lower left panel: normalized dynamic spectrum of the data in the upper panel. This was constructed by forming sequential spectra of short ( $≃$ 102 s) unwindowed segments of the data. Each data segment is 75% overlapped with its neighbors. The heavy white line indicates the two-way light time at the beginning of the data set. The plotted quantity is power at a given point in (frequency-time) divided by a local estimate of the average power at that (frequency, time) point, a nondimensional measure of the contrast of the Fourier power at that point relative to an estimated background. The color code runs from black (low values) through green (higher values) to red (very high values). Points with this estimated contrast ratio $>$ 10 are marked with white circles. If two high-contrast features are at the same frequency and separated by a two-way light time, they are connected with a thin white line. The FTS glitch shown also in Figure 8

is clearly evident in both the time series and in T₂-separated bands of high-contrast Fourier power in the lower plot. Additional features paired at T₂ in the earlier, lower-frequency part of the data are also detected in the normalized dynamic spectrum.