Periodogram

Methods of analysis Fourier periodograms calculated in the range between 0 and 20 d 1. Automatic search for periodicities with the application of prewhitening and detrending. The photometry of selected objects was examined visually. 

Whenever the frequency / becomes close to a strong periodic component of the measured signal, the terms in parentheses tend to add up and the periodogram shows a power peak around /. Between these peaks, the terms are not correlated, their sign and amplitude tend to be random, and the sum will be small. Still with reference to Fourier analysis, it is common practice to plot the power P(f) as AS(/)/2. 

For simplicity, we will assume that depth below the sea bottom varies linearly with time. The first step in the calculation consists in removing the long-term drift over the whole period by fitting the data with a parabola and determining a periodogram out of the residuals from this fit. Depth has also been scaled to 1 in order to minimize round-off errors. Applying the method shown above, the best-fit parabola is obtained for... 

Figure 5.8 Periodogram of the reduced data shown in Figure 5.7. The strong peak at ca./ = 3.5 and its harmonics at ca. 7.0 and 10.5 correspond to a ca. 105 year periodic component.

This suppression rule is derived by analogy with the well-known Wiener filtering formula replacing die power spectral density of die noisy signal by its periodogram estimate. 

Consequences of the random nature of the attenuation. In the previous section we deliberately left apart a major problem the fact that the attenuation is a random quantity. The randomness of the attenuation comes from the fact that it is (in general) determined as a function of the relative signal level which in turn involves the short-time transform of the noisy signal. This aspect plays a key role in STSA because the relative signal level is estimated by the periodogram (at least in the STFT case) characterized by a very high variance. 

A phaseogram was produced for each independent frequency confined to the wide frequency range between 26.0 and 32.0 Hz. The independent Fourier spacing (IFS) between this range of independent frequencies is known to be 1/T (periodogram bin size), where T hours is the observation time (2 hours). 

We have previously reported that, although most storms do not alter the bottom-water resultant flow vectors, the probability of large water speeds over the bottom is increased (Bokuniewicz et ai, 1975b) when the wind stress is high. We now present further evidence on this effect. A periodogram calculated for a current meter record obtained in the western end of Long Island Sound is presented in Fig. 6. Three tidal peaks are... 

Fig 6. Periodogram calculated from a current meter record made in western Long Island Sound (7.4 km north of Eatons Neck) showing the dominant semidiurnal tide and two shallow water constituents (CPH, cycles/hr). 

Figure 5. Cumulative periodogram of a rotational growth series.

Figure 8. Cumulative periodogram of an elongational residual series.

Two model-checking methods are common, and they are often applied to a "new" set of data. First, the residuals are analyzed for lack of correlation or "whiteness." White noise has the characteristic that all frequencies contained in it should be present in equal strength. The cumulative periodogram is the integral of a spectrum estimate therefore, for white noise, it should be the Integral of a constant or a linear ramp function. 

The "whiteness" of the residuals can be analyzed by plotting their cumulative periodogram and observing closeness to the ideal ramp line. Another common diagnostic technique is to determine the statistical efficiency of adding one or two parameters (and terms) to the model. This will check if the minimal parametric model chosen is truly parsimonious. The statistical F-test based on the extra-sum-of-squares principle is useful for this efficiency test. 

FIG. 4. Periodogram of data after fast Fourier transform. A strong peak indicates a significant period (t) of approximately 3.5 days. 

Figure 5.4 Example plot of power speetral density distribution (periodogram) for a control input...

This is the noncausal Wiener filter for the clean signal with an adjustable noise level determined by the constraint a in Eq. (19.81). This filter is commonly implemented using estimates of the two power spectral densities. Let fy 9) andfy,(9) denote the estimates of fy 9) and /,y(0), respectively. These estimates could, for example, be obtained from the periodogram or the smoothed periodogram. In that case, the filter is implemented as... 

Periodogram and Its Use in Frequency-Domain Analysis of Time Series... 

The most common way to use the Fourier transform is to construct a periodogram that shows all the identifiable frequencies and their amplitudes. When using the fast Fourier transform to obtain the periodogram, only half of the values are plotted, since the other half is a mirror image (about /= 0). The periodogram is constructed as follows ... 

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