The periodogram

Given a geochemical variable y, m measurements at times tu t2. tm produce the unevenly spaced time series yl5 y2. ym, which we lump together as the vector y. In order to find out eventual periodicities, Lomb (1976) suggests fitting the data by a sine wave using a least-square criterion. For any arbitrary frequency /, the fitting function is written 

The solution becomes particularly simple and time-invariant if r is chosen in such a way that the cross-terms vanish. Using basic trigonometric identities, we cancel the off-diagonal term 

The reduction in the sum of squares is a concept that may a priori look surprising (Lomb, 1976 Scargle, 1982). Nevertheless, its use is supported by the convergence between the reduction in the sum of squares and the familiar power spectrum in Fourier analysis when the data become equally spaced. It is simply the difference AS(f) in the sum of squares before the fit and after the fit for one particular frequency 

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 

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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. 

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... 

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 ... 

To plot half of the periodogram, set the x-axis equal to F and the y-axis to 2q C - l)/n. The coefficient of 2 augments the amplitude to take into consideration the fact that only half of the periodogram was plotted. 

To plot the frequency in the original units, multiply F by the sampling rate to give cycles per unit time, that is F = F x /sampling- F would then be used in place of F when plotting the periodogram. 

Consider the Edmonton temperature series that is fully described in Sect. A5.1. Plot the periodograms for the spring, summer, and winter mean temperature series. Also, plot the periodogram for the differenced summer temperature series. What are some of the salient features ... 

Figure 5.20 shows the periodogram for the once-differenced summer temperature time series. Unlike in the previous periodogram, there are now a series of peaks clustered in the area around 2.5-3 years/cycle. Also, there is a secondary peak around 4 years/cycle followed by a rather weak peak in the 8 years/cycle region. All these values seem to be multiples of each other suggesting that they represent a single feature rather than separate features. 

If a peak at/= 0.25 cycles/sample is observed on the periodogram, then it can be concluded that the process has a seasonal component, such that s = 0.25. 

Custom-built function that creates the periodogram for a given signal %Inputs ... 

With a modest assumption on the periodogram of the input signal, we can obtain a simplified expression for the diagonal elements of the correlation matrix. 

Theorem 4.3 If we assume that the periodogram of the input signal is approximately equal to a constant value ( C/(e )p U) over the narrow frequency region from to then... 

To increase the magnitude of a diagonal element corresponding to a particular value of r, the periodogram of the input signal has to be increased in the vicinity of the centre frequency of the rth frequency sampling filter. 

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