Geoscience Reference
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A popular way to overcome the problem of spectral leakage is by
windowing, in which the sequence of data is simply multiplied by a smooth
bell-shaped curve with positive values. Several window shapes are available,
e.g., Bartlett (triangular), Hamming (cosinusoidal) and Hanning (slightly
dif erent cosinusoidal) (Fig. 5.4). h e use of these windows slightly modii es
the equation for the Blackman-Tukey auto-spectrum to
where w ( k ) is the windowing function. h e Blackman-Tukey method
therefore performs auto-spectral analysis in three steps: calculation of the
autocorrelation sequence corr xx ( k ), windowing and, i nally, computation of
the discrete Fourier transform. MATLAB allows power spectral analysis to be
performed with a number of modii cations to the above method. One useful
modii cation is the Welch method (Welch 1967) (Fig. 5.5). h is method
involves dividing the time series into overlapping segments, computing the
power spectrum for each segment, and then averaging the power spectra.
h e advantage of averaging the spectra is obvious: it simply improves the
signal-to-noise ratio of a spectrum. h e disadvantage is a loss of resolution
in the spectra.
Cross-spectral analysis correlates two time series in the frequency domain.
b
a
Fig. 5.4 Spectral leakage. a h e amplitudes of the side lobes relative to that of the main lobe
are reduced by multiplying the corresponding time series by b a smooth bell-shaped window
function. A number of dif erent windows with advantages and disadvantages are available for
use instead of the default rectangular window, including Bartlett (triangular) and Hanning
(cosinusoidal) windows. Graph generated using the function wvtool .
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