Geology Reference
In-Depth Information
The Rayleigh spacing determines the spacing between spectral estimates
along the frequency axis of the FFT, and sets the frequency resolution of the
FFT. Synonyms for Rayleigh spacing include the terms “elementary band-
width,” “frequency bins,” and “sampling frequency.” The remaining sampled
frequencies from N/2 + 2 → N map as negative frequencies from N/2 → 0;
real-valued time series will produce identical Fourier coefficients as the
positive frequencies (in reverse order) (Figure 4.8). In practice, only the
first N/2 Fourier coefficients are considered for applications involving
real-valued data. For noise modeling (Section 4.3.6), it is important to
remember that the first N/2 Fourier coefficients represent only half the
variance of the input time series.
4.3.5.2 Direct Spectral Estimators, Leakage, and Tapers
The squared modulus of the FFT in Figure 4.8b is a direct spectral estimator
and is known as the “Schuster,” “classical,” or “unsmoothed” periodogram:
2
−
∑
N1
−
i2 nf
π
S(f)
=
x(n)D(n)e
D
n0
=
The exponential is shorthand for writing cosine and sine from Euler's
formula: e
iy
= cosy + isiny. D(n) is a data window, also known as a “lag
window,” which multiplies the data time series x(n), i.e., which restricts it to
a finite length. Technically speaking, the unsmoothed periodogram is mul-
tiplied by a rectangular (“Dirichlet”) data window defined by the value of 1
over the time interval occupied by the time series and 0 elsewhere. The
unsmoothed periodogram does not estimate the spectrum of the time
series in Figure 4.8a very accurately: the time series has two sinusoids with
equal amplitudes, but the FFT mesh imposed by evaluating the Fourier
transform at the Δf = 1/512 spacing does not coincide with either frequency,
and power is leaked into neighboring frequency bins.
Multiplication of the data window D(n) with the time series x(n) is
equivalent to convolving the Fourier transform of D(n) with that of the time
series. The Fourier transform of the Dirichlet window, called the Dirichlet
“spectral window,” is the sinc function, consisting of a “central lobe” of width
Δf surrounded by an infinite series of sidelobes with decreasing power. This
“ringing” is caused by the corners at the start and end of the Dirichlet window.
The net result of convolving the sinc function with spectral peaks of the
Fourier-transformed (infinite) time series is power leakage of approximately
10% (Table 1 of Durrani and Nightingale (1972) into the sidelobes).
To reduce this leakage, tapered data windows have been developed to
smooth the window corners (Harris 1978). In Figure 4.9, the Dirichlet spectral
window is compared with the spectral windows of the well-known Bartlett
and Hann
tapers
. The “central lobes” of the spectral windows appear as half
lobes centered on f = 0, and define the frequency resolution of the spectral esti-
mator. The Dirichlet spectral window has the narrowest central lobe; the
Bartlett and Hann central lobes are wider. In practice, central lobe width is
defined in terms of “equivalent noise bandwidth” with 1.0Δf (Dirichlet),
1.333Δf (Bartlett), and 1.5Δf (Hann).
