The astronomical theory and data compared - Ice Ages and Interglacials: Measurements, Interpretation, and Models - page 304

Many sophisticated procedures have been developed for estimating H ð f Þ

from the dataset (t j ; F j ) for j ¼ 1toN. However, M&M described a simple brute

force procedure that works very transparently even though it is not ecient.
Nevertheless, with modern desktop computers, it is fast enough for many

purposes. The procedure involves the following steps:

(1) Choose a value of f .
(2) Calculate the sum over all j:
H c ð f Þ¼ X F j cos ð 2

ft j Þ

(3) Calculate the sum over all j:
H s ð f Þ¼ X F j sin ð 2

ft j Þ

(4) The ''strength'' of frequency f in representing the function F ð t Þ is:

2 2

S ¼ð H s ð f ÞÞ þð H c ð f ÞÞ

(5) Repeat steps 1-4 for many values of f , and plot the strength vs. f to determine
which frequencies contribute the most to F ð t Þ .

This process works in the following way. As M&M pointed out, this
procedure takes your set of data, multiplies it by a sine wave, and then sums the
results. If the data oscillate in phase with the sine wave, so that they are positive
together and negative together, then all the terms in the sum are positive and the
Fourier amplitude is large. If they drift into phase and out of phase, then roughly
half of the values in the product will be positive and half will be negative, and the
sum will be small. The sum is not particularly sensitive to sharp changes in
the data (e.g., sudden terminations); it is more sensitive to the bulk behavior of
the data (e.g., are most of the data points positive when the sine wave is positive?).
Sharp changes in F vs. t lead to a broad range of frequencies that contribute.
Regular oscillations in F ð t Þ lead to narrow peaks in the plot of strength vs.

frequency.
M&M provided the examples of a simple sine wave with a 100,000-year period
and a sawtooth wave with a 100,000-year period. The resultant spectra are given
in Figure 10.17 .
A few additional simple examples are now given. Consider Figure 10.18 . In
this figure we not only plot the function F ð t Þ¼ cos ð 2t Þ at 17 discrete points but
two further modifications of this function with some noise added.
Let us pretend we know nothing about the three functions and we desire to
undertake spectral analysis to determine which frequencies contribute most to the
underlying functions. We use the above procedure and the results are as shown in
Figure 10.19 . The prime frequency is 0.32 cycles per unit time, corresponding to a
period of 1/0.32 2

/2 time units. Adding noise broadens the strength vs.
frequency curve but the main peak remains close to the original frequency without
noise.