Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 31

The three expressions in equation (3) are respectively obtained by multiplying both sides of
equation (2) by 1 (i.e. cos 0t), cos nt and sin nt and integrating with respect to t over the
period length 2π. We then use the orthogonality properties of sine and cosine functions to
eliminate some of the expressions.
The Fourier series in equation (2) will exactly represent the function f(t) only when an
infinite number of terms are included in the summation, i.e. when n runs from 1 to infinity
( ∞ ). If a finite number of terms is taken, the sum will shoot beyond the value of f(t) in the
neighbourhood of a discontinuity (boundaries of the function). The overshoot oscillates
about the value with a decreasing amplitude as we move away from the discontinuity.
Increasing the number of terms does not influence the error magnitude at the discontinuity,
but only leads to a better approximation for the continuous part of f(t). The overshoot and
oscillatory behaviour of f(t) at the vicinity of a discontinuity is known as Gibb's
phenomenon.
If f(t) is an even function, then equation (2) becomes

()=

+∑ ( cos)

(4)

Since b n = 0 for even f(t), Similarly for odd f(t), a 0 , a n = 0 and so

( ) = ∑ b sin)

(5)

To solve many physical problems, it is necessary to develop a Fourier series that will be
valid over a wider interval. To obtain an expansion valid in the interval [-T, T], we let
()=

and determine such that f(t) = f(t + 2T). In this case,

+∑ ( cos+ sin)

=

, hence we obtain:

()=

+∑ ( cos + sin

)

, -T≤ t ≤ T (6)

If f(t) satisfies the Dirichlet conditions in this interval, then the Fourier coefficients a 0 , a n and
b n can be computed in a similar fashion as in equation (3).
The complex form of the Fourier series in equation (6) can be obtained by expressing cos

and sin

in the exponential using the Euler identity cos θ + sin θ = e iθ , where i = √-1 is a
complex number. Thus the complex form of the Fourier series can be written as

( ) = ∑

[-T, T] (7)

Where C 0 = a 0 /2 = F(0), C n = (a n - ib n )/2 = F(n) and C -n = (a n + ib n )/2 = F(-n)
Equation (7) is the complex form of the Fourier series. On multiplying both sides of this
equation by e -inπt/T and integrating with respect to t, we obtain

=

()

(8)

Note that the amplitude spectrum, | () | = | (−) | = is symmetrical. From the

identity in complex number representation: a + i b = re iθ , with =√ + and tan θ = b/ a ,
we have ()=|()| () , (−)=|(−)| () with [()]= , [(−)]=