Basic Fourier Definitions
R. A. Johnson, Department of Geosciences, University of Arizona
Continuous Periodic Functions
Fourier Series. This can be used to represent any continuous periodic function, and in effect gives the Fourier Transform
of continuous periodic functions. The Fourier series itself can be thought of as the inverse transform, whereas the coefficients an and bn represent the forward transform because they give the amplitudes of the various frequency components.
f HtL =
a0
+ ‚ Han Cos@2 π fn tD + bn Sin@2 π fn tDL
2
∞
n=1
where
2
2
=
f HtL Cos@2 π fn tD t
T ‡− T
T
an
8n = 0, 1, 2, ...<
and
2
2
2
f HtL Sin@2 π fn tD t
T ‡− T
T
bn =
8n = 1, 2, ...<
2
and fn = n f0 = n ê T (T = period; f0 = fundamental frequency).
Another representation of the Fourier Series (essentially the same thing as above, but admitting negative frequencies) is:
f HtL = ‚ Cn
∞
+ 2 π fn t
HInverseL
8n = .. −1, 0, 1, ...<
HForwardL
8n = .. −1, 0, 1, ...<
n=−∞
2
1
f HtL
T ‡− T
T
Cn =
− 2 π fn t
t
2
Note that the amplitudes of the Cn terms are half those calculated using the first definition for an and bn because the Cn
terms exist for both positive and negative n. These last two expressions are in a form that more closely resembles the
inverse and forward Fourier Transforms below. Because the function f(t) is periodic, the frequencies are discrete (i.e., just
points, not continuous functions). Continuous periodic functions transform into discrete non-periodic functions.
Fourier-Definitions-2006.nb
2
Discrete Periodic Functions (Number Series)
Discrete Fourier Transform. This can be used to give the forward Fourier Transform of the discrete periodic time series
gk or the inverse Fourier Transform of the discrete periodic frequency series Gn . The coefficients Gn represent the
forward transform of the time series gk and give the amplitudes of the discrete frequencies contained in gk . The forward
transform of a time series is a frequency series that repeats with a period of ÅÅÅÅ1ÅÅ where Dt is the sample interval in time.
Dt
Note that N samples represent the function in both the time and frequency domains, and that discrete periodic functions
transform to discrete periodic functions.
1
‚ gk
N
N−1
Gn =
− 2πnk
N
HForward L
JPeriod =
1
= NN
∆t
HInverseL
JPeriod =
1
= NN
∆f
k=0
gk = ‚ Gn
N−1
+ 2πnk
N
n=0
Continuous, Non-Periodic Functions
General Fourier Transform. This is used to give the Fourier Transform of continuous functions that are non-periodic
(never repeat). These expressions constitute a Fourier Transform pair, and give the forward and inverse transformations
to represent the continuous frequency or time functions in the other domain. Continuous non-periodic functions transform
to continuous non-periodic functions.
F HfL = ‡ f HtL
∞
− 2πft
t
HForwardL
f
HInverseL
−∞
f HtL = ‡
∞
−∞
F HfL
+ 2πft