## 9.5. Fourier Analysis

The Fourier Analysis is used to transform real or complex data, which is assumed to be in the time domain, into the frequency domain. Once a series has been transformed, various further transformations can be carried out. Then the series can be transformed back into the original scale using the Inverse Fourier Transform procedure. A filtering effect can be achieved in this way.

The Fourier Transform is based on an expansion of a complex periodic function of time into a sum of sine and cosine waves.

where:

y_{p}
is the p^{th} complex-valued output in the frequency domain, p = 0,…,
n - 1,

x_{t} is the t^{th}
complex-valued input in the time domain and

n is the number of observations.

This formula requires n^{2} computations and applies
to any number of observations. For large values of n though, it may take a very
long time to compute. Instead, the Fast Fourier Transform (FFT) method is
employed here, which requires only n(Log_{2}(n)) computations (see
Elliott, D. F. & K. R. Rao 1982). A restriction brought by the FFT method is
that it only works with a number of observations which is a power of 2. If the
number of points is not a power of 2, then UNISTAT extends the series by its mean
so that it has a number of cases which is equal to the next power of 2.

You may select a column containing the real part by clicking
on [__R__eal] and / or a second column containing the imaginary part by
clicking on [Ima__g__inary]. One of real or imaginary components or both of
them can be selected. In most cases only the real part will need to be
selected. When this is the case, the real and imaginary parts of the output
will only contain n / 2 distinct values (or
(n + 1) / 2 if n is odd), values being symmetric about the
midpoint of the series (the Nyquist frequency). For the Inverse Fourier Transform both the real and imaginary parts will usually be
needed.