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.
yp is the pth complex-valued output in the frequency domain, p = 0,…, n - 1,
xt is the tth complex-valued input in the time domain and
n is the number of observations.
This formula requires n2 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(Log2(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 [Real] and / or a second column containing the imaginary part by clicking on [Imaginary]. 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.