Fourier transforms provide information about the frequencies contained in a signal. They also have many other applications in science and engineering. Here we give a quick overview of the discrete Fourier transform of a real valued signal, possibly the most common case.
We start with a signal . For example, the function could be a voltage varying with time. Suppose we sample it at a series of times for , separated by a constant time interval . For definiteness, let's take . This gives us a sequence of sampled signal values .
For a concrete example of sampling, suppose the signal is a pure
cosine with angular frequency .
The sampling domain in time is finite, since we have
. That is, we sample over a total time interval
. When we sample over a finite domain, we need to say what
we mean by a signal oscillating purely at only one frequency. We
require that over the length of the sampling interval we have an
integer number of complete oscillations. That is
Of course we can do a Fourier transform of any signal, no matter whether its oscillations come out even. Here we are just describing what we mean by a component with a single pure frequency over a finite interval in time.
The discrete Fourier transform is given by
We say that we have transformed a signal from the time domain to the frequency domain.
The same formalism applies to the Fourier transform of data
depending on a spatial coordinate . In this case we sample at a
series of equally spaced points
, giving us, again
the discretely sampled values . The Fourier transform is
for the wavenumber
Another way to look at the Fourier transform is to separate it into
its real and imaginary parts. We use the identity
The Fourier transform is complex. The power spectrum is the
square of the norm of the Fourier transform:
The signal can be reconstructed from its Fourier transform using
It is easy to show that the discrete Fourier transform is
periodic with period . That is
Now we can answer the question, how many independent Fourier components are there? We start with real signal components in the time domain. But complex Fourier components in the frequency domain count as real numbers if we consider both real and imaginary components. The identity above constrains them so that there are only independent real values involved.
Often programs that compute the discrete Fourier transform use the constraint to pack the Fourier transform into a vector with the same number of real components as the signal vector. For example, for the Fourier transform in the Gnu Scientific Library packs the Fourier transform this way:
The Fast Fourier Transform algorithm in Numerical Recipes by Press et al uses a slightly different order for even . It puts the two real components and in the first and second positions and then continues in the same order as above.
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