Discrete Fourier Transform
Overview
The discrete fourier transform for a sequence of numbers {% f_1,f_2,...,f_N %}
is given by
{% F_n = \sum_{k=1}^N f_k \frac{exp[-2\pi i kn/N]}{\sqrt{2 \pi}} %}
The sequence of numbers can be viewed as the output of a function {% f %} that is sampled at a fixed frequency, that is,
the points are equally spaced apart. Viewed this way, the discrete fourier transform can be shown to be equivalent
to the following approximation to the
fourier transform
{% \int_0^T dt \frac{exp[-i \omega_n t]}{\sqrt{2 \pi}} f(t) %}
where this integral is approximated by using the
Trapezoid rule for integration.
Fourier Series Expansion
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