Discrete Fourier Transform

Define Fourier modes:

$$q_k$$

$$= \frac{1}{\sqrt{N}} \sum_{n=1}^{N} q_n \, e^{-i 2\pi k n /N},$$

$$p_k$$

$$= \frac{1}{\sqrt{N}} \sum_{n=1}^{N} p_n \, e^{-i 2\pi k n /N},$$

(14)

with $k=-N/2+1,\dots,N/2$ and reality conditions

$$q_{-k} = q_k^*, \quad p_{-k} = p_k^*.$$ (15)

The inverse transform is

$$q_n = \frac{1}{\sqrt{N}} \sum_k q_k e^{i 2\pi k n /N}, \quad p_n = \frac{1}{\sqrt{N}} \sum_k p_k e^{i 2\pi k n /N}.$$ (16)

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