Cosine coefficients:

$$F_{\cos}[k] = \frac{c_k}{N} \sum_{n=0}^{N-1} x[n] \cos\left(\frac{2\pi kn}{N}\right)$$

Sine coefficients:

$$F_{\sin}[k] = \frac{c_k}{N} \sum_{n=0}^{N-1} x[n] \sin\left(\frac{2\pi kn}{N}\right)$$

where $c_k = 1$ for $k = 0, N/2$ and $c_k = 2$ otherwise

$k = 0, 1, \ldots, N/2$ and $N$ is the number of samples

$$x[n] = \sum_{k=0}^{N/2} F_{\cos}[k] \cos\left(\frac{2\pi kn}{N}\right) + F_{\sin}[k] \sin\left(\frac{2\pi kn}{N}\right)$$

This reconstructs the original signal from its frequency components

Each frequency $k$ contributes:

$$A_k \cos\left(\frac{2\pi kn}{N} - \phi_k\right)$$

where:

• $A_k = \sqrt{F_{\cos}[k]^2 + F_{\sin}[k]^2}$ (amplitude)
• $\phi_k = \arctan\left(\frac{F_{\sin}[k]}{F_{\cos}[k]}\right)$ (phase)