Fourier Transform

Discrete Fourier Transform (DFT)
Converts a time-domain signal to a frequency-domain representation and back via the Inverse DFT (IDFT):

$$\begin{array}{rl}{X}_k& =\sum _{n=0}^{N-1}{x}_n{e}^{-i2\pi kn/N},k=0,1,\dots ,N-1,\\ x_n& =\frac{1}{N}\sum _{k=0}^{N-1}{X}_k{e}^{i2\pi kn/N},n=0,1,\dots ,N-1.\end{array}$$

Key Points:

1. Bijection: The DFT maps an $N$-dimensional time-domain signal to an $N$-dimensional frequency-domain signal, ensuring invertibility.
2. Frequency Bins: The index $k$ represents frequencies $f_k=\frac{k\cdot F_s}{N}$ Hz, where $F_s$ is the sampling rate.
3. Example: If $F_s=N$ (e.g., 1000 Hz sampling for 1000 samples), then $f_k=k$ Hz.
4. Generally, frequencies depend on $F_s$ and $N$, not just $k$.