Discrete Fourier Transform

Discrete Fourier Transform (DFT) Equations

The Discrete Fourier Transform of a finite time sequence \( x(n) \), with \( n = 0, 1, 2, \ldots, N-1 \), is defined as:

$$ X(k) = \sum_{n=0}^{N-1} x(n) \cdot e^{-j \frac{2\pi}{N} k n} $$

Where:

• \( X(k) \): Frequency component \( k \).
• \( N \): Total number of points in the transform.
• \( j \): Imaginary unit.

The magnitude of the frequency is calculated as:

$$ |X(k)| = \sqrt{\text{Re}(X(k))^2 + \text{Im}(X(k))^2} $$

Frequency Calculation

The frequency corresponding to each index \( k \) is calculated as:

$$ f(k) = \frac{k}{N} \cdot f_s $$

Where \( f_s \) is the sampling frequency of the original signal.

Normalized Frequency:

In many cases, normalized frequency is used:

$$ f(k) = \frac{k}{N} $$