The Fourier transform of a function $ f(t) $ is defined as $ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt $. Its inverse transform is $ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega $, forming a transform pair. This allows analysis of functions in the frequency domain instead of time domain. The Fourier transform is widely used in various fields like Antennas, Optics, Acoustics, Geophysics, Linear Systems, and Quantum Physics. It is also related to the characteristic function in Probability Theory. Substituting $ F(\omega) $ into the inverse transform yields the Fourier's Integral Theorem, which states that the integral of the Fourier transform over all frequencies reconstructs the original function. The theorem holds under certain conditions, including continuity, monotonicity, and bounded variation of $ f(t) $. To compute integrals in the Fourier transform, discretization is necessary due to the rapid oscillation of exponentials. The Discrete Fourier Transform (DFT) replaces the infinite integral with a finite sum over equally spaced points. The DFT is given by $ F(\omega_n) = \Delta t \sum_{k=0}^{N-1} f(t_k) e^{-i\omega_n t_k} $, where $ \Delta t $ is the spacing between points. This discretized form approximates the continuous Fourier transform and allows efficient computation using numerical methods. The DFT is a key tool in signal processing and data analysis.The Fourier transform of a function $ f(t) $ is defined as $ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt $. Its inverse transform is $ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega $, forming a transform pair. This allows analysis of functions in the frequency domain instead of time domain. The Fourier transform is widely used in various fields like Antennas, Optics, Acoustics, Geophysics, Linear Systems, and Quantum Physics. It is also related to the characteristic function in Probability Theory. Substituting $ F(\omega) $ into the inverse transform yields the Fourier's Integral Theorem, which states that the integral of the Fourier transform over all frequencies reconstructs the original function. The theorem holds under certain conditions, including continuity, monotonicity, and bounded variation of $ f(t) $. To compute integrals in the Fourier transform, discretization is necessary due to the rapid oscillation of exponentials. The Discrete Fourier Transform (DFT) replaces the infinite integral with a finite sum over equally spaced points. The DFT is given by $ F(\omega_n) = \Delta t \sum_{k=0}^{N-1} f(t_k) e^{-i\omega_n t_k} $, where $ \Delta t $ is the spacing between points. This discretized form approximates the continuous Fourier transform and allows efficient computation using numerical methods. The DFT is a key tool in signal processing and data analysis.