In Chapter 10, the Fast Fourier Transform (FFT) is introduced, focusing on the Fourier integral and its inverse transform. The Fourier transform of an integrable function \( f(t) \) is defined as \( F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt \), and its inverse is \( f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega \). These equations form a Fourier transform pair, allowing the analysis of functions in the frequency domain instead of the time domain. The Fourier transform is widely used in various fields such as antennas, optics, acoustics, geophysics, and quantum physics. The Fourier's Integral Theorem, which states that the Fourier transform and its inverse are reciprocal, is discussed, with conditions for its validity including continuity and bounded variation of \( f(t) \). For practical computation, the infinite integral is discretized by dividing the domain into a finite number of subintervals, leading to the Discrete Fourier Transform (DFT). 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 length of each subinterval. This discretized form allows for numerical computation and inversion of the Fourier transform.In Chapter 10, the Fast Fourier Transform (FFT) is introduced, focusing on the Fourier integral and its inverse transform. The Fourier transform of an integrable function \( f(t) \) is defined as \( F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt \), and its inverse is \( f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega \). These equations form a Fourier transform pair, allowing the analysis of functions in the frequency domain instead of the time domain. The Fourier transform is widely used in various fields such as antennas, optics, acoustics, geophysics, and quantum physics. The Fourier's Integral Theorem, which states that the Fourier transform and its inverse are reciprocal, is discussed, with conditions for its validity including continuity and bounded variation of \( f(t) \). For practical computation, the infinite integral is discretized by dividing the domain into a finite number of subintervals, leading to the Discrete Fourier Transform (DFT). 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 length of each subinterval. This discretized form allows for numerical computation and inversion of the Fourier transform.