Fourier transform technique with refinement in the frequency domain for the synthesis of optical thin films

The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...

The application of Fourier Transform properties in radio broadcasting is multifaceted, enabling significant advancements in the way signals are transmitted and received. Key areas where these properties are utilized include simultaneous multi-channel transmission, audio clip speed adjustments, live broadcast delays for different time zones, audio frequency adjustments, and signal demodulation.
In radio broadcasting, multiple audio signals often need to be transmitted simultaneously. The Fourier...

The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...

Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...

The Fast Fourier Transform (FFT) is a computational algorithm designed to compute the Discrete Fourier Transform (DFT) efficiently. By breaking down the calculations into smaller, manageable sections, the FFT significantly reduces the computational complexity involved. Direct computation of an N-point DFT requires N2 complex multiplications, whereas the FFT algorithm needs only (N/2)log⁡2N multiplications, offering a much faster performance.
The computational efficiency of the FFT becomes...

The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...