MATLAB Code Implementation of Fast Fourier Transform (FFT)

Fast Fourier Transform (FFT) is an efficient algorithm in signal processing used to convert time-domain signals into frequency-domain representations. In MATLAB, this transformation can be easily implemented using the built-in `fft` function, which utilizes the Cooley-Tukey algorithm to reduce computational complexity from O(n²) to O(n log n).

Implementation Approach Data Preparation: Start by preparing experimental time-domain signal data, typically represented as discrete time-series samples stored in a vector or array. FFT Computation: Call the `fft` function with the signal data as input to compute the frequency-domain representation. The function returns complex-valued frequency components. Frequency Axis Adjustment: Since FFT output defaults to a two-sided spectrum, use MATLAB's `fftshift` function and frequency vector generation (e.g., `f = fs*(-N/2:N/2-1)/N`) to properly display a single-sided spectrum. Magnitude and Phase Analysis: Obtain the magnitude spectrum by applying `abs` function to FFT results, and extract phase information using the `angle` function, which returns phase angles in radians. Visualization: Use `plot` or `stem` functions to display the original signal and spectrum graphs, with optional labeling using `xlabel`, `ylabel`, and `title` functions for clear analysis.

Extended Applications FFT finds widespread applications in signal processing, communication systems, and audio analysis. Key implementations include digital filter design using frequency response analysis, spectral leakage evaluation, and integration with window functions (e.g., Hanning window via `hann` function) to reduce spectral leakage effects. Advanced techniques involve zero-padding for frequency resolution improvement and inverse FFT (`ifft`) for signal reconstruction.

Through MATLAB's FFT implementation, users can efficiently explore signal frequency characteristics without manually programming complex butterfly operation logic, while benefiting from built-in optimization for different data lengths including prime-number handling.