Code Implementation for FFT Transformation and Spectrum Analysis

FFT (Fast Fourier Transform) is a fundamental algorithm in signal processing used to convert time-domain signals into frequency-domain representations. In MATLAB, the built-in FFT function enables efficient implementation of this transformation, facilitating comprehensive spectrum analysis.

### Core Logic of FFT Transformation Signal Acquisition: Begin by obtaining time-domain signal data, typically represented as uniformly sampled discrete point sequences. FFT Computation: Utilize the `fft()` function to transform the signal, generating complex-valued frequency-domain data. Spectral Magnitude: Extract magnitude values by applying the absolute value function (`abs()`) to the complex results, representing the amplitude of each frequency component. Frequency Axis Generation: Calculate corresponding frequency axis scales based on the sampling frequency (Fs) and signal length (N), ensuring accurate representation of actual frequencies along the horizontal axis using `f = (0:N-1)*(Fs/N)` for full spectrum or `f = (0:N/2)*(Fs/N)` for single-sided spectrum.

### Key Steps in Spectrum Analysis DC Offset Removal: If the signal contains DC components (zero frequency), preprocess by subtracting the mean value using `signal = signal - mean(signal)`. Window Function Application: Apply window functions (such as Hanning window via `hanning(N)`) to truncated signals to minimize spectral leakage effects through element-wise multiplication. Normalization: Adjust magnitude values according to the window function and FFT points to maintain accurate physical meaning of spectral amplitudes, typically dividing by the window's coherent gain. Single-Sided Spectrum Conversion: For real-valued signals exhibiting spectral symmetry, display only the positive frequency portion (0~Fs/2) by taking the first half of the FFT result and doubling the magnitudes (except DC component).

### Result Validation Peak Detection: Identify prominent peaks in the spectrum plot corresponding to dominant frequency components using `findpeaks()` function or threshold-based analysis. Noise Assessment: Evaluate baseline noise levels to determine signal quality or filtering requirements through statistical analysis of non-peak regions.

Using MATLAB's plotting capabilities (such as `plot()` for continuous displays or `stem()` for discrete representations), users can visually compare time-domain waveforms and frequency-domain spectra, facilitating analysis of signal frequency characteristics. This methodology finds extensive applications in audio processing, vibration analysis, communication systems, and various engineering domains.