# 采样定理与混叠可视化

奈奎斯特-香农采样定理和混叠现象的交互式可视化

> 标准页面: https://elysiatools.com/zh/visualizations/sampling-theorem

- **分类:** Math

## 概述

Comprehensive interactive visualization of the Nyquist-Shannon sampling theorem and aliasing phenomenon in signal processing. Features core theorem: f_s >= 2f_max for perfect reconstruction, sampling process: x[n] = x(n/f_s), and aliased frequency: f_alias = |f - k*f_s|. Real-time visualization includes: (1) Main signal canvas showing original continuous signal (blue), sample points (red dots), and reconstructed signal (green dashed) with adjustable sampling rate f_s, signal frequency f, amplitude, and phase. (2) Frequency spectrum display showing original spectrum, spectral replicas from sampling, and aliased components highlighted in orange. (3) Nyquist indicator bar with color-coded status (green for adequate, orange for warning, red for aliasing). (4) Reconstruction methods comparison: Zero-Order Hold (staircase), Linear Interpolation, Sinc Interpolation (ideal Whittaker-Shannon), and Cubic Spline. (5) Real-time metrics: sampling rate, Nyquist rate, max frequency, aliased frequency, and reconstruction MSE. Multiple signal types: sine wave, composite (multi-frequency), square wave (demonstrating harmonics), and custom frequency addition. Preset scenarios: Perfect (f_s = 4f), Critical (f_s = 2f), Aliasing (f_s = 1.5f), Severe (f_s = 1.1f), and Wagon Wheel effect demo. Animation mode shows scanning line moving across signal with real-time sample capture. Educational content covers sampling theorem statement, Nyquist condition, aliasing mechanism with wagon wheel effect analogy, reconstruction method formulas, frequency folding around f_s/2, and real-world applications: CD audio (44.1 kHz), video frame rates (24/30/60 fps), image pixel sampling, and digital communications ADC/DAC. Uses KaTeX for formula rendering including f_s >= 2f_max, x[n] = x(nT_s), f_alias = |f - k*f_s|, and sinc interpolation formula. Multi-language support (zh, en, es, fr, de, ru, pt).

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