# 卷积交互式可视化

卷积的交互式演示：翻转、滑动、相乘、求和过程，包含时域和频域

> 标准页面: https://elysiatools.com/zh/visualizations/convolution

- **分类:** Math

## 概述

Interactive visualization of convolution demonstrating the flip-slide-multiply-sum process in both time and frequency domains. Features continuous formula: (f*g)(t) = ∫ f(τ)·g(t-τ) dτ, discrete formula: (x*h)[n] = Σ x[k]·h[n-k], and convolution theorem: FFT(x*h) = FFT(x)·FFT(h). Time domain visualization shows: (1) Input signal x[n] (static), (2) Flipped kernel h[-k] showing horizontal mirror, (3) Shifted kernel h[n-k] sliding to current position n, (4) Product view showing x[k]·h[n-k] multiplication at overlapping samples, (5) Output y[n] building up progressively with current position marker. Frequency domain view demonstrates: (1) FFT magnitude of input signal |FFT(x[n])|, (2) FFT magnitude of kernel |FFT(h[n])|, (3) Product of FFTs |FFT(x)·FFT(h)|, (4) FFT of output |FFT(y[n])| showing convolution theorem. Multiple signal types: rectangular pulse, triangular pulse, Gaussian, sinc function, and custom drawable signal. Kernel options: rectangular (moving average), Gaussian (smoothing), derivative (edge detection), sinc (low-pass filter), Sobel (edge detection). Adjustable parameters: kernel size (3-21), normalization toggle, sampling rate/resolution, animation speed, and manual position control. Animation controls: play/pause, step forward/backward, reset, position slider, and speed control. Real-time metrics display: current position n, output value y[n], sum of products, overlapping samples count, and detailed calculation breakdown showing each term. Educational content covers four-step process (flip, slide, multiply, sum), key concepts (commutative, associative, identity element, convolution theorem), and practical applications: image filtering (blur, sharpen, edge detection), audio processing (reverb, EQ), neural network convolution layers, probability distributions, and signal smoothing. Color coding: blue for input signal, red for kernel, green for output, purple for product, orange for overlap. Custom drawing mode allows hand-drawn signals with mouse/touch input. Multi-language support (zh, en, es, fr, de, ru, pt).

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