# 小波变换之小波构建

## 1. 公式构建小波 (crude wavelet)

1. 点间距固定，沿用相同的公式，在尾部加点，时域总距离增加
2. 点间距固定，但使用不同的公式，相当于插值

### 1.2 Adding More Points by Interpolation for Lower Cutoff Frequency

stretching the Sinc function by 2 divides the cutoff frequency by 2.

### 1.3 以 Morlet Wavelet 为例

Morlet 小波的数学公式如下: $\text {morl} (t) = \exp (-t^2/2) \ast \cos(5t)$

## 2. 从基本长度滤波器扩展

The original points are upsampled by a factor of 2, convolve the data with a lowpass filter, and use a constant of multiplication (常数相乘) later to normalize (adjust for the change of magnitude due to the extra points).

Db4 wavelet function filter (本质是带通滤波器) 为: $$[-0.1294 \quad -0.2241 \quad 0.8365 \quad -0.4830]$$

$[-0.1294 \quad 0 \quad -0.2241 \quad 0 \quad 0.8365 \quad 0 \quad -0.4830] \rightarrow \\\\ [-0.0625 \quad -0.1082 \quad -0.1372 \quad -0.1707 \quad 0.3538 \\\\ \quad 0.7287 \quad -0.0458 \quad -0.5123 \quad -0.1082 \quad 0.0625]$

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