# 希尔伯特变换的实现——数据分析漫谈3

HT在实际应用中会遇到两个具体的问题：i）其滤波器在零点是奇异的，不易于实际计算；ii）其不能压制高频噪声，因而不利于求包络。本博客提供一个低通HT，即 Low-passing HT (LPHT)LPHT能够可以很好地解决上述问题，因此可以作为HT的实际应用版本。

$H{ s} = s(t) * h(t) = frac{i}{pi }int {frac{{s(tau )}}{{t - tau }}} dtau$                                                 (1)

$h(t) = frac{i}{{pi t}}$                                                                (2)

$hat h(omega ) = sign(omega ) = left{ {begin{array}{*{20}{c}} {1,} & {omega > 0} \ {0,} & {omega = 0} \ { - 1,} & {omega < 0} \ end{array}} right.$                                        (3)

$|h(0)| = infty$                                                                          (4)

${H_varpi }{ s} = s(t) * {h_varpi }(t) = frac{i}{pi }int {s(tau)frac{{1 - cos (varpi (t - tau ))}}{{t - tau }}} dtau$                                                                        (5)

${h_varpi }(t) = frac{i}{{pi t}}(1 - cos varpi t)$                                    (6)

${{hat h}_varpi }(omega ) = left{ {begin{array}{*{20}{c}} {sign(omega ),} & {|omega | le varpi } \ {0,} & {|omega | > varpi } \ end{array}} right.$                                    (7)

${h_varpi }(0) = 0$                                                                    (8)

l         LPHT也可以应用到HHT的计算；

l         如果大家觉得LPHT很实用，别忘了引用本博客哦。

https://m.sciencenet.cn/blog-634454-608204.html

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