希尔伯特变换作为无限矩阵Hilbert transform as an infinite matrix
文章将希尔伯特变换重新表述为无限矩阵运算，连接了其傅里叶级数表示与线性代数视角。具体推导表明：若函数f(x)的傅里叶系数为cₙ，则其希尔伯特变换的系数为(-i sgn(n))cₙ，其中sgn为符号函数。这一数学视角简化了信号处理中的卷积计算，尤其适用于非平稳信号分析。作者强调，矩阵形式为数值实现提供了新思路。
John
The previous post linked to a post I wrote a few years ago about the Hilbert transform and Fourier series. That post says that if the Fourier series of a function is
then the Fourier series of its Hilbert transform is
When I looked back at that post I thought about how if you thought of the Fourier coefficients as elements of an infinite vector then the Hilbert transform can be represented as multiplying by an infinite block matrix.
I rarely see infinite matrices except in older math books. Apparently they were more fashionable a few decades ago than they are now. I suppose the notation falls between two stools, too concrete for some tastes and not concrete enough for others. The former folks would prefer something like H and the latter would prefer the sum above.
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