计算机网络(computer networks)2.1.1 节 Fourier 分析，没看懂

[koebehshian]

In the early 19th century, the French mathematician Jean-Baptiste Fourier proved that any reasonably behaved periodic function, g(t) with period T, can be constructed as the sum of a (possibly infinite) number of sines and cosines:

g(t) = (1/2)c + Σ(an sin(2πnft), n=1->∞) + Σ(bn cos(2πnft), n=1->∞) (2-1)

where f = 1/T is the fundamental frequency, an and bn are the sine and cosine amplitudes of the nth harmonics (terms), and c is a constant. Such a decomposition is called a Fourier series. From the Fourier series, the function can be reconstructed. That is, if the period, T, is known and the amplitudes are given, the original function of time can be found by performing the sums of Eq. (2-1).

A data signal that has a finite duration, which all of them do, can be handled by just imagining that it repeats the entire pattern over and over forever (i.e., the interval from T to 2T is the same as from 0 to T, etc.).

The an amplitudes can be computed for any given g(t) by multiplying both sides of Eq. (2-1) by sin(2πkft) and then integrating from 0 to T. Since

∫(sin(2πkft)sin(2πnft)dt, t=0->T) = { 0 for k ≠ n T/2 for k = n }

only one term of the summation survives: an. The bn summation vanishes completely. Similarly, by multiplying Eq. (2-1) by cos(2πkft) and integrating between 0 and T, we can derive bn. By just integrating both sides of the equation as it stands, we can find c. The results of performing these operations are as follows:

an = (2/T)∫(g(t)sin(2πnft)dt, t=0->T) bn = (2/T)∫(g(t)cos(2πnft)dt, t=0->T) c = (2/T)∫(g(t)dt, t=0->T)

公式(2-1)两边乘上 sin(2πkft)，再对 t 从 0 到 T 积分，到底是怎么算出 an 的，公式(2-1)中还有常量 c，是怎么消掉的，原来还有 1 到无穷的求和，怎么也没了

[byaiu]

这个是通信原理里的东西吧？找一本信号系统就都来了。
简单来说周期相差整数倍的 sin cos 相乘积分为 0。

[koebehshian]

@byaiu 那常量 c 是怎么消掉的

[byaiu]

c 并没有消掉呀😅
c 是信号中的直流分量
这段只是介绍了一下信号在频域上的表示，我猜这章是讲物理层的吧，ee 的会学这个
cs 学这个太硬核了

[misaka19000]

哈哈，想起了大学的时候被三大变换所支配的恐惧~

[silhouette]

来一本信号与系统

[wheeler]

可以看看这里。
http://www.sosmath.com/fourier/fourier1/fourier1.html

[wheeler]

这本 https://book.douban.com/subject/21359219/ 写得非常棒。

PS.
网络不应该把重点放在这里呀。

[feather12315]

大兄弟你是不是考虑放张图片上来？你这公式写得不想让人看啊

[lzhCoooder]

不用看信号系统，你自己查查三角函数的正交性就明白了
c 是一个常数 乘以一个三角函数然后积分一个周期不管怎样都是 0...

大一上的内容....

[koebehshian]

@wheeler 我知道重点不是物理层，只是我比较喜欢探究原理。

[koebehshian]

@feather12315 本来有几个换行符的，被吞了。图片我还不会贴，就复制粘贴文字了，求和与积分，我扁平化成函数形式，逗号分隔。

[koebehshian]

@lzhCoooder 首先，公式 2-1 原来有个从 1 到无穷求和的，这个怎么没了

[lzhCoooder]

@koebehshian
我不知道你到底哪里不懂，我突然发现你连三角函数正交性都不需要查，∫(sin(2πkft)sin(2πnft)dt, t=0->T) = { 0 for k ≠ n T/2 for k = n }

书上已经写明白了，只有 k 等于 n 剩下 其他根据三角函数正交性都为 0，无穷多项都没用的