Definition
Example
- a 3-dim Euclid space \(R^{3} \vec{a}=(1,0,0) \vec{b}=(0,1,0)\vec{c}=(0,0,1)\)
- define n-dim space \(V\)
\(X(t)=\Sigma^{\inf}_{k=-\inf}e^{jk\omega _{0}t}\)
\(S=\{x(t)|x(t)=x(t+t_{0})\forall \}\) \(T_{0}=\frac{2\pi}{\omega_{0}}\)
comment: <2\(T_{0}\)
\(B=\{\phi_{k}t|\phi_{k}(t)=e^{jkw_{0}t} k\in Z\}\) \(B\subset C\)
We have:
The equation above can be deducted then:
各个弦波都是标准正交积。
Fourier's Idea
Example
composition in graph
Periodic signals & Fourier Series Expansion cont.
Some simple theory
| time | frequency |
|---|---|
| period | discrete |
| non-period | continuous |
deduction
\(a_{k}\) is denoted as Fourier coefficient.
example
系数\(a_{k}\) -> \(Fourier expression\)
前后均为实系数偶函数
误差
Convergence
决对可积
狄利克雷性质-收敛
只有第三个满足条件。
条件
频率域和时间域的 性质等价
性质8
example
把任何信号变成标准正交积
