傅里叶公式推导(三)
文章目录
- 周期 2L
- 周期T
周期 2L
周期 T = 2 L T=2L T=2L 的傅里叶变换
即
f ( t ) + f ( t + 2 L ) f(t) + f(t+2L) f(t)+f(t+2L)
| x | t |
|---|---|
| 2 π \pi π | 2 L 2L 2L |
原公式
f ( x ) = a 0 2 + ∑ n = 1 ∞ [ a n cos n x + b n sin n x ] a 0 = 1 π ∫ − π π f ( x ) d x a n = 1 π ∫ − π π f ( x ) cos n x d x b n = 1 π ∫ − π π f ( x ) sin n x d x \begin{array}{l} f(x) = \frac{a_0}{2} +\sum_{n=1}^{\infty }[a_n\cos{nx} + b_n\sin{nx}] \\ a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\mathrm{d}x \\ a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos{nx}\mathrm{d}x \\ b_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin{nx}\mathrm{d}x \end{array} f(x)=2a0+∑n=1∞[ancosnx+bnsinnx]a0=π1∫−ππf(x)dxan=π1∫−ππf(x)cosnxdxbn=π1∫−ππf(x)sinnxdx
令
t = L π x x = π L t ω = 2 π T = π L t=\frac{L}{\pi}x \\ x=\frac{\pi}{L}t \\ \omega = \frac{2\pi}{T} = \frac{\pi}{L} t=πLxx=Lπtω=T2π=Lπ
换元得，（周期T=2L）
f ( t ) = a 0 2 + ∑ n = 1 ∞ [ a n cos n ω t + b n sin n ω t ] a 0 = 1 L ∫ − L L f ( t ) d t a n = 1 L ∫ − L L f ( t ) cos n ω t d t b n = 1 L ∫ − L L f ( t ) sin n ω t d t \begin{array}{l} f(t) = \frac{a_0}{2} +\sum_{n=1}^{\infty }[a_n\cos{n\omega{t}} + b_n\sin{n\omega{t}}] \\ a_0 = \frac{1}{L}\int_{-L}^{L}f(t)\mathrm{d}t \\ a_n = \frac{1}{L}\int_{-L}^{L}f(t)\cos{n\omega{t}}\mathrm{d}t \\ b_n = \frac{1}{L}\int_{-L}^{L}f(t)\sin{n\omega{t}}\mathrm{d}t \end{array} f(t)=2a0+∑n=1∞[ancosnωt+bnsinnωt]a0=L1∫−LLf(t)dtan=L1∫−LLf(t)cosnωtdtbn=L1∫−LLf(t)sinnωtdt
周期T
统一周期T的写法如下。
f ( t ) = a 0 2 + ∑ n = 1 ∞ [ a n cos n ω t + b n sin n ω t ] a 0 = 2 T ∫ 0 T f ( t ) d t a n = 2 T ∫ 0 T f ( t ) cos n ω t d t b n = 2 T ∫ 0 T f ( t ) sin n ω t d t \begin{array}{l} f(t) = \frac{a_0}{2} +\sum_{n=1}^{\infty }[a_n\cos{n\omega{t}} + b_n\sin{n\omega{t}}] \\ a_0 = \frac{2}{T}\int_{0}^{T}f(t)\mathrm{d}t \\ a_n = \frac{2}{T}\int_{0}^{T}f(t)\cos{n\omega{t}}\mathrm{d}t \\ b_n = \frac{2}{T}\int_{0}^{T}f(t)\sin{n\omega{t}}\mathrm{d}t \end{array} f(t)=2a0+∑n=1∞[ancosnωt+bnsinnωt]a0=T2∫0Tf(t)dtan=T2∫0Tf(t)cosnωtdtbn=T2∫0Tf(t)sinnωtdt
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