訊號與系統/非週期訊號與傅立葉級數

非週期訊號與傅立葉級數

無論是三角傅立葉級數或指數傅立葉級數，級數和所代表的訊號均為ㄧ週期訊號。

對於一個非週期訊號基本上是無法用傅立葉級數表示。

但是，如果只考慮某ㄧ時間範圍，則可用傅立葉級數來表示。

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範例4.20

試用傅立葉級數來表示非週期訊號$x(t)=e^{-t/2}$。僅考慮時間範圍$0<t<\pi$

...........圖 ©B. P. Lathi, Signal Processing and Linear Systems, Berkeley-Cambridge Press, 1998

【解】因為僅考慮時間範圍$0<t<\pi$ ，故$T_0=\pi$令 $\Rightarrow {\omega }_0=\frac{2\pi }{T_0}=2$

三角傅立葉級數

$\mathrm{\Phi }(t)=a_0+\sum _{n=1}^{\mathrm{\infty }}(a_n\cos 2nt+b_n\sin 2nt)$

$a_0=\frac{1}{T_0}\underset{T_0}{\overset{}{\int }}x(t)dt=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}{e}^{-t/2}dt=0.504$

$a_n=\frac{2}{T_0}\underset{T_0}{\overset{}{\int }}x(t)\cos 2ntdt=\frac{2}{\pi }\underset{0}{\overset{\pi }{\int }}{e}^{-t/2}\cos 2ntdt=0.504\left(\frac{2}{1+16n^2}\right)$

$b_n=\frac{2}{T_0}\underset{T_0}{\overset{}{\int }}x(t)\sin 2ntdt=\frac{2}{\pi }\underset{0}{\overset{\pi }{\int }}{e}^{-t/2}\sin 2ntdt=0.504\left(\frac{8n}{1+16n^2}\right)$

$\mathrm{\Phi }(t)=0.504[1+\sum _{n=1}^{\mathrm{\infty }}\frac{2}{1+16n^2}(\cos 2nt+4n\sin 2nt)]$

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...............圖

©B. P. Lathi, Signal Processing and Linear Systems, Berkeley-Cambridge Press, 1998.

注意：傅立葉級數$\varphi$為ㄧ週期訊號而原函數$x(t)$為非週期訊號。 但$\varphi (t)=x(t)0<t<\pi$當

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參考資料

B. P. Lathi, Signal Processing and Linear Systems, Berkeley-Cambridge Press, 1998. G. E. Carlson, Signal and Linear System Analysis, 2nd ed., John Wiley & Sons, 1998. 余兆棠、李志鵬，信號與系統， 滄海書局，2007。 Edward W. Kamen and Bonnie S. Heck, Fundamentals of Signals and System Using the Web and Matlab, 2nd ed., Prentice Hall International, 2000. Rodger E. Ziemer, William H. Tranter, D. Ronald Fannin, Signals & Systems: Continuous and Discrete, 4th ed., Prentice Hall International, 1998. Charls L. Phillips, John M. Parr, Eve A. Riskin, Signals, Systems, and Transforms, 3rd ed., Pearson Education, 2003. Rodger E. Ziemer and William H. Tranter, Principles of Communications, John Wiley & Sons, 2002. Simon Haykin, Communication Systems, 4th ed., John Wiley & Sons, 2001. John G. Proakis and M. Salehi, Communication Systems Engineering, 2nd ed., Prentice Hall International, 2002.

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