查看“︁訊號與系統/能量與功率”︁的源代码

{{noteTA
|G1=Communication
|G2=Math
}}

== 能量與功率 ==
若 x(t) 為能量訊號，其總能量：

<math>E_x = \int_{-\infty}^{\infty} {|x(t)|}^2 \, dt</math>

若 x(t) 為功率訊號，其平均功率：

<math>P_x = \lim_{T \to \infty}\frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} {|x(t)|}^2 \, dt</math>

若 x(t) 為週期訊號且基本週期為   ，其平均功率：

<math>P_x = \frac{1}{T_0} \int_{-\infty}^{} {|x(t)|}^2 \, dt</math>

上述公式是由時域(time-domain)求訊號的能量及功率。

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== 在頻域求訊號能量 ==
(1)由時域求能量：

<math>E_x = \int_{-\infty}^{\infty} {|x(t)|}^2 \, dt</math>

(2)根據傅立葉逆轉換公式

<math>x(t) = \int_{-\infty}^{\infty} X(f)e^{j2\pi ft} \, df</math>


(3)將(2)代入(1)

<math>E_x = \int_{-\infty}^{\infty} {|x(t)|}^2 \, dt = \int_{-\infty}^{\infty} x(t)x^*(t) \, dt</math> = (<math>\Rightarrow</math>積分順序對調)<math>\int_{-\infty}^{\infty} [ \int_{-\infty}^{\infty} X(f)e^{j2\pi ft} \, df ] x^*(t)\, dt = \int_{-\infty}^{\infty} X(f) [ \int_{-\infty}^{\infty} x^*(t)e^{j2\pi ft} \, dt ] \, df = \int_{-\infty}^{\infty} X(f)X^*(f) \, df = \int_{-\infty}^{\infty} {|X(f)|}^2 \, df</math>

(4)故訊號的能量

<math>E_x = \int_{-\infty}^{\infty} {|x(t)|}^2 \, dt = \int_{-\infty}^{\infty} {|X(f)|}^2 \, df</math>(<math>\Rightarrow</math>稱為Parseval定理)

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== 能量密度頻譜 ==
<math>E_x =\int_{-\infty}^{\infty} {|X(f)|}^2 \, df</math>

定義:能量密度頻譜(energy density spectrum)

<math>G_x(f)\equiv {|X(f)|}^2</math>其中<math>X(f)</math> = <math>\Im</math> {<math>x(t)</math>}

<math>G_x(f)</math>描述訊號能量在頻譜上的分佈情形。

訊號 x(t) 在頻帶範圍<math>f_1<f_2<f_3</math>內的總能量為

<math>E_12 = \int_{-f1}^{f2} G_x(f) \, df + \int_{f2}^{f1} G_x(f) \, df</math>

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== 相關函數 ==
假設存在函數<math>r_x(\tau)</math>使得

<math>r_x(\tau) \leftrightarrow G_x(f) \Rightarrow r_x(\tau) = </math><math>\Im ^{-1}</math> {<math>G_x(f)</math>} = <math>\Im ^{-1}</math> {<math>X(f)|^2</math>} = <math>\Im ^{-1}</math> {<math>X(f)X^*(f)</math>} =(<math>\Rightarrow</math>旋積定理)<math>\Im ^{-1}</math> {<math>X(f)</math>} * <math>\Im ^{-1}</math> {<math>X^*(f)</math>} =(<math>\Rightarrow</math>共軛複數及時間反轉定理) <math>x(\tau)*x^*(-\tau)</math> = <math>\int_{-\infty}^{\infty} x(\lambda)x^*(\lambda - \tau) \, d\lambda</math> =(<math>\Rightarrow</math>令<math>\lambda - \tau = t , d\lambda = dt</math>)<math>\int_{-\infty}^{\infty} x(t)x^*(t + \tau) \, dt</math>

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== 自相關函數 ==
訊號 x(t) 的自相關函數(autocorrelation function)

<math>r_x(\tau)\equiv \int_{-\infty}^{\infty} x(t)x^*(t + \tau) \, dt = x(\tau) * x^*(-\tau)</math>

<math>r_x(0) = \int_{-\infty}^{\infty} x(t)x^*(t) \, dt = \int_{-\infty}^{\infty} {|x(t)|}^2 \, dt = E_x</math>

x(t)的自相關函數<math>r_x(\tau)</math> 取傅立葉轉換可得x(t)的能量密度頻譜<math>G_x(f)</math>

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== 在頻域求平均功率 ==
已知一功率訊號x(t) ，定義<math>x_T(t) = x(t)rect(\frac{t}{T})</math>如下圖：

<math>x_T(t)</math>為一能量訊號，

<math>_T(t) \leftrightarrow X_T(f)</math>

原功率訊號x(t)的平均功率

<math>P_x = \lim_{T \to \infty}\frac{E_{xT}}{T}</math>其中<math>E_{xT}</math>為<math>x_T(t)</math>的總能量

=<math>\lim_{T \to \infty}\frac{1}{T}\int_{-\infty}^{\infty} {|x_T(t)|}^2 \, dt = \lim_{T \to \infty}\frac{1}{T}\int_{-\infty}^{\infty} {|X_T(f)|}^2 \, df = </math><math>\int_{-\infty}^{\infty}</math><math>\lim_{T \to \infty}\frac{1}{T} {|X_T(f)|}^2 df</math>

x(t)的功率密度頻譜<math>S_x(f)</math>定義如下：

<math>S_x(f)\equiv \lim_{T \to \infty}\frac{1}{T} {|X_T(f)|}^2</math>

功率訊號X(t)在頻帶範圍<math>f_1<f_2<f_3</math>內的平均功率有

<math>R_{12} = \int_{-f_2}^{-f_1} S_x(f) \, df + \int_{f_2}^{f_1} S_x(f) \, df</math>

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== 週期訊號的功率密度頻譜 ==
週期訊號x(t) ，基本週期<math>T_0</math> ，

<math>x(t) = \sum_{n=-\infty}^\infty X_ne^{j2\pi f_0t}</math>其中<math>f_0 = \frac{1}{T_0}</math>

根據第四章所提週期訊號的Parseval定理知

<math>P_x = \sum_{n=-\infty}^\infty {|X_n|}^2</math>

故x(t)的功率密度頻譜

<math>S_x(f) = \sum_{n=-\infty}^\infty {|X_n|}^2 \delta (f-nf_0)</math>

注意：第四章所講週期訊號的功率頻譜為線頻譜；本處為功率密度頻譜，故用脈衝函數表示。

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== 功率訊號的相關函數 ==
比照能量訊號，定義功率訊號的自相關函數(autocorrelation function)如下：

<math>r_x(\tau) \equiv \lim_{T \to \infty}\frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} x^*(t)x(t+\tau) \, dt </math>

當<math>\tau = 0</math>

<math>r_x(0) \equiv \lim_{T \to \infty}\frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} x^*(t)x(t) \, dt = \lim_{T \to \infty}\frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} {|X_n|}^2 \, dt = P_x...r_x(0)</math>代表訊號的平均功率

x(t)的功率密度頻譜

<math>S_x(f) \equiv </math><math>\Im</math> {<math>r_x(\tau)</math>}<math>\Rightarrow r_x(\tau) = \int_{-\infty}^{\infty} S_x(f)e^{j2\pi f\tau} \, df  \Rightarrow r_x(0) = \int_{-\infty}^{\infty} S_x(f) \, df  = P_x</math>

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== 週期訊號的自相關函數 ==
週期訊號為功率訊號，其自相關函數

<math>r_x(\tau) = \lim_{T \to \infty}\frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} x^*(t)x(t+\tau) \, dt = \lim_{k \to \infty}\frac{1}{kT_0} \int_{\frac{-kT_0}{2}}^{\frac{kT_0}{2}} x^*(t)x(t+\tau) \, dt</math>(<math>\gets T_0</math>為基本週期) = (<math> \Rightarrow </math>x(t)為週期訊號，積分k個週期相當於積分1個週期再乘上k倍)<math>\lim_{k \to \infty}\frac{k}{kT_0} \int_{\frac{-T_0}{2}}^{\frac{T_0}{2}} x^*(t)x(t+\tau) \, dt = \lim_{k \to \infty}\frac{1}{T_0} \int_{\frac{-T_0}{2}}^{\frac{T_0}{2}} x^*(t)x(t+\tau) \, dt</math> = (<math> \Rightarrow </math>式子中已經沒有k ，故<math>\lim_{k \to \infty}</math>可略去。)<math>\frac{1}{T_0} \int_{\frac{-T_0}{2}}^{\frac{T_0}{2}} x^*(t)x(t+\tau) \, dt</math>

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== 週期訊號的功率密度頻譜 ==
週期訊號x(t)，基本週期<math>T_0</math>

<math>x(t) = \sum_{n=- \infty}^\infty X_ne^{j2\pi nf_0}</math>其中<math>f_0 = \frac{1}{T_0}</math>

<math>r_x(\tau) = \frac{1}{T_0} \int_{\frac{-T_0}{2}}^{\frac{T_0}{2}} x^*(t)x(t+\tau) \, dt = \frac{1}{T_0} \int_{\frac{-T_0}{2}}^{\frac{T_0}{2}}[\sum_{n=- \infty}^\infty X_n^*e^{-j2\pi nf_0t}][\sum_{m=- \infty}^\infty X_me^{j2\pi mf_0(t+\tau)}] \, dt </math>

<math>= \frac{1}{T_0} \int_{\frac{-T_0}{2}}^{\frac{T_0}{2}} \sum_{n=-\infty}^\infty \sum_{m=- \infty}^\infty X_n^*X_m e^{j2\pi mf_0 \tau}e^{j2\pi (m-n)f_0t} \, dt </math>

<math>= \sum_{n=-\infty}^\infty \sum_{m=- \infty}^\infty X_n^*X_m e^{j2\pi mf_0 \tau} \frac{1}{T_0} \int_{\frac{-T_0}{2}}^{\frac{T_0}{2}} e^{j2\pi (m-n)f_0t} \, dt </math>

<math>= \sum_{n=-\infty}^\infty {|X_n|}^2 e^{j2\pi nf_0 \tau}</math>

<math>= \begin{cases} 1, & \mbox{m=n} \\ o, & other \end{cases}</math>

<math>S_x(f)</math> = <math>\Im</math> {<math>r_x(\tau)</math>} = <math>\sum_{n=-\infty}^\infty {|X_n|}^2 \delta (f-nf_0)</math>

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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