[系統理論] (弱)平穩隨機過程特性 與 線性非時變系統 (2)- Jointly wide-sense stationary and Frequency domain property

回憶對於 具有脈衝響應為 $h(t)$ 的 LTI 系統 而言，若輸入 $X_t$ 為 WSS 隨機過程，則輸出 $Y_t$ 必定也為一 WSS 隨機過程，且 $Y_t$ 與 $X_t$ 的輸入輸出關係為 convolution integral：
\[{Y_t} = \int_{ - \infty }^\infty  h (t - \tau ){X_\tau }d\tau  = \int_{ - \infty }^\infty  {{X_{t - \tau }}h} (\tau )d\tau
\]且其 輸出隨機過程 $Y_t$ 的 mean function $m_Y(t)$ 與 autocorrelation function $R_Y(\tau)$為
\[\begin{array}{l}
{m_Y}\left( t\right) = E[{Y_t}] = E[\int_{ - \infty }^\infty  {{X_{t - \theta }}h} (\theta )d\theta ]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_{ - \infty }^\infty  {\underbrace {E[{X_{t - \theta }}]}_{ = {m_X}\left( {t - \theta } \right)}h} (\theta )d\theta \\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_{ - \infty }^\infty  {{m_X}\left( {t - \theta } \right)h} (\theta )d\theta
\end{array}\]與
\[\begin{array}{l}
{R_Y}\left( \tau  \right) = E[{Y_{t + \tau }}{Y_t}]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = E\left[ {\int_{ - \infty }^\infty  {{X_{t + \tau  - \beta }}h} (\beta )d\beta \int_{ - \infty }^\infty  {{X_{t - \theta }}h} (\theta )d\theta } \right]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = E\left[ {\int_{ - \infty }^\infty  {\left( {\int_{ - \infty }^\infty  {{X_{t + \tau  - \beta }}h(\beta )} d\beta } \right){X_{t - \theta }}h(\theta )d\theta } } \right]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_{ - \infty }^\infty  {\left( {\int_{ - \infty }^\infty  {\underbrace {E[{X_{t + \tau  - \beta }}{X_{t - \theta }}]}_{ = {R_X}\left( {\left( {t + \tau  - \beta } \right) - \left( {t - \theta } \right)} \right)}} h(\beta )d\beta } \right)h(\theta )d\theta } \\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_{ - \infty }^\infty  {\left( {\int_{ - \infty }^\infty  {{R_X}\left( {\tau  - \left( {\beta  - \theta } \right)} \right)} h(\beta )d\beta } \right)h(\theta )d\theta }  \ \ \ \ \ (*)
\end{array}\]

現在我們定義 jointly WSS 隨機過程
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Definition (Jointly wide-sense stationary, J-WSS)
考慮 $X_t$ 與 $Y_t$ 為兩 WSS 隨機過程，現給定任意兩時刻 $t_1, t_2$， 若其 cross-crorelation $E[X_{t_1}Y_{t_2}]$ 只與 $t_1 - t_2$ 有關，則我們稱 $X_t$ 與 $Y_t$ 為 Jointly wide-sense stationary。
============================

現在回憶 WSS隨機過程之 auto-correlation function 可以用單變數改寫，同樣的，J-WSS 的 $X_t$ 與 $Y_t$ 亦可改寫成單變數 univariate cross-correlation function ；亦即，令 $t_1 := t + \tau$ 且 $t_2 := t$ 則我們有
\[
R_{XY}(t_1 - t_2) = E[X_{t_1} Y_{t_2}] = E[X_{t+ \tau} Y_{t}] = R_{XY}(t+ \tau - t) = R_XY(\tau)
\]亦即 J-WSS 有 univariate cross-correlation function 定義如下：
\[
R_XY(\tau) := E[X_{t+\tau}Y_t]
\]

上述討論我們可知，若一個 WSS 隨機過程 輸入到 LTI 系統，則輸入與輸出為 J-WSS 且其 cross-correlation function 可寫為
\[\begin{array}{*{20}{l}}
{R_{XY}}\left( \tau  \right) = E\left[ {{X_{t + \tau }}{Y_t}} \right] = E\left[ {{X_{t + \tau }}\int_{ - \infty }^\infty  {{X_{t - \gamma }}h} (\gamma )d\gamma } \right]\\
{\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_{ - \infty }^\infty  {\underbrace {E[{X_{t + \tau }}{X_{t - \gamma }}]}_{ = {R_X}\left( {\left( {t + \tau } \right) - \left( {t - \gamma } \right)} \right)}h} (\gamma )d\gamma }\\
{\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = \int_{ - \infty }^\infty  {{R_X}\left( {\tau  + \gamma } \right)h} (\gamma )d\gamma  \ \ \ \ \ \ (\star)}
\end{array}\] 上式已經非常接近 convolution 形式，但並非 convolution 形式；因為上式積分中的 integrand 為 $R_X(t+\gamma)h(\gamma)$ 但我們要的形式是 $R_X(t - \text{something})h(\text{something})$ ，故我們現在用變數變換：令 $\gamma := - \alpha$ 可得
\[ \Rightarrow {R_{XY}}\left( \tau  \right) = \int_{ - \infty }^\infty  {{R_X}\left( {\tau  - \alpha } \right)h} ( - \alpha )d\alpha \]此即具備 convolution 形式

另外觀察上式 $(\star)$ 與之前計算的 $Y_t$ 之auto-correlation function $(*)$ ：
\[{R_Y}\left( \tau  \right) = \int_{ - \infty }^\infty  {\underbrace {\left( {\int_{ - \infty }^\infty  {{R_X}\left( {\tau  - \left( {\beta  - \gamma } \right)} \right)} h(\beta )d\beta } \right)}_{ = {R_{XY}}\left( {\tau  - \beta } \right)}h(\gamma )d\gamma } \]亦即
\[{R_Y}\left( \tau  \right) = \int_{ - \infty }^\infty  {{R_{XY}}\left( {\tau  - \beta } \right)h(\gamma )d\gamma }  \]也就是說 $R_Y$ 為 $h$ 與 $R_{XY}$ 的 convolution。

現在我們將手邊結果總結如下 $(\star \star)$：
\[\left\{ \begin{array}{l}
{R_Y}\left( \tau  \right) = \int_{ - \infty }^\infty  {h(\gamma ){R_{XY}}\left( {\tau  - \beta } \right)d\gamma } \\
{R_{XY}}\left( \tau  \right) = \int_{ - \infty }^\infty  {h( - \alpha ){R_X}\left( {\tau  - \alpha } \right)} d\alpha
\end{array} \right.\]回憶之前我們曾定義對 correlation function 定義 Fourier transform 如下：
\[\begin{array}{l}
{S_X}\left( f \right): = \int_{ - \infty }^\infty  {{R_X}\left( \tau  \right){e^{ - j2\pi f\tau }}d\tau } \\
H\left( f \right): = \int_{ - \infty }^\infty  {h\left( \tau  \right){e^{ - j2\pi f\tau }}d\tau }
\end{array}\]且由 Fourier transform 的 multiplication property：時域convolution = 頻域相乘。故對 $(\star \star ) $ 取 Fourier transform 可得
\[\left\{ \begin{array}{l}
{R_Y}\left( \tau  \right) = \int_{ - \infty }^\infty  {h(\gamma ){R_{XY}}\left( {\tau  - \beta } \right)d\gamma } \\
{R_{XY}}\left( \tau  \right) = \int_{ - \infty }^\infty  {h( - \alpha ){R_X}\left( {\tau  - \alpha } \right)} d\alpha
\end{array} \right.\mathop  \Rightarrow \limits^{F\left\{ . \right\}} \left\{ \begin{array}{l}
{S_Y}\left( f \right) = H(f){S_{XY}}\left( f \right)\\
{S_{XY}}\left( \tau  \right) = {H^*}\left( f \right){S_X}\left( f \right)
\end{array} \right.\] (上述轉換用了一個FACT: $\cal{F}\{ h(-t) \}$ $= H^*(f)$ 我們會在後面給出證明)。

故輸入與輸出的頻域關係為
\[{S_Y}\left( f \right) = H(f){S_{XY}}\left( f \right) = H(f){H^*}\left( f \right){S_X}\left( f \right) = \left| {H\left( f \right)} \right|^2{S_X}\left( f \right)\]亦即
\[
S_Y(f) = |H(f)|^2 S_X(f)
\]

Comments:
1. $S_X(f)$ 稱為 power spectral density。 (nonnegative, real, and even function)
2. $S_{XY}(f)$ 稱為 cross power spectral density。

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FACT:
若 LTI 系統脈衝響應 $h(t)$ 之 Fourier transform 為 $H(f)$，則 $h(-t)$之 Fourier transform 為 $H^*(f)$：
=================
Proof
由Fourier transform 定義
\[H\left( f \right) = \int_{ - \infty }^\infty  {h(\tau ){e^{ - j2\pi f\tau }}d\tau } \]現在對兩邊同取 complex conjugate可得
\[{H^*}\left( f \right) = {\left( {\int_{ - \infty }^\infty  {h(\tau ){e^{ - j2\pi f\tau }}d\tau } } \right)^*} = \int_{ - \infty }^\infty  {{h^*}(\tau ){e^{j2\pi f\tau }}d\tau } \]又因為 LTI 系統脈衝響應 $h(t)$ 為 real function 故 $h^*(t) = h(t)$ 我們可改寫上式
\[{H^*}\left( f \right) = \int_{ - \infty }^\infty  {h(\tau ){e^{j2\pi f\tau }}d\tau } \]做變數變換令 $\theta := - \tau$ 可得
\[{H^*}\left( f \right) = \int_{ - \infty }^\infty  {h(\tau ){e^{j2\pi f\tau }}d\tau }  = \int_{ - \infty }^\infty  {h( - \theta ){e^{ - j2\pi f\theta }}d\theta }  = H\left( { - f} \right)\]亦即 $h(-t)$之 Fourier transform 為 $H(-f) = H^*(f)$。 $\square$。