# 隨機程序的時頻分析

## 廣義的穩態

${\displaystyle E[x(t)]=m}$

${\displaystyle R_{xx}(t_{1},t_{2})=R_{xx}(\tau )}$

## 廣義的穩態隨機程序的時頻分析

### 韋格納分布

${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}d\tau }$

${\displaystyle R_{xx}(t_{1},t_{2})=R_{xx}(\tau )}$

${\displaystyle E[W_{x}(t,f)]=\int _{-\infty }^{\infty }E[x(t+\tau /2)x^{*}(t-\tau /2)]e^{-j2\pi \tau \,f}d\tau }$

${\displaystyle =\int _{-\infty }^{\infty }R_{xx}(t+\tau /2,t-\tau /2)e^{-j2\pi \tau \,f}d\tau }$
${\displaystyle =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-j2\pi \tau \,f}d\tau }$
${\displaystyle =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-j2\pi \tau \,f}d\tau }$
${\displaystyle =FT[R_{xx}(\tau )]}$
${\displaystyle =S_{xx}(f)}$

### 模稜函數

${\displaystyle E[A_{x}(\eta ,\tau )]=\int _{-\infty }^{\infty }E[x(t+\tau /2)x^{*}(t-\tau /2)]e^{-j2\pi t\eta }dt}$

${\displaystyle =\int _{-\infty }^{\infty }R_{xx}(t+\tau /2,t-\tau /2)e^{-j2\pi t\eta }dt}$
${\displaystyle =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-j2\pi t\eta }dt}$
${\displaystyle =R_{xx}(\tau )\delta (\eta )}$

## 參考文獻

• Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2012.
• W. Martin, “Time-frequency analysis of random signals”, ICASSP’82, pp. 1325-1328, 1982.