噪声和系统响应

系统响应

  • 关于确定信号的时频分析,可参见频率响应和采样率
  • 确定信号通过线性时不变系统x(t) \to y(t),系统响应为

        \[ y(t) = (x * h)(t). \]

    随机信号通过线性时不变系统X_i(t) \to X_o(t),系统响应为

        \[ X_o(t) = (X_i * h)(t). \]

    这里,h(t)为脉冲信号\delta(t)的系统响应,H(\omega)为相应的频率响应
  • 对于随机信号,我们考虑平稳过程
    • 均值

          \[ \mu_o(t) = \mathbb{E}[X_o(t)] = \int_{-\infty}^{+\infty} \mathbb{E}[X_i(t - \tau)] \cdot h(\tau)d\tau = \mu_i(0)H(0). \]

    • 自相关函数

          \begin{equation*}\begin{split}  R_o(\tau) &= \mathbb{E}[X_o(t + \tau)X_o(t)] \\ &= \mathbb{E}\bigg[\int_{-\infty}^{+\infty} X_i(t + \tau - \tau_1)h(\tau_1)d\tau_1\int_{-\infty}^{+\infty} X_i(t - \tau_2)h(\tau_2)d\tau_2\bigg] \\ &= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \mathbb{E}[X_i(t + \tau - \tau_1)X_i(t - \tau_2)] \cdot h(\tau_1)h(\tau_2)d\tau_1d\tau_2 \\ &= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} R_i(\tau - \tau_1 + \tau_2) \cdot h(\tau_1)h(\tau_2)d\tau_1d\tau_2.  \end{split}\end{equation*}

    • 功率谱密度

          \[ S_o(\omega) = \int_{-\infty}^{+\infty} R_o(\tau)e^{-j\omega\tau}d\tau. \]

      积分的结果为

          \begin{equation*}\begin{split}  &\mathrel{\phantom{=}} \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} R_i(\tau - \tau_1 + \tau_2) \cdot h(\tau_1)h(\tau_2)e^{-j\omega\tau}d\tau d\tau_1d\tau_2 \\ &= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} R_i(\tau') \cdot h(\tau_1)h(\tau_2)e^{-j\omega(\tau' + \tau_1 - \tau_2)}d\tau'd\tau_1d\tau_2. \\ &= \int_{-\infty}^{+\infty} R_i(\tau')e^{-j\omega\tau'}d\tau'\int_{-\infty}^{+\infty} h(\tau_1)e^{-j\omega\tau_1}d\tau_1\int_{-\infty}^{+\infty} h(\tau_2)e^{j\omega\tau_2}d\tau_2 \\ &= S_i(\omega) \cdot |H(\omega)|^2.  \end{split}\end{equation*}

噪声通过系统

  • 白噪声S_i(\omega) = A通过低通滤波器

        \[ H(\omega) = 1_{[-2\pi W, 2\pi W]}, \]

    可得带宽有限的白噪声

        \[ S_o(\omega) = A1_{[-2\pi W, 2\pi W]}. \]

  • 如果噪声的频谱位于|\omega| \leq 2\pi W,那么通过低通滤波器之后,功率谱密度保持不变

        \[ S_o(\omega) = S_i(\omega) \cdot 1_{[-2\pi W, 2\pi W]} = S_i(\omega). \]

    在时域上,与类似于如下的sinc函数卷积之后

        \[ sinc(x) = \frac{sin(x)}{x}, \]

    自相关函数保持不变

        \[ R_o(\tau) = R_i(\tau) * \bigg[\frac{\sin(2\pi W\tau)}{\pi\tau}\bigg] = R_i(\tau). \]

  • Fourier级数的收敛性中,Dirichlet核是sinc函数的离散版本,

        \[ D_N(x) = \frac{\sin[(2N + 1)\pi x]}{\sin(\pi x)}. \]

    Fejér核K_N的频谱位于|\omega| \leq 2\pi N。在时域上,与Dirichlet核卷积之后,Fejér核保持不变

        \[ D_N * K_N = K_N. \]