离散时间信号的频域分析¶
DTFT 的性质¶
时域卷积¶
\[ \begin{aligned} x[n] * y[n] \overset{\mathcal{F}}{\longrightarrow} &\sum_{k = -\infty}^{+\infty} x[k] y[n - k] \overset{\mathcal{F}}{\longrightarrow} \\ &\sum_{k = -\infty}^{+\infty} x[k] \left( \frac{1}{2\pi} \int_{2 \pi} Y(e^{j\omega}) e^{j \omega (n - k)} \, \mathrm{d} \omega \right) \\ =& \frac{1}{2\pi} Y(e^{j\omega}) e^{j \omega n} \left( \sum_{k = -\infty}^{+\infty} x[k] e^{-j \omega k} \right) \\ =& X(e^{j\omega}) Y(e^{j\omega}) \end{aligned} \]
调制¶
若 \(x[n] \overset{\mathcal{F}}{\longrightarrow} X(e^{j\omega}), \, y[n] \overset{\mathcal{F}}{\longrightarrow} Y(e^{j\omega})\)，则
\[ \begin{aligned} x[n] y[n] \overset{\mathcal{F}}{\longrightarrow} &\frac{1}{2\pi} \int_{2 \pi} X(e^{j\theta}) Y(e^{j(\omega - \theta)}) \, \mathrm{d} \theta \\ =& \frac{1}{2\pi} X(e^{j\omega}) \circledast Y(e^{j\omega}) \end{aligned} \]
圆卷积的性质
以 \(T\) 为周期的信号 \(x(t), y(t)\)，圆卷积定义为
\[ x(t) \circledast y(t) = \int_T x(\tau) y(t - \tau) \, \mathrm{d} \tau \]
- 圆卷积也是以 \(T\) 为周期的函数
- 简易计算
- 先算 \(x(t) * y(t)\)（正常的卷积）
- 再将结果以 \(T\) 为周期进行延拓并叠加
Parseval 定理¶
\[ \sum_{n = -\infty}^{+\infty} \Big| x[n] \Big|^2 = \frac{1}{2\pi} \int_{2 \pi} \Big| X(e^{j\omega}) \Big|^2 \, \mathrm{d} \omega \]
证明：
\[ \begin{aligned} \sum_{n = -\infty}^{+\infty} \Big| x[n] \Big|^2 &= \sum_{n = -\infty}^{+\infty} x[n] x^*[n] \\ &= \sum_{n = -\infty}^{+\infty} x[n] \left( \frac{1}{2\pi} \int_{2 \pi} X(e^{j\omega}) e^{j \omega n} \, \mathrm{d} \omega \right)^* \\ &= \sum_{n = -\infty}^{+\infty} x[n] \left( \frac{1}{2\pi} \int_{2 \pi} X^*(e^{j\omega}) e^{-j \omega n} \, \mathrm{d} \omega \right) \\ &= \frac{1}{2\pi} \int_{2 \pi} X^*(e^{j\omega}) \left( \sum_{n = -\infty}^{+\infty} x[n] e^{-j \omega n} \right) \,\mathrm{d} \omega \\ &= \frac{1}{2\pi} \int_{2 \pi} X^*(e^{j\omega}) X(e^{j\omega}) \, \mathrm{d} \omega = \frac{1}{2\pi} \int_{2 \pi} \Big| X(e^{j\omega}) \Big|^2 \, \mathrm{d} \omega \end{aligned} \]