周期信号傅里叶级数公式

最新推荐文章于 2025-09-24 20:06:39 发布

翻译最新推荐文章于 2025-09-24 20:06:39 发布 · 5.8k 阅读

信号处理专栏收录该内容

10 篇文章

订阅专栏

连续周期信号

x (t) = \sum - \infty + \infty a k e j k w 0 t = \sum - \infty + \infty a k e j k (2 π / T) t a k = 1 T \int T x (t) e - j k w 0 t d t = 1 T \int T x (t) e - j k (2 π / T) t d t

$x(t) = \sum_{-\infty}^{+\infty}a_ke^{jkw_0t}= \sum_{-\infty}^{+\infty}a_ke^{jk(2π/T)t} \\ a_k = \frac{1}{T} \int_{T}x(t)e^{-jkw_0t}dt =\frac{1}{T} \int_{T}x(t)e^{-jk(2π/T)t}dt$

离散周期信号

x (n) = \sum k = (N) a k e j k w 0 n = \sum k = (N) a k e j k (2 π / N) n a k = 1 N \sum n = (N) x [n] e - j k w 0 n = 1 N \sum n = (N) x [n] e - j k (2 π / N) n

$x(n) = \sum_{k=(N)}a_ke^{jkw_0n}= \sum_{k=(N)}a_ke^{jk(2π/N)n} \\ a_k = \frac{1}{N} \sum_{n=(N)}x[n]e^{-jkw_0n}= \frac{1}{N} \sum_{n=(N)}x[n]e^{-jk(2π/N)n}$