已已知 F(w)=sin(wπ)·cos(2w) 是f(t)傅里叶变换,求 f(t)
时间: 2024-05-23 20:11:32 浏览: 148
根据傅里叶变换的定义,有:
$$f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(w)e^{jwt}dw$$
代入 $F(w)$,得到:
$$\begin{aligned} f(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty}\sin(w\pi)\cos(2w)e^{jwt}dw \\ &= \frac{1}{4\pi i}\int_{-\infty}^{\infty}(\sin(w\pi)\cos(2w)e^{jwt}-\sin(-w\pi)\cos(-2w)e^{-jwt})dw \\ &= \frac{1}{4\pi i}\int_{-\infty}^{\infty}(\sin(w\pi)\cos(2w)e^{jwt}+\sin(w\pi)\cos(2w)e^{-jwt})dw \\ &= \frac{1}{2\pi}\int_{-\infty}^{\infty}\sin(w\pi)\cos(2w)\cos(wt)dw \\ &= \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{1}{2}[\sin((3w+t)\pi)+\sin((w-t)\pi)]\cos(2w)dw \end{aligned}$$
根据正弦函数的奇偶性和周期性,上式可化为:
$$f(t) = \frac{1}{4}\left[\delta(t-3)+\delta(t+1)\right]+\frac{1}{4}\left[\delta(t-1)+\delta(t+3)\right]$$
其中 $\delta(t)$ 为单位冲激函数。因此,$f(t)$ 的表达式为:
$$f(t) = \frac{1}{2}\left[\delta(t-3)+\delta(t-1)+\delta(t+1)+\delta(t+3)\right]$$
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