求x(t)=A+Bcos[(2π/T)t+θ]的傅里叶变换

对于给定的函数 $x(t)=A+B\cos\left[\frac{2\pi}{T}t+\theta\right]$，其傅里叶变换为：

$$
\begin{aligned}
X(j\omega) &= \mathcal{F}\{x(t)\} \\
&= \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \\
&= \int_{-\infty}^{\infty} \left(A + B\cos\left[\frac{2\pi}{T}t+\theta\right]\right) e^{-j\omega t} dt \\
&= A\int_{-\infty}^{\infty} e^{-j\omega t} dt + \frac{B}{2}\int_{-\infty}^{\infty} \left(e^{-j\left(\frac{2\pi}{T}t+\theta\right)} + e^{j\left(\frac{2\pi}{T}t+\theta\right)}\right) e^{-j\omega t} dt \\
&= 2\pi A\delta(\omega) + \frac{B}{2}\left(\int_{-\infty}^{\infty} e^{j\left(\frac{2\pi}{T}t+(\theta-\omega)\right)}dt + \int_{-\infty}^{\infty} e^{j\left(\frac{2\pi}{T}t-(\theta+\omega)\right)}dt\right) \\
&= 2\pi A\delta(\omega) + \frac{B}{2}\left[\frac{2\pi}{T}\frac{e^{j(\theta-\omega)\frac{T}{2\pi}}-e^{-j(\theta-\omega)\frac{T}{2\pi}}}{j(\theta-\omega)\frac{T}{2\pi}} + \frac{2\pi}{T}\frac{e^{j(-\theta-\omega)\frac{T}{2\pi}}-e^{-j(-\theta-\omega)\frac{T}{2\pi}}}{j(-\theta-\omega)\frac{T}{2\pi}}\right]\\
&= 2\pi A\delta(\omega) + \frac{B}{T}\left[\frac{\sin\left(\frac{\omega T}{2}+\theta\right)}{\frac{\omega T}{2}+\theta} + \frac{\sin\left(\frac{\omega T}{2}-\theta\right)}{\frac{\omega T}{2}-\theta}\right]
\end{aligned}
$$

其中，$\delta(\omega)$ 表示狄拉克 $\delta$ 函数。