**Appendix B. Expected Value and Variance of the Trigonometric Function of a Variable with a Normal Distribution**

Suppose *<sup>θ</sup>* <sup>∼</sup> *<sup>N</sup>* - 0, *σ*<sup>2</sup> . Then, consider another variable,

$$e^{j\theta} = \cos\theta + j\sin\theta\tag{A14}$$

where *j* is the imaginary unit. In the meantime, the moment-generating function of *θ* with a normal distribution is

*Remote Sens.* **2019**, *11*, 1663

$$\mathbb{E}\left[e^{j\theta}\right] = e^{0 + \left(j\sigma\right)^2/2} = e^{-\sigma^2/2} \tag{A15}$$

which means

$$\mathbb{E}\left[\cos\theta\right] = e^{-v^2/2} \tag{A16}$$

$$\mathbb{E}\left[\sin\theta\right] = 0\tag{A17}$$

The variance of cos *θ* and sin *θ* is

$$\begin{aligned} \text{Var}\left[\cos\theta\right] &= \mathbb{E}\left[\cos^2\theta\right] - \mathbb{E}\left[\cos\theta\right]^2 \\ &= \frac{1}{2}\left(1 + \mathbb{E}\left[\cos 2\theta\right]\right) - \left(e^{-\sigma^2/2}\right)^2 \\ &= \frac{1}{2}\left(1 - e^{-\sigma^2}\right)^2 \\ \text{Var}\left[\sin\theta\right] &= \mathbb{E}\left[\sin^2\theta\right] - \mathbb{E}\left[\sin\theta\right]^2 \end{aligned} \tag{A18}$$

$$\begin{aligned} \text{Var}\left[\sin\theta\right] &= \mathbb{E}\left[\sin^2\theta\right] - \mathbb{E}\left[\sin\theta\right]^2\\ &= \frac{1}{2}\mathbb{E}\left[1 - \cos 2\theta\right] \\ &= \frac{1}{2}\left(1 - e^{-2\sigma^2}\right) \end{aligned} \tag{A19}$$

For a new variable *<sup>z</sup>* <sup>∼</sup> *<sup>N</sup>* - *μ*, *σ*<sup>2</sup> , *θ* = *z* − *μ*,

$$\begin{aligned} \mathbb{E}\left[\cos z\right] &= \mathbb{E}\left[\cos \left(\theta + \mu\right)\right] \\ &= \mathbb{E}\left[\cos \theta \cos \mu - \sin \theta \sin \mu\right] \\ &= \cos \mu \to \left[\cos \theta\right] - \sin \mu \to \left[\sin \theta\right] \\ &= e^{-\sigma^2/2} \cos \mu \\ \mathbb{E}\left[\sin z\right] &= \mathbb{E}\left[\sin \left(\theta + \mu\right)\right] \\ &= \mathbb{E}\left[\sin \theta \cos \mu + \cos \theta \sin \mu\right] \\ &= \cos \mu \to \left[\sin \theta\right] + \sin \mu \to \left[\cos \theta\right] \\ &= e^{-\sigma^2/2} \sin \mu \end{aligned} \tag{A21}$$

The variance of cos *z* and sin *z* is

$$\begin{aligned} \text{Var}\left[\cos z\right] &= \mathbb{E}\left[\cos^2 z\right] - \mathbb{E}\left[\cos z\right]^2 \\ &= \frac{1}{2} + \frac{1}{2} \mathbb{E}\left[\cos\left(2\theta + 2\mu\right)\right] - \mathbb{E}\left[\cos z\right]^2 \\ &= \frac{1}{2} + \frac{1}{2}e^{-2r^2}\cos 2\mu - e^{-r^2}\cos^2 \mu \\ \text{Var}\left[\sin z\right] &= \mathbb{E}\left[\sin^2 z\right] - \mathbb{E}\left[\sin z\right]^2 \\ &= \frac{1}{2} - \frac{1}{2}\mathbb{E}\left[\cos\left(2\theta + 2\mu\right)\right] - \mathbb{E}\left[\sin z\right]^2 \\ &= \frac{1}{2} - \frac{1}{2}e^{-2r^2}\cos 2\mu - e^{-r^2}\sin^2 \mu \end{aligned} \tag{A23}$$
