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Phase Noise in Wireless Communication systems

Phase Noise Noise introduced due to random fluctuations in the phase of carrier signal generated by Osciallator in wireless communication systems is termed as phase noise. An Ideal Oscillator generates a pure sine wave but in practical design, Oscillator generates a multitude of sine waves around the desired center frequency with decaying spectral power and this behavior of Oscillator causes phase noise with uniform density characteristics.

Say $X(t) = e^{j\phi(t)}$ and $Y(t) = e^{-j\phi(t)} e^{-j\theta(t)}$ are incoming signal and locally generated carrier signal respectively, are inputs to mixer in quadrature demoudlation receiver.

$\phi(t)$ be the carrier signal.

$\theta(t)$ is carrier phase error with uniform density in the interval $\left[-\Theta ~\Theta\right]$.

Considering a matched filter operation on $R(t) = e^{-j\theta(t)}$ over one symbol duration $T$. $$R_T = \frac{1}{T} \int_{T} R(t) ~dt = \frac{1}{T} \int_{T} e^{-j\theta(t)} ~dt$$

Mean squared error due to phase fluctuations can be computed by, $$\begin{split} mse\{er\} &= \left\lvert 1 - R_T\right\rvert^{2} = \left\lvert 1 - \frac{1}{T} \int_{T} e^{-j\theta(t)} ~dt\right\rvert^{2}\\ &= 1 + \left[\frac{1}{T} \int_{T} e^{-j\theta(t)} ~dt\right]\left[\frac{1}{T} \int_{T} e^{-j\theta(t)} ~dt\right]^* - \frac{2}{T} \int_{T} \cos\left(\theta(t)\right) ~dt \end{split}$$

It is fair to assume that $\theta(t)$ are i.i.d for all $t$ and as $\theta(t)$ is defined by uniform density $$\begin{split} E_{\Theta}\left[mse\{er\}\right] &= E_{\Theta}\left[1 + \left[\frac{1}{T} \int_{T} e^{-j\theta(t)} ~dt\right]\left[\frac{1}{T} \int_{T} e^{-j\theta(t)} ~dt\right]^* - \frac{2}{T} \int_{T} \cos\left(\theta(t)\right) ~dt\right]\\ &= 1 + E_{\Theta}\left[\left[\frac{1}{T} \int_{T} e^{-j\theta(t)} ~dt\right]\left[\frac{1}{T} \int_{T} e^{-j\theta(t)} ~dt\right]^*\right] - E_{\Theta}\left[\frac{2}{T} \int_{T} \cos\left(\theta(t)\right) ~dt\right] \end{split}$$

applying the i.i.d nature of $\theta(t)$ in the above expression gives rise to, $$\begin{split} E_{\Theta}\left[mse\{er\}\right] &= 1 + E_{\Theta}\left[\frac{1}{T^2} \int_{T} \int_{T} \left\lvert e^{-j\theta(t)} \right\rvert^2 ~dt~dt\right] - E_{\Theta}\left[\frac{2}{T} \int_{T} \cos\left(\theta(t)\right) ~dt\right]\\ &= 1 + E_{\Theta}\left[1\right] - \frac{2}{T} \int_{T} E_{\Theta}\left[\cos\left(\theta(t)\right)\right] ~dt\\ &= 2 - \frac{2}{T} \int_{T} E_{\Theta}\left[\cos\left(\theta(t)\right)\right] ~dt\\ \end{split}$$

Using the density function of $\theta(t)$, $E_{\Theta}\left[\cos\theta\right]$ can be computed as, $$E_{\Theta}\left[\cos\theta\right] = \int_{-\Theta}^{\Theta} \cos\theta . \frac{1}{2\Theta} d\theta = \frac{1}{2\Theta} \sin\theta \Big\rvert _{-\Theta}^{\Theta} = \frac{\sin \Theta}{\Theta} \begin{split} \end{split}$$

Updating $E_{\Theta}\left[mse\{er\}\right]$ with $E_{\Theta}\left[\cos\theta\right]$, $$\begin{split} E_{\Theta}\left[mse\{er\}\right] &=2 - \frac{2}{T} \int_{T} \frac{\sin \Theta}{\Theta} ~dt\\ &= 2 - \frac{2\sin \Theta}{\Theta} \\ \end{split}$$

Below presented figure describe the impact of phase noise on signal power