Assume that the input to a perfect second-order PLL is an unmodulated sinusoid plus band limited AWGN. In other words, the PLL input is represented by

X_c(t) = A_c cos(2πf_ct + θ) + n_c(t) cos(2πf_ct + θ) - n_s(t) sin(2πf_ct + θ)

Also assume that the SNR at the loop input is large so that the phase jitter (error) is sufficiently small to justify use of the linear PLL model. Using the linear model, derive an expression for the variance of the loop phase error due to noise in terms of the standard PLL parameters defined in Chapter 4. Show that the probability density function of the phase error is Gaussian and that the variance of the phase error is inversely proportional to the SNR at the loop input.