Consider a discrete-time WSS random process \( X(n) \) with autocorrelation function given by: \[ R_{x x}(k)=2 e^{-|k|} \] Then find the power spectral density of \( X(n) \).

To find the power spectral density \( S_{XX}(f) \) of the discrete-time wide-sense stationary (WSS) random process \( X(n) \), we can utilize the relationship between the autocorrelation function and the power spectral density through the Fourier transform. Given the autocorrelation function \( R_{XX}(k) = 2 e^{-|k|} \), we can compute the power spectral density by taking the discrete-time Fourier transform (DTFT) of \( R_{XX}(k) \): \[ S_{XX}(f) = \sum_{k=-\infty}^{\infty} R_{XX}(k) e^{-j 2 \pi f k} \] We can compute this sum by recognizing that \( R_{XX}(k) \) is an even function due to \( e^{-|k|} \), which allows us to break the sum into two parts: \[ S_{XX}(f) = \sum_{k=-\infty}^{-1} R_{XX}(k)e^{-j 2 \pi f k} + R_{XX}(0) + \sum_{k=1}^{\infty} R_{XX}(k)e^{-j 2 \pi f k} \] Substituting \( R_{XX}(k) \): \[ = \sum_{k=1}^{\infty} 2 e^{-k} e^{-j 2 \pi f k} + 2 + \sum_{k=1}^{\infty} 2 e^{-k} e^{j 2 \pi f k} \] This simplifies to: \[ = 2 + 2 \sum_{k=1}^{\infty} e^{-k(1+j 2 \pi f)} + 2 \sum_{k=1}^{\infty} e^{-k(1-j 2 \pi f)} \] Now, we recognize that \( \sum_{k=1}^{\infty} r^k = \frac{r}{1 - r} \) for \( |r| < 1 \). Therefore, \[ \sum_{k=1}^{\infty} e^{-k(1 \pm j 2 \pi f)} = \frac{e^{-(1 \pm j 2 \pi f)}}{1 - e^{-(1 \pm j 2 \pi f)}} = \frac{e^{-1}}{e^{-1 \pm j 2 \pi f} - 1} \] Substituting this back and simplifying yields: \[ S_{XX}(f) = 2 + 2 \left( \frac{e^{-1}}{1 - e^{-1 + j 2 \pi f}} + \frac{e^{-1}}{1 - e^{-1 - j 2 \pi f}} \right) \] After further manipulation and simplification, we arrive at the final expression for the power spectral density: \[ S_{XX}(f) = \frac{2 e^{-1}}{1 - e^{-1} e^{-j 2 \pi f}} + \frac{2 e^{-1}}{1 - e^{-1} e^{j 2 \pi f}} = \frac{2 e^{-1}}{1 - 2 e^{-1} \cos(2 \pi f) + e^{-2}} \] Thus, the power spectral density \( S_{XX}(f) \) of the random process \( X(n) \) is given by: \[ S_{XX}(f) = \frac{2 e^{-1}}{1 - 2 e^{-1} \cos(2 \pi f) + e^{-2}} \]