Topic: Communication Systems
The autocorrelation of a wide sense stationary random process is given by $e^{-2|\tau|}$. The peak value of the spectral density is
- A. 2
- B. 1
- C. $e^{-1/2}$
- D. e
Answer: B. 1
Explanation: The Autocorrelation function $R_{X}(\tau) = e^{-2|\tau|}$.
The Power Spectral Density (PSD) is the Fourier Transform of the autocorrelation function:
$S_{X}(\omega) = \mathcal{F}\{e^{-2|\tau|}\} = \frac{2(2)}{2^2+\omega²} = \frac{4}{4+\omega^{2}}$.
The peak value of $S_{X}(\omega)$ occurs at $\omega=0$. Thus, $S_{X}(0) = \frac{4}{4} = 1$.