1. **Problem Statement:** Find the power spectral density (PSD) of a random process whose autocorrelation function is given by: $$ R(\tau) = \begin{cases} \lambda^2 & |\tau| > c \\ \lambda^2 + \lambda \left(1 - \frac{|\tau|}{c}\right) & |\tau| \leq c \end{cases} $$ 2. **Formula and Explanation:** The power spectral density $S_x(\omega)$ is the Fourier transform of the autocorrelation function $R(\tau)$: $$ S_x(\omega) = \int_{-\infty}^{\infty} R(\tau) e^{-j \omega \tau} d\tau $$ Since $R(\tau)$ is an even function (depends on $|\tau|$), we use the cosine transform: $$ S_x(\omega) = 2 \int_0^{\infty} R(\tau) \cos(\omega \tau) d\tau $$ 3. **Break the integral into two parts:** $$ S_x(\omega) = 2 \left( \int_0^c \left[ \lambda^2 + \lambda \left(1 - \frac{\tau}{c} \right) \right] \cos(\omega \tau) d\tau + \int_c^{\infty} \lambda^2 \cos(\omega \tau) d\tau \right) $$ 4. **Simplify the first integral:** $$ \int_0^c \left[ \lambda^2 + \lambda - \frac{\lambda \tau}{c} \right] \cos(\omega \tau) d\tau = \int_0^c (\lambda^2 + \lambda) \cos(\omega \tau) d\tau - \frac{\lambda}{c} \int_0^c \tau \cos(\omega \tau) d\tau $$ 5. **Evaluate integrals:** - $\int_0^c \cos(\omega \tau) d\tau = \frac{\sin(\omega c)}{\omega}$ - $\int_0^c \tau \cos(\omega \tau) d\tau = \frac{1}{\omega^2} \left( \sin(\omega c) - \omega c \cos(\omega c) \right)$ 6. **Substitute back:** $$ \int_0^c \left[ \lambda^2 + \lambda \left(1 - \frac{\tau}{c} \right) \right] \cos(\omega \tau) d\tau = (\lambda^2 + \lambda) \frac{\sin(\omega c)}{\omega} - \frac{\lambda}{c} \cdot \frac{1}{\omega^2} \left( \sin(\omega c) - \omega c \cos(\omega c) \right) $$ 7. **Evaluate the second integral:** $$ \int_c^{\infty} \lambda^2 \cos(\omega \tau) d\tau $$ This integral does not converge in the usual sense because $\cos(\omega \tau)$ oscillates indefinitely. However, since $R(\tau) \to \lambda^2$ as $|\tau| \to \infty$, the process has a constant offset $\lambda^2$ which corresponds to a delta function at zero frequency in the PSD: $$ 2 \pi \lambda^2 \delta(\omega) $$ 8. **Final expression for PSD:** $$ S_x(\omega) = 2 \left[ (\lambda^2 + \lambda) \frac{\sin(\omega c)}{\omega} - \frac{\lambda}{c \omega^2} \left( \sin(\omega c) - \omega c \cos(\omega c) \right) \right] + 2 \pi \lambda^2 \delta(\omega) $$ This expression gives the power spectral density of the process. **Summary:** The PSD consists of a continuous part given by the Fourier transform of the piecewise autocorrelation minus the constant offset, plus a delta function at zero frequency due to the constant term $\lambda^2$ in $R(\tau)$.