1. **State the problem:** We want to analyze the function $$S_{xx}(\omega) = \frac{\omega^2 + 9}{\omega^4 + 5\omega^2 + 4}$$ which represents a power spectral density function. 2. **Understand the function:** This is a rational function where the numerator is $$\omega^2 + 9$$ and the denominator is $$\omega^4 + 5\omega^2 + 4$$. 3. **Factor the denominator:** To simplify or analyze, factor the denominator: $$\omega^4 + 5\omega^2 + 4 = (\omega^2 + 4)(\omega^2 + 1)$$ 4. **Rewrite the function:** Now the function is $$S_{xx}(\omega) = \frac{\omega^2 + 9}{(\omega^2 + 4)(\omega^2 + 1)}$$ 5. **Analyze behavior:** Since all terms are sums of squares plus positive constants, the denominator never equals zero for real $$\omega$$, so the function has no real poles. 6. **Limits at infinity:** As $$\omega \to \infty$$, $$S_{xx}(\omega) \approx \frac{\omega^2}{\omega^4} = \frac{1}{\omega^2} \to 0$$ 7. **Evaluate at zero:** At $$\omega = 0$$, $$S_{xx}(0) = \frac{0 + 9}{0 + 0 + 4} = \frac{9}{4} = 2.25$$ 8. **Summary:** The function is positive for all real $$\omega$$, smooth, with a maximum near zero and approaches zero as $$\omega$$ grows large. **Final answer:** $$S_{xx}(\omega) = \frac{\omega^2 + 9}{(\omega^2 + 4)(\omega^2 + 1)}$$ with no real poles and positive values for all real $$\omega$$.