Topic: Signals and Systems
Consider the system defined by $\frac{d^2y}{dt²} + (a+b)\frac{dy}{dt} + ab = x(t)$; $a>0$, $b>0$. can be realized using which impulse response function.
- A. $h(t)=(e^{-at}+e^{-bt})u(t)$
- B. $h(t)=(e^{-at} \times e^{-bt})u(t)$
- C. $h(t)=(e^{at}+e^{-bt})u(t)$
- D. $h(t)=e^{-(a+b)t}u(t)$
Answer: A. $h(t)=(e^{-at}+e^{-bt})u(t)$
Explanation: Taking the Laplace transform of the given differential equation (as interpreted in the source solution): $s² Y(s) + (a+b)s Y(s) + ab = X(s)$. This rearranges to $Y(s) = \frac{X(s) - ab}{s² + (a+b)s}$. For the impulse response, $X(s) = 1$, giving $H(s) = \frac{1-ab}{s[s+(a+b)]}$. Applying partial fractions yields $h(t) = \frac{1-ab}{a+b}[1 - e^{-(a+b)t}]u(t)$, noting the source solution indicates a discrepancy with the exact provided options.