Match the LIST-I with LIST-II Choose the correct answer from the options given below: LIST-I LIST-II A. $e^{-at} u(t)$ I. $\frac{\omega_0}{s^2+\omega_0^2}$; $Re[s]>0$ B. $-e^{-at} u(-t)$ II. $\frac{1}{s+a}$; $R_e[s]>-a$ C. $Cos\omega_0 t \ u(t)$ III. $\frac{1}{s+a}$; $R_e[s]<-a$ D. $Sin\omega_0 t \ u(t)$ IV. $\frac{s}{s^2+\omega_0^2}$; $R_e[s]>0$
Laplace Transform Matching: Signals and Systems
This solution explains the matching of time-domain functions (LIST-I) with their corresponding Laplace transforms and Region of Convergence (ROC) (LIST-II).
Matching Logic
-
A. $e^{-at} u(t)$
- This is a standard form for a causal exponential function.
- Its Laplace transform is known to be $\frac{1}{s+a}$.
- The ROC for this causal signal is $Re[s] > -a$.
- This matches item II in LIST-II.
-
B. $-e^{-at} u(-t)$
- This represents an anti-causal exponential function.
- The Laplace transform of $e^{-at} u(-t)$ is $\frac{1}{s+a}$ with ROC $Re[s] < -a$.
- Multiplying by -1 does not change the transform function or its ROC.
- This matches item III in LIST-II.
-
C. $Cos\omega_0 t \ u(t)$
- This is the standard Laplace transform pair for a causal cosine function.
- Its Laplace transform is $\frac{s}{s^2+\omega_0^2}$.
- The ROC for this causal signal is $Re[s] > 0$.
- This matches item IV in LIST-II.
-
D. $Sin\omega_0 t \ u(t)$
- This is the standard Laplace transform pair for a causal sine function.
- Its Laplace transform is $\frac{\omega_0}{s^2+\omega_0^2}$.
- The ROC for this causal signal is $Re[s] > 0$.
- This matches item I in LIST-II.
Summary of Matches
Based on the standard Laplace transforms and ROC properties:
- A matches with II
- B matches with III
- C matches with IV
- D matches with I
Therefore, the correct option is A-II, B-III, C-IV, D-I.
Important Questions from Laplace Transform
Which of the following is the final value of the impulse response of the system whose transfer function is
(2s + 1)/(s 4 + 8s 3 + 16s 2 + s)
Find the Laplace transform for the following time domain.
y(t) = -2te -t + 4e -t - 4e -2t
Match List I with List II
List – I
List – II
f(t)
F(S)
A.
e -at
I.
\(\rm \frac{s}{s^2+ \omega^2}\)
B.
te at
II.
\(\rm \frac{\omega}{s^2+ \omega^2}\)
C.
sinωt
III.
\(\rm \frac{1}{(s- a)^2}\)
D.
cosωt
IV.
\(\rm \frac{1}{(s+ a)}\)
Choose the correct answer from the options given below:
The Laplace transform of sin h (at) is
The unilateral Laplace transform of f(t) is \(\frac{1}{s^2+s+1}\). The unilateral Laplace transform of t f(t) is
