Question

Consider the following statements

1. The necessary condition for convergence of the Laplace transform is the absolute integrability of f(t) e^-σt

2.  The impulse response of an LTI system that is continuous is H (t) = e^-|t|. The system is Stable and causal.

3.  A series RC circuit excited by voltage V is a memory-less system.

Which of the following statements is/are correct?

1. only 1 
2. All
3. Only 1 and 2
4. Only 2 and 3

Answer 1

Correct Answer - Option 1 : only 1 

The necessary condition for convergence of the Laplace transform is the absolute integrability of f(t)e^-σt. Mathematically, this can be stated as

\(\int_{-\infty}^\infty\) | f(t) e-σt |< ∞

Laplace transform exists only for signals which satisfy the above equation in the given region.

For t < 0,

H (t) ≠ 0

Therefore the system is not causal

Again:

\(\int_{-\infty}^\infty\)|h(t)|dt =1/2 < ∞

∴ The system is stable.

Dynamic systems are those systems that consist of memory.

In the series RC circuit excited by voltage V, the capacitor C is an energy-storing element that acts as a memory for the circuit.

Therefore since the system has the memory it is not a memoryless system.

Also, a causal system depends only on the past and present value.

But since the future value of the charge is also under consideration in this type of circuit, so the system is not causal.

Since charge moves about in the circuit due to the applied voltage V, hence the system is not a static system.

Therefore the system is a dynamic system.