Let h(t) denote the impulse response of a causal system with transfer function $${1 \over {s + 1}}.$$ Consider the following three statements: S1 : The system is stable. S2 : $${{h\left( {t + 1} \right)} \over {h\left( t \right)}}$$ independent of t for t > 0. S3 : A non-causal system with the same transfer function is stable. For the above system,

The correct answer is A. Only S1 and S2 are true.

S1 is true because the system is stable if and only if the poles of the transfer function lie within the unit circle. In this case, the pole of the transfer function is at $s=-1$, which lies within the unit circle.

S2 is true because the impulse response of a causal system is zero for negative time. In this case, the impulse response is given by

$$h(t) = \begin{cases} 1 & t \ge 0 \\ 0 & t < 0 \end{cases}$$

Therefore,

$$\frac{h(t+1)}{h(t)} = \begin{cases} 1 & t+1 \ge 0 \\ 0 & t+1 < 0 \\ = 1 & t \ge 0 \\ = 0 & t < 0 \end{cases}$$

which is independent of $t$ for $t>0$.

S3 is false because a non-causal system is not stable if it has poles outside the unit circle. In this case, the transfer function has a pole at $s=-1$, which lies outside the unit circle. Therefore, the non-causal system is not stable.