Networks, Signals and Systems · LTI Systems · 2 marks · Numerical answer
A causal LTI system has transfer function $H(s)=1/s$. Its input is $x(t)=[\sin(t)/(\pi t)]u(t)$, where u(t) is the unit step. The output y(t) as $t\to\infty$ is ________.
The causal impulse response corresponding to $1/s$ is $h(t)=u(t)$, so the system integrates its input: $$y(t)=\frac1\pi\int_0^t\frac{\sin\tau}{\tau}\,d\tau.$$ The Dirichlet integral $\int_0^\infty\sin\tau/\tau\,d\tau=\pi/2$ gives the limit.
Therefore $y(\infty)=1/2=$ **0.50**, within the official interval 0.45–0.55. The integrator's zero-frequency pole does not by itself imply that every input produces an unbounded output.
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