Question

b) [5] Let $h(n) = (\frac{1}{2})^n u(n)$ be the impulse response of a LTI system. i) Show that this system is stable. ii) Show that the LTI system with impulse response $h'(n) = \delta(n) - \frac{1}{2}\delta(n-1)$ is an inverse of the system with impulse response $h[n]$ given above. c) [5] Let $h_1(t)$, $h_2(t)$, $h_3(t)$, and $h_4(t)$ be the impulse responses of LTI systems. Construct a system with impulse response $h(t)$, which uses $h_1(t)$, $h_2(t)$, $h_3(t)$, and $h_4(t)$ as subsystems. Draw the interconnection of systems for the following case: $h(t) = h_1(t) \cdot (h_2(t) + h_3(t) \cdot h_4(t))$ d) [5] Let the impulse response of an LTI system be $h(t) = u(t+1)$. Find the output $y(t)$ if the input is $x(t) = e^{-t}$, all $t$.

          b) [5] Let $h(n) = (\frac{1}{2})^n u(n)$ be the impulse response of a LTI system.

i) Show that this system is stable.
ii) Show that the LTI system with impulse response $h'(n) = \delta(n) - \frac{1}{2}\delta(n-1)$ is an
inverse of the system with impulse response $h[n]$ given above.
c) [5] Let $h_1(t)$, $h_2(t)$, $h_3(t)$, and $h_4(t)$ be the impulse responses of LTI systems. Construct a
system with impulse response $h(t)$, which uses $h_1(t)$, $h_2(t)$, $h_3(t)$, and $h_4(t)$ as
subsystems. Draw the interconnection of systems for the following case:
$h(t) = h_1(t) \cdot (h_2(t) + h_3(t) \cdot h_4(t))$
d) [5] Let the impulse response of an LTI system be $h(t) = u(t+1)$. Find the output $y(t)$ if
the input is $x(t) = e^{-t}$, all $t$.

b) [5] Let h(n) = ((1)/(2))^n u(n) be the impulse response of a LTI system.

i) Show that this system is stable.
ii) Show that the LTI system with impulse response h'(n) = δ(n) - (1)/(2)δ(n-1) is an
inverse of the system with impulse response h[n] given above.
c) [5] Let h1(t), h2(t), h3(t), and h4(t) be the impulse responses of LTI systems. Construct a
system with impulse response h(t), which uses h1(t), h2(t), h3(t), and h4(t) as
subsystems. Draw the interconnection of systems for the following case:
h(t) = h1(t) · (h2(t) + h3(t) · h4(t))
d) [5] Let the impulse response of an LTI system be h(t) = u(t+1). Find the output y(t) if
the input is x(t) = e^-t, all t.

Added by Eva S.

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