6.3: Exercises

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Exercise 6.1

Suppose the output \(y(t)\) of a system is related to the input \(u(t)\) via the following relation:

\[y(t)=\int_{0}^{\infty} e^{-(t-s)} u(s) d s\nonumber\]

Verify that the model is linear, time-varying, non-causal, and not memoryless.

Exercise 6.2

Suppose the input-output relation of a system is given by

\[y(t)=\left\{\begin{array}{ll}
u(t) & \text { if }|u(t)| \leq 1 \\
\frac{u(t)}{\mid u(t)} \mid & \text { if }|u(t)|>1
\end{array}\right.\nonumber\]

This input-output relation represents a saturation element. Is this map nonlinear? Is it memoryless?

Exercise 6.3

Consider a system modeled as a map from \(u(t)\) to \(y(t)\), and assume you know that when

\[u(t)=\left\{\begin{array}{ll}
1 & \text { for } 1 \leq t \leq 2 \\
0 & \text { otherwise }
\end{array}\right.\nonumber\]

the corresponding output is

\[y(t)=\left\{\begin{array}{ll}
e^{t-1}-e^{t-2} & \text { for } t \leq 1 \\
2-e^{1-t}-e^{t-2} & \text { for } 1 \leq t \leq 2 \\
e^{2-t}-e^{1-t} & \text { for } t \geq 2
\end{array}\right.\nonumber\]

In addition, the system takes the zero input to the zero output. Is the system causal? Is it memoryless?

A particular mapping that is consistent with the above experiment is described by

\[y(t)=\int_{-\infty}^{\infty} e^{-|t-s|} u(s) d s\ \tag{6.24}\]

Is the model linear? Is it time-invariant?

Exercise 6.4

For each of the following maps, determine whether the model is (a) linear, (b) timeinvariant, (c) causal, (d) memoryless.

\[y(t)=\int_{0}^{t}(t-s)^{3} u(s) d s\nonumber\]
\[y(t)=1+ \int_{0}^{t}(t-s)^{3} u(s) d s\nonumber\]
\[y(t)=u^{3}\tag{t}\]
\[y(t)=\int_{0}^{t} e^{-t s} u(s) d s\nonumber\]