LTI Systems Quiz 2

Choose the statements that are always true for the situation described in the question. Don't make extra assumptions.

Question 1: Consider the system $\mathcal{B}$ defined by the state space representation \[ \sigma x = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} x, \quad y = \begin{bmatrix} 1 & 0 \end{bmatrix} x. \] The output $y\in\mathcal{B}$ is

sum of two exponetials.

Incorrect.
$y$ is a constant.

Incorrect.
$y$ is a linear function.

Correct!

Question 2: If $\mathcal{B} = \{\, (u,y) \ | \ y = h \star u \,\}$, then

there are polynomials $p(z)$ and $q(z)$, such that $\mathcal{B} = \{\, (u,y) \ | \ y = Z^{-1} \big(q(z) / p(z)\big) Z(u) \,\}$, where $Z$ is the Z-transform. ✓ ✗

This option is incorrect.

there are matrices $A$, $B$, $C$, $D$, such that $\mathcal{B} = \{\, (u,y) \ | \ \sigma x = Ax + Bu, \ y = Cx + Du \,\}$. ✓ ✗

This option is incorrect.

there is a polynomial matrix $R(z)$, such that $\mathcal{B} = \text{ker}\big(R(\sigma)\big)$. ✓ ✗

This option is incorrect.

Question 3: If $\mathcal{B} = \{\, (u,y) \ | \ y = Z^{-1} \big(q(z) / p(z)\big) Z(u) \,\}$, where $Z$ is the Z-transform, then

there is a sequence $h$, such that $\mathcal{B} = \{\, (u,y) \ | \ y = h \star u \,\}$. ✓ ✗

This option is correct.

there are matrices $A$, $B$, $C$, $D$, such that $\mathcal{B} = \{\, (u,y) \ | \ \sigma x = Ax + Bu, \ y = Cx + Du \,\}$. ✓ ✗

This option is correct.

there is a polynomial matrix $R(z)$, such that $\mathcal{B} = \text{ker}\big(R(\sigma)\big)$. ✓ ✗

This option is correct.

Question 4: The output of a signle-input single-output linear time-invariant system to a nonzero input can not be a zero signal.

True

Incorrect.
False

Correct!

Question 5: The output of a signle-input single-output linear time-invariant system to $\text{sin}(\omega_1t)$ input can not be $\text{sin}(\omega_2t)$, where $\omega_1\neq\omega_2$.

True

Incorrect.
False

Correct!