PYQs Based test 6: Signals & Systems

The output of a continuous-time, linear time-invariant system is denoted by $T\left\{x\left(t\right)\right\}$ where $x\left(t\right)$ is the input signal. A signal $z\left(t\right)$ is called eigen-signal of the system , when $T\left\{z\left(t\right)\right\}=\gamma z\left(t\right)$, where $\gamma$ is a complex number, in general, and is called an eigenvalue of $T$. Suppose the impulse response of the system $T$ is real and even. Which of the following statements is TRUE?

Consider a discrete time signal given by

$x\left[n\right]={\left(-0.25\right)}^{n}u\left[n\right]+{\left(0.5\right)}^{n}u\left[-n-1\right]$

The region of convergence of its Z – transform would be

The signal energy of the continuous-time signal

X(t) = [(t – 1) u(t - 1)] – [(t - 2)u(t - 2)] – [(t - 3)u(t – 3)] + [(t - 4)]u(t – 4)] is

Let $X\left(z\right)=\frac{1}{1-z^{-3}}$ be the z-transform of a causal signal x[n].

Then, the values of x[2] and x[3] are

The function shown in the figure can be represented as