List of top Signals and Systems Questions on Digital Signal Processing

A continuous time periodic signal \( x(t) \) is given by: \[ x(t) = 1 + 2\cos(2\pi t) + 2\cos(4\pi t) + 2\cos(6\pi t) \] If \( T \) is the period of \( x(t) \), then evaluate: \[ \frac{1}{T} \int_0^T |x(t)|^2 \, dt \quad {(round off to the nearest integer).} \]

Consider a continuous-time signal \[ x(t) = -t^2 \left\{ u(t+4) - u(t-4) \right\} \] where \( u(t) \) is the continuous-time unit step function. Let \( \delta(t) \) be the continuous-time unit impulse function. The value of \[ \int_{-\infty}^{\infty} x(t)\delta(t+3) \, dt \] is:

Selected data points of the step response of a stable first-order linear time-invariant (LTI) system are given below. The closest value of the time-constant, in sec, of the system is:

Time (sec)0.61.62.610∞
Output0.781.652.182.983

The Nyquist plot of a strictly stable \( G(s) \), having the numerator polynomial as \( (s - 3) \), encircles the critical point \(-1\) once in the anti-clockwise direction. Which one of the following statements on the closed-loop system shown in the figure is correct?

The open-loop transfer function of the system shown in the figure is: \[ G(s) = \frac{K s (s + 2)}{(s + 5)(s + 7)} \] For \( K \geq 0 \), which of the following real axis point(s) is/are on the root locus? 

Let continuous-time signals \( x_1(t) \) and \( x_2(t) \) be defined as:

\[ x_1(t) = \begin{cases} 1, & t \in [0, 1] \\ 2 - t, & t \in [1, 2] \\ 0, & \text{otherwise} \end{cases} \quad \text{and} \quad x_2(t) = \begin{cases} t, & t \in [0, 1] \\ 2 - t, & t \in [1, 2] \\ 0, & \text{otherwise} \end{cases} \]

Consider the convolution \( y(t) = x_1(t) * x_2(t) \). Then

\[ \int_{-\infty}^{\infty} y(t)\,dt =\ ? \]

Let \( G(s) = \frac{1}{(s+1)(s+2)} \). Then the closed-loop system shown in the figure below is:

The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system \( \mathcal{S} \). The output is observed to be the continuous-time unit step signal \( u(t) \). Which one of the following statements is true?
A continuous time periodic signal \( x(t) \) is given by: \[ x(t) = 1 + 2\cos(2\pi t) + 2\cos(4\pi t) + 2\cos(6\pi t) \] If \( T \) is the period of \( x(t) \), then evaluate: \[ \frac{1}{T} \int_0^T |x(t)|^2 \, dt \quad \text{(round off to the nearest integer).} \]