List of top Signals and Systems Questions asked in GATE Electrical Engineering

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?

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

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

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

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: \[ \frac{1}{T} \int_0^T |x(t)|^2 \, dt \quad \text{(round off to the nearest integer).} \]

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?

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?

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)	0.6	1.6	2.6	10	∞
Output	0.78	1.65	2.18	2.98	3

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:

Consider the discrete-time systems $ T_1 $ and $ T_2 $ defined as follows:
$ [T_1x][n] = x[0] + x[1] + \dots + x[n], $
$ [T_2x][n] = x[0] + \frac{1}{2}x[1] + \dots + \frac{1}{2^n}x[n]. $
Which of the following statements is true?

If the Z-transform of a finite-duration discrete-time signal $ x[n] $ is $ X(z) $, then the Z-transform of the signal $ y[n] = x[2n] $ is:

Consider a vector $ \Bar{u} = 2\hat{x} + \hat{y} + 2\hat{z} $, where $ \hat{x}, \hat{y}, \hat{z} $ represent unit vectors along the coordinate axes $ x, y, z $ respectively. The directional derivative of the function $ f(x, y, z) = 2 \ln(xy) + \ln(yz) + 3 \ln(xz) $ at the point $ (x, y, z) = (1, 1, 1) $ in the direction of $ \mathbf{u} $ is:

The input $ x(t) $ and the output $ y(t) $ of a system are related as \[ y(t) = e^{-t} \int_{-\infty}^t e^t x(\tau) d\tau, \quad -\inftyLt;tLt;\infty. \] The system is: