List of top Signals and Systems Questions asked in GATE Electronics and Communication Engineering

Consider the discrete-time system below with input \( x[n] \) and output \( y[n] \). In the figure, \( h_1[n] \) and \( h_2[n] \) denote the impulse responses of LTI Subsystems 1 and 2, respectively. Also, \( \delta[n] \) is the unit impulse, and \( b>0 \).
Assuming \( h_2[n] \neq \delta[n] \), the overall system (denoted by the dashed box) is _________.
\includegraphics[width=0.5\linewidth]{17.png}

Consider a part of an electrical network as shown below. Some node voltages, and the current flowing through the \( 3\,\Omega \) resistor are as indicated. 
The voltage (in Volts) at node \( X \) is _________. 


 

Consider a continuous-time, real-valued signal \( f(t) \) whose Fourier transform \[ F(\omega) = \int_{-\infty}^{\infty} f(t) \exp(-j \omega t) \, dt { exists.} \] Which one of the following statements is always TRUE?
For a vector $\bar{x} = [x[0], x[1], ..., x[7]]$, the 8–point DFT is $\bar{X} = DFT(\bar{x}) = [X[0], X[1], ..., X[7]]$ where \[ X[k] = \sum_{n=0}^{7} x[n]\exp\left(-j\,\frac{2\pi}{8}nk\right). \] If $\bar{x} = [1, 0, 0, 0, 2, 0, 0, 0]$ and $\bar{y} = DFT(DFT(\bar{x}))$, the value of $y[0]$ is __________ (rounded off to one decimal place).
The outputs of four systems $(S_1, S_2, S_3, S_4)$ corresponding to the input signal $\sin(t)$, for all time $t$, are shown. Based on the given information, which of the four systems is/are definitely NOT LTI (linear and time-invariant)?
Let $x_1(t)=e^{-t}u(t)$ and $x_2(t)=u(t)-u(t-2)$. If $y(t)$ is the convolution $x_1(t)x_2(t)$, find \[ \lim_{t\to\infty} y(t) \] (rounded to one decimal place).
The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure.
Statement I: The system is necessarily a pure delay system for inputs which are bandlimited to $-\alpha \le f \le \alpha$.
Statement II: For any wide-sense stationary input process with power spectral density $S_X(f)$, the output power spectral density $S_Y(f)$ obeys $S_Y(f) = S_X(f)$ for $-\alpha \le f \le \alpha$.
Which one of the following combinations is true?
The Fourier transform $X(j\omega)$ of the signal \[ x(t)=\frac{t}{(1+t^{2})^{2}} \] is ________________.