List of top Communications Questions asked in GATE EC

For the Boolean function: \[ F(A, B, C, D) = \Sigma m(0, 2, 5, 7, 8, 10, 12, 13, 14, 15), \] the essential prime implicants are:

In a number system of base \(r\), the equation \(x^2 - 12x + 37 = 0\) has \(x = 8\) as one of its solutions. The value of \(r\) is \(\_\_\_\_\).

In the network shown below, maximum power is to be transferred to the load \(R_L\). The value of \(R_L\) (in \(\Omega\)) is \(\_\_\_\_\). \begin{center} \includegraphics[width=8cm]{62.png} \end{center}

Let \(X(t) = A \cos(2\pi f_0 t + \theta)\) be a random process, where amplitude \(A\) and phase \(\theta\) are independent of each other, and are uniformly distributed in the intervals \([-2, 2]\) and \([0, 2\pi]\), respectively. \(X(t)\) is fed to an 8-bit uniform mid-rise type quantizer. Given that the autocorrelation of \(X(t)\) is: \[ R_X(\tau) = \frac{2}{3} \cos(2\pi f_0 \tau), \] the signal-to-quantization noise ratio (in dB, rounded off to two decimal places) at the output of the quantizer is \(\_\_\_\_\).

In the circuit below, the opamp is ideal. If the circuit is to show sustained oscillations, the respective values of \(R_1\) and the corresponding frequency of oscillation are \(\_\_\_\_\). \begin{center} \includegraphics[width=8cm]{50.png} \end{center}

Consider a system \(S\) represented in state space as: \[ \frac{dx}{dt} = \begin{bmatrix} 0 & -2
1 & -3 \end{bmatrix} x + \begin{bmatrix} 1
0 \end{bmatrix} r, \quad y = \begin{bmatrix} 2 & -5 \end{bmatrix} x. \] Which of the state space representations given below has/have the same transfer function as that of \(S\)?

Consider two continuous-time signals \(x(t)\) and \(y(t)\) as shown below. If \(X(f)\) denotes the Fourier transform of \(x(t)\), then the Fourier transform of \(y(t)\) is \(\_\_\_\_\). \begin{center} \includegraphics[width=8cm]{44.png} \end{center}

A full-scale sinusoidal signal is applied to a 10-bit ADC. The fundamental signal component in the ADC output has a normalized power of 1 W, and the total noise and distortion normalized power is \(10 \, \mu {W}\). The effective number of bits (rounded off to the nearest integer) of the ADC is \(\_\_\_\_\).

The propagation delay of the \(2 \times 1\) MUX shown in the circuit is \(10 \, {ns}\). Consider the propagation delay of the inverter as \(0 \, {ns}\). If \(S\) is set to 1, then the output \(Y\) is \(\_\_\_\_\). \begin{center} \includegraphics[width=4cm]{41.png} \end{center}

Suppose \(X\) and \(Y\) are independent and identically distributed random variables that are distributed uniformly in the interval \([0,1]\). The probability that \(X \geq Y\) is \(\_\_\_\_\).

An amplitude modulator has output (in Volts): \[ s(t) = A \cos(400\pi t) + B \cos(360\pi t) + B \cos(440\pi t). \] The carrier power normalized to \(1 \, \Omega\) resistance is 50 Watts. The ratio of the total sideband power to the total power is \(1/9\). The value of \(B\) (in Volts, rounded off to two decimal places) is \(\_\_\_\_\).

For a causal discrete-time LTI system with transfer function: \[ H(z) = \frac{2z^2 + 3}{(z + \frac{1}{3})(z - \frac{1}{3})}, \] which of the following statements is/are true?

A causal and stable LTI system with impulse response \(h(t)\) produces an output \(y(t)\) for an input signal \(x(t)\). A signal \(x(0.5t)\) is applied to another causal and stable LTI system with impulse response \(h(0.5t)\). The resulting output is:

In the circuit below, assume that the long channel NMOS transistor is biased in saturation. The small signal transconductance of the transistor is \(g_m\). Neglect body effect, channel length modulation, and intrinsic device capacitances. The small signal input impedance \(Z_{in}(j\omega)\) is: \begin{center} \includegraphics[width=6cm]{17.png} \end{center}

In the context of Bode magnitude plots, \(40 \, {dB/decade}\) is the same as:

A digital transmission system uses a (7,4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code \( (m_i ; c_i) \), where \( c_i \) is the codeword corresponding to the \( i^{th} \) message \( m_i \), are known to be \[ (1 1 0 0 ; 0 1 0 1 1 0 0), \quad (0 0 1 1 1 1 0 ; 0 1 1 1 0 ; 1 0 0 0 1 1 0), \] then which of the following is a valid codeword in this code?