GATE EC 2025 Question Paper (15-Feb-2025) (Shift 2)

A source transmits symbol S that takes values uniformly at random from the set $\{-2,0,2\}$. The receiver obtains $Y=S+N$, where N is a zero-mean Gaussian random variable independent of S. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol S.
Suppose the probability of symbol estimation error $P_e$ is expressed as follows:
$P_e=\alpha P(N>1)$,

where $P(N>1)$ denotes the probability that N exceeds1. What is the value of $\alpha$?

Consider a real-valued random process

$f(t)={\sum }_{n=1}^N{a}_{n}p(t\text{–}nT)$,

where $T>0$ and N is a positive integer. Here, $p(t)=1$ for $t\in [0,0.5T]$ and 0 otherwise. The coefficients $a_n$ are pairwise independent, zero-mean unit-variance random variables.
Read the following statements about the random process and choose the correct option.
(i) The mean of the process $f(t)$ is independent of time $t$.
(ii) The autocorrelation function $E[f(t)f(t+\tau )]$ is independent of time $t$ for all $\tau$.
(Here, $E[\cdot ]$ is the expectation operation.)

The identical MOSFETs $M_1$ and $M_2$ in the circuit given below are ideal and biased in the saturation region. $M_1$ and $M_2$ have a transconductance $g_m$ of 5 mS. The input signals (in Volts) are:
$V_1=2.5+0.01\sin \omega t$
$V_2=2.5\text{–}0.01\sin \omega t$
The output signal $V_3$ (in Volts) is ________.

A 10-bit analog-to-digital converter (ADC) has a sampling frequency of 1 MHz and a full scale voltage of 3.3 V. For an input sinusoidal signal with frequency 500 kHz, the maximum SNR (in dB, rounded off to two decimal places) and the data rate (in Mbps) at the output of the ADC are
________, respectively.

A positive-edge-triggered sequential circuit is shown below. There are no timing violations in the circuit. Input $P0$ is set to logic '0' and $P1$ is set to logic '1' at all times. The timing diagram of the inputs $SEL$ and $S$ are also shown below. The sequence of output $Y$ from time $T_0$ to $T_3$ is __________.