GATE ECE · Communication Systems

Random Signals And Noise Practice Questions

Generate GATE-level questions on Random Processes and Noise. Focus on: 1. Random Processes: Autocorrelation, Power Spectral Density, and WSS properties. 2. Filtered Random Processes: Response of LTI systems to random inputs. 3. White Noise and Gaussian Noise characteristics. 4. Random Variables: PDFs, Expectations, and Variance calculations.

73 questions · 20 PYQs · 0 AI practice · GATE ECE 2027

Q21GATE 2016NAT

Consider a random process X(t) = 3V(t) - 8, where V(t) is a zero mean stationary random process with autocorrelation $R_{v}(\tau )=4e^{-5|\tau |}$ . The power in X(t) is ________

🚀 Solve in Practice Mode

📖 Explanation

\begin{aligned} X(t) &=3 V(t)-8 \quad \text { and } \quad E[V(t)]=0 \\ R_{v}(\tau) &=4 e^{-5|\tau|} \\ \text { Power of } X(t) &=E\left[X^{2}(t)\right]\$ \\ &=E\left[9 V^{2}(t)\right]\$+64-48 E[V(t)] \\ &=9 E\left[V^{2}(t)\right]\$+64-48 E[V(t)] \\ E\left[V^{2}(t)\right]\$ &=R_{v}(0)=4 \\ \text { Power of } X(t) &=((9 \times 4)+64) \\ &=100 \end{aligned}

Q22GATE 2016MCQ

An antenna pointing in a certain direction has a noise temperature of 50 K. The ambient temperature is 290 K. The antenna is connected to a pre-amplifier that has a noise figure of 2 dB and an available gain of 40 dB over an effective bandwidth of 12 MHz. The effective input noise temperature $T_{e}$ for the amplifier and the noise power $P_{a0}$ at the output of the preamplifier, respectively, are

🚀 Solve in Practice Mode

📖 Explanation

\begin{aligned} (i) \quad T_{e}&=(F-1) T_{0}=\left(10^{2 / 10}-1\right) 290 \\ &=169.6 \mathrm{K} \\ (ii) N_{t}&=k\left(T_{a x}+T_{e}\right) B \\ &=1.38 \times 10^{-23} \times (50+169.6) \times 12 \times 10^{6} \\ &=3.63 \times 10^{-14} \mathrm{W} \\ N_{0} &=N_{i} \times \text { Gain } \\ &=3.63 \times 10^{-14} \times 10^{4} \\ &=3.63 \times 10^{-10} \mathrm{W} \end{aligned}

Q23GATE 2016NAT

An information source generates a binary sequence $\{\alpha_{n} \}. \; \alpha_{n}$ can take one of the two possible values -1 and +1 with equal probability and are statistically independent and identically distributed. This sequence is precoded to obtain another sequence $\{\beta_{n}, \; as \; \beta _{n}=\alpha_{n}+k \alpha_{n-3}\}$ . The sequence { $\beta _{n}$ } is used to modulate a pulse g(t) to generate the baseband signal $X(t)=\sum_{n=-\infty }^{\infty }\beta _{n}g(t-nT),$ where

g(t)=\left\{\begin{matrix} 1, &0\leq t\leq T \\ 0,&otherwise. \end{matrix}\right.

If there is a null at $f=\frac{1}{3T}$ in the power spectral density of X(t), then k is ________

🚀 Solve in Practice Mode

📖 Explanation

Power spectral density of $x(t)=S_{X}(t)$

\begin{aligned} &S_{X}(f)=\frac{|G(f)|^{2}}{T} \sum_{n=-\infty }^{\infty } R_{b}(\tau) e^{j 2 \pi fn \tau} \\ &R_{b}(\tau)=E\left[\beta_{n} \beta_{n-\tau}\right] \\ &=E\left[\left(\alpha_{n}+k \alpha_{n-3}\right)\left(\alpha_{n-\tau}+k \alpha_{n-\tau-3}\right)\right] \\ &=E\left[\alpha_{n} \alpha_{n-\tau}\right]+k E\left[\alpha_{n-3} \alpha_{n-7}\right]+k E\left[\alpha_{n} \alpha_{n-\tau-3}\right] \\ &+k^{2} E\left[\alpha_{n-3} \alpha_{n--3}\right] \\ &=E\left[\alpha_{n} \alpha_{n-\tau}\right]+k E\left[\alpha_{n-3} \alpha_{n-\tau-3+3}\right]+k E\left[\alpha_{n} \alpha_{n-\tau-3}\right] \\ &\quad +k^{2} E\left[\alpha_{n-3} \alpha_{n-\tau-3}\right] \\ &= A(\tau)+k R(\tau-3)+k R(\tau+3)+k^{2} R(\tau) \\ &=\left(1+k^{2}\right) R(\tau)+k R(\tau+3)+k R(\tau-3) & \end{aligned}

Hence autocorrection function can be defined

R_{b}(\tau)=\left\{\begin{array}{cc} 1+k^{2} & \tau=0 \\ k & \tau=\pm 3 \\ 0 & \text { otherwise } \end{array}\right\}

Power spectral density

\begin{aligned} s_{b}(f)&=1+k^{2}+2 k \cos (2 \pi f 3 T)\\ \text{Null will occur at }f&=\frac{1}{3 T} \\ f&=\frac{1}{3 T}\\ s_{b}(f)&=1+k^{2}+2 k \cos (2 \pi)=0\\ \Rightarrow\quad 1+k^{2}+2 k&=0 \\ \Rightarrow\quad (k+1)^{2}&=0 \\ \therefore \quad k&=-1 \end{aligned}

Q26GATE 2015MCQ

$\left \{ X_{n} \right \}_{n=-\infty }^{n=\infty }$ is an independent and identically distributed (i.i.d.) random process with $X_{n}$ equally likely to be +1 or -1. $\left \{ Y_{n} \right \}_{n=-\infty }^{n=\infty }$ is another random process obtained as $Y_{n}=X_{n}+0.5 X_{n-1}.$ The autocorrelation function of $\left \{ Y_{n} \right \}_{n=-\infty }^{n=\infty }$ , denoted by $R_{Y}[k]$ ,is