Practice Questions on Communications

 Q1. A message signal $m(t)=\cos 200\pi t+4\cos \pi t$  modulates the carrier $c(t)=\cos 2\pi f_{c} t$ where $f_{c}=$ 1 MHz to produce an $AM$ signal.  For demodulating the generated $AM$ signal using an envelope detector, the time constant $RC$ of the detector circuit should satisfy,
 A. $0.5\text{ms}<\text{RC}<1 \text{ms}$ B. $1\mu s<< \text{RC} <<0.5 \text{ms}$ C. $\text{RC}<<\mu s$ D. $\text{RC}>>0.5\text{ms}$

 Q2. $x(t)$ is a stationary random process with auto-correlation function.$R_{x}(\tau)=e^{\pi r^2}$ This process is passed through the system shown below.  The power spectral density of the output process $y(t)$ is:
 A. $(4\pi^2f^2+1)e^{-\pi f^2}$ B. $(4\pi^2f^2-1)e^{-\pi f^2}$ C. $(4\pi^2f^2+1)e^{-\pi f}$ D. $(4\pi^2f^2-1)e^{-\pi f}$

 Q3. A source generates 4 symbols with probabilities $\dfrac{1}{2},\;\dfrac{1}{4},\;\dfrac{1}{8},\;\dfrac{1}{8}$. The source uses a practical coding scheme has an average code word length of 2 bits/symbol. the efficiency of the code is:
 A. $1$ B. $7/8$ C. $1/2$ D. $1/4$

 Q4. Consider the low pass filter given in the diagram.  A white noise process, $x(t)$ of zero-mean and PSD $\text N_{0}/2$ is applied to the input of the filter. What is the autocorrelation function of the random process $y(t)$ at the output of the filter.
 A. $\dfrac{\text N_{0}R}{L}e^{-(R/L)|\tau|}$ B. $\dfrac{\text N_{0}}{3}\dfrac{R}{L}e^{-(R/L)|\tau|}$ C. $\dfrac{\text N_{0}}{2}\dfrac{R}{L}e^{-(R/L)|\tau|}$ D. $\dfrac{\text N_{0}}{4}\dfrac{R}{L}e^{-(R/L)|\tau|}$

 Q5. A noise signal $n_{i}(t)$ with power spectral density $S(\omega)=\text K$ is applied at input of an ideal differentiator. The power spectral density of the output $n_{0}(t)$ is:
 A. $\text K^2$ B. $\omega \text K^2$ C. $\omega \text K$ D. $\omega^2\text K$

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