GATE ECE 2004 | Question: 77

Question

Two sinusoidal signals of same amplitude and frequencies $10 \; \mathrm{kHz}$ and $10.1 \; \mathrm{kHz}$ are added together. The combined signal is given to an ideal frequency detector. The output of the detector is

• A. $0.1 \; \mathrm{kHz}$ sinusoid
• B. $20.1 \; \mathrm{kHz}$ sinusoid
• C. a linear function of time
• D. a constant

Answer 1

$$
\begin{aligned}
x(t)= & \cos (2 \pi \times 10 k t)+\cos (2 \pi \times 10.1 k t) \\
= & \cos (2 \pi \times 10 k t)+\cos [2 \pi(10 k+0.1 k) t] \\
= & \cos (2 \pi \times 10 k t)+\cos (2 \pi \times 10 k t) \cos (200 \pi t) - \sin (2 \pi \times 10 k t) \sin (200 \pi t) \\
= & \cos (2 \pi \times 10 k t)[1+\cos (200 \pi t)]  -\sin (2 \pi \times 10 k t) \sin (200 \pi t) \\
= & \sqrt{[1+\cos (200 \pi t))^{2}+\sin ^{2}(200 \pi t)} \cos [2 \pi \times 10 k t+\phi(t)] \\
\phi(t)= & \tan ^{-1}\left[\frac{\sin (200 \pi t)}{1+\cos (200 \pi t)}\right] \\
A_{c} \cos \left[2 \pi f_{c} t+\phi(t)\right] & \begin{array}{c}  \text { → Frequency Detector }
\end{array} \text { → Output }{ } \propto \frac{d \phi(t)}{d t}
\end{aligned}
$$Output $\propto \frac{d \phi(t)}{d t}=\frac{1}{1+\left[\frac{\sin (200 \pi t)}{1+\cos (200 \pi t)}\right]^{2}} \times\left[\frac{1+\cos (200 \pi t)](200 \pi \cos 200 \pi t)}{[1+\cos (200 \pi t)]^{2}}\right. \left.-\frac{\sin (200 \pi t) [-200 \pi \sin 200 \pi t)}{[1+\cos (200 \pi t)]^{2}}\right]$

\begin{aligned}
& \left.\propto \frac{200 \pi \cos (200 \pi t)+200 \pi}{\left.[1+\cos (200 \pi t)]^{2} +\sin ^{2}(200 \pi t)\right]}\right] \\
& =\frac{\left.200 \pi[1+\cos (200 \pi t)]\right]}{2[1+\cos (200 \pi t)]}
\end{aligned}

Output $\propto 100 \pi$ (constant)