Question

Let $x\left(t\right)=2cos\left(800\pi t\right)+cos\left(1400\pi t\right).$ x(t) is sampled with the rectangular pulse train shown in the figure. The only spectral components (in kHz) present in the sampled signal in the frequency range 2.5 kHz to 3.5 kHz are

• A. 2.7 , 3.4
• B. 2.7 , 3.3
• C. 3.3 , 3.6
• D. 2.6 , 2.7 , 3.3 , 3.4 , 3.6

Answer 1

The correct option is B 2.7 , 3.3

From Fourier series expansion,
$C_n=\frac{1}{T_o}{\int }_{-T_o/6}^{T_o/6}A.e^{-jn{\omega }_{0}t}dt=\frac{A}{\pi n}sin\left(\frac{n\pi }{3}\right)$

$\therefore$ From $C_n$ its clear that 1, 2, 3, 4, 5, 7.... harmonics are present.

Frequency of p(t) corresponding to 1, 2, 3, 4, 5, 7.... are ${10}^3,2\times {10}^3,4\times {10}^3....$

x(t) has frequency components 0.7 k and 0.4 k

$\therefore p\left(t\right)\times x\left(t\right)gives1\pm 0.7)k,(2\pm 0.7)k,(4\pm 0.7)k,(1\pm 0.4)k,(2\pm 0.4)k....$

$\text{Frequency present in range of 2.5 k to 3.5 k are 2.7 , 3.3.}$