Question

A signal $x\left(t\right)=cos\left(200\pi t\right)+2cos\left(320\pi t\right)$ is ideally sampled at 300Hz. If the sampled signal is passed through an ideal low-pass filter with cut-off frequency of 250 Hz. Highest frequency component will appear in the output is ____ Hz.

• A. 200

Answer 1

The correct option is A 200
The Fourier transform X(f) of the given signal x(t) is

$X\left(f\right)=\frac{1}{2}\left[\delta \right(f-100)+\delta (f+100\left)\right]+\left[\delta \right(f-160)+\delta (f+160\left)\right]$

$\text{With sampling frequency}f=300Hz\text{the Fourier transform of the sampled signal is given as}$

$X_{\delta }\left(f\right)=f_s{\sum }_{n=-\infty }^{\infty }X(f-n{f}_s)=300{\sum }_{n=-\infty }^{\infty }X(f-300n)$

$X_{\delta }\left(f\right)=.....+300X(f+300)+300X\left(f\right)+300X(f-300)+....$

$X_{\delta }\left(f\right)=...+150\delta (f+200)+150\delta (f+400)+300\delta (f+140)+300\delta (f+460)+150\delta (f-100)+150\delta (f+100)+300\delta (f-160)+300\delta (f+160)+150\delta (f-400)+150\delta (f-200)+300\delta (f-460)+300\delta (f-140)+....$

The sampled signal spectrum $X_{\delta }\left(f\right)$ is shown in figure below,



The frequency components that will appear at the output of the low-pass filter are 100, 140, 160 and 200 Hz.

$\therefore$ The highest frequency component present at the output is 200 Hz.