Question

A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period T_s. In which one of the following cases is the sampled signal periodic?
1. \(T = \sqrt 2 \;{T_s}\)
2. T = 1.2 T_s
3. Always
4. Never

Answer 1

Correct Answer - Option 2 : T = 1.2 T_s

Consider x(t) = cos (ω₀t)

If x(t) is sampled with a sampling period T_s

x(n) = cos Ω₀n is obtained

\(\frac{{{{\rm{\Omega }}_0}}}{{{T_s}}} = {\omega _0}\)

Here \(\frac{{2\pi \;m}}{{{N_0}\;{T_s}}} = \frac{{2\pi }}{{{T_0}}}\) 

\(\therefore \frac{{{T_0}}}{{{T_s}}} = \frac{{{N_0}}}{m}\)

For the sampled signal to be periodic, ‘m’ should be such that an above expression is a rational number.

Thus \(\frac{T}{{{T_s}}} = \frac{{12}}{{10}} = \frac{6}{5}\)

T = 1.2 T_s