Question

A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period Tₛ. In which one of the following cases is the sampled signal periodic?

A. T = √2 Tₛ
B. T = √1.2 Tₛ
C. Always
D. Never

Answer 1

Using rational relationships between sampling intervals, the question determines that the sampled signal would never be periodic when T is an irrational multiple of Tₛ, which aligns with option D.

Step-by-step explanation:

The sampled signal is periodic if the sampling period Tₛ is an integer multiple of the original period T. In other words, if Tₛ = nT, where n is an integer, then the sampled signal will be periodic.

A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period Ts. For the sampled signal to be periodic, there must be a rational relationship between T and Ts, such that T can be expressed as the product of Ts and some integer. In the cases provided:

• T = √2 Ts
• T = √1.2 Ts

Both √2 and √1.2 are irrational numbers and would not result in a periodic sampled signal since it would not satisfy the necessary rational relationship. Thus, the sampled signal is never periodic when T is an irrational multiple of Ts. The correct answer is option D. Never.