Let $x_c(t)$ be any continuous-time periodic signal with period $T$. It is sampled uniformly with a sampling period $T_s$ where $T_s \neq T$, resulting in the discrete sequence $x[n] = x_c(nT_s)$, where $n$ is an integer. Which one of the following statements is correct about $x[n]$?
Signal Periodicity Condition for Sampling
Let $x_c(t)$ be a continuous-time periodic signal with fundamental period $T$. It is sampled with a period $T_s$ to obtain the discrete-time sequence $x[n] = x_c(nT_s)$.
A discrete-time sequence $x[n]$ is periodic if there exists a positive integer $N$ such that $x[n] = x[n+N]$ for all integers $n$.
Applying this condition to the sampled signal:
- $x[n] = x[n+N]$ means $x_c(nT_s) = x_c((n+N)T_s)$.
- Since $x_c(t)$ is periodic with period $T$, the equality $x_c(t_1) = x_c(t_2)$ holds if $t_1 = t_2 + k T$ for some integer $k$.
- Therefore, we must have $(n+N)T_s = nT_s + k T$ for some integer $k$.
- Simplifying the equation gives $N T_s = k T$.
Rearranging this equation, we get:
$ \frac{T}{T_s} = \frac{N}{k} $This shows that for $x[n]$ to be periodic with an integer period $N$, the ratio of the continuous-time period $T$ to the sampling period $T_s$ must be a rational number ($N/k$). Conversely, if $T/T_s$ is a rational number, say $P/Q$, then we can choose $N=P$ and $k=Q$ (or related integers) to satisfy $N T_s = k T$, ensuring the sampled sequence $x[n]$ is periodic.
Thus, the discrete sequence $x[n]$ will be periodic if and only if the ratio $T/T_s$ is a rational number.
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