(Solved): 8. Let x(t) be the continuous-time complex exponential signal x(t)=ej0t, with fundamental freq ...

8. Let $x(t)$ be the continuous-time complex exponential signal $x(t)=e^{j{?}_{0}t},$ with fundamental frequency ${?}_0$ and fundamental period $T_0=2?/{?}_0$. Consider the discrete-time signal obtained by taking equally spaced samples of $x(t)$-that is, $x[n]=x(nT)=e^{j{?}_{0}nT}.$ (a) Show that $x[n]$ is periodic if and only if $T/T_0$ is a rational number-that is, if and only if some multiple of the sampling interval exactly equals a multiple of the period of $x(t)$. (b) Suppose that $x[n]$ is periodic-that is, that $\frac{T}{T_0}=\frac{p}{q},$ where $p$ and $q$ are integers. What are the fundamental period and fundamental frequency of $x[n]$ ? Express the fundamental frequency as a fraction of ${?}_{0}T$. (c) Again assuming that $T/T_0$ satisfies Eq. (1), determine precisely how many periods of $x(t)$ are needed to obtain the samples that form a single period of $x[n]$.