(Solved): 1.5) Sampling a complex exponential [15 points]. Let \( x(t) \) be the CT complex exponential sign ...

1.5) Sampling a complex exponential [15 points]. Let \( x(t) \) be the CT complex exponential signal \( x(t)=e^{j \omega_{0} t} \) with fundamental frequency \( \omega_{0} \) and fundamental period \( T_{0}=2 \pi / \omega_{0} \). Consider the DT signal obtained by taking equally spaced sampled of \( x(t) \), i.e.,\( x[n]=x(n T)=e^{j \omega_{0} n T} \). a) [5 points] 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 \( x(t) \). Homework 1 - Due \( 01 / 18 / 2023 \) \( 1-3 \) b) [5 points] Suppose that \( x[n] \) is periodic so that \( T / T_{0}=p / q \) for integers \( p \) and \( q \). What is the fundamental period of \( x[n] ? \) c) \( [5 \) points \( ] \) Suppose that \( x[n] \) is periodic so that \( T / T_{0}=p / q \) for integers \( p \) and \( q \). Determine how many periods of \( x(t) \) are needed to obtain the samples that form a single period of \( x[n] \).