$20$ pts$)$: A continuous time function, x$($t$),$ is sampled at a sampling period of

$$t $=0\mathrm{.}2$ milliseconds to form a discrete time signal, x$[$n$],$ which is stored in a

buffer that holds $128$ samples.

a$.)$ If a discrete Fourier transform $($DFT$)$ is taken of the samples in the buffer to

estimate the spectrum with the vector, X$[$k$],$ what is the frequency in Hz repre$-$

sented by the spacing between consecutive frequency samples?

b$.)$ If

X$[$k$]=38\mathrm{.}4$e$$j$37$o

for k $=15$

X$[$k$]=38\mathrm{.}4$e$+$j$37$o

for k $=113$

and for all other values of k in the range: $0$ k $$ N $1$

X$[$k$]=0$

what is x$[$n$]?$

c$)$ Assuming the original time signal, x$($t$),$ was oversampled to get x$[$n$],$ what was

x$($t$)$ before it was sampled?