$1.)(20$ pts$)$: A continuous time function, x$($t$),$ is sampled at a sampling period of

$$t $=3\mathrm{.}125$ milliseconds to form a discrete time signal, x$[$n$],$ which is stored in a

buffer that holds $1024$ 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 the X$[$k$]=25606-123$o

for k $=128$ and X$[$k$]=25606123$o

for k $=896.$ For

all other values of k in the range $0$ k $$ N $1,$ X$[$k$]=0.$ What is x$[$n$]?$