$1.)(20$ pts$)$: A continuous time function, x$($t$),$ is sampled at a sampling period of $$t $=50$ nanoseconds to form a discrete time signal, x$[$n$],$ which is stored in a buffer that holds $64$ 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$.)$ IftheX$[$k$]=40030$o fork$=19$andX$[$k$]=400-30$o fork$=45.$ Forall other values of k in the range $0$ k $$ N $1,$ X$[$k$]=0.$ What is x$[$n$]?$