A continuous time function, x(t), is sampled at a sampling period of At = 25 microseconds to form a discrete time signal, x[n], which is stored in a buffer that holds 256 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] = 320/j for k = 17 and X[k] = -320/j for k= 239. 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?

black-box Syst G Consider an input signal s4) that is passed thiough d to obtain an oud put signal rt), St) <(t) Buppose the Fourier transtorms of st) and rCt) are reed to each other as: K215 (is2) + R liz) otherwise. Is de an LTI Systemm? Justify your answer,

2) Describe: a) various types of Signals, b) The idea behind Fourier Transform, c) Analog to Digital Conversion, Nyquist Sampling Theorem, d) Errors associated with A to D conversion that are related to Nyquist Sampling Theorem.