Consider a time-limited continuous-time signal xc(t) whose duration is 100 ms. Assume th...

Consider a time-limited continuous-time signal x_c(t) whose duration is 100 ms. Assume that this signal has a bandlimited Fourier transform such that 2π(10,000) rad/s; i.e., assume that aliasing is negligible. We want to compute samples of . This can be done with a

4000-point DFT. Specifically, we want to obtain a 4000-point sequence x[n] for which the 4000-point DFT is related to by:

where α is a known scale factor. The following method is proposed to obtain a 4000-point sequence whose DFT gives the desired samples of . First, x_c(t) is sampled with a sampling period of T = 50μs. Next, the resulting 2000-point sequence is used to form the sequence x [n] as follows:

Finally, the 4000-point DFT of this sequence is computed. For this method, determine how is related to . Indicate this relationship in a sketch for a “typical” Fourier transform .Explicitly state whether or not ˆX[k] is the desired result, i.e., whether equals X[k] as specified in Eq. (P8.47-1).