A continuous-time finite-duration signal xc(t) is sampled at a rate of 20,000 samples/s,...

A continuous-time finite-duration signal xc(t) is sampled at a rate of 20,000 samples/s, yielding a 1000-point finite-length sequence x[n] that is nonzero in the interval 0 ≤ n ≤ 999. Assume for this problem that the continuous-time signal is also bandlimited such that = 0 for | | ≥ 2π(10,000); i.e., assume that negligible aliasing distortion occurs in sampling. Assume also that a device or program is available for computing 1000-point DFTs and inverse DFTs.

(a) If X[k] denotes the 1000-point DFT of the sequence x[n], how is X[k] related to ?What is the effective continuous-time frequency spacing between DFT samples? The following procedure is proposed for obtaining an expanded view of the Fourier transform in the interval | | ≤ 2π(5000), starting with the 1000-point DFT X[k].

Step 1. Form the new 1000-point DFT

Step 2. Compute the inverse 1000-pointDFTofW[k], obtainingw[n] for n = 0, 1, . . . , 999.

Step 3. Decimate the sequence w[n] by a factor of 2 and augment the result with 500 consecutive zero samples, obtaining the sequence

Step 4. Compute the 1000-point DFT of y[n], obtaining Y [k].

(b) The designer of this procedure asserts that

where α is a constant of proportionality. Is this assertion correct? If not, explain why not.