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

Consider a real time-limited continuous-time signal x_c(t) whose duration is 100 ms. Assume that this signal has a bandlimited Fourier transform such that = 0 for | | ≥ 2π(10, 000) rad/s; i.e., assume that aliasing is negligible. We want to compute samples of ≤ 2π(10,000). 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 X[k] = αXc(j2π · 5 · k), k = 0, 1, . . . , 1999, where α is a known scale factor. Three methods are proposed to obtain a 4000-point sequence whose DFT gives the desired samples of .

METHOD 1: x_c(t) is sampled with a sampling period T = 25 μs; i.e., we compute X₁[k], the DFT of the sequence

Since x_c(t) is time limited to 100 ms, x₁[n] is a finite-length sequence of length 4000 (100 ms/25 μs).

METHOD 2: x_c(t) is sampled with a sampling period of T = 50 μs. Since x_c(t) is time limited to 100 ms, the resulting sequence will have only 2000 (100 ms/50 μs) nonzero samples; i.e.,

In other words, the sequence is padded with zero-samples to create a 4000-point sequence for which the 4000-point DFT X₂[k] is computed.

METHOD 3: x_c(t) is sampled with a sampling period of T = 50 μs, as in Method 2. The resulting 2000-point sequence is used to form the sequence x₃[n] as follows:

The 4000-point DFT X₃[k] of this sequence is computed.

For each of the three methods, determine how each 4000-point DFT is related to . Indicate this relationship in a sketch for a “typical” Fourier transform .