Consider two sequences x[n] and h[n], and let y[n] denote their ordinary (linear) convol...

Consider two sequences x[n] and h[n], and let y[n] denote their ordinary (linear) convolution, y[n] = x[n] ∗ h[n]. Assume that x[n] is zero outside the interval 21 ≤ n ≤ 31, and h[n] is zero outside the interval 18 ≤ n ≤ 31.

(a) The signal y[n] will be zero outside of an interval N₁ ≤ n ≤ N₂. Determine numerical values for N₁ and N₂.

(b) Now suppose that we compute the 32-point DFTs of

(i.e., the zero samples at the beginning of each sequence are included). Then, we form the product Y₁[k] = X₁[k]H₁[k]. If we define y₁[n] to be the 32-point inverse DFT of Y₁[k], how is y₁[n] related to the ordinary convolution y[n]? That is, give an equation that expresses y₁[n] in terms of y[n] for 0 ≤ n ≤ 31.

(c) Suppose that you are free to choose the DFT length (N) in part (b) so that the sequences are also zero-padded at their ends. What is the minimum value of N so that y₁[n] = y[n] for 0 ≤ n ≤ N − 1?