Consider two sequences x [ n ] and h [ n ], and let y [ n
Question:
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 ? ? 31, and h[n] is zero outside the interval 18 ? ? 31.
(a)?The signal y[n] will be zero outside of an interval N1 ? ? 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 ? ? 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?? 1?
Introduction to Real Analysis
ISBN: 978-0471433316
4th edition
Authors: Robert G. Bartle, Donald R. Sherbert