Digital & Kalman Filtering by S. M. Bozic

Author:S. M. Bozic
Language: eng
Format: epub, azw3
Publisher: Courier Publishing
Published: 1994-12-15T16:00:00+00:00

which is called the circular convolution, while equation 5.27 is called the linear convolution. Filtering by the circular convolution involves the same four steps as the linear convolution: folding, shifting, multiplying and summing. However, x(k – i) involves time values outside the range 0 ≤ k ≤ N – 1. Therefore, the modulo N indexing mechanism has to be introduced in order to refer to the sequence elements within the periodic range. Since the linear convolution has the length L + M – 1, the DFT has to be of size N ≥ L + M – 1, in order to obtain the correct sequence for y(k).

Example 5.3

For h(k) = {1,2,3} and x(k) = {1,2,2, 1}, equation 5.27 produces the result y(k) = {1, 4, 9, 11, 8, 3}. One can obtain the same result by using DFT of size N = 6, but it is simpler to use 8-point DFT. Then equation 5.18 for T = 1 gives us:

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