Suppose that we have a periodic signal x(n) = x(n + N) with period N = 8 defined as follows :

 

This signal can be arranged into a vector X which is defined as :

X = [x(1) x(2) · · · x(8)]^T

In this problem, we would like to perform a decomposition of this periodic signal X using different set of bases q1, q2, · · · , q8 so that X can be written as :

X = a1q1 + a2q2 + · · · + a8q8

Let us consider 3 different set of bases q1, q2, · · · , q8 as described below.

   

1. Determine the value of a1, a2, · · ·, a8 for each set of bases!
2. Based on the result of Fourier decomposition, which frequency components are contained in signal x(n)?
3. If we use these 3 sets of bases to construct a file compression algorithm, which set of bases will produce the smallest file size for this particular signal x(n)?

n x(n) 1 2 2 -1 42 3 -1 2 2 -1 5 6 7 8 -1 NO 2