7. Given a signal as the column vector x = (3.0 0.5 2.0 7.0)". The pyramid algorithm (for Haar wavelets) is as follows: The first two entries (3.0 0.5)T in the signal give an average of (3.0 + 0.5)/2 = 1.75 and a difference average of (3.0 – 0.5)/2 = 1.25. The second two entries (2.0 7.0) give an average of (2.0 + 7.0)/2 = 4.5 and a difference average of (2.0 – 7.0)/2 = -2.5. Thus we end up with a vector %3D (1.75 1.25 4.5 – 2.5)". Now we take the average of 1.75 and 4.5 providing (1.75 +4.5)/2 = 3.125 and the difference average (1.75 – 4.5)/2 = -1.375. Thus we end up with the vector y = (3.125 - 1.375 1.25 – 2.5)". (i) Find a 4 x 4 matrix A such that 3.0 3.125 0.5 -1.375 = Ay = A 2.0 1.25 7.0. -2.5 (ii) Show that the inverse of A exists. Then find the inverse.
7. Given a signal as the column vector x = (3.0 0.5 2.0 7.0)". The pyramid algorithm (for Haar wavelets) is as follows: The first two entries (3.0 0.5)T in the signal give an average of (3.0 + 0.5)/2 = 1.75 and a difference average of (3.0 – 0.5)/2 = 1.25. The second two entries (2.0 7.0) give an average of (2.0 + 7.0)/2 = 4.5 and a difference average of (2.0 – 7.0)/2 = -2.5. Thus we end up with a vector %3D (1.75 1.25 4.5 – 2.5)". Now we take the average of 1.75 and 4.5 providing (1.75 +4.5)/2 = 3.125 and the difference average (1.75 – 4.5)/2 = -1.375. Thus we end up with the vector y = (3.125 - 1.375 1.25 – 2.5)". (i) Find a 4 x 4 matrix A such that 3.0 3.125 0.5 -1.375 = Ay = A 2.0 1.25 7.0. -2.5 (ii) Show that the inverse of A exists. Then find the inverse.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:Given a signal as the column vector
x= (3.0 0.5 2.0 7.0)".
The pyramid algorithm (for Haar wavelets) is as follows: The first two entries
(3.0 0.5)" in the signal give an average of (3.0 +0.5)/2 = 1.75 and a difference
average of (3.0-0.5)/2 = 1.25. The second two entries (2.0 7.0) give an average
of (2.0 + 7.0)/2 = 4.5 and a difference average of (2.0 – 7.0)/2 =-2.5. Thus we
end up with a vector
(1.75 1.25 4.5 – 2.5)".
Now we take the average of 1.75 and 4.5 providing (1.75 + 4.5)/2 = 3.125 and
the difference average (1.75 – 4.5)/2 = -1.375. Thus we end up with the vector
%3D
у 3 (3.125 — 1.375 1.25 — 2.5)T.
(i) Find a 4 x 4 matrix A such that
3.0
3.125
0.5
-1.375
= Ay = A
2.0
1.25
7.0,
-2.5
(ii) Show that the inverse of A exists. Then find the inverse.
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