booklet format - inaf iasf bologna

A.A. 2011/2012
Temporal Data Analysis
2.4 Approximating the triangular function — Page 22
The triangular function
has the mean squared “signal”:
1
T
∫ +T /2
−T /2
⎧
⎪⎨
f (t) =
⎪⎩
f 2 (t)dt = 2 T
The most coarse approximation is
1 + 2t
T
1 − 2t
T
∫ +T /2
0
for − T /2 ≤ t ≤ 0
for 0 ≤ t ≤ +T /2
f 2 (t)dt = 2 T
∫ +T /2
0
(
1 + 2t ) 2
dt = 1 T 3
S 0 = 1 2
The next approximation results in
=⇒ δ 2 0 = 1 3 − 1 4 = 1
12 = 0.0833...
S 1 = 1 2 + 4 π 2 cosωt =⇒ δ2 1 = 1 3 − 1 4 − 1 ( ) 4 2
2 π 2 = 0.0012...
For δ 2 3
we get 0.0001915. . . , the approximation of the partial sum to the “triangle” quickly gets
better and better.
M.Orlandini 119