Answer and Step-by-step explanation:
Solution:
(a)
The waveform has even symmetry.
(b) Obtain its cosine/sine Fourier series representation.
Given:
Period = T = 4 s
W0 = 2π/ T = π / 2 rad/s
F(t) = { -5t( -2≤t≤5t 0s≤ t ≤2s )}
a0= 1/ T ∫_(-t/2)^(t/2)f(t)dt
= 1/4 [ ∫_(-2)^0〖-5tdt+ ∫_0^25tdt〗]
= 1/4 [5/2 x 4 + 5/2 x 4]
= 1/4 [20 /2 + 20 / 2]
= 1/4 [10 + 10]
= 1/4 (20)
= 5
An= 2/T ∫_(-t/2)^(t/2)〖f(t)cos(nw0t)dt〗
= 1/2 [∫_(-2)^0〖-5tcos(nπ/2)tdt+ ∫_0^2〖5tcos(nπ/2)tdt〗〗]
= 20 / n2π2 [ cos(nπ) – 1]
Bn = 0
(Even symmetry)
F(t) = 5 + Ʃn=1∞ 20 / n2π2 [ cos(nπ) -1] (cos (nπ/2)t)
