Determine if the waveform has dc, even, or odd symmetry. (b) Obtain its cosine/sine Fourier series representation.

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)

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