A continuous-time periodic signal x(t) is real valued and has a fundamental period T = 8. The nonzero Fourier series coefficients for x(t) are specified as a_1 = a*_-1 = j, a_5 = a_-5 = 2.
Express x(t) in the form
x(t) = sigma^infinity_k=0 A_k cos(W_k t + phi_k).

x(t) = −2 cos (  − ) + 4 cos ( )

Explanation:

Given:

Fundamental period of real valued continuous-time periodic signal x(t) = T = 8

Non-zero Fourier series coefficients for x(t) :

a₁ = = j

a₅ = = 2  

To find:

Express x(t) in the form

               ∞

x(t) =        ∑    A cos ( w t + φ )

Solution:

Compute fundamental frequency of the signal:

w₀ = 2 π / T

     = 2 π / 8               Since T = 8

w₀  = π / 4

                   ∞

                    ∑

x(t) =          k=⁻∞

     =    

     =  

     = −2 sin ( ) + 4 cos ( )

     = −2 cos (  − ) + 4 cos ( )

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