Question

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).

Answer 1

x(t) = −2 cos (
`(\pi )/(4)t` −
`(\pi )/(2)` ) + 4 cos (
`(5\pi )/(4)t` )

Step-by-step explanation:

Given:

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

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

a₁ =
`a^(*)_(-1)` = j

a₅ =
`a_(-5)` = 2

To find:

Express x(t) in the form

∞

x(t) = ∑ A
`_(k)` cos ( w
`_(k)` t + φ
`_(k)` )

`_(k=0)`

Solution:

Compute fundamental frequency of the signal:

w₀ = 2 π / T

= 2 π / 8 Since T = 8

w₀ = π / 4

∞

∑
`a_(k)e^{jw_(0)t }`

x(t) = k=⁻∞

=
`a_(1)e^{jw_(0) t} + a_(-1)e^{-jw_(0) t} + a_(5)e^{5jw_(0) t}+a_(-5)e^{-5jw_(0) t}`

=
`je^(j(\pi/4)t) - je^(-j(\pi/4)t) +2e^((5\pi/4)t)+2e^(-(5\pi/4) t)`

= −2 sin (
`(\pi )/(4)t` ) + 4 cos (
`(5\pi )/(4)t` )

= −2 cos (
`(\pi )/(4)t` −
`(\pi )/(2)` ) + 4 cos (
`(5\pi )/(4)t` )