Question

Consider a discrete time periodic signal x[n] = sin(πn/s). Let a_k be the complex Fourier series coefficients of x[n]. The coefficients {a_k} are non-zero when k = Bm  ± 1, where m is any integer. The value of B is_________. 

Answer 1

Given x[n] = sin(πn/5); N = 10

⇒ Fourier series co-efficients are also periodic with period N = 10

x[n] = 1/2je^(j(2π/10)n)-1/2je^{(-i(2π/10)n)}

a₁ = 1/2j; a_-1 = - 1/2j  ⇒ a_-1 = a_-1 + 10 = a₉ = -1/2j

(a₁ = a₁ + 10), (a_-1 = a_-1 + 10)} or a₁ = a₁ + 20

a_-1 = a_-1 + 20

⇒ k = 10m + 1 or k = 10m - 1 ⇒ B = 10