GATE Electrical Engineering (EE) 2023 Solved Paper

The discrete-time Fourier transform of a signal 𝑥[𝑛] is 𝑋(Ω) = (1 + 𝑐𝑜𝑠Ω)𝑒^−𝑗Ω. Consider that 𝑥_𝑝[𝑛] is a periodic signal of period N = 5 such that
𝑥_𝑝 [𝑛] = 𝑥[𝑛], for 𝑛 = 0, 1 ,2
= 0, for 𝑛 = 3, 4
Note that ${\mathit{x}}_{\mathit{p}}\left[\mathit{n}\right]={\mathit{\Sigma }}_{\mathit{k}=0}^{\mathit{N}-1}{\mathit{a}}_{\mathit{k}}{\mathit{e}}^{\mathit{j}\frac{2\mathit{\pi }}{\mathit{N}}\mathit{k}\mathit{n}}$. The magnitude of the Fourier series coefficient 𝑎₃ is _______________ (Round off to 3 decimal places).