Question

The discrete time Fourier coefficient of cos^2(\(\frac{π}{8}\) n) is ______________

(a) \(\frac{π}{2}\)(δ(k+1] + 2δ[k] + δ[k-1])

(b) \(\frac{1}{4}\)j(δ(k+1] + 2δ[k] + δ[k-1])

(c) \(\frac{1}{4}\)(δ(k+1] + 2δ[k] + δ[k-1])

(d) \(\frac{π}{4}\)(δ(k+1] + 2δ[k] + δ[k-1])

The question was asked in homework.

This intriguing question originated from Fourier Series Coefficients topic in division Fourier Series of Signals and Systems

Answer 1

Right choice is (c) \(\frac{1}{4}\)(δ(k+1] + 2δ[k] + δ[k-1])

The best I can explain: N=8, ω = \(\frac{2π}{8} = \frac{π}{4}\)

X[n] = cos^2 (\(\frac{π}{8}\) n) = \(\frac{1}{4}(e^{j(\frac{π}{8})n} + e^{-j(\frac{π}{8})n})^2\)

\(= \frac{1}{4}(e^{j(\frac{π}{8})n} + 2 + e^{-j(\frac{π}{8}n})^2\)

Or, X[k] = \(\frac{1}{4}\)(δ(k+1] + 2δ[k] + δ[k-1]).