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]).