determine autocorrelation of given block signal

Question

How are these autocorrelations determined?

Answer 1

For a periodic discrete zero-mean even signal $u$ of period $T$, the autocorrelation $R_u$ is computed as :

$$R_u(\tau)=\lim_{N \to \infty}\frac{1}{2N+1}\sum_{i=-N}^{N}u(i)u(i-\tau)=\lim_{N \to \infty}\frac{1}{NT}\sum_{i=0}^{NT-1}u(i)u(i-\tau)$$

$$=\lim_{N \to \infty}\frac{1}{NT}\left(N\sum_{i=0}^{T-1}u(i)u(i-\tau)\right)$$ $$=\frac{1}{T}\sum_{i=0}^{T-1}u(i)u(i-\tau)$$

Hence, $R_u(1)=\frac{1+1-1+1+1-1}{6}=\frac{1}{3}$

$u$ looks like a sampling of a rect signal so it is consistent that its autocorrelation looks like a sampling triangle signal which is the autocorrelation of a rect signal.