Are there any signals with brickwall autocorrelation?
It is not possible for a (WSS) random process $\{X(t)\}$to have an autocorrelation function of the form
  $$R_X(\tau) = E[X(t)X(t+\tau)] = \begin{cases}\sigma^2, &amp; |\tau|\leq T,\\0, &amp; |\tau| &gt; T.
\end{cases}$$


Assuming a zero-mean process for simplicity, note that $R_X(0) = E[X^2(t)] = \sigma^2$ where $\sigma^2$ is the common variance of the random variables. Suppose that the highlighted statement above is true and that $T = 2$ and thus $R_X(1) = E[X(t)X(t+1)] = \operatorname{cov}(X(t), X(t+1))$ also has the same value $\sigma^2$. Then, $X(t)$ and $X(t+1)$ are perfectly correlated random variables and thus are equal in the mean-square sense -- their difference has variance $0$:
\begin{align}\operatorname{var}(X(t) - X(t+1)) 
&amp;= \operatorname{var}(X(t))+\operatorname{var}(X(t+1))
- 2\operatorname{cov}(X(t), X(t+1))\\ &amp;= \sigma^2+\sigma^2-2\sigma^2\\
&amp;=0.\end{align} and thus $X(t) = X(t+1)$ in the mean-square sense.
But then, since $t$ is arbitrary, we have that
\begin{align}X(t+1) &amp;= X((t+1) + 1) = X(t+2)\\
X(t+2) &amp;= X((t+2) + 1) = X(t+3).
\end{align} 
So, $X(t) = X(t+1) = X(t+2) = X(t+3)$ so that $R_X(3) = E[X(t)X(t+3)]=\sigma^2$ also, contrary to the hypothesis that 
$R_X(\tau) = 0$ when $|\tau| &gt; 2$.

In summary,
$R_X(\tau)$ cannot have a fixed nonzero value in a finite-length interval including the origin and value $0$ outside this finite-length interval.