An information source generates a binary sequence $\left \{ \alpha _{n} \right \}$. $\alpha _{n}$ can take one of the two possible values $-1$ and $+1$ with equal probability and are statistically independent and identically distributed. This sequence is precoded to obtain another sequence $\left \{ \beta _{n} \right \}$, as $\beta _{n}=\alpha _{n}+k\alpha _{n-3}.$
The sequence $\left \{ \beta _{n} \right \}$ is used to modulate a pulse $g(t)$ to generate the baseband signal
$X(t)=\displaystyle{} \sum_{n=-\infty}^{\infty} \beta _{n}g(t-nT), \text{ where }g(t)=\begin{cases} 1, &0\leq t\leq T \\ 0,& \text{ otherwise. }\end{cases}$
If there is a null at $f=\frac{1}{3T}$ in the power spectral density of $X(t)$, then $k$ is ________