GATE Electronics and Communications (EC) 2016 Shift 2 Solved Paper

An information source generates a binary sequence {𝛼_𝑛}. 𝛼_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 {𝛽_𝑛}, as ${\mathrm{𝛽}}_{\mathrm{𝑛}}={\mathrm{𝛼}}_{\mathrm{𝑛}}+\mathrm{𝑘}{\mathrm{𝛼}}_{\mathrm{𝑛}-3}$ . The sequence {𝛽_𝑛} is used to modulate a pulse $\mathit{X}\left(\mathit{t}\right)$ to generate the baseband signal
$\mathit{X}\left(\mathit{t}\right)=\sum _{\mathit{n}=-\mathit{\infty }}^{\mathit{\infty }}{\mathit{\beta }}_{\mathit{n}}\mathit{g}\left(\mathit{t}-\mathit{n}\mathit{T}\right),$ where $\mathit{g}\left(\mathit{t}\right)=\{\begin{array}{ccc}1& & 0\le \mathit{t}\le \mathit{T}\\ 0& & \mathit{o}\mathit{t}\mathit{h}\mathit{e}\mathit{r}\mathit{v}\mathit{i}\mathit{s}\mathit{e}.\\ \end{array}.$
If there is a null at 𝑓 = $\frac{1}{3\mathit{T}}$ in the power spectral density of $\mathit{X}\left(\mathit{t}\right)$, then 𝑘 is ________