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GATE EC 2016 Set 2 — Question 58

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NAT+2 / -0HardAutocorrelation & PSDRandom ProcessesCommunicationsBaseband Pulse TransmissionSource Coding & Baseband Transmission

Communications → Random Processes → Autocorrelation & PSD

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Question

An information source generates a binary sequence {αn}\{\alpha_n\}. αn\alpha_n can take one of the two possible values −1-1 and +1+1 with equal probability and are statistically independent and identically distributed. This sequence is precoded to obtain another sequence {βn}\{\beta_n\}, as βn=αn+kαn−3\beta_n = \alpha_n + k \alpha_{n-3}. The sequence {βn}\{\beta_n\} is used to modulate a pulse g(t)g(t) to generate the baseband signal
X(t)=∑n=−∞∞βng(t−nT)X(t) = \sum_{n=-\infty}^{\infty} \beta_n g(t - nT), where g(t)={1,0≤t≤T0,otherwise.g(t) = \begin{cases} 1, & 0 \leq t \leq T \\ 0, & \text{otherwise.} \end{cases}
If there is a null at f=13Tf = \frac{1}{3T} in the power spectral density of X(t)X(t), then kk is ________
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