An iid binary sequence $\alpha_n$ takes ±1 equiprobably. It is precoded as $\beta_n=\alpha_n+k\alpha_{n-3}$. The baseband signal is $$X(t)=\sum_{n=-\infty}^{\infty}\beta_n g(t-nT),$$ where $g(t)=1$ for $0\le t\le T$ and zero otherwise. If the power spectral density has a null at $f=1/(3T)$, k is ________.
The precoder frequency response is $1+ke^{-j2\pi f(3T)}$. Since the input is independent and zero mean, the output spectrum includes its squared magnitude.
At f=1/(3T), the exponential is 1. The rectangular pulse spectrum is nonzero there, so the null must come from the precoder: $|1+k|^2=0$.
Therefore **k=−1**.
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