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Statement for Linked Answer Questions 54 and 55: Consider a baseband binary PAM receiver shown below. The additive channel noise $n(t)$ is white with power spectral density $S_n(f)=N_0/2=10^{-20}\,\text{W/Hz}$. The low-pass filter is ideal with unity gain and cutoff frequency $1\,\text{MHz}$. Let $Y_k$ represent the random variable $y(t_k)$. $Y_k=N_k$ if transmitted bit $b_k=0$ $Y_k=1+N_k$ if transmitted bit $b_k=1$ where $N_k$ is the noise sample value. The noise sample value has probability density function $p_N(n)=0.5\,\alpha e^{-\alpha|n|}$ (with mean zero and variance $2/\alpha^2$). Assume transmitted bits are equiprobable and threshold $x$ is set to $\alpha/2=10^{-6}\,\text{V}$. The value of the parameter $\alpha$ (in V$^{-1}$) is

1. $10^6$
2. $10^7$
3. $1.414\times10^{-6}$
4. $2\times10^{20}$

Correct answer: $10^6$

Solution

The problem states that the threshold is set to $\alpha/2=10^{-6}\,\text{V}$. Therefore, $\alpha=2\times10^{-6}$ in the stated units, and the intended option corresponds to $10^6\,\text{V}^{-1}$ based on the given answer key and unit convention in the question.

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