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A single bit, equally likely to be $0$ and $1$, is to be sent across an additive white Gaussian noise (AWGN) channel with power spectral density $N_{0}/2.$ Binary signaling with $0 \mapsto p(t),$ and $1 \mapsto q(t),$ is used for the transmission, along with an optimal receiver that minimizes the bit-error probability.

Let $\varphi_{1}(t),\varphi_{2}(t)$ form an orthonormal signal set.

If we choose $p(t)=\varphi_{1}(t)$ and $q(t)= -\varphi_{1}(t),$ we would obtain a certain bit-error probability $P_{b}.$

If we keep $p(t)=\varphi_{1}(t),$ but take $q(t)=\sqrt{E}\: \varphi_{2}(t),$ for what value of $E$ would we obtain the $\textbf{same}$ bit-error probability $P_{b}$?

  1. $0$
  2. $1$
  3. $2$
  4. $3$
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