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 N₀/2. Binary signaling, with 0 → p(t) and 1 → q(t), is used for the transmission, along with an optimal receiver that minimizes the bit-error probability.
Let ϕ₁(t), ϕ₂(t), form an orthonormal signal set. If we choose p(t) = ϕ₁(t) and q(t) = -ϕ₁(t), we would obtain a certain bit-error probability P_b. If we keep p(t) = ϕ₁(t), but take q(t) = $$\sqrt {\text{E}} $$ ϕ₂(t), for what value of E would we obtain the same bit-error probability P_b?