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) = √E φ₂(t), for what value of E would we obtain the same bit-error probability P_b?