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.

Q. 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?

(A) 0

(B) 1

(C) 2

(D) 3

Ans: 3