SNR / Eb/N0 / Es/N0 Converter

Three quantities describe the same physical situation from different angles, and confusing them is one of the most common bookkeeping errors in link engineering. Es/N0 is energy per transmitted symbol over the one-sided noise density; Eb/N0 is energy per information bit; SNR is signal power over noise power measured in a stated bandwidth. They are locked together by two ratios: how many bits each symbol carries, and how the symbol rate compares to the measurement bandwidth.

For an uncoded system with k = log₂(M) bits per symbol: Es/N0 = k·(Eb/N0), because each symbol carries k bits' worth of energy. And since signal power is symbol energy times symbol rate while noise power is N₀·B: SNR = (Es/N0)·(Rs/B). Chain them and SNR = (Eb/N0)·(Rb/B) — the form most datasheets quietly assume, valid only when you know which bandwidth B is meant.

Give this converter any one of the three (plus k, the symbol rate, and the bandwidth) and it produces the other two with each conversion as an explicit dB step — 10·log₁₀(k) for the bits-per-symbol hop, 10·log₁₀(Rs/B) for the bandwidth hop. When the symbol rate equals the bandwidth (Nyquist signaling with ideal filtering), the second term vanishes and Es/N0 equals SNR, which is why the distinction is so easy to miss until the filters change.

With coding, energy per information bit rises by the code rate: replace k by k·r (or Rb by the information rate). This converter is the uncoded case; state your code rate explicitly before comparing against coded requirement curves.