Energy per Bit to Noise Power Spectral Density Ratio

Daniel Weibel
Created 17 Jun 2016

The energy per bit to noise power spectral density ratio $E_b/N_0$ can be seen as a signal-to-noise ratio (SNR) per individual bit.


Energy per bit.

Unit: J (Joule), Ws (Watt-second)

Calculated as received signal power (in Watts) divided by bit rate (in $\frac{1}{\text{s}}$):

\[E_b = \frac{\text{W}}{\frac{1}{\text{s}}} = \text{W}\, \frac{\text{s}}{1} = \text{Ws} = \text{J}\]

Remember: Energy/Work (J, Ws, kWh) vs. Power, i.e. energy/work per time (W)


Noise power spectral density.

Unit: $\frac{\text{W}}{\text{Hz}}$

\[N_0 = \frac{\text{W}}{\text{Hz}} = \frac{\text{W}}{\frac{1}{\text{s}}} = \text{W}\, \frac{\text{s}}{1} = \text{Ws} = \text{J}\]

Important: the noise power spectral density $N_0$ is assumed to be constant. That is, white noise, in particular additive white Gaussian noise (AWGN), is assumed. For frequency-selective noise, the $E_b/N_0$ formula does not work.


Energy per bit to noise power spectral density ratio.

Unit: dimensionless (often expressed in dB)

\[\frac{E_b}{N_0} = \frac{\text{J}}{\text{J}} = \cdot\]

$E_b/N_0$ can be seen as a “normalised SNR”, in particular a “SNR per bit”. This is because the traditional SNR is always relative to a certain bandwidth.