# Relation between Eb/N0 and SNR

http://www.gaussianwaves.com/2008/11/relation-between-ebn0-and-snr-2/

I have seen a lot of questions being posted in several forums on the relationship between Eb/N0 and SNR. Like, whether SNR and Eb/No are same or not. In fact , the relationship is very easier to understand.

Lets start with the basic equation and try to verify its authenticity.

$\frac{S}{R_{b}}=E_{b}\;\;\;\;\; \rightarrow (1)$

where,
Rb = bit rate in bits/second
Eb = Energy per bit in Joules/bit
S = Total Signal power in Watts

We all know from fundamental physics that Power = Energy/Time. Using SI units, lets verify the above equation.

$\frac{S}{R_{b}}=E_{b}$

$\frac{Watts}{\frac{bits}{second}} = \frac{Joules}{bit}$

$\Rightarrow Watts = \frac{Joules}{second}$

This verifies the power, energy relationship between S and Eb. Now, introducing the noise power $N_{0}$ in equation (1)

$\Rightarrow \frac{Eb}{N0} = \frac{S}{ \left( Rb*N0 \right )}$

$\Rightarrow SNR = \frac{Rb*Eb}{N0}\;\;\;\;\; \rightarrow (2)$

This equation implies that the SNR will be more than $\frac{Eb}{N_{0}}$ by a factor of $R_{b}$ (if Rb > 1 bit/second)

Increasing the data rate will increase the SNR, however , increasing Rb will also cause more noise and noise term also increases ( due to ISI – intersymbol interference , since more bits are packed closer and sent through the channel).

So we cannot increase SNR by simply increasing Rb. We must strike a compromise between the data rate and the amount of noise our receiver can handle.

Always engineering is about where and when we want to compromise …

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