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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)

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{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 …



\frac{S}{N} = \frac{\left( Rb*Eb \right)}{B*N0}

where B is the noise bandwidth. Here, I have taken an unity noise bandwidth (ie. B=1) for simplification.

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See also:

[1] Colored Noise Generation in Matlab
[2] Sampling Theorem – Baseband Sampling
[3] Sampling Theorem – Bandpass or Intermediate or Under Sampling
[4] Window Functions – An Analysis
[5] FFT and Spectral Leakage
[6] Raised Cosine Filter
[7] Moving Average Filter ( MA filter )

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– Founder and Author @ gaussianwaves.com which has garnered worldwide readership. He is a masters in communication engineering and has 7 years of technical expertise in channel modeling and has worked in various technologies ranging from read channel design for hard drives, GSM/EDGE/GPRS, OFDM, MIMO, 3GPP PHY layer and DSL.
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