If you are looking for a function to generate a signal with given SNR, please see this latest post.

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,

\(R_b\) = bit rate in bits/second

\(E_b\) = 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 \(E_b\). 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 R_{b} > 1 bit/second)

Increasing the data rate will increase the SNR, however , increasing \(R_b\) 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 \(R_b\). We must strike a compromise between the data rate and the amount of noise our receiver can handle.

**Note:**

**Actually,**

**$$ \frac{S}{N} = \frac{\left( R_b*E_b \right)}{B*N_0}$$**

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

### 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 )