## AWGN Channel

### Section Overview

An AWGN channel adds white Gaussian noise to the signal that passes through it. You can
create an AWGN channel in a model using the `comm.AWGNChannel`

System object™, the AWGN Channel block, or the `awgn`

function.

The following examples use an AWGN Channel: QPSK Transmitter and Receiver and Estimate Symbol Rate for General QAM Modulation in AWGN Channel.

### AWGN Channel Noise Level

Typical quantities used to describe the relative power of noise in an AWGN channel include:

Signal-to-noise ratio (SNR) per sample. SNR is the actual input parameter to the

`awgn`

function.Ratio of bit energy to noise power spectral density (

*E*_{b}/*N*_{0}). This quantity is used by Bit Error Rate Analysis app and performance evaluation functions in this toolbox.Ratio of symbol energy to noise power spectral density (

*E*_{s}/*N*_{0})

Use the `convertSNR`

function to convert between these ratios.

#### Relationship Between *E*_{s}/*N*_{0}
and
*E*_{b}/*N*_{0}

The relationship in dB between
*E*_{s}/*N*_{0}
and
*E*_{b}/*N*_{0}
is:

$${E}_{s}/{N}_{0}\text{(dB)}={E}_{b}/{N}_{0}\text{(dB)}+10{\mathrm{log}}_{10}(k)$$

where *k* is the number of information bits per symbol.

In a communications system, the modulation alphabet size and code rate of an
error-control code influence *k*. For example, in a system using a rate
1/2 code and 8-PSK modulation, the number of information bits per symbol
(*k*) is the product of the code rate and the number of coded bits per
modulated symbol. Specifically, (1/2)log_{2}(8) = 3/2. In such a system, three information bits correspond to six coded bits,
which in turn correspond to two 8-PSK symbols.

#### Relationship Between *E*_{s}/*N*_{0}
and SNR

The relationship in dB between
*E*_{s}/*N*_{0}
and SNR is:

$${E}_{s}/{N}_{0}\text{(dB)}=10{\mathrm{log}}_{10}\left({T}_{sym}/{T}_{samp}\right)+SNR\text{}\text{(dB)forcomplexinputsignals}$$

where *T*_{sym} is the symbol period of the
signal and *T*_{samp} is the sampling period of the
signal.
*T*_{sym}/*T*_{samp}
computes to *Samples/Symbol*.

You can derive the relationship between
*E*_{s}/*N*_{0}
and SNR for complex input signals as follows:

$$\begin{array}{c}{E}_{s}/{N}_{0}\text{(dB)}=10{\mathrm{log}}_{10}\left((S\cdot {T}_{sym})/(N/{B}_{n})\right)\\ =10{\mathrm{log}}_{10}\left(({T}_{sym}{F}_{s})\cdot (S/N)\right)\\ =10{\mathrm{log}}_{10}\left({T}_{sym}/{T}_{samp}\right)+SNR\text{}\text{(dB)}\end{array}$$

where

*S*= Input signal power, in watts*N*= Noise power, in watts*B*_{n}= Noise bandwidth in Hertz =*F*_{s}= 1/*T*_{samp}.*F*_{s}= Sampling frequency in Hertz

For a complex baseband signal oversampled by a factor of 4, the
*E*_{s}/*N*_{0}
exceeds the corresponding SNR by 10log_{10}(4).

#### Behavior for Real and Complex Input Signals

These figures illustrate the difference between the real and complex cases by showing the noise power spectral densities of a real bandpass white noise process and its complex lowpass equivalent.