## 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 (EbN0). 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 (EsN0)

#### Relationship Between EsN0 and EbN0

The relationship between EsN0 and EbN0, both expressed in dB, is as follows:

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

where k is the number of information bits per symbol.

In a communications system, *k* might be influenced by the size of
the modulation alphabet or the code rate of an error-control code. 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 EsN0 and SNR

The relationship between EsN0 and SNR, both expressed in dB, is as follows:

$$\begin{array}{l}{E}_{s}/{N}_{0}\text{(dB)}=10{\mathrm{log}}_{10}\left({T}_{sym}/{T}_{samp}\right)+SNR\text{}\text{(dB)forcomplexinputsignals}\\ {E}_{s}/{N}_{0}\text{(dB)}=10{\mathrm{log}}_{10}\left(0.5{T}_{sym}/{T}_{samp}\right)+SNR\text{}\text{(dB)forrealinputsignals}\end{array}$$

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*.

For a complex baseband signal oversampled by a factor of 4, the EsN0 exceeds the
corresponding SNR by 10 log_{10}(4).

**Derivation for Complex Input Signals. **You can derive the relationship between EsN0 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

**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.