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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 General QAM Modulation in an AWGN Channel.

The relative power of noise in an AWGN channel is typically described by quantities such as

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

`awgn`

function.Ratio of bit energy to noise power spectral density (EbNo). This quantity is used by

`BER Analyzer`

Tool and performance evaluation functions in this toolbox.Ratio of symbol energy to noise power spectral density (EsNo)

The relationship between EsNo and EbNo, 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 communication system, k might be influenced by the size of the modulation
alphabet or the code rate of an error-control code. For example, if a system uses a
rate-1/2 code and 8-PSK modulation, then the number of information bits per symbol (k) is
the product of the code rate and the number of coded bits per modulated symbol: (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.

The relationship between EsNo 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 signal's symbol period and
T_{samp} is the signal's sampling period.

For example, if a complex baseband signal is oversampled by a factor of 4, then EsNo
exceeds the corresponding SNR by 10 log_{10}(4).

**Derivation for Complex Input Signals. **You can derive the relationship between EsNo 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}= Sampling frequency, in Hertz

Note that *B*_{n}=
*F*_{s} =
1/*T*_{samp}.

**Behavior for Real and Complex Input Signals. **The following figures illustrate the difference between the real and complex cases
by showing the noise power spectral densities S_{n}(f) of a real
bandpass white noise process and its complex lowpass equivalent.

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