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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 BERTool 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 HertzF

_{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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