# Documentation

## 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 General QAM Modulation in an AWGN Channel.

### AWGN Channel Noise Level

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)

#### Relationship Between EsNo and EbNo

The relationship between EsNo and EbNo, both expressed in dB, is as follows:

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) log2(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 EsNo and SNR

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

where Tsym is the signal's symbol period and Tsamp 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 log10(4).

Derivation for Complex Input Signals.  You can derive the relationship between EsNo and SNR for complex input signals as follows:

where

• S = Input signal power, in watts

• N = Noise power, in watts

• Bn = Noise bandwidth, in Hertz

• Fs = Sampling frequency, in Hertz

Note that Bn= Fs = 1/Tsamp.

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 Sn(f) of a real bandpass white noise process and its complex lowpass equivalent.