## 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 (Eb/N0). 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 (Es/N0)

Use the `convertSNR` function to convert between these ratios.

#### Relationship Between Es/N0 and Eb/N0

The relationship in dB between Es/N0 and Eb/N0 is:

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)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 Es/N0 and SNR

The relationship in dB between Es/N0 and SNR is:

where Tsym is the symbol period of the signal and Tsamp is the sampling period of the signal. Tsym/Tsamp computes to Samples/Symbol.

You can derive the relationship between Es/N0 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 = 1/Tsamp.

• Fs = Sampling frequency in Hertz

For a complex baseband signal oversampled by a factor of 4, the Es/N0 exceeds the corresponding SNR by 10log10(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.