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 (E_b/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 (E_s/N₀)

Use the convertSNR function to convert between these ratios.

Relationship Between E_s/N₀ and E_b/N₀

The relationship in dB between E_s/N₀ and E_b/N₀ is:

$$E_s/N_0\text{ (dB)}=E_b/N_0\text{ (dB)}+10{\log }_{10}(k)$$

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 E_s/N₀ and SNR

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

$$E_s/N_0\text{ (dB)}=10{\log }_{10}\left(T_{sym}/T_{samp}\right)+SNR\text{}\text{ (dB) for complex input signals}$$

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.

You can derive the relationship between E_s/N₀ and SNR for complex input signals as follows:

$$\begin{array}{c}{E}_s/N_0\text{ (dB)}=10{\log }_{10}\left((S\cdot T_{sym})/(N/B_n)\right)\\ =10{\log }_{10}\left((T_{sym}{F}_s)\cdot (S/N)\right)\\ =10{\log }_{10}\left(T_{sym}/T_{samp}\right)+SNR\text{}\text{ (dB)}\end{array}$$

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

For a complex baseband signal oversampled by a factor of 4, the E_s/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.