AWGN Channel :: Channels (Communications Toolbox™)

AWGN Channel

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Section Overview Describing the Noise Level of an AWGN Channel

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Describing the Noise Level of an AWGN Channel

Section Overview

An AWGN channel adds white Gaussian noise to the signal that passes through it. To model an AWGN channel, use the awgn function. Several examples that illustrate the use of awgn are in Getting Started. The following demos also use awgn: basicsimdemo, vitsimdemo, and scattereyedemo.

Describing the Noise Level of 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 (E_b/N₀). This quantity is used by BERTool and performance evaluation functions in this toolbox.
Ratio of symbol energy to noise power spectral density (E_s/N₀)

Relationship Between E_s/N₀ and E_b/N₀

The relationship between E_s/N₀ and E_b/N₀, 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 E_s/N₀ and SNR

The relationship between E_s/N₀ and SNR, both expressed in dB, is as follows:

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 E_s/N₀ exceeds the corresponding SNR by 10 log₁₀(4).

Derivation for Complex Input Signals. You can derive the relationship between E_s/N₀ and SNR for complex input signals as follows:

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.