AWGN Channel

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 (EbN0). 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 (EsN0)

The relationship between EsN0 and EbN0, both expressed in dB, is as follows:

Es/N0 (dB)=Eb/N0 (dB)+10log10(k)

where k is the number of information bits per symbol.

In a communications system, k might be influenced by the size of the modulation alphabet or the code rate of an error-control code. 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.

The relationship between EsN0 and SNR, both expressed in dB, is as follows:

Es/N0 (dB)=10log10(Tsym/Tsamp)+SNR​ (dB)   for complex input signalsEs/N0 (dB)=10log10(0.5Tsym/Tsamp)+SNR​ (dB) for real input signals

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

For a complex baseband signal oversampled by a factor of 4, the EsN0 exceeds the corresponding SNR by 10 log10(4).

Derivation for Complex Input Signals.  You can derive the relationship between EsN0 and SNR for complex input signals as follows:

Es/N0 (dB)=10log10((S⋅Tsym)/(N/Bn))=10log10((TsymFs)⋅(S/N))=10log10(Tsym/Tsamp)+SNR​ (dB)

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

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.