Communication Technology

 

 

 

AWGN, AWGN vs SNR

AWGN stands for Additive WhiteGaussianNoise. Actually this name itself has all the detailed information in it as follows. AWGN is also the channel that every link budget starts from, because it is the one impairment that is always present. A receiver adds thermal noise of its own however clean the transmission was. Each of the three letters names a property, and each property is a modelling choice that can be relaxed.

What does each letter of AWGN mean?

Each letter fixes one property of the noise, and each one has an opposite that appears elsewhere in engineering. Reading them one at a time is the fastest way to see what the model assumes, and what it quietly leaves out.

  •   Additive : This means that the noise is ADDED to the original signal. (Is there any other type (Non-additive) noise ? Yes, for example, you might have heard of multiplicative in some other area like image processing)
  •   White : This means that it contains all the frequency components with equal distribution (Is there any other type (non-White)  ? Yes, for example, you might have heard of pink noise (1/f noise) in other area like semiconductor design)
  •   Gaussian : This means that it has Gaussian (Normal) Distribution.  (Is there any other type (non-Gaussian) ? Yes, for example, you might have heard of Log Normal Noise)

Those three properties are what make the model easy to work with. Additive noise means the receiver sees signal plus noise, so linear methods can separate the two. White noise has a flat power spectral density, so the noise power a receiver collects grows in proportion to the bandwidth it opens. Gaussian noise follows from the central limit theorem. Thermal noise is the sum of a very large number of independent electron movements, and such a sum tends towards a normal distribution.

  • Additive is what makes the model tractable : the receiver sees signal plus noise, rather than signal multiplied by it.
  • White is a statement about bandwidth : a flat spectrum means every extra hertz of receiver bandwidth collects more noise power.
  • Gaussian is not an assumption of convenience : it follows from the central limit theorem, because the noise is a sum of very many independent contributions.

How does SNR set the size of the noise?

The three letters say what kind of noise it is, and they say nothing about how much. That number comes from SNR. The whole of the relationship fits into one equation with a single branch in it, and the listing further down implements that equation line by line.

Some examples of the signal that are impacted by AWGN are as follows.

As you see here, the overall distribution of AWGN varies depending on SNR (Signal To Noise Ratio). Following equation shows the relationship between AWGN and SNR.

 

< List 1 >

    clear all;

     

    % Number of symbols

    N = 10000;

     

    % Specify SNR in dB. Try setting various different value here and see how the result changes

    SNR_dB = 10;

     

    % Create a randomly generated QAM symbols

    s = (2*randi([0 1],1,N)-1)+j*(2*randi([0 1],1,N)-1);

     

    % Calculate Symbol Energy

    Eavg = sum(abs(s) .^ 2)/N;

     

    % Convert SNR (in dB) to SNR (in Linear)

    SNR_lin = 10 .^ (SNR_dB/10);

     

    % Calculate the Sigma (Standard Deviation) of AWGN

    awgnSigma = sqrt(Eavg/(2*SNR_lin));

     

    % Generate a sequence of noise with Normal Distribution and rescale it with the sigma

    awgn = awgnSigma*(randn(1,N)+j*randn(1,N));

     

    % Add the noise to the original signal

    y = s + awgn;

     

    % Plot the noised signal and the original signal

    plot(real(y),imag(y),'ko','MarkerFaceColor',[0 0 0],'MarkerSize',1);

    axis([-2 2 -2 2]);

    title(strcat('SNR=', num2str(SNR_dB), ' dB'));

    hold on;

    plot(real(s),imag(s),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',8);

    axis([-2 2 -2 2]);

    hold off;

Two things in that equation are worth separating. N(0,1) is a standard normal distribution with mean 0 and variance 1, and it never changes. The sigma in front of it is the only part that depends on SNR, so sigma is the entire answer to how much noise there is.

The branch in the equation is about the symbol, not about the noise. When the symbol is real, sigma is the square root of Es over SNR. When the symbol is complex, the same expression carries a 2 in the denominator. A complex symbol carries noise on the real and the imaginary axis alike, and each axis takes half of the noise power.

The listing takes the complex branch, because it builds a complex symbol. Its awgnSigma line divides by 2*SNR_lin. The line after it multiplies that sigma by randn(1,N)+j*randn(1,N), and each of those two draws is an independent standard normal. The two halves therefore add back up to the noise power that the SNR asked for.

Es is the last term left to define. The equation defines it as the mean of the squared magnitude across the symbols, and the Eavg line computes exactly that. Every symbol in this constellation sits at plus or minus one on both axes, so Es comes out at 2.

The three constellation panels are what this listing produces, at three settings of SNR_dB. The red markers are the transmitted symbols and the black cloud is what arrives. At 30 dB the cloud sits tight around each red point. At 10 dB the four clouds have grown until they almost touch.

  • Sigma carries the whole SNR dependence : N(0,1) is fixed, so only the scale factor in front of it moves.
  • The 2 belongs to the symbol, not to the noise : a complex symbol splits the noise power across two axes, so each axis takes half.
  • Es is measured, not assumed : the listing averages the squared magnitude of the symbols it actually generated.
  • SNR is visible as cloud radius : the same constellation at 30, 20 and 10 dB shows the cloud growing while the red reference points stay where they are.

What does AWGN not model?

AWGN is the easy channel, and that is exactly why it is the reference. Every real link adds impairments on top of it, so a result quoted in AWGN is closer to an upper bound than to a prediction.

Fading is the first thing missing. AWGN adds noise to a signal whose amplitude never changes, while a real radio channel varies both the amplitude and the phase as the terminal or the reflectors move. A link can work at a given SNR in AWGN and still fail at the same average SNR in fading. The instantaneous value spends part of its time far below the average.

Interference is the second. The noise in this model comes from thermal sources and from the receiver itself, and it carries no structure. Interference from another transmitter does carry structure, and a receiver can sometimes subtract it when it knows that structure. Nothing can be subtracted from AWGN.

Two more absences are worth naming. Impulsive noise, from switching equipment or from ignition systems, arrives in short bursts and is not Gaussian at all. Phase noise from the local oscillator rotates the constellation rather than spreading it, so it appears as a smeared arc instead of a round cloud.

  • An AWGN result is a best case : every impairment a real channel adds sits on top of the noise this model describes.
  • Average SNR is not enough once there is fading : the instantaneous value spends part of its time far below the average, and that is where the errors happen.
  • Interference has structure and noise does not : that difference is what lets a receiver cancel one and not the other.
  • Not every impairment makes a round cloud : phase noise smears the constellation into an arc, and impulsive noise arrives in bursts.

 

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