June – Gaussian vs. impulsive noise

Noise is often modeled to have a Gaussian waveform. However, it might be unrealistic in some circumstances if the outlier events are more likely than in the Gaussian distribution. In this case, we might have to model the noise with fat-tailed distributions with significant mass in the tail distribution.

The $\alpha$-stable distributions are used as noise models in various applications, including economics and electromagnetic communications. Also, the normal distribution belongs to the $\alpha$-stable family of $\alpha$-stable distributions ( $\alpha = 2$), the only distribution in the family with a finite variance. With values $\alpha \leq 1$, even the mean is infinite. The qualitative difference to the normal distribution is clear: imagine investing money in a share that has a fixed expected profit value – say, 10 euros in a year – versus investing in a share whose value fluctuates so rapidly that the expected profit (mean profit over time) can be arbitrary. Other impulsive noise distribution models are the Middleton distributions representing the envelope of non-Gaussian electromagnetic noise.

In the following, we compare the Gaussian and $\alpha$-stable noises. The in-phase and quadrature components are distributed as Gaussian distribution or $\alpha$-stable distribution with $\alpha =1$, accordingly. The following GNU Octave code produces plots of bandwidth-limited analog Gaussian and $\alpha$-stable noises with $\alpha =1$.

pkg load statistics

clear all;
close all;

%%Function returns the values at times T of the analog signal of the corresponding digital signal sampled at speed Fs.
function xb = digitaltoanalog(T, digitalsignal, Fs)
  xb = [];
  for t = T
    xb =  [xb sum(digitalsignal.*sinc(Fs*t - (0 : length(digitalsignal) - 1)))];
  end
end


%%Inverse of the alpha stable CDF with alpha = 1.
invalphaCDF = @(x) tan(0.5*(-1 + 2*x)*pi);
%%Generate N Alpha-stable samples.
N = 100;
alphasamples = invalphaCDF(unifrnd(0,1,1,N));

%%Generate normal samples.
normalsamples = normrnd(0,1,1,N);
%%Convert to analog signal.
T = 1 : 0.1 : N;
alphasamples = digitaltoanalog(T, alphasamples, 1);
normalsamples = digitaltoanalog(T, normalsamples, 1);

alphasamples = alphasamples/(mean(abs(alphasamples))); %Normalize by mean instantaneous amplitude.
normalsamples = normalsamples/(mean(abs(normalsamples)));
figure(1)
plot(T, alphasamples)
legend('Alpha-stable noise')
axis([0 100 -15 15])
figure(2)
plot(T, normalsamples)
legend('Gaussian noise')
axis([0 100 -15 15])

Comparing the figures, one can easily understand where the term “impulsive noise” comes from.

It is also interesting to compare the audiolized noise signals: my Soundcloud.

References:

• Samorodnitsky, Gennady., and Murad S. Taqqu. Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance. New York: Chapman & Hall, 1994. Print.
• Middleton, David. Statistical-Physical Models of Man-made and Natural Radio Noise parts I - , 1976 - .
• Tse, David., and Pramod. Viswanath. Fundamentals of Wireless Communication. Cambridge, U.K. ;: Cambridge University Press„ 2005. Print.