Communication Systems Simulation - III Harri Saarnisaari

Communication Systems Simulation - III Harri Saarnisaari Part ofSimulations and Tools for Telecommunication Course

Random Number Generation • Many simulators include random number generators • you can use those • These should be validated • If not already accepted by your community • But you always can questionnaire those, especially new generators • Here we consider some aspects of validation

Random Number Generation • Complex additive white Gaussian noise is usually met in communication simulations • Used to model thermal noise • Properties: • I and Q channels (real and imaginary part) • Zero mean • I and Q independent • White • Flat power spectrum density (PSD) • Impulsive autocorrelation (impulse at zero flag) • I and Q are Gaussian

RNG example • We use MATLAB to do some tests • MATLAB has randn m-file that generates zero mean, unit variance random variables • RNGs use an initial value (state, seed) from which they start • This should be randomized • In MATLAB (5) you may add command RANDN('state',sum(100*clock))to your startup file to set the generator to a different state each time you start

RNG example %create complex white Gaussian signal with N elements N=10000; n=randn(N,1)+j*randn(N,1); • Is it zero mean? %check mean MEAN=mean(n) plot([real(n) imag(n)]) legend('real','imag') title('mean of real and imaginary part')

The larger N is the close zero the mean is N=1000, mean 0.0177 + 0.0263i N=10 000, mean 0.0044 - 0.0017i However, if you repeat the trial several times and calculate the average the result is zero.

RNG example %check variance, should be 2 now since %we have a sum of two Gaussian processes with var 1 VAR=var(n) • N=1000, VAR=1.9617 • N=10 000, VAR=2.0292 • The larger N is, the closer VAR is 2

RNG example • I and Q independent, how it is measured? • Cross correlation should be zero • Cross correlation coefficient should be zero %check independency of I and Q %since maximum is N (if fully correlated sequences %the result is scaled by N plot(xcorr(real(n),imag(n))/N) %same is verified by calculating correlation coefficient, where %results are normalized by standard deviations XCORR=corrcoef(real(n),imag(n))

N=1000 XCORR = 1.0000 -0.0600 -0.0600 1.0000 Correlation 6 % N= 10 000 XCORR = 1.0000 0.0296 0.0296 1.0000 Correlation 3 % The larger N the Closer independency I and Q are

RNG example • Whiteness? • Autocorrelation impulsive • PSD flat %check whiteness, %maximum autocorrelation 2N plot(abs(xcov(n))) title('autocorrelation') psd(n) %uses Welch method title('PSD')

Quite impulsive

Quite flat

RNG example • Gaussianity? • Histogram • Statistical tests (not considered herein) %check Gaussianity hist([real(n) imag(n)],50) %50 bars legend('real','imag') title('histograms')

Quite Gaussian shapes

Fading channels • To create fading channels you have to understand what they are • In fading channels signal and possibly its delayed versions are multiplied by random (tap) coefficients • Rayleigh fading channel • Taps are zero mean, complex Gaussian variables with a variance set so that SNR is what is wanted • Equally, amplitude is Rayleigh distributed and phase is uniformly distributed between 0 and 2

Fading channels • Rician fading channels • Taps are non-zero mean complex Gaussian variables with a variance and amplitude set so that SNR is what is wanted • Alternatively, amplitude is Rician distributed and phase is uniformly distributed between 0 and 2 • Also other fading channels exist

Fading channels • Changing mean and variance of a zero mean unit variance complex Gaussian variance x to m and 2 • Multiply by  and add m, i.e., new x = x+m • In fading channels one is usually interested the amplitude of the tap, not the phase • Create taps as rayleigh * exp(j*2*pi*rand) • Or rician * exp(j*2*pi*rand) • Some MATLAB toolboxes include generation of Rayleigh and Rician variables, otherwise you may create them from two Gaussian variables

Communication Systems Simulation - III Harri Saarnisaari