Communication Systems Simulation - III

Harri SaarnisaariPart of Simulations and Tools for Telecommunication

Course

2

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

3

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

4

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

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RNG example

%create complex white Gaussian signal with N elementsN=10000;n=randn(N,1)+j*randn(N,1);

• Is it zero mean?%check meanMEAN=mean(n)plot([real(n) imag(n)])legend('real','imag')title('mean of real and imaginary part')

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The larger N is the close zero the mean isN=1000, mean 0.0177 + 0.0263iN=10 000, mean 0.0044 - 0.0017i

However, if you repeat the trial several times andcalculate the average the result is zero.

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RNG example

%check variance, should be 2 now since%we have a sum of two Gaussian processes with

var 1VAR=var(n)

– N=1000, VAR=1.9617– N=10 000, VAR=2.0292– The larger N is, the closer VAR is 2

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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 Nplot(xcorr(real(n),imag(n))/N)%same is verified by calculating correlation coefficient,

where%results are normalized by standard deviationsXCORR=corrcoef(real(n),imag(n))

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N=1000XCORR =

1.0000 -0.0600 -0.0600 1.0000

Correlation 6 %

N= 10 000XCORR =

1.0000 0.0296 0.0296 1.0000

Correlation 3 %

The larger N the Closer independency I and Q are

10

RNG example

• Whiteness?– Autocorrelation impulsive– PSD flat

%check whiteness, %maximum autocorrelation 2Nplot(abs(xcov(n)))title('autocorrelation')psd(n) %uses Welch methodtitle('PSD')

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Quite impulsive

12Quite flat

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RNG example

• Gaussianity?– Histogram– Statistical tests (not considered herein)

%check Gaussianityhist([real(n) imag(n)],50) %50 barslegend('real','imag')title('histograms')

14Quite Gaussian shapes

15

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

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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

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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