random signals. sinusoid of random amplitude num_real=4; simulation_length=1024;...

Random Signals

Sinusoid of Random Amplitude

NUM_REAL=4;

SIMULATION_LENGTH=1024;

t=0:(1/SIMULATION_LENGTH):(1-1/SIMULATION_LENGTH);

realizations=zeros(NUM_REAL,SIMULATION_LENGTH);

figure(1);

clf;

for n=1:NUM_REAL realizations(n,:)=random('unid',4,1,1)*cos(2*pi*4*t);

subplot(NUM_REAL,1,n);

plot(t,realizations(n,:));

end

subplot(NUM_REAL,1,1);

title('Realizations of Sinusoid of discrete random amplitude (PAM)')

( ) cos(2 4 ) uniform [1,4]X t A t Ap=

Sinusoid of Random Phase

NUM_REAL=4;




figure(1);

clf;

for n=1:NUM_REAL realizations(n,:)=cos(2*pi*4*t+random('unif',-pi,pi,1,1));



end


title('Realizations of Sinusoid of cont. random phase')

( ) cos(2 4 ) uniform [ , ]X t tp p p= +Q Q -

Sinusoid of Random Frequency

NUM_REAL=4;




figure(1);

clf;

for n=1:NUM_REAL realizations(n,:)=cos(2*pi*random('unid',4,1,1)*t);



end


title('Realizations of Sinusoid of discrete random frequency (FSK)')

( ) cos(2 ) uniform [1,4]X t ft fp=

Sinusoid of Random Amp, Freq, Phase

NUM_REAL=4;




figure(1);

clf;

for n=1:NUM_REAL realizations(n,:)=random('unid',4,1,1)*cos(2*pi*random('unid',4,1,1)*t+random('unif',-pi,pi,1,1)); subplot(NUM_REAL,1,n);


end


title('Realizations of Sinusoid of cont. random amp, freq, phase')

( ) cos(2 ) uniform [1,4] uniform [1,4] uniform [ , ]X t A ft A fp p p= +Q Q -

White Gaussian Random Process

NUM_REAL=4;




figure(1);

clf;

for n=1:NUM_REAL realizations(n,:)=randn(1,SIMULATION_LENGTH);



end


title('Realizations of WGN process')

Noisy Random Sinusoid

NUM_REAL=4;




figure(1);

clf;

for n=1:NUM_REAL realizations(n,:)=random('unid',4,1,1)*cos(2*pi*random('unid',4,1,1)*t+random('unif',-pi,pi,1,1))+0.1*randn(1,SIMULATION_LENGTH);



end


title('Realizations of noisy random sinusoid')

( ) cos(2 )X t A ft Np= +Q +

Poisson Arrival Process

NUM_REAL=4;




lambda=0.01;

figure(1);

clf;

for n=1:NUM_REAL arrivals=random('poiss',lambda,1,SIMULATION_LENGTH); realizations(n,:)=cumsum(arrivals);



end


2 1[ ( )]2 12 1

[ ( )][ ( ) ( ) ] 0,1,2,...

!

kt tt t

P Q t Q t k e kk

ll - --- = = =

Picking a RV from a Random Process

NUM_REAL=10000;




figure(1);

clf;

for n=1:NUM_REAL realizations(n,:)=randn(1,SIMULATION_LENGTH);

end

x=realizations(:,3);

hist(x,30);

A Gaussian RV of mean 0 and std 1

Autocorrelation

function [Rxall]=Rx_est(X,M)

N=length(X);

Rx=zeros(1,M+1);

for m=1:M+1,

for n=1:N-m+1,

Rx(m)=Rx(m)+X(n)*X(n+m-1);

end;

Rx(m)=Rx(m)/(N-m+1);

end;

for i=1:M,

Rxall(i)=Rx(M+2-i);

end

Rxall(M+1:2*M+1)=Rx(1:M+1);

1

1( ) 0,1,...,

N m

x n n mn

R m X X m MN m

-

+=

= =- å

Autocorrelation of Gaussian Random Process

N=1000;

X=randn(1,N);

M=50;

Rx=Rx_est(X,M);

plot(X)

title('Gaussian Random Process')

pause

plot([-M:M],Rx)

title('Autocorrelation function')

Autocorrelation of Gauss-Markov Random Process

rho=0.95;

X0=0;

N=1000;

Ws=randn(1,N);

X(1)=rho*X0+Ws(1);

for i=2:N,

X(i)=rho*X(i-1)+Ws(i);

end;

M=50;

Rx=Rx_est(X,M);

plot(X)

title('Gauss-Markov Random Process')

pause

plot([-M:M],Rx)

title('Autocorrelation function')

[ ] 0.95 [ 1] [ ], [ ] ~ (0,1)

[0] 0

X n X n w n w n N

X

= - +

=

random signals. sinusoid of random amplitude num_real=4; simulation_length=1024;...

Documents