Dr. Uri Mahlab
Dr. Uri Mahlab
Generation of Random Variable
•Most computer software libraries include a
uniform random number generator.
•Such a random number generator a numb
between 0 and 1 with equal probability.
•We call the output of the random number
generator a random variable.
•If A denotes such a random variable, its
range is the interval o  A  1
Dr. Uri Mahlab
f ( x)
Is called the probability density function
F(A) is called the probability distribution function
Where:
Dr. Uri Mahlab
F(A)
f(A)
1
1
0
1
1
2 (a)
A
0
Fig 1
1
(b)
Figure: Probability density function f(A)
and the probability distribution function F(A)
of a uniformly distributed random variable A.
Dr. Uri Mahlab
A
If we wish to generate uniformly distributed noise in an
interval (b,b+1), it can be accomplished simply by using
the
output A of the random number generator and shifting it
by an
amount b. Thus a new random variable B can be defined
as
Which now has a mean
value
Dr. Uri Mahlab
F(B)
f(B)
1
1

2
0
(a)
1
1
2
B
1

2
0
(b)
1
2
B
Fig 2
Figure : Probability density function and the probability
distribution function of zero- mean uniformly
distributed
random variable
Dr. Uri Mahlab
B  Ab
F(C)
1
F(C)  A
C  F 1 (A )
A=f(C)
C 0
C
Fig.3
Figure : Inverse mapping from the uniformly distributed
random variable A to the new random variable C
Dr. Uri Mahlab
Example 2.1 Generate a random variable C that has the
linear probability density function shown in Figure (a);I.e.,
f(C)
1
0
1
 C, 0  C  2
f (C )   2
0,
otherwise
c
2
(a) 2
F(C)
1 1
4
C
Fig - 4
0
C2
(b) 2
C
Figure : linear probability density function and the
corresponding probability distribution function
Dr. Uri Mahlab
Answer
ip_02_01
•Thus we generate a random variable C with
probability function F(C) ,as shown in Figure 2.4(b). In
Illustartive problem 2.1 the inverse1mapping C=
(A) was simple. In some cases itFis not.
Gaussian Normal Noise
Lets try to generate random numbers that have a
normal distribution function.
•Noise encountered in physical systems is often
characterized by the normal,or Gaussian probability
distribution, which is illustrated in Figure 2.5. The
probability density function is given by
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Where  2 is the variance of C, which is a measure of the spread of the
probability density function f(c). The probability
distribution function F(C) is the area under f(C)
over the range(-,C). Thus
f(C)
1
F(C)
1
2
Dr. Uri Mahlab
0
(a)
C Fig 5
0
(b)
Gaussian probability density function
and the corresponding probability distribution function.
C
Unfortunately,the integral
cannot be expressed in
terms of simple functions. Consequently the inverse
mapping is difficult to achieve.
A way has been found to circumvent this problem.
From probability theory it is known that a Rayleigh
distributed random variable R, with probability
distribution function
Dr. Uri Mahlab
Is related to a pair of Gaussian random variables C
and D through the transformation
where  is uniformly distribute d vaiable in the interval(0 ,2).
The parameter  2 is the variance of C and D. since f(R) is
easily inverted, we have
Dr. Uri Mahlab
Where A is a uniformly distributed random variable in
the interval(0,1). Now, if we generate a second
uniformly distributed random variable B and define
Then from transformation,we obtain two statistically
independent Gaussian distributed random variables C
and D.
Dr. Uri Mahlab
Gaussian Normal Noise
u=rand;
% a uniform random variable in (0,1)
z=sgma*(sqrt(2*log(1/(1-u)))); % a Rayleigh distributed random variable
u=rand;
% another uniform random variable in (0,1)
gsrv1=m+z*cos(2*pi*u);
gsrv2=m+z*sin(2*pi*u);
Gngauss.m
Dr. Uri Mahlab
Example 2.2:
Generation of samples of a multivariate gaussian proce
Generate samples of a multivariate Gaussian
random process X(t) having a specified mean
value
Dr. Uri Mahlab
mx
and a covariance Cx
Answer
ip_02_02
Example 2.3:
generate a sequence of 1000(equally spaced)
samples of gauss Markov process from the recursive
x  0.95x  w relation
n
n 1
n,
n  1,2,...,1000
where x0  0 and {w n } is a sequence of zero mean
and unit variance i.i.d. a Gaussian random variables.
plot the sequence{x n, 1  n  1000} as a function of
the time index n and the autocorrelation.
Dr. Uri Mahlab
Answer
ip_02_03
Power spectrum of random processes and
white processes
A stationary random process X(t) is characterized in
Sx ( f )
the frequency domain by its power spectrum
which is the fourier transform of the autocorrelation
Rx ( t )
function
of the random process.that is,
Rx ( t )
Conversely, the autocorrelation function
of a
Sx ( f )
stationary process X(t) is obtained from the power
spectrum
by means of the inverse fourier
transform;I.e.,
Dr. Uri Mahlab
Definition:A random process X(t) is called a white
S x ( f )if
process if it has a flat power spectrum, I.e.,
is a constant for all f frequency
Dr. Uri Mahlab
Example 2.4:
1) Generate a discrete-time sequence of N=1000 i.i.d.
uniformly distributed random in interval (-1/2,1/2) and
compute the autocorrelation of the sequence {Xn} defined as
1 N m
Rx (m) 
X n X n  m,

N  m n 1
N
1

X n X nm,

N  m n m
m  0,1,..., M
m  1,2,..., M
2) Determine the power spectrum of the sequence {Xn} by
computing the discrete Fourier transform (DFT) of Rx(m) The
DFT,which is efficiently computed by use of the fast Fourier
transform (FFT) algorithm, is defined as
M
Sx ( f ) 
Dr. Uri Mahlab
 R (m)e
m  M
x
 j 2 fm /( 2 m 1)
Answer
Matlab:
M-file
ip_02_04
Example 2.5:
compute the auto correlation Rx(t) for the random
process whose power spectrum is given by
 N0
 , f B
Sx ( f )   2
0,
f B
Dr. Uri Mahlab
Answer
Matlab: M-file
ip_02_05
Dr. Uri Mahlab
Linear Filtering of Random
Processes
Suppose that a stationary random process X(t) is
passed through a liner time-invariant filter that is
characterized in the time domain by its impulse
response h(t) and in the frequency domain by its
frequency response.
Dr. Uri Mahlab
The auto correlation function of y(t) is
In the Frequency domain
Dr. Uri Mahlab
Example 2.6
suppose that a white random process X(t) with
power spectrum Sx(f)= 1 for all f excites a linear filter
with impulse response
e , t  0
h( t )  
0, t  0
-t
Determine the power spectrum Sy(f)of the filter
output.
Dr. Uri Mahlab
Answer
Matlab: M-file
ip_02_06
Example 2.7
Compute the autocorrelation function Ry(t)
corresponding to Sy(f)in the Illustrative problem 2.6
for the specified Sx(f)=1
Answer
Matlab: M-file
ip_02_07
Dr. Uri Mahlab
Example 2.8
Suppose that a white random process with
samples{X(n)} is passed through linear filter with
impulse response
(0.95) , n  0
h( n)  
n0
0,
n
Determine the power spectrum of the output
process{Y(n)}
Dr. Uri Mahlab
Answer
Matlab: M-file
ip_02_08
Lowpass and Bandpass processes
•Definition:A random process is called lowpass if its
power spectrum is large in the vicinity of f=0 and small
(approaching 0) at high frequencies.
•In other words, a lowpass random process has most
of its power concentrated at low frequencies .
•Definition: A lowpass random process x (t) is band
limited if the power spectrum Sx(f)=0 for Sx(f)>B.
The parameter B is called the bandwidth of the
random process
Dr. Uri Mahlab
Example # 9:consider the problem of generating samples of a
lowpass random process by passing a white noise sequence
{Xn}through a lowpass filter. The input sequence is an I.I.d.
sequence of uniformly distributed random variables on the
interval
(-0.5,0.5 ). The lowpass filter hasnthe impulse response.
(0.9) , n  0
h( n)  
n0
0,
And is characterized by the input-output
recursive(difference)equation.
yn  0.9 yn 1  xn ,
n 1
y 1  0
Compute the output sequence{yn} and determine the
autocorrelation function Rx(m) and Ry(m),as indicated in
problem 2.4 Determine the power spectra Sx(f) and Sy(f) by
computing the DFT of Rx(m) and Ry(m)
Answer
Dr. Uri Mahlab
Matlab: M-file
ip_02_09
N=1000;
% The maximum value of n
M=50;
Rxav=zeros(1,M+1);
Ryav=zeros(1,M+1);
Sxav=zeros(1,M+1);
Syav=zeros(1,M+1);
for i=1:10,
% take the ensemble average ove 10 realizations
X=rand(1,N)-(1/2);
% Generate a uniform number sequence on (1/2,1/2)
Y(1)=0; %should be x(1) !!!!!
for n=2:N,
Y(n) = 0.9*Y(n-1) + X(n);
% note that Y(n) means Y(n-1)
end;
Rx=Rx_est(X,M);
% Autocorrelation of {Xn}
Ry=Rx_est(Y,M);
% Autocorrelation of {Yn}
Sx=fftshift(abs(fft(Rx)));
% Power spectrum of {Xn}
Sy=fftshift(abs(fft(Ry)));
% Power spectrum of {Yn}
Rxav=Rxav+Rx;
Ryav=Ryav+Ry;
Sxav=Sxav+Sx;
Syav=Syav+Sy;
end;
Rxav=Rxav/10;
Ryav=Ryav/10;
Sxav=Sxav/10;
Syav=Syav/10;
% Plotting commands follow
»
Dr. Uri Mahlab
•Definition:
•A random process is called bandpass if its power
spectrum is large in a band of frequencies centered in
the neighborhood of a central frequency fo and
relatively small outside of this band of frequencies.
•A random process is called narrowband if its
bandwidth B<< fo
•Bandpass processes are suit for representing modulated sign
In communication the information-bearing signal is usually a
lowpass random process that modulates a carrier for
transmission over a bandpass communication channel. Thus
the modulated signal is bandpass random process.
Dr. Uri Mahlab
Example # 10:[Generation of samples a bandpass
random process] Generate samples of a bandpass
random process by first generating samples of two
statistically independent random processes XC(t) and
f 0t
XS(t) and using these to modulate the quadrature
f 0t 2
carriers cos(
)
and sin 2
,as shown in figure 7.


+
-
Answer
Matlab: M-file
ip_02_10
Figure 7: generation of a bandpass random process
Dr. Uri Mahlab
N=1000;
% number of samples
for i=1:2:N,
[X1(i) X1(i+1)]=gngauss;
[X2(i) X2(i+1)]=gngauss;
end;
% standard Gaussian input noise
processes
A=[1 -0.9];
% lowpass filter parameters
B=1;
Xc=filter(B,A,X1);
Xs=filter(B,A,X2);
fc=1000/pi;
% carrier frequency
for i=1:N,
band_pass_process(i)=Xc(i)*cos(2*pi*fc*i)-Xs(i)*sin(2*pi*fc*i);
end;
% T=1 is assumed
% Determine the autocorrelation and the spectrum of the band-pass
process
M=50;
bpp_autocorr=Rx_est(band_pass_process,M);
bpp_spectrum=fftshift(abs(fft(bpp_autocorr)));
% plotting commands follow
Dr. Uri Mahlab