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Dr. Uri Mahlab
Dr. Uri Mahlab
Dr. Uri Mahlab
Generation of Random Variables•Most computer software libraries include a uniform random number generator.•Such a random number generator a number 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)
1
0 1A
F(A)
1
0 1 A
(b)
1
2 (a)
Figure: Probability density function f(A)
and the probability distribution function F(A)
of a uniformly distributed random variable A.
Fig 1
Dr. Uri Mahlab
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)1
0B
F(B)
1
0
(b)1
2(a)
1
2
B1
2
1
2
Figure : Probability density function and the probability
distribution function of zero- mean uniformly distributed
random variable B A b
Fig 2
Dr. Uri Mahlab
CC 0
A=f(C)
1
F(C) F C A
C F A
( )
( )
1
Figure : Inverse mapping from the uniformly distributed random variable A to the new random variable C
Fig.3
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 CC
( ),
1
2 0 C 2
0, otherwisef(C)
1
0 2 C
c
2
(a)
F(C)
1
0 2 C
1
42C
(b)
Figure : linear probability density function and the corresponding probability distribution function
Answerip_02_01
MATLAB.lnk
Fig - 4
Dr. Uri Mahlab
•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 mapping C= (A) was simple. In some cases it is not.
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
F 1
Gaussian Normal Noise
Dr. Uri Mahlab
Where 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
2
f(C)
0C
F(C)1
0
(b)
1
2
(a)
CFig 5
Gaussian probability density function and the corresponding probability distribution function.
Dr. Uri Mahlab
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
have weinverted,easily
is f(R) since D. and C of variance theisparameter The
).,2interval(0 in the vaiableddistributeuniformly is where 2
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
MATLAB.lnk
Gngauss.m
Gaussian Normal Noise
u=rand; % a uniform random variable in (0,1) z=sgma*(sqrt(2*log(1/(1-u)))); % a Rayleigh distributed random variableu=rand; % another uniform random variable in (0,1)gsrv1=m+z*cos(2*pi*u);gsrv2=m+z*sin(2*pi*u);
Dr. Uri Mahlab
Generate samples of a multivariate Gaussian random
process X(t) having a specified mean value mx
and a covariance Cx
Answerip_02_02MATLAB.lnk
Example 2.2: [Generation of samples of a multivariate gaussian process]
Dr. Uri Mahlab
Example 2.3: generate a sequence of 1000(equally spaced) samples of
gauss Markov process from the recursive relation
x x wn n n 0 95 1. , n 1 2 1000, ,...,
where x and {w is a sequence of zero mean
and unit variance i.i.d. a Gaussian random variables.
plot the sequence{x 1 n 1000} as a function of
the time index n and the autocorrelation.
n
n,
0 0
}
Answerip_02_03
MATLAB.lnk
Dr. Uri Mahlab
Power spectrum of random processes and white processes
A stationary random process X(t) is characterized in the frequency domain by its power spectrum which is the fourier transform of the autocorrelation function of the random process.that is,
S fx ( )R tx ( )
Conversely, the autocorrelation function of a stationary process X(t) is obtained from the power spectrum by
means of the inverse fourier transform;I.e.,
R tx ( )S fx ( )
Dr. Uri Mahlab
Definition:A random process X(t) is called a white process if it has a flat power spectrum, I.e., if is a
constant for all f frequencyS fx ( )
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
R mN m
X Xx nn
N m
n m( ) ,
1
1m M0 1, ,...,
1
N mX Xn
n m
N
n m, m M 1 2, ,...,
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
S f R m ex xj fm m
m M
M
( ) ( ) /( )
2 2 1
MATLAB.lnk
AnswerMatlab: M-file
ip_02_04
Dr. Uri Mahlab
Example 2.5:
compute the auto correlation Rx(t) for the random process whose power spectrum is given by
S f
NB
Bx ( )
0
2, f
0, f
MATLAB.lnk
AnswerMatlab: M-file
ip_02_05
Dr. Uri Mahlab
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
h t( )
e , t
0, t 0
-t 0
Determine the power spectrum Sy(f)of the filter output.
MATLAB.lnk
AnswerMatlab: M-file
ip_02_06
Dr. Uri Mahlab
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
MATLAB.lnk
AnswerMatlab: M-file
ip_02_07
Dr. Uri Mahlab
AnswerMatlab: M-file
ip_02_08MATLAB.lnk
Example 2.8
Suppose that a white random process with samples{X(n)} is passed through linear filter with impulse response
h n( )
(0.95) , n
0, n 0
n 0
Determine the power spectrum of the output process{Y(n)}
Dr. Uri Mahlab
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 has the impulse response.
AnswerMatlab: M-file
ip_02_09MATLAB.lnk
h n( )
(0.9) , n
0, n 0
n 0
And is characterized by the input-output recursive(difference)equation.
y y xn n n 0 9 1. , 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)
Dr. Uri Mahlab
N=1000; % The maximum value of nM=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 signals.In communication the information-bearing signal is usually a lowpass random process that modulates a carrier for transmission over a bandpass communication channel. Thusthe 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 XS(t) and using these to modulate the quadrature carriers cos( 2 ) and sin 2 ,as shown in figure 7.f t0
f t0
+-
Figure 7: generation of a bandpass random process
AnswerMatlab: M-file
ip_02_10
MATLAB.lnk
Dr. Uri Mahlab
N=1000; % number of samplesfor i=1:2:N, [X1(i) X1(i+1)]=gngauss; [X2(i) X2(i+1)]=gngauss;end; % standard Gaussian input noise processesA=[1 -0.9]; % lowpass filter parametersB=1;Xc=filter(B,A,X1); Xs=filter(B,A,X2);fc=1000/pi; % carrier frequencyfor 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 processM=50;bpp_autocorr=Rx_est(band_pass_process,M);bpp_spectrum=fftshift(abs(fft(bpp_autocorr)));% plotting commands follow
