16 ELE4606 – Communication systems Problem Cellular telephony has forever changed the way people...

Question

16 ELE4606 – Communication systems Problem Cellular telephony has forever changed the way people communicate with one another, being one of those exciting technologies that has directly and significantly influenced our everyday life. The characteristics of wireless signal changes as it travels from the transmitter antenna to the receiver antenna. These characteristics depend upon the distance between the two antennas, the path(s) taken by the signal, and the environment (buildings and other objects) around the path. The profile of received signal can be obtained from that of the transmitted signal if we have a model of the medium between the two. This model of the medium is called channel modeling. Rayleigh fading is a statistical model for the effect of a propagation environment on a radio signal, such as that used by wireless devices. Rayleigh fading is a reasonable model when there are many objects in the environment that scatter the radio signal before it arrives at the receiver. Rayleigh fading is exhibited by the assumption that the real and imaginary parts of the response are modelled by independent and identically distributed zero-mean Gaussian processes so that the amplitude of the response is the sum of two such processes. Question 1 Random variable R is Rayleigh distributed if 2 2 R ? ? X Y , where 2 X N ? (0, ) ? and 2 Y N ? (0, ) ? are independent normal random variables. Derive theoretically the probability density function (PDF) and cumulative density function (CDF) of the Rayleigh distribution as well as its amplitude, and plot the figures in MATLAB. Question 2 If a channel is not changing with time, it does not fade and instead remains at some particular level. Separate instances of the channel in this case will be uncorrelated with one another, owing to the assumption that each of the scattered components fades independently. Once relative motion is introduced between any of the transmitter, receiver, and scatterers, the fading becomes correlated and varying in time. The normalised autocorrelation function of a Rayleigh faded channel with motion at a constant velocity is a zeroth-order Bessel function of the first kind: XXXXXXXXXXR o d ?? ? ? J f at delay ? when the maximum doppler shift is df . The Jakes model is a well-known and popularly used channel model in simulating a Rayleigh fading channel. The Jakes model has the following characteristics: ? approximate the Rayleigh fading process by summing a set of complex sinusoids; ? the sinusoids are weighted so as to produce an accurate approximation of desired channel Doppler spectrum; © University of Southern Queensland ELE4606 – Communication systems 17 ? Jakes shows that the theoretical Doppler spectrum for the isotropic scattering mobile radio channel can be well approximated by a summation of relatively small number of sinusoids with frequencies and relative phases of the sinusoids set according to a specific formulation. Write a MATLAB program to simulate a wireless channel based upon the Jakes model with Doppler shifts of 10 Hz, 100 Hz, and 1000 Hz. You should plot the amplitude of the wireless fading and compare with the theoretical results in Question 1. Comment on the comparison. Allocation of marks Question 1 50 Question 2 150 Total 200 NB: the discussion should demonstrate that you can interpret the waveforms. Additional resources You may find the following references useful in completing this assignment: [1] http://en.wikipedia.org/wiki/Rayleigh_fading#Jakes.27_model [2] Rapport, TS 2002, Wireless Communications Principle and Practice, Prentice Hall, New Jersey. [3] Turin, W, Rittwik, J, Martin, C & Winters, J 2001, Modeling Wireless Channel Fading, IEEE , 2001. [4] Steele, R & Hanzo, L 1999, Mobile Radio Communications, Wiley. [5] Patzold, M 2002, Mobile Fading Channels, John Wiley. [6] Jakes, WC 1974, Microwave Mobile Communications, Wiley, New York. ________________ End of Assignment © University of Southern Queensland

David · Accepted Answer

Question  1 
Random variable R is Rayleigh distributed if 22 YXR   , where  X~N(0, σ2 ) 
andY ~N(0, σ2 ) are independent normal random variables. Derive theoretically 
the probability density function (PDF) and cumulative density function (CDF) of 
the Rayleigh distribution as well as its amplitude, and plot the figures in 
MATLAB. 
Solution 
 
 
Probability Density Function for random variable X  is 
X ~ (0, 2)= 
2
2
2
2
1
)( 

x
X exf

  
And PDF for random variableY  is 
Y ~  (0, 2)= 
2
2
2
2
1
)( 

y
Y eyf

  
As the random variables X & Y are independent to each other there joint PDF is to be given 
by 
2
2
2
2
1
),( 

x
XY eyxf

 .
2
2
2
2
1


y
e

 
 22
22
1
22
1
),(
yx
XY eyxf


 

 
Joint probability that the RV X  lies in between x  and dxx   and RV Y lies in between y  
and dyy   is given by 
 
dxdyedyyYydxxXxP
yx 22
22
1
22
1
),(


 

 
Substituting  


Sinry
Cosrx
.&
.


                                      









x
y
yxr
1
22
tan  
As we know that area ddrrdydx ...   (Conversion from Cartesian Coordinate System to 
Polar Coordinate) which implies that we can have 
 
),( dyyYydxxXxP  = ),(  ddrrRrP   
 
So now we  have 


  drde
r
ddrrRrP
r
2
2
2
22
),(

  
The equivalent joint PDF can be given as  
2
2
2
2, 2
),( 

r
R e
r
rf

  
And as random variables R and   are independent to each other  their independent PDFs 
can be given as 


2
1
)( f     
2
2
2
2
)( 

r
R e
r
rf

        0, r  
 
Cumulative Density Function (CDF) of random variable is given by 
 






r
r
R
r
Circle
R
R
drde
r
rF
drde
r
rF
rYXPrRPrF
0
2
0
2
2
2
2
22
2
2
2
2
2
)(
2
)(
)()()(











 

2
00
2
2 2
1
.)(
2
2
ddre
r
rF
r r
R
2
2
21)( 
r
R erF

 0, r  
 
 MATLAB code for generating Rayleigh PDF 
 
x=0:0.1:6; 
y=raylpdf (x,1); 
plot (x,y); 
title ('RAYLEIGH PDF'),xlabel('Random Values'),ylabel('f(r)'); 
text(3,0.66,'Scale Parameter=1'); 
grid on;
Output: 
 
 
 Figure-1
MATLAB code for generating Rayleigh CDF 
 
x=0:0.1:10; 
y=raylcdf (x,1); 
plot (x,y); 
title ('RAYLEIGH CDF'),xlabel('Random Values'),ylabel('F(r)'); 
text(6.3,0.66,'Scale Parameter=1'); 
grid on;
Output: 
 
 
 Figure-2 
 
  
Question  2 If a channel is not changing with time, it does not fade and instead remains at 
some particular level.

16 ELE4606 – Communication systems Problem Cellular telephony has forever changed the way people communicate with one another, being one of those exciting technologies that has directly and...

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