Correlation Consider independent, zero mean, same variance  2.

 Correlation Consider independent, zero mean, same variance  2.

Correlation Consider independent, zero mean, same variance  2.

Correlation1. Multipah channel models2. HW Channel Parameters3 System Design

Consider independent, zero mean, same variance  2 Gaussian random variables ( ) i x k for
every k and i=1,2. Define:( 1) 1 2 ( ) for 1 ( )
( ) for 1i i
ii
y k y k x k kx k k
         
for i=1 or 2 and for every k1 1
22 12 1 12 2
( ) ( )( ) ( ) 1 ( )
k y kk k yk
   
 
   for every k
a.) Find the mean and variance of ( ) i  k for i=1 or 2 and for every k and briefly proveyour results.
b.) Find the pdf of ( ) i  k for i=1 or 2 and for every k.c.) Find  ( ) ( ) i j E  k  l for i=j and k  l. Briefly prove your results.
d.) Find  ( ) ( ) i j E  k  l for i  j and k=l. Briefly prove your results.e.) Find  ( ) ( ) i j E  k  l for i  j and k  l. Briefly prove your results.
Sol:

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2 CholeskyConsider independent, zero mean, same variance  2 Gaussian random variables
( ) i x m for every m and i=1,2,…,I . Define vectors1 2 ( ) [ ( ), ( ),…, ( )] I x m  x m x m x m ,
1 2 ( ) [ ( ), ( ),…, ( )] I y m  y m y m y m1 2 ( ) [ ( ), ( ),…, ( )] I  m   m  m  m
where( 1) 1 2 ( ) for m 1 ( )
( ) for m 1i i
ii
y m y m x mx m
         
for every m and i=1,2,…,I (m)  y(m)R for every m and Rt R is a Hermition matrix.
a.) Find the mean and variance of ( ) i  m for every i and every m and briefly prove yourresults.
b.) Find the pdf of ( ) i  m for every i and every m.c.) Find  ( ) ( ) i j E  k  l for i=j and k  l. Briefly prove your results.
d.) Find  ( ) ( ) i j E  k  l for i  j and k=l. Briefly prove your results.e.) Find  ( ) ( ) i j E  k  l for i  j and k  l. Briefly prove your results.
Sol:

3 Probabilistic Transmission Loss Model (Optional)Prob.a:
Generate one realization of the following path lossPL(d)= PL1(d)+ PL2(d)+ PL3(d) for d=1 + 0.2*k with k=0,1,2,…..,2500
wherePL1(d)= PLref+10 n log10(d) with PLref=10dB and Slop n=5
PL2(d)= s(d)PL3(d)= 10 log10(y(d))
Shadowing:s(d) zero mean Gaussian random variable with s=2dB
Correlation between s(d) and s(d+) is exp(-/D)Where the correlation distance D is 100m.
Fast fading:the pdf of y is (e-y/T)/T when y0. T=10.
Here y(d) and y(d+) are not correlated.Plot PL(d)and PL3(d) for k=0,1,2,…..,200.
Plot PL1(d)and PL2(d) for k=0,1,2,…..,2500.Prob. b:
Assume that you don’t know any of the parameters, develop a procedure to findapproximate values for PL1(d), PL2(d) and PL3(d)from PL(d) if it is possible. (Hint: You
don’t need to find the environmental parameters: PLref, n, s, D and T. Just find PL1(d),
PL2(d) and PL3(d).)Prob. c:
Use the results from Parb. a to generate 100 realizations of PL(d) with d=1 + 0.2*k withk=0,1,2,…..,2500. Assume that you don’t know any of the parameters, develop a
procedure to derive the environmental parameters: PLref, n, s, D and T.

A transmitter transmits a signal s(t) with carrier frequency fc. The signal reaches the
receiving antennas through three paths. The path gains are a1, a2, and a3. The path delaysare t1, t2, and t3. The Doppler shifts at the carrier frequency are f1, f2, and f3. What is the
received signal? What is the transfer function characterizing the channel between thereceiving and transmitting antenna?
Ans:

Carrier frequency is 1 GHz. Moving speed is 2 m/sec. In the following, what are the
reasonable numbers in indoors for the following parametersa.) coherent bandwidth:
b.) coherent time:c.) delay spread:
d.) Doppler spread:Choose (just pick one value for each parameter) your answers from the following values:
1.) 1 sec; 2.) 0.1 sec; 3.) 0.01 sec; 4.) 1 msec; 5.) 0.1 msec; 6.) 0.01 msec; 7.) 1 sec; 8.)0.1 sec; 9.) 0.01 sec; 10.) 1 nsec; 11.) 1 GHz; 12.) 0.1 GHz; 13.) 0.01 GHz; 14.) 1
MHz; 15.) 0.1 MHz; 16.) 0.01 MHz; 17.) 1 KHz; 18.) 0.1 KHz; 19.) 0.01 KHz; 20.) 1 Hz;Sol:

The delay spread in a given environment TD is 5 micro seconds.
The maximum Doppler spread FD is 200Hz.We would like to design a digital communication system in this environment such that
1. Distortion per data symbol is negligible2. The time variation during a symbol period is negligible
What is the appropriate range of the symbol duration in this communication system?Ans: