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Exercise 514 Consider a MISO slow fading channel 1 Verify that the Alamouti scheme radiates energy in an isotropic manner 2 Show that a transmit diversity scheme radiates energy in an is

Exercise 5.14 Consider a MISO slow fading  channel.

1.    Verify that the Alamouti scheme radiates energy in an isotropic manner.

2.    Show that a transmit diversity scheme radiates energy in an isotropic manner if and only if the signals transmitted from the antennas have the same power and are uncorrelated.

Exercise 5.15 Consider the MISO channel with transmit antennas and channel gain vector [h1,..., hL]. The noise variance is N0 per symbol and the total power constraint across the transmit antennas is  P.

1.    First, think of the channel gains as fixed. Suppose someone uses a transmission strategy for which the input symbols at any time have zero mean and a covariance matrix K. Argue that the maximum achievable reliable rate of communication under this strategy is no larger than

 

(

log   1 +

 

hK\

N0

 

 

bits/symbol.                        (5.126)

 

 

2.    Now suppose we are in a slow fading scenario and is random and i.i.d. Rayleigh. The outage probability of the scheme in part (1) is given by

 

 

pout (R) = T

 

     (

log   1 +

 

hK\

N0

 

 .                  (5.127)

 

Show that correlation never improves the outage probability: i.e., given a total power constraint P, one can do no worse by choosing Kto be diagonal. Hint: Observe  that  the  covariance  matrix  Kx   admits  a  decomposition  of  the   form

diag{P1,..., PL}U∗.

 

 

 

 

Jul 31 2020 View more View Less

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