Home / Questions / Exercise 514 Consider a MISO slow fading channel 1 Verify that the Alamouti scheme ra...
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 L transmit antennas and channel gain vector h = [h1,..., hL]t . 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 Kx . Argue that the maximum achievable reliable rate of communication under this strategy is no larger than
(
log 1 +
ht Kx h \
N0
bits/symbol. (5.126)
2. Now suppose we are in a slow fading scenario and h is random and i.i.d. Rayleigh. The outage probability of the scheme in part (1) is given by
pout (R) = T
(
log 1 +
ht Kx h \
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 Kx to be diagonal. Hint: Observe that the covariance matrix Kx admits a decomposition of the form
U diag{P1,..., PL}U∗.
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