The received signal vector in a frequency-nonselective AWGN MIMO channel with NT transmit antennas and NR receive antennas is given by Equation 15.2–7 as
a. Use the SVD to transform the received signal vector to the form
where Σ is a diagonal matrix of rank r with the nonzero diagonal elements equal to the singular values of the channel matrix H.
b. Show that if the elements of η are statistically iid, zero-mean, complex-valued Gaussian random variables, then the elements of η are also iid zero-mean complex-valued Gaussian random variables.
c. Show that the capacity of the AWGN MIMO channel may be expressed as
where P1, P2,…, Pr are the allocated powers based on the water-filling criterion with the total power constraint