TABLE OF CONTENTS The optimal coding and decoding matrices have been obtained using a completely general setting. The optimal solutions will be now particularized to specific operative conditions, in order to associate a physical meaning with the solutions found before.
In case of SISO time-invariant channels, using block transmissions, with blocks of size N, and a CP of length at least equal to the channel order, the channel matrix assumes the structure of (3.17). Then, it admits an eigen-decomposition as in (3.18), with eigenvalues equal to the values of the channel transfer function. In case of white noise, we have
| (6.90) |  |
where H( k) = 
h( l) e - j 2 ?kl / N.
Equation (6.90) allows us to interpret the power allocation 
resulting from the optimization strategies studied before as a power distribution function of the discrete frequency k/ N. In the limit as the block length increases, the discrete power allocation 
tends to a function P( f) of the continuous variable f that assumes the meaning of a frequency. Hence, using (6.90), we can write the asymptotic (as the blocklength goes to infinity) power spectral density P( f) of the transmitted signals, according to different optimization criteria as follows.
MMSE with average power constraint
From (6.49), we get
| (6.91) |  |
where [ x] + ? max( x, 0), and the coefficient ?
