Assignment:
In orthogonal frequency division multiplexing (OFDM) communication systems such as those used for digital subscriber lines (ADSL, HDSL, VDSL), the channel can be thought of as the parallel of N independent subchannels. Each subchannel m is AWGN with its own bandwidth Bi, power gain gi and noise PSD Ni/2, and has a capacity Ci that depends on its transmit power Mi. A possible aim in designing such systems is to distribute the total transmit power MTx among the subchannels so that the total channel capacity is maximized, that is
max {Mi≥0} ΣNi=1 Ci, under the constraint ΣNi=1 Mi ≤ MTx
a) For N = 2 subchannels find the optimal values of M1 and M2 as a function of the parameters {Bi, gi, Ni} and total power MTx. Plot the capacity C as a function of M1 for
B1 = 4MHz, B2 = 1 MHz, g1 = -40 dB, g2 = -45 dB, N1= N2 10-10 V2/Hz, MTx = 1V2.
Repeat the plot in the case the gains are exchanged,
g1 = -45dB, g2 = -40dB.
b) For general N, prove that if all subchannels have the same bandwidth Bi = B and
Ni/gi ≤ P0, with P0 = 1/N (MTx/B + ΣNi=1 Ni/gi)
the optimal solution is
Mi = B (P0 - Ni/gi)