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Iterative processing for linear matrix channels, aka turbo equalization, turbo demodulation, or turbo code-division multiple access (CDMA), has traditionally been addressed as the concatenation of conventional error control codes with the linear (matrix) channel. However, in several situations, such as CDMA, multiple-input-multiple-output (MIMO) channels, orthogonal frequency-division multiplexing (OFDM), and intersymbol-interference (ISI) channels, the channel itself either contains inherent signal redundancy or such redundancy can readily be introduced at the transmitter. For such systems, iterative demodulation of the linear channel exploiting this redundancy using simple iterative cancellation demodulators, followed by conventional feedforward error control decoding, provides a low-complexity, but extremely efficient decoding alternative. This two-stage demodulator/decoder outperforms more complex turbo CDMA methods for equal power modes (users). Furthermore, it is shown that arbitrary numbers of modes can be supported if an unequal power distribution is adopted. These power distributions are nested, which means that additional modes can be added without disturbing an existing mode population. The main result shows that these nested power distributions enable the two-stage receiver to approach the Shannon capacity of the channel to within less than one bit for any signal-to-noise ratio (SNR).