The capacity of bandlimited direct-detection channels is challenging to compute or approach due to the receiver non-linearity. A generalized vector approximate message passing (GVAMP) detector is designed to achieve high rates at a reasonable level of complexity. The rates increase by using multi-level coding and successive interference cancellation. The methods are applied to fiber-optic channels with intersymbol interference caused by spectrally efficient pulse shapes, chromatic dispersion, and receiver sampling at twice the baud rate. Bipolar modulation operates within 0.26 bits per channel use (bpcu) of the real-alphabet coherent capacity for optically amplified links, reducing the best-known theoretical gap of 1 bpcu. Remarkably, bipolar modulation achieves 6 dB and 3 dB of power gain over unipolar modulation with and without optical amplification, respectively. Simulations with polar-coded modulation confirm the gains. The GVAMP complexity, measured in multiplications per information bit (mpib), is proportional to the number of iterations and to the logarithm of the block length, and is substantially lower than that of other equalizers. For example, a system with 64-ary bipolar modulation and a root-raised cosine pulse with a 1% roll-off factor was simulated over 4 km of optically amplified standard single-mode fiber in the C-band. The GVAMP receiver requires 93 mpib to achieve 5 bpcu at 300 gigabaud.

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