This paper studies a generalization of sparse superposition codes (SPARCs)
for communication over the complex additive white Gaussian noise (AWGN)
channel. In a SPARC, the codebook is defined in terms of a design matrix, and
each codeword is a generated by multiplying the design matrix with a sparse
message vector. In the standard SPARC construction, information is encoded in
the locations of the non-zero entries of the message vector. In this paper we
generalize the construction and consider modulated SPARCs, where information in
encoded in both the locations and the values of the non-zero entries of the
message vector. We focus on the case where the non-zero entries take values
from a phase-shift keying (PSK) constellation. We propose a computationally
efficient approximate message passing (AMP) decoder, and obtain analytical
bounds on the state evolution parameters which predict the error performance of
the decoder. Using these bounds we show that PSK-modulated SPARCs are
asymptotically capacity achieving for the complex AWGN channel, with either
spatial coupling or power allocation. We also provide numerical simulation
results to demonstrate the error performance at finite code lengths. These
results show that introducing modulation to the SPARC design can significantly
reduce decoding complexity without sacrificing error performance
