  Integrated sensing and communications (ISAC) is a spectrum-sharing paradigm
that allows different users to jointly utilize and access the crowded
electromagnetic spectrum. In this context, intelligent reflecting surfaces
(IRSs) have lately emerged as an enabler for non-line-of-sight (NLoS) ISAC.
Prior IRS-aided ISAC studies assume passive surfaces and rely on the
continuous-valued phase-shift model. In practice, the phase-shifts are
quantized. Moreover, recent research has shown substantial performance benefits
with active IRS. In this paper, we include these characteristics in our
IRS-aided ISAC model to maximize the receive radar and communications
signal-to-noise ratios (SNR) subjected to a unimodular IRS phase-shift vector
and power budget. The resulting optimization is a highly non-convex unimodular
quartic optimization problem. We tackle this problem via a bi-quadratic
transformation to split the design into two quadratic sub-problems that are
solved using the power iteration method. The proposed approach employs the
M-ary unimodular sequence design via relaxed power method-like iteration
(MaRLI) to design the quantized phase-shifts. Numerical experiments employ
continuous-valued phase shifts as a benchmark and demonstrate that our
active-IRS-aided ISAC design with MaRLI converges to a higher value of SNR with
an increase in the number of IRS quantization bits.
