  This paper proposes a min-max design of noise-shaping delta-sigma modulators.
We first characterize the all stabilizing loop-filters for a linearized
modulator model. By this characterization, we formulate the design problem of
lowpass, bandpass, and multi-band modulators as minimization of the maximum
magnitude of the noise transfer function (NTF) in fixed frequency band(s). We
show that this optimization minimizes the worst-case reconstruction error, and
hence improves the SNR (signal-to-noise ratio) of the modulator. The
optimization is reduced to an optimization with a linear matrix inequality
(LMI) via the generalized KYP (Kalman-Yakubovich-Popov) lemma. The obtained NTF
is an FIR (finite-impulse-response) filter, which is favorable in view of
implementation. We also derive a stability condition for the nonlinear model of
delta-sigma modulators with general quantizers including uniform ones. This
condition is described as an H-infinity norm condition, which is reduced to an
LMI via the KYP lemma. Design examples show advantages of our design.
