  The creation of composite quantum gates that implement quantum response
functions $\hat{U}(\theta)$ dependent on some parameter of interest $\theta$ is
often more of an art than a science. Through inspired design, a sequence of $L$
primitive gates also depending on $\theta$ can engineer a highly nontrivial
$\hat{U}(\theta)$ that enables myriad precision metrology, spectroscopy, and
control techniques. However, discovering new, useful examples of
$\hat{U}(\theta)$ requires great intuition to perceive the possibilities, and
often brute-force to find optimal implementations. We present a systematic and
efficient methodology for composite gate design of arbitrary length, where
phase-controlled primitive gates all rotating by $\theta$ act on a single spin.
We fully characterize the realizable family of $\hat{U}(\theta)$, provide an
efficient algorithm that decomposes a choice of $\hat{U}(\theta)$ into its
shortest sequence of gates, and show how to efficiently choose an achievable
$\hat{U}(\theta)$ that for fixed $L$, is an optimal approximation to objective
functions on its quadratures. A strong connection is forged with
\emph{classical} discrete-time signal processing, allowing us to swiftly
construct, as examples, compensated gates with optimal bandwidth that implement
arbitrary single spin rotations with sub-wavelength spatial selectivity.
