  An optimal dynamical decoupling of a quantum system coupled to a noisy
environment must take into account also the imperfections of the control
pulses. We present a new formalism which describes, in a closed-form
expression, the evolution of the system, including the spectral function of
both the environment and control noise. We show that by measuring these
spectral functions, our expression can be used to optimize the decoupling pulse
sequence. We demonstrate this approach with an ensemble of optically trapped
ultra-cold Rubidium atoms, and use quantum process tomography to identify the
effect of the environment and control noise. Our approach is applicable and
important for any realistic implementation of quantum information processing.
