  The fundamental assumption of dynamical decoupling based noise spectroscopy
is that the coherence decay rate of qubit (or qubits) driven with a sequence of
many pulses, is well approximated by the environmental noise spectrum spanned
on frequency comb defined by the sequence. Here we investigate the precise
conditions under which this commonly used spectroscopic approach is
quantitatively correct. To this end we focus on two representative examples of
spectral densities: the long-tailed Lorentzian, and finite-ranged
Gaussian---both expected to be encountered when using the qubit for nano-scale
nuclear resonance imaging. We have found that, in contrast to Lorentz spectrum,
for which the corrections to the standard spectroscopic formulas can easily be
made negligible, the spectra with finite range are more challenging to
reconstruct accurately. For Gaussian line-shape of environmental spectral
density, direct application of the standard dynamical decoupling based
spectroscopy leads to erroneous attribution of long-tail behavior to the
reconstructed spectrum. Fortunately, artifacts such as this, can be completely
avoided with the simple extension to standard reconstruction method.
