In electronic devices, the 'signal filter' allows the signal at specific passband while blocks the signal and noise at other frequency bands. The filter function formalism explicitly shows how the control pulse sequences perform as signal filters against the environmental noise employed on the quantum systems.
Works for sufficiently weak noise, as long as noise is sufficiently weak for low-order approximations to be viable.
Considering the general Hamiltonian to make problem explicit:
$$ H = \left(\begin{matrix} h+\delta h(t) & J+\delta J(t)\\ J+\delta J(t) & -h-\delta h(t) \end{matrix}\right), $$
which expresses a two-level quantum system (single-qubit) without external drive control ($H_{d}=0$) but undergoing time-dependent random noise. The $\delta J(t)$ and $\delta h(t)$ are terms of relaxation and pure dephasing, respectively.
How to enhance dephasing time in superconducting qubits
$$ \ket{0}=\left(\begin{matrix} 1\\ 0 \end{matrix}\right), \space \ket{1}=\left(\begin{matrix} 0\\ 1\end{matrix}\right)
$$
The frequency is varied as the vibration of the noisy environment:
$$ H = -\frac{1}{2}\left[\Omega+\beta(t)\right]\sigma_{z}, $$
where $\beta(t)$ is the classical stochastic noise term. This term implies that qubits couple to a random classical temporally fluctuating field. Such noise is, in fact, the major source of quantum dephasing in superconducting qubits.
Initial state: $\ket{\psi(0)}=a\ket{0}+b\ket{1}$.