Perfusion imaging of the brain via Dynamic Susceptibility Contrast MRI (DSC-MRI) allows tissue perfusion characterization by recovering the tissue impulse response function and scalar parameters such as the cerebral blood flow (CBF), blood volume (CBV), and mean transit time (MTT). However, the presence of bolus dispersion causes the data to reflect macrovascular properties, in addition to tissue perfusion. In this case, when performing deconvolution of the measured arterial and tissue concentration time-curves it is only possible to recover the effective, i.e. dispersed, response function and parameters. We introduce Dispersion-Compliant Bases (DCB) to represent the response function in the presence and absence of dispersion. We perform in silico and in vivo experiments, and show that DCB deconvolution outperforms oSVD and the state-of-the-art CPI+VTF techniques in the estimation of effective perfusion parameters, regardless of the presence and amount of dispersion. We also show that DCB deconvolution can be used as a pre-processing step to improve the estimation of dispersion-free parameters computed with CPI+VTF, which employs a model of the vascular transport function to characterize dispersion. Indeed, in silico results show a reduction of relative errors up to 50% for dispersion-free CBF and MTT. Moreover, the DCB method recovers effective response functions that comply with healthy and pathological scenarios, and offers the advantage of making no assumptions about the presence, amount, and nature of dispersion.