Direct numerical simulation (DNS) is carried out to study turbulent flow over irregular rough surfaces in periodic minimal channels. A passive scalar transport equation is solved to study heat transfer over rough surfaces with the Prandtl number $Pr$= 0.7. The generation of irregular roughness is based on a mathematical randomization algorithm, in which the power spectrum (PS) of the roughness height function along with its probability density function (PDF) can be directly prescribed. The hydrodynamic and thermal properties of the roughness, particularly the roughness function ($\Delta U^+$) as well as the temperature profile offset ($\Delta \Theta^+$), are compared with those obtained from a full span DNS for 6 types of roughness topographies with systematically varied PDF and PS configurations at Re$\tau \approx$ 500. The comparison confirms the ability of the minimal channel approach to perform characterization of irregular rough surfaces providing excellent agreement (within 5$\%$) in $\Delta U^+$ and $\Delta \Theta^+$ across various types of roughness topographies. Results also indicate that random realizations of roughness, with a fixed PS and PDF, translate to similar prediction with a narrow scatter. ... mehrFinally , the impact of systematically varied roughness PDF and PS to both velocity and temperature profile is shown. It is also demonstrated that the influence of varying PDF and PS on the velocity and temperature fields is different.