This paper provides a generic analysis of the relationship between time- and frequency-domain disturbance observer (DOB) design methodology. It is discovered that the traditional frequency-domain DOBs using a low-pass filter with unity gain can only handle disturbances satisfying matching condition, while the traditional time-domain DOBs always generate an observer with a high-order. A functional disturbance observer (FDOB) is proposed to improve the existing results together with its design guideline, frequency analysis, and existence condition. Compared with the existing frequency-domain DOBs, the proposed FDOB can handle more classes of disturbances, while compared with the existing time-domain DOBs, the proposed FDOB can generate an observer with a lower-order. Numerical examples are presented to illustrate the main findings of this paper including a rotary mechanical system of nonminimum phase.