The empirical mode decomposition is a powerful tool for analyzing nonlinear and nonstationary data. In this method, one of the main conceptual innovations is the introduction of intrinsic mode functions (IMFs), which arise as basic modes from the application of the empirical mode decomposition to signals. The beauty of this decomposition method is in its simplicity, adaptivity, and extraordinary effectiveness in many important and diverse settings. Because of its original algorithmic, recent works have contributed to its theoretical framework. Up to now, it is still challenging questions to establish the mathematical formulation for IMFs and answer whether the instantaneous frequency is always positive everywhere. In view of this, some developments have been made in recent years. Following these works, in this paper, we give the precise mathematical representation and characterization for the structures of some typical IMFs, the so‐called weak‐IMFs and those with constant amplitudes. Some necessary and sufficient conditions of IMFs are provided. Finally, it is answered that the instantaneous frequency of such an IMF is positive everywhere. Copyright © 2012 John Wiley & Sons, Ltd.