The present invention relates generally to separation of ions based on their transport properties. More particularly, the invention relates to separation of ionic mixtures and characterization of ions in gases using higher-order differential ion mobility spectrometry (HODIMS) enabled by asymmetric waveforms of fundamentally new types. The invention discloses a method and apparatus for separation of ionic mixtures and characterization, identification, or quantification of ions in a gas based substantially on the terms of third or higher order in a series expansion of ion mobility as a function of electric field intensity. This is achieved using a periodic, time-dependent electric field with novel waveform profiles that cancel or substantially reduce the contributions to time-averaged ion motion of the leading n (where n.gtoreq.2) terms of that expansion, thereby achieving ion separations based substantially on the (n+1)th term. Separations using HODIMS with different n are expected to be highly orthogonal, enabling multidimensional separations employing HODIMS analyzers of different orders. The expected high orthogonality between HODIMS and mass spectrometry or ion mobility spectrometry would make HODIMS/MS and HODIMS/IMS combinations powerful analytical tools of broad utility.