Bearing　weak　fault　feature　extraction　under　time-varying　speed　conditions　is　a　challenging　task.　The　classic　time-frequency　analysis　(TFA)　based　ridge　detection　algorithms　cannot　work　well　for　detecting　time-varying　fault　characteristic　frequency　(FCF)　harmonics　of　the　bearing　due　to　the　limitation　of　time-frequency　resolution　and　noise　interference.　As　such,　a　novel　frequency　matching　demodulation　transform　(FMDT)　technique　is　developed　by　extending　the　generalized　demodulation　transform　for　bearing　weak　fault　feature　extraction　and　diagnosis　under　variable　speeds.　The　main　novelties　of　this　article　are　concluded　that:　the　FMDT　can　first　estimate　FCF　under　strong　noise　conditions　by　generating　a　series　of　demodulation　parameters　to　match　FCF　with　the　criterion　that　the　maximal　peak　can　be　captured　at　a　specific　frequency　from　the　spectra　of　the　demodulated　signals,　and　then　calculate　rotational　frequency　(RF)　harmonics;　and　the　proposed　FMDT-based　method　does　not　require　the　measured　bearing　RF,　and　can　directly　determine　bearing　fault　type　using　the　estimated　FCF　and　RF　harmonics.　The　effectiveness　of　the　FMDT-based　bearing　fault　diagnosis　method　is　verified　by　both　simulated　and　experimental　results.　Comparison　analysis　shows　that　the　proposed　method　can　generate　much　better　results　than　the　classic　TFA-based　ridge　detection　algorithms.