In　this　paper,　a　wavelet-based　periodic　group-sparse　signal　denoising　approach　is　proposed　for　detecting　faults　in　rotary　machines.　The　proposed　approach　exploits　group　sparsity　in　the　wavelet　domain.　For　this　purpose,　a　periodicity-induced　overlapping　group　shrinkage　technique　is　utilized　to　threshold　the　wavelet　coefficients.　The　wavelet　coefficients　are　obtained　by　using　the　tunable　Q-factor　wavelet　transform　to　decompose　the　measured　vibration　signals.　The　proposed　approach　is　constrained　to　promote　sparsity　more　strongly　than　convex　regularization　for　estimating　periodic　group-sparse　signals　in　noise,　while　avoiding　nonconvex　optimization.　In　addition,　this　maximally　sparse　convex　approach　has　the　advantage　of　preserving　the　oscillatory　behavior　of　the　useful　fault　features.　A　simulated　signal　is　formulated　to　verify　the　performance　of　the　proposed　approach　in　periodic　feature　extraction.　The　detection　performance　of　the　proposed　approach　is　compared　with　that　of　the　comparative　methods　via　root　mean　square　error　values.　Finally,　the　proposed　approach　is　applied　to　fault　diagnosis　of　both　experimental　cases　and　engineering　application.　The　processed　results　demonstrate　that　the　proposed　feature　extraction　technique　can　effectively　detect　the　fault　features　from　heavy　background　noise.