Time　optimal　control　(TOC)　based　tracking　differentiator　(TD)　was　first　proposed　by　Han　as　a　practical　solution　to　avoid　setpoint　jump　in　active　disturbance　rejection　control.　In　practice,　the　discrete-time　optimal　control　(DTOC)　is　implemented　in　the　form　of　state　feedback　for　a　double-integral　system,　which　is　widely　used　to　design　controllers,　observers　and　noise-tolerant　differentiators.　The　convergence　of　the　DTOC-TD,　however,　has　not　been　fully　understood.　This　article　provides　a　rigorous　full　convergence　analysis　of　the　DTOC-TD.　It　then　illustrates　the　frequency-domain　characteristics　of　this　DTOC-TD　in　signal-tracking　filtering　and　differentiation　acquisition,　giving　a　rule　of　thumb　for　regulating　the　parameters.　Finally,　the　case　studies　including　comparison　simulations　and　experiments　on　processing　gap　sensor＇　signals　in　the　suspension　system　of　maglev　train　are　carried　out　to　verify　the　effectiveness　of　the　DTOC-TD.