Adjacent stages of a shift register are rising-edge-triggered D flip-flops with Q_i…

Adjacent stages of a shift register are rising-edge-triggered D flip-flops with \(Q_i\) connected to \(D_{i+1}\) through a routing delay \(t_{\mathrm{rt}}=0.40\,\mathrm{ns}\) and no other logic. Flip-flop parameters are \(t_{\mathrm{co,max}}=2.80\,\mathrm{ns}\), \(t_{\mathrm{co,min}}=1.10\,\mathrm{ns}\), \(t_{\mathrm{su}}=1.50\,\mathrm{ns}\), and \(t_{\mathrm{h}}=1.20\,\mathrm{ns}\). Clock skew is zero. (a) Determine the minimum clock period and the maximum clock frequency from the setup constraint. (b) Determine the hold slack with the given \(t_{\mathrm{rt}}\). (c) If the routing delay is reduced to \(0\), determine the new hold slack and state whether the register still operates correctly. (d) Determine the smallest routing delay \(t_{\mathrm{rt,min}}\) that restores a hold slack of zero when \(t_{\mathrm{h}}=1.20\,\mathrm{ns}\) and \(t_{\mathrm{co,min}}=1.10\,\mathrm{ns}\).

A shift-register stage is a pipeline with routing as its only logic delay; hold is the usual risk when that delay is too small.