The novel architecture of the center-fed pinching antenna system (C-PASS) is investigated, where the waveguide-fed signal is divided into two propagation directions through controllable power splitting. By doing so, a doubled degree of freedom (DoF) is achieved compared to conventional PASS. Based on the new designed basic signal model of C-PASS, three practical operating protocols for C-PASS are proposed, namely power splitting (PS), direction switching (DS), and time switching (TS). Then, the sum-rate maximization problem for the joint optimization of transmit and pinching beamforming is formulated for each of the proposed protocols. 1) For PS, the highly coupled non-convex problem is first transformed into a tractable form via the weighted minimum mean square error reformulation and solved using the alternating optimization framework; 2) For DS, the above approach is subsequently extended to solve the mixed-integer constraints inherent for DS via the penalty-based algorithm; 3) For TS, the optimization problem can be decomposed into two subproblems and solved using the similar iterative techniques, while its optimal time allocation ratio is derived in closed form. Finally, numerical results reveal that TS is superior in the low-power regime, while PS and DS achieve significantly higher rates in the high-power regime due to the enhanced DoF.