In this paper, we derive state equations for linearized discrete-time models of forth-order charge-pump phase-locked loops. We solve the differential equations of the loop filter by using the initial conditions and the boundary conditions in a period. The solved equations are linearized and rearranged as discrete-time state equations for checking stability conditions. Some behavioral simulations are performed to verify the proposed method. By examining the stability of loops with different conditions, we also propose an expression between the lower bound of the reference frequency, the open loop unit gain bandwidth, and the phase margin.

Using a pair of matched square-root-raised-cosine (SRRC) filters in the transmitter and the receiver in a band-limited digital communication system can theoretically achieve zero inter-symbol interference (ISI). In reality, the ISI cannot be zero when both SRRC filters are approximately implemented because of some numerical precision problems in the design phase as well as in the implementation phase. In this paper, the author proposes an iterative method to design the coefficients of SRRC FIR filters. The required ISI of the system can be specified such that both ISI and frequency domain specifications are monitored in the design phase. Since the ISI can be specified beforehand, the tradeoff between performance and the filter length becomes possible in the proposed design algorithm.

This article presents a three-step method for designing linear phase FIR filters with signed-powers-of-two (SPT) coefficients. In Step one, a prototype optimal FIR filter is designed by the Remez exchange algorithm. In Step two, a scaling factor is selected by employing simple ad-hoc rules. In Step three, each coefficient of the prototype filter is scaled by the scaling factor and is quantized coarsely as the canonic-signed-digit (CSD) representation. Then, a mixed-integer-linear-programming (MILP) algorithm is applied to three least significant digits (LSDs) of the filter's coefficients to reduce the number of SPT terms. Design examples demonstrate that the proposed algorithm is able to produce linear phase fixed-point FIR filters using fewer SPT terms than the existing methods under the same normalized peak ripple magnitude (NPRM) specification.