This paper describes a digital delay-lock Loop (DLL) to which delta-sigma (Δ Σ) modulation technique is applied in order to reduce circuit elements. The DLL is evaluated in both transient and steady-state behavior by theoretical analysis, computer simulations and circuit experiments. Not deteriorated by the internally generated Δ Σ-modulation noise, the DLL shows good tracking performance in transient response and steady-state RMS jitter of phase error against additive white Gaussian noise. Using the proposed DLL most parts of receiving circuits are realized by digital integrated circuits. After realizing the circuit, power-line communication system with spread spectrum is possibly expected in a small size with low cost.

We propose a method to search all the zeros of a complex function in a given compact region D Cn. The function f: Cn Cn to be considered is assumed to consist of polynomial and transcendental terms and to satisfy f (x) Rn for any x Rn. Using the properties of such a complex function, we can compute the number of zeros and determine the starting points of paths on the boundary of D, which attain all the zeros of f in D without encountering a singular point. A piecewiselinear approximation of the function on a triangulation is used for both computing the number of zeros and following the paths.

It is important to analyze a tracking or synchronizing process in Spread Spectrum (SS) receiving system. The most common SS tracking system considered here consists of pseudorandom (PN) generator, Lowpass Filter (LPE) and Voltage Controlled Oscillator (VCO). The SS receiver is to track or synchronize its local PN generator to the received PN waveform by VCO. The fundamental equation of the system is known by a second order nonlinear differential equation in terms of phase difference between local PN generator and received PN waveform. The differential equation is nonautonoumous due to PN function of time t with period T. Picking up the gain of VCO as the main parameter in the system we show that the system has bifurcation from the normal oscillation through subharmonic oscillation to finally chaos. In the final case, chaos is confirmed by investigating maximum Liapunov number and both stable and unstable manifolds.