Physical realizable circuit structure for adaptive frequency Hopf oscillator

Abstract

Abstract-This paper presents a novel structure for the adaptive frequency Hopf oscillator where the nonlinear function is modified to make the system realizable using analog circuit components. Mathematical model is derived and it is shown using VHDL-AMS model that despite using a new nonlinear function, the oscillator exhibits the same characteristics as the original one. Our simulation results show the same learning behavior of this oscillator with improved learning time. Subsequently, an equivalent circuit model and transistor level implementation for the oscillator is suggested and the mathematical model is confirmed with system and circuit level simulations. I. INTRODUCTIONOver the years the theory of Nonlinear Oscillators has been used as a mathematical tool for modeling several scientific phenomena [1]- [7]. Despite complexity of the nonlinear oscillators, their wide range of applicability has made them a serious suggestion for future generation system design in electronics [8]. Recently, in [9] Buchli et al. developed a new type of nonlinear oscillator by modifying the original Hopf oscillator where, under an additive perturbation, the intrinsic frequency of the system evolves towards the frequency of the perturbation. For nonstationary signals it works in the analogous way to wavelet or Short Time Fourier Transform (STFT), which are essential techniques in time-frequency analysis methods. The present work is motivated by this work where we look at the proposed nonlinear oscillator from a circuit design point of view.In terms of circuit implementation, the oscillator proposed in [9] has two fundamental difficulties. Firstly, it still uses complex nonlinear functions which are extremely hard to realize in practice and secondly, the intrinsic frequency of the oscillator is considered as a system state, which in analog circuit design has to be represented as capacitor voltage (or inductance current). This linear relationship between the frequency and voltage implies that the voltage needs to be impractically high to represent a frequency greater than a few Hz meaning that the frequency basin will be too small for practical applications. These two practical problems inspire us to develop a novel circuit realizable nonlinear oscillator while staying within the framework of the same applications. Through simulations we show that the proposed function not only produces the same oscillation characteristics as the original one but also adapts the frequency at a faster rate. The second novelty of our work is that, we develop a method for extending the frequency basin significantly. Subsequently, we

On the VLSI Implementation of Adaptive-Frequency Hopf Oscillator

“…values in the Table I show that, in terms of jitter and convergence, differences between oscillators in [18] and our present work are negligible.…”

“…From this understanding of the system behavior, we propose to change the first parts of the both equations in (1) as (4) where is the new measure for the oscillation amplitude/ energy and is the sign value of and . Comparing with the oscillator presented in [18], two multipliers in the equation are replaced by sign functions, which has a very simple physical realization in analog circuit design (a comparator). The statespace representation of (4) is shown in Fig.…”

“…6 briefly compares the frequency jitters of two different versions of implementable oscillators (presented in [18] and here). It is observable from figures that oscillators presented here and in [18] have a similar jitter pattern. Jitter values are also calculated in Table I for these oscillators.…”

“…Some parts of this study have been reported in [18] in a short form, but here we present a detailed analysis on the limit cycle characteristics and nature of the convergence for the proposed oscillator. We have also proposed a new nonlinear function which is a modification to that presented in [18] to make the proposed oscillator more implementation friendly compared to those in [16] and [18].…”

“…We have also proposed a new nonlinear function which is a modification to that presented in [18] to make the proposed oscillator more implementation friendly compared to those in [16] and [18]. In addition a novel method for amplitude extraction of the constituent frequency components with their corresponding system/circuit level simulation results are presented.…”