Multi-tone Excitation Research Articles

On the Cyclostationary Noise Analysis in Large RF Integrated Circuits

This paper examines the issue of noise analysis in RF integrated circuits (RFICs). The complexity of the RFIC grows constantly, with transistor counts now into the thousands or tens of thousands. Since RFICs are driven by large-signal multitone excitations, the stochastic noise process in these circuits is cyclostationary. This combination of large device count and noise cyclostationarity makes noise analysis in RFICs a time-consuming task. Simulation algorithms that minimize computer memory consumption and computation cost are thus necessary. There is a consensus now that harmonic balance is the most appropriate circuit equation solution method for multitone analysis of RF circuits. The paper thus gives a comprehensive description of a harmonic balance-based algorithm for computing the circuit noise response, showing how it effectively accounts for all types of cyclostationary noise sources supported by modern compact device models. The algorithm has been implemented in a commercial simulator and shows good capabilities for the analysis of full transceiver circuits within reasonable computer memory occupancy and simulation times.

Calibrated non-linear vector network measurement without using a multi-harmonic generator

The current generation of non-linear vector network analysers (NVNA) relies on a multi-harmonic generator to establish an absolute phase reference. Such a reference may not be readily available or, more importantly, may impose limitations on tone spacing in multi-tone testing for one type of NVNA based on a four-port VNA. The latter is a severe limitation when characterising memory behaviour in RF non-linear circuits. Here a hybrid NVNA is proposed that combines a sampling oscilloscope, a four-port VNA and phase-locked signal generators. A signal generator, phase locked to generators that are combined as the multi-tone excitation, provides the relative phase reference during swept power measurements using the VNA. The high dynamic range VNA-based measurements are corrected in absolute phase using the oscilloscope-based NVNA measurement at a specific power level. The hybrid NVNA has high dynamic range and the lower bound on tone spacing is determined by the memory depth of the oscilloscope, the windowing effect of the VNA receivers and the phase noise of the multi-tone sources. At 2 GHz, the hybrid NVNA has a dynamic range of 40 dB at 200 Hz tone spacing for a two-tone signal, increasing to more than 80 dB for tone spacings of 200 kHz and greater.

Parameter estimation of Duffing oscillator using Volterra series and multi-tone excitation

Use of Volterra series in nonlinear system identification is well established now. The series represents response of a nonlinear system in a functional series form consisting of convolution integrals involving higher-order impulse response functions known as Volterra kernels. Multi-dimensional Fourier transforms of these Volterra kernels give the higher-order frequency response functions (FRFs). The measurement of these FRFs under harmonic excitation and their relationship with the first-order FRFs provide a basis for estimation of the nonlinear parameters. However, most of the methods employ single-tone excitation, which provide limited FRF measurement in a single experiment. In the present study, a novel procedure based on multi-tone excitation is presented for a typical Duffing oscillator and it is demonstrated that accurate estimation of both nonlinear and linear parameters is possible with fewer number of experiments.

Impact of Group Delay on the Load Emulation in Multi-Tone Active Envelope Load-Pull System

Power amplifiers used in modern day wireless communication systems need to be characterised under complex excitations, into variable load impedances, for the determination of optimum matching conditions. Envelope load-pull technique provides the most realistic possibility for impedance emulation under complex modulation. The system developed for continuous wave modulation, when used under multi-tone excitation, experiences group delay in the feedback loop. This paper explains the impact of group delay on the emulated load and a proposal to overcome the inherent group delay in the feedback loop.

Analytical prediction of spectral regrowth and correlated and uncorrelated distortion in multicarrier wireless transmitters exhibiting memory effects

Measurement of non-linearity effects in wireless transmitters had been always difficult under complex digitally modulated signals. On the other hand, blind time domain simulation of these effects is generally time consuming and insufficient. In this study, the authors present new frequency domain closed-form formulas for predicting the spectral regrowth in wireless communication systems/subsystems with fifth-order non-linearity exhibiting memory effects. Particularly, the suggested formulas estimate the power spectral density, as well as the undesirable adjacent-channel and co-channel emissions of the modelled device, under phase-aligned and randomly phase-modulated multitone excitations. In addition, they are able to predict the correlated and uncorrelated distortion powers separately. The efficiency and robustness of the proposed approach has been demonstrated by predicting the output of a laterally diffused metal oxide semiconductor Doherty-based power amplifier. The obtained results have been compared to measurements and those of time-domain simulators and have revealed good accuracy and time efficiency.

On the convergence of multiple excitation sources to a two-tone excitation class: Implications for a global optimum excitation in active sensing for structural health monitoring

We showed in a recent study that the parameters of a system of ordinary differential equations (ODEs) may be adjusted via an evolutionary algorithm (EA) to produce deterministic excitations that improve damage detection in a simple computational spring–mass system. In that study, frequency considerations were found to be dominant, but the exact mechanism by which detection improvement is effected was not immediately apparent. In this work, an EA is used to shape bandlimited noise excitations in the frequency domain such that improvement is seen in the sensitivity of the detection feature. Tailored excitations generated from three bandlimited excitation sources, a multi-tone (linear sum of sinusoids) excitation source, and an ODE excitation source complement results from the previous study and help explain what excitation frequency characteristics are important for improved damage discernment. The convergence of all excitation source optimizations to similar solutions suggests the existence of a class of excitations that are globally optimal for improved damage detection sensitivity in the model application. Copyright © 2008 John Wiley & Sons, Ltd.

A Combined MoM-AOM Approach for Frequency Domain Analysis of Nonlinearly Loaded Antennas in the Presence of a Lossy Ground

A modeling technique based on the frequency-domain spectral balance method is proposed for analyzing nonlinearly loaded antennas in the presence of a lossy ground. The modeling involves two stages. First, the problem is transformed into a nonlinear microwave equivalent circuit with the circuit parameters of the antenna extracted by the method of moments. Then, the nonlinear microwave equivalent circuit is treated using the arithmetic operator method. To efficiently perform basic arithmetic operations in the frequency-domain, a linear matrix transformation of the spectra is introduced. The main feature of the proposed technique is its efficiency in dealing with problems involving strongly nonlinear loads, the presence of a lossy ground, and multitone excitations. It is also suited for the analysis of array antennas, considering the mutual effects between array elements. The accuracy of the proposed model is confirmed by comparing the simulation results of several case studies with those obtained using the conventional methods available in the literature. Mutual coupling effects, multi-frequency excitation, strong nonlinear loads, and the presence of a lossy ground are included in the simulations to cover all of the asserted aspects of the paper.

N-dimensional Harmonic Balance Method extended to non-explicit nonlinearities

The harmonic balance method is widely used for the analysis of strongly nonlinear problems under periodic excitation. The concept of hypertime allows for the generalization of the usual formulation to multi-tone excitations. In this article, the method is applied to a system involving a nonlinearity which cannot be explicitly expressed in the multi-frequency domain in terms of harmonic coef_cients. The direct and inverse Discrete Fast Fourier Transforms are then necessary to alternate between time and frequency domains in order to take into account this nonlinearity. The results show the efficiency and the precision of the method.

On the steady-state analysis of nonlinear circuits with frequency dependent parameters in wavelet domain

This letter introduces a new method for steady state analysis of nonlinear circuits with frequency dependent parameter components. The method enables treatment of distributed parameter components in a wavelet domain steady state circuit simulator. The method features thresholding of the convolution matrix in wavelet domain in order to reduce computational cost and is suitable for highly nonlinear circuits with distributed parameters under multitone excitations.

New Closed-Form Expressions for the Prediction of Multitone Intermodulation Distortion in Fifth-Order Nonlinear RF Circuits/Systems

This paper presents a rigorous analytic approach for the prediction of the in-band and out-of-band intermodulation distortion of fifth-order memoryless nonlinear RF circuits/systems modeled using a Taylor series and driven by phase-aligned or random phase multitone excitation. Nonlinear distortion figures-of-merit such as intermodulation ratio (IMR), adjacent channel power ratio, co-channel power ratio, and noise-to-power ratio, as well as the output power density can be straightforward computed using newly developed closed-form expressions. Simulation results of output power density obtained using the developed expressions for an L-band commercial amplifier demonstrates the time efficiency and robustness of the proposed approach when compared to averaged data obtained using numerical simulators such as Agilent ADS. The comparison of the computed nonlinearity figures-of-merit with those previously published shows the importance of considering the fifth order when modeling nonlinear RF circuits/systems. The proposed analytical approach explicitly highlights the dependency of the normalized figures-of-merit relative to the standard two-tone Mm (IMR/sub 2/) to the input power and to the coefficients of the Taylor model contrary to third-order-based approaches.

Incremental harmonic balance method with multiple time variables for dynamical systems with cubic non‐linearities

AbstractThe incremental harmonic balance method with multiple time variables is developed for analysis of almost periodic oscillations in multi‐degree‐of‐freedom dynamical systems with cubic non‐linearities, subjected to the external multi‐tone excitation. The method is formulated to treat non‐autonomous as well as autonomous dynamical systems. The almost periodic oscillations, which coexist with periodic oscillations in a rotating system model with cubic restoring force and an electromagnetic eddy‐current damper are analysed. The closed form solutions based on generalized Fourier series containing two incommensurate frequencies are obtained in the case of small non‐dimensional stiffness ratio. Almost periodic oscillations of a rotating system model in dependence on variable parameters are also analysed, where solutions are computed through an augmentation process including a greater number of harmonics and combination frequencies involved. Copyright © 2003 John Wiley & Sons, Ltd.

Using a low‐voltage intermodulation distortion sweet‐spot for controlling gain in HEMT amplifiers

AbstractIn this paper, a low‐voltage high‐linearity sweet‐spot, associated with an adjustable transconductance, is proposed for implementing a simple and highly linear HEMT‐based gain control strategy. The proposed approach, based on a selective gate and drain biasing technique, is compared with the conventional solution under two‐tone and multitone excitations. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 39: 67–70, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11129

Volterra-mapping-based behavioral modeling of nonlinear circuits and systems for high frequencies

Presents and validates a discrete-time/frequency-domain approach to the problem of Volterra-series-based behavioral modeling for high-frequency systems. The proposed technique is based on the acquisition of samples of the input/output data, both of which are sampled at the Nyquist rate corresponding to the input signal. The method is capable of identifying the time-/frequency-domain Volterra kernels/transfer functions of arbitrary causal time-invariant weakly nonlinear circuits and systems operating at high frequencies subject to essentially a general random or multitone excitation. The validity and efficiency of the proposed modeling approach has been demonstrated by several examples in high-frequency applications and good agreement has been obtained between results calculated using the proposed model and results measured or simulated with commercial simulation tools.

STIFFNESS NON-LINEARITY CLASSIFICATION THROUGH STRUCTURED RESPONSE COMPONENT ANALYSIS USING VOLTERRA SERIES

Most non-linear analysis problems, consider only the Duffing oscillator as a representative case. In engineering analysis, it is however, also important to recognise the type of non-linearity actually influencing the system. A procedure, involving structured higher-order FRF analysis based on Volterra theory is suggested in the present work, to distinguish a polynomial form of non-linearity from other possible forms. Volterra theory provides concepts of linear, bilinear, trilinear, etc. kernels, which upon convolution with the excitation force and subsequent summation can be employed to represent the response of a non-linear system. The kernels of the system are understood as multidimensional unit impulse response functions. The Volterra series response representation is employed in this work to facilitate its processing in a structured manner, to extract characteristic features, which can help in placing the system non-linearity in an appropriate class. The Volterra series platform is also employed to make a distinction between symmetric and asymmetric forms of the restoring force function. A multi-tone excitation procedure is further suggested, through which higher-order kernels of the system can be constructed for identification of the structure of the polynomial representing the restoring force. The procedures are illustrated through numerical simulation.

Analysis of large nonlinearly loaded antenna arrays under multitone excitation

In this paper, the responses of large nonlinearly loaded antenna arrays under multitone excitation are analyzed using the inexact Newton approach and infinite periodic structure Green's function. Numerical operations are shown to be very efficient. The analysis is suitable for large, nonlinearly loaded antenna arrays under multitone excitation. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 25: 319–323, 2000.

Cyclostationary noise analysis of large RF circuits with multitone excitations

This paper introduces a new, efficient technique for analyzing noise in large RF circuits subjected to true multitone excitations. Noise statistics in such circuits are time-varying, hence cyclostationary stochastic processes, characterized by harmonic power spectral densities (HPSDs), are used to describe noise. HPSDs are used to devise a harmonic-balance-based noise algorithm with the property that required computational resources grow almost linearly with circuit size and nonlinearity. Device noises with arbitrary spectra (including thermal, shot, and flicker noises) are handled, and input and output correlations, as well as individual device contributions, can be calculated. HPSD-based analysis is also used to establish the nonintuitive result that bandpass filtering of cyclostationary noise can result in stationary noise. Results from the new method are validated against Monte Carlo simulations. A large RF integrated circuit (>300 nodes) driven by a local oscillator (LO) tone and a strong RF signal is analyzed in less than two hours. The analysis predicts correctly that the presence of the RF tone leads to noise folding, affecting the circuit's noise performance significantly.

Nonlinear processing of digitally modulated carriersby the inexact-Newton harmonic-balance technique

The Newton-iteration based harmonic-balance (HB) technique is generally acknowledged as the principal method for the simulation of nonlinear circuits driven by multitone excitations. The authors propose a new implementation of the HB principle, named the inexact-Newton harmonic balance (INHB). This method provides a well-behaved and efficient solution to the problem of HB simulation with very large numbers of closely spaced spectral lines.

IMD prediction techniques for active filters

Two single-domain techniques for evaluating the output nonlinear distortion of active systems with multitone excitations are described as a means of circumventing the time-frequency transformations of standard harmonic balance methods. These two techniques are compared with respect to accuracy, computational efficiency, and flexibility for a band-pass active filter containing a saturation-type nonlinearity.

Numerical steady state analysis of electronic circuits driven by multi-tone signals

Characteristics of analogue circuits such as intermodulation distortion and transfer characteristics can often be received from the steady state behavior. This paper presents a unified approach for the simulation of non-autonomous circuits with multi-tone excitation. The steady state is here regarded as the solution of a partial differential-algebraic equation. A suitable numerical method for its solution is a variational method with trigonometric basis functions. The Harmonic Balance technique based either on the multi-dimensional Fourier transformation or the Artificial Frequency Map technique can be interpreted as a special variant of this method.

On the harmonic balance analysis of almost-periodic autonomous nonlinear circuits with dynamic nonlinear characteristics

Analytical expressions for the efficient computation of sensitivities with respect to fundamental periods in dynamic nonlinear characteristics under multitone excitation are presented. Almost-periodic oscillations in the forced Van der Pol oscillator are computed to illustrate the capability and efficiency of the resulting harmonic balance technique.