Linear Periodically Time-varying Research Articles

Reciprocity and Inter-Reciprocity: A Tutorial— Part I: Linear Time-Invariant Networks

We present a systematic treatment of reciprocity and inter-reciprocity in linear electrical networks, without resorting to the use of Tellegen’s theorem. In Part I, we discuss linear time-invariant networks. In Part II, we expand the applications of our approach to linear periodically time-varying (LPTV) networks.

A Detailed Model of Cyclostationary Noise in Switched-Resistor Circuits

The Switched-Resistor (S-R) technique is becoming more and more interesting to implement low-voltage low-power active- RC filters with high tuning range and front-end amplifiers for biomedical circuits and systems. This approach exploits MOS switches driven by a duty-cycle-controlled clock signal to achieve tunability of the equivalent resistance and, hence, of the RC time constant. Since S-R circuits can be considered as linear periodically time variant (LPTV) systems with sampled outputs, we exploit the theory of the adjoint (inter-reciprocal) network in order to develop a detailed model of the cyclostationary noise involved in this kind of circuits. The proposed model allows to gain insight into the different noise sources and transfer functions involved, highlighting the circuit parameters on which the cyclostationary noise depends. Validation of the proposed model has been carried out by comparing analytical results against periodic noise simulations referring to a commercial 130nm CMOS process. The validation activity has confirmed the good accuracy of the proposed noise model and provided some useful design guidelines to optimize the noise performance of S-R circuits.

A Variant of Extended LMI: Performance Analysis with Static Multiplier for Discrete-Time Lifted Systems

This note revisits extended Linear Matrix Inequality (LMI) formulation of performance analysis, which incorporates static multiplier, for parametrically multi-affine discrete-time Linear Time-Invariant Parameter-Dependent (LTIPD) and Linear Periodically Time-Varying (LPTV) systems using lifting technique. Almost the same analysis conditions for parametrically affine discrete-time LTIPD/LPTV systems as the ones shown in this note have already been proposed in the literature, we thus focus on the simple derivations of the conditions using only Elimination lemma. To this end, a variant of extended LMI formulation, whose dimensional sizes are slightly larger than the conventional formulation with additional auxiliary matrices, is proposed in this note. It is also shown that our derived variant directly leads to a parametrically multi-affine extended LMI formulation for our addressed problem.

A 10-MHz 85.1-dB SFDR 1.1-mW continuous-time Delta–sigma modulator employing calibration-free SC DAC and passive front-end low-pass filter

This article demonstrates the circuit technique of a highly power-efficient continuous-time (CT) Delta–sigma modulator (DSM) based on a switched-capacitor (SC) feedback digital-to-analog converter (DAC) and a passive front-end low-pass filter (LPF). The technique addresses the clock sensitivity issue of conventional DACs while maintaining the alias rejection. It is validated by a fourth-order CT DSM prototype fabricated in 65-nm CMOS technology. The input stage of the DSM is an active-RC integrator merged with a passive front-end LPF. A resonator follows and realizes the cascade of resonators with feedforward (CRFF) structure. The accurate feedback loop employs a highly linear and calibration-free 5-level SC DAC. Based on the linear periodically time-varying (LPTV) system model, an exact analysis is given to explain the effects of the passive LPF and SC DAC combo on the signal transfer function (STF), and hence, on the alias rejection. The prototype DSM achieves a 70.2-dB signal to noise and distortion ratio (SNDR) and an 85.1-dB spurious-free dynamic range (SFDR) in a 10-MHz bandwidth, while dissipating 1.1 mW with a 1-V supply. The modulator achieves a 65.4-dB alias rejection and confirms the closed-form analytical results.

A Commutated-LC RF Broadband Delay Circuit

This article presents a commutated-inductor–capacitor (commutated- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$LC$ </tex-math></inline-formula> ) or switched- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$LC$ </tex-math></inline-formula> circuit that acts as a radio frequency (RF) delay line. Thanks to its linear-periodically time-varying (LPTV) operation and fully passive implementation, it concurrently achieves long maximum delays, fine delay tuning steps, and wide instantaneous bandwidths while being low loss and highly linear. Unlike existing LPTV switched-capacitor broadband delays, the introduction of inductors in the proposed commutated- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$LC$ </tex-math></inline-formula> delay circuit provides a new degree of freedom, allowing it to operate at a much higher RF with a wider instantaneous bandwidth. A proof-of-concept prototype in a 65-nm CMOS process demonstrates a measured 1.3-GHz 3-dB bandwidth around a 4.3-GHz RF, i.e., a 30% fractional bandwidth, when clocked at 250 MHz. The measured maximum delay is 1.4 ns with a 23-dB loss and noise figure; this loss or noise is orders of magnitude lower compared with fully passive linear-time-invariant RF delay lines operating at a similar frequency with the same delay. The measured IIP3 is +16 dBm.

Generalized framework for computing the [formula omitted]-induced norm of sampled-data systems

This paper develops a generalized framework for computing the L∞-induced norm of sampled-data systems, by which we mean those consisting of a continuous-time linear time-invariant (LTI) plant and a discrete-time LTI controller, and such systems have a linear periodically time-varying (LPTV) nature. In the authors’ preceding studies, the input/output relation of sampled-data systems was described in an operator-theoretic fashion, and the kernel and input functions of an input operator and the hold function of an output operator were approximated by piecewise lth order polynomials with l=0,1. This leads to an approximate computation of this norm, where the computation errors were shown to converge to 0 in the order of 1/Ml+1 as the approximation parameter M becomes larger. Along this line, we aim at improving the computation performance compared to the preceding studies by considering the freedom in the point around which the corresponding functions are expanded to a Taylor series. This leads to a generalization of the piecewise lth order approximations, and we show that taking the central point for the Taylor expansion leads to quantitatively improved accuracy than that of our preceding results that take an edge point.Finally, the effectiveness of the developed method is verified through a numerical example.

A generalized analysis of high‐order N‐path filters

AbstractIn N‐path filter analysis, assumptions are used to simplify their analysis to avoid the complexity of the calculations. These assumptions limit the number of examinable circuits. This paper analyzes a general system including an arbitrary number of N‐path filters to generalize the analysis. In the proposed method, due to the calculation of harmonic transfer functions for each independent capacitor in each time interval, it is possible to examine N‐path filters in the presence of common capacitors between multiple branches, leakage resistors, and branches with asymmetric switching. The absence of discontinuous jumps during the switching moments and switching operation in a non‐overlapping fashion with a specified periodicity are the only assumptions of the proposed method. As a result, more LPTV (linear periodically time‐varying) systems can be analyzed using the calculations provided. After presenting the method, we show the proposed method's ability by analyzing a recently published impedance‐transforming N‐path filter and examining its details.

The generalized [formula omitted] controller synthesis problem of sampled-data systems

This paper tackles the generalized H2 controller synthesis problem of sampled-data systems, which is associated with the controller minimizing the induced norm from L2 to L∞. To alleviate the difficulty of the linear periodically time-varying (LPTV) nature of sampled-data systems, we first take an operator-based approach to sampled-data systems via the lifting treatment. We next develop a framework for piecewise constant approximation in the context of the generalized H2 controller synthesis problem after further applying the so-called fast-lifting treatment. An optimal controller for the approximate treatment is also shown to achieve the generalized H2 performance for the sampled-data system that is close enough to its optimal generalized H2 performance, if the fast-lifting parameter N is large enough. This is established by deriving, in a fashion suitable for controller synthesis, upper and lower bounds on the resulting sampled-data generalized H2 performance, where their gap tends to 0 at the rate of 1/N. We further introduce a discretization method of the continuous-time plant, with which the controller synthesis in the approximate fashion can actually be carried out through an equivalent discrete-time counterpart of the generalized H2 controller synthesis problem. Finally, numerical examples are given to validate the overall arguments.

Analysis and Modeling of N-Path Circuits Peak Frequency Shift Caused by Switch Parasitics

The N-path circuit serves both as a mixer and as a radio frequency (RF) filter. Certain performance aspects, such as noise, linearity, and out-of-band (OOB) rejection dictate large switches size. However, such increment raises the value of the parasitic RF capacitance, which causes further in-band (IB) loss and peak frequency shift. In this brief, we derive simplified equations for the assessment of the mentioned non-ideal properties, using the novel linear periodically time-variant (LPTV) state-space analysis, by making derivations and approximations based on realistic circuit parameters. We present closed-form equations for the voltage gain peak frequency shift and a revised linear time-invariant (LTI) model for the assessment of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{11}$ </tex-math></inline-formula> dip frequency shift.

Real Time Computationally Efficient MIMO System Identification Algorithm

MIMO system identification is a fundamental concern in a variety of applications. Various iterative and recursive MIMO system identification algorithms exist in the literature. The iterative algorithms suffer from high computational cost due to large matrix computations and recursive algorithms suffer from slow convergence speed. This paper proposes a fast recursive exact least squares algorithm for MIMO system identification with fast convergence speed and low computational cost. The recursions of the algorithm give rise to a lattice structure. The lattice structure is less sensitive to round-off errors, coefficients variations and can also be used for model order reduction. Although in this work we estimate an FIR (finite impulse response) model of the MIMO system, the framework can also be used for IIR (infinite impulse response) models, and for estimating linear periodically time varying (LPTV) and multivariable systems. The theory proposed is validated using simulation results.

Multi-Watt, 1-GHz CMOS Circulator Based on Switched-Capacitor Clock Boosting

There has been significant recent progress in the implementation of integrated non-reciprocal components based on linear periodically time-varying (LPTV) circuits. Nevertheless, integrated circulators still require a leap forward in power handling, clock power consumption, and insertion loss (IL) to become compelling compared with ferrite circulators or integrated reciprocal alternatives, such as the electrical-balance duplexer (EBD). This article introduces three innovations: 1) a new switched-capacitor clock-boosting scheme; 2) high-Bragg-frequency quasi-distributed transmission lines based on periodically loaded inductors; and 3) a new gyrator based on switched partially reflecting t-lines—which enable significant performance improvement for integrated circulators and for LPTV circuits more broadly. These are showcased in a 1-GHz 180-nm SOI CMOS circulator that exhibits 2.1-/2.6-dB TX-ANT/ANT-RX IL (0.3 dB better than prior art), +34-dBm TX-ANT P1 dB (2.5 $\times $ or 4 dB better), and 40% lower chip area, all at 39-mW power consumption (4.4 $\times $ lower).

Design of a SAW-less, noise-canceling receiver using an LPTV analysis of a general system with an arbitrary number of N-path filters

This paper, first, presents a Linear Periodically Time-Variant (LPTV) analysis of a general system with an arbitrary number of n-path filters, that the results of mathematical derivations are verified through simulation. Then, a noise-canceling receiver is presented with 3 stages of the 8-path filter. In the structure of this receiver, two filters are used for noise-canceling, and the third filter is used as a Miller bandpass filter to increases the selectivity. The proposed circuit is designed with 65 nm RF CMOS technology and the simulation results show > 44 dB gain, 3–4.5 dB NF, and 0 dBm out-of-band IIP3 at the 50 MHz offset, while in 0.3–3 GHz functional frequency, consuming just 15.52mW of power.

Analysis of the Effect of Switch Parasitic Resistance and Capacitance on N-Path Filters Using State Space Representation

This brief suggests a new method for analyzing the effects of nonideal properties of CMOS switches on N-path filters by expanding the linear periodically time-variant (LPTV) method to state space equations. The proposed technique provides an algorithm for analytical derivation of the harmonic transfer functions (HTF) and the noise figure (NF) of the N-path filter across all frequencies and harmonics. The analysis results show that for constant switch resistance and shunt capacitance product, a trade-off exists between out of band rejection, NF and insertion loss.

An N-Path Band-Pass Filter With Parametric Gain-Boosting

This paper presents a gain-boosting technique for two-port N-path band-pass filters (BPFs) based on the principle of discrete-time parametric amplification in MOS devices. The proposed technique also acts as a noise figure (NF) reduction technique due to the low-noise amplification before the output commutation. Analytical expressions for gain, NF, and input impedance of the proposed parametrically gain-boosted N-path BPF are derived using linear periodically time-variant (LPTV) network analysis and have been verified through simulations. The measurement results of a pseudo-differential eight-path BPF, implemented in a 180 nm CMOS technology, are presented. The center frequency of the implemented filter is tunable from 0.05 GHz to 0.4 GHz with a Q varying from 4 to 18. The gain of the filter is programmable with the input common-mode voltage. At 0.4 GHz, the filter achieves a variable gain of 2–8 dB over the input common-mode range of 0.55 V to 0.85 V. At every tuned frequency, the filter has a peak gain ≥8 dB, NF 0 dBm. It is also shown in simulations that the proposed gain-boosting technique reduced the NF of the filter by >2.2 dB while consuming a dynamic power in the range of 3.8-31 mW per filter.

Unified Analysis, Modeling, and Simulation of Chopping Artifacts in Continuous-Time Delta-Sigma Modulators

Chopping is an efficient way of mitigating the effect of flicker noise in continuous-time delta-sigma modulators (CT ${\Delta \Sigma }$ Ms). Unfortunately, chopping causes the demodulation of shaped quantization noise into the signal band. Prior works have analyzed noise-folding effects in chopped integrators that use single-stage and two-stage feedforward-compensated OTAs. These works use restrictive assumptions, such as settling of the OTA internal nodes at the chopping instants, and an NRZ feedback DAC waveform. This paper gives a general model for aliasing of shaped noise in a chopped integrator that incorporates arbitrary OTAs, arbitrary DAC pulse-shapes, and incomplete settling. Using the theory of linear periodically time-varying (LPTV) systems, we derive a simple simulation test-bench that can be used to estimate the parameters of our model. Thanks to this, the effects on chopping on the performance of the modulator can be rapidly estimated without running long transient simulations. Simulation and experimental results that support our theory are given.

A Generalized Model for Linear-Periodically-Time-Variant Circulators

Magnetic-free non-reciprocity based on linear-periodically-time-variant (LPTV) circuits has received significant research and commercial attention since it could revolutionize wireless communications. LPTV circuits are formed by two main components: linear-time-invariant (LTI) networks and periodically-modulated switches. The modulated switches are the core elements to break the reciprocity of LTI networks. To understand and design LPTV circulators, a universal and intuitive analytical model is required. However, such model does not exist as it is extremely challenging to accurately model and fully understand the LPTV behaviour of energy storage networks. To address this limitation, this work introduces a novel analysis method, which is broadly applicable to any LPTV networks, and validates it experimentally. The novelty of this methodology comes from two main contributions: (1) modelling of the switch as a resistor in parallel with a current-controlled current source; (2) the decomposition of the LPTV network into the linear superposition of two LTI networks. We apply this technique to model the exact behaviour of an LPTV circulator in the frequency domain.

Cycling-Based Synthesis of Robust Output Estimators for Uncertain LPTV Systems

This paper studies a method of designing robust output estimators for discrete-time linear periodically time-varying (LPTV) systems with uncertainties. The key idea is to use not the well-known lif...

Cyclic Signals and Systems in Power Line Communications

This paper provides a comprehensive discussion about the characteristics of power line communication (PLC) channels related to its synchronization with the mains period. This fact constitutes an interesting and distinguishing feature of these kind of channels. Under the mains voltage presence, the behavior of some loads connected to the power network allows representing the transmission medium as a linear periodically time varying (LPTV) system. Some components in the received noise exhibit periodicity as well, in terms of impulsive waveforms synchronized with the mains period and other cyclostationary (CS) signals. This fact has been taken into account in the design of PLC systems, for instance by defining medium access control (MAC) frames synchronized with the mains period and by adapting the modulation to the cyclic signal-to-noise ratio (SNR) to improve the system performance. Hence, in PLC, it is of particular interest to describe the general relations of LPTV systems and CS signals when both have a common period, which is the purpose of this work. It is not aimed at providing new mathematical tools for the analysis of these cyclic signals and systems, which have been extensively developed over the last five decades, but to serve the readers of the PLC community more accessible mathematical relations that can be applied to the practical problems they have to face. A review of recent works dealing with the cyclic properties of PLC channels and transmission techniques aware of these strategies is included as well.

Impedance Matching and Reradiation in LPTV Receiver Front-Ends: An Analysis Using Conversion Matrices

Linear periodically time-varying (LPTV) circuits are finding increased prominence in reconfigurable transceiver applications. Impedance matching of such circuits to the antenna impedance is essential for fully on-chip receiver topologies. However, deriving the $S_{11}$ of an LPTV circuit is non-trivial in general. Moreover, reflections at harmonic offsets from input frequencies can also be expected due to the time-varying nature of the circuit. This paper utilizes the conversion matrix-based analysis technique to compute the $S_{11}$ of a general LPTV circuit. Furthermore, it is shown that signal reradiation at harmonic shifts from input frequencies is also obtained as a natural consequence of this analysis. Examples related to mixer-first receivers, switched-capacitor receiver front-ends (with reset), and Filtering-by-Aliasing (FA)-based receivers are considered to verify the analysis. Design guidelines are provided to attain optimal $S_{11}$ in a few such scenarios. Explicit analytical expressions are derived for reradiation in ideal $N$ -path mixer-first receivers and FA-based receivers, while guidelines are provided for proper choice of conversion matrix expressions/sizes in case of numerical computation.

Design and Analysis of a Programmable Receiver Front End Based on Baseband Analog-FIR Filtering Using an LPTV Resistor

This paper presents a programmable receiver frontend that uses periodically time-varying components. A sharp programmable filtering response was achieved at RF using a linear periodically time-varying (LPTV) resistor in conjunction with a passive mixer and an integrate-and-dump circuit. The LPTV resistor along with the integrate-and-dump circuit is shown to achieve a programmable finite-impulse response (FIR) baseband filter. The baseband filter is then upconverted with the passive mixer to achieve sharp bandpass filtering at the desired RF center frequency. Further, it is shown that an additional S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sub> constraint can be imposed on the FIR filter design to allow for impedance matching to the antenna impedance. The implemented receiver achieved high close-in linearity with 17 dBm of IIP <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> at only 1.2× bandwidth frequency offset, while achieving a wideband impedance match with S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sub> better than -10 dB throughout the local oscillator range of 0.1-1 GHz.