## Introduction to RF Simulation and Its Application (1999)

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### BibTeX

@MISC{Kundert99introductionto,

author = {Kenneth S. Kundert},

title = {Introduction to RF Simulation and Its Application},

year = {1999}

}

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### Abstract

Radio-frequency (RF) circuits exhibit several distinguishing characteristics that make them difficult to simulate using traditional SPICE transient analysis. The various extensions to the harmonic balance and shooting method simulation algorithms are able to exploit these characteristics to provide rapid and accurate simulation for these circuits. This paper is an introduction to RF simulation methods and how they are applied to make common RF measurements. It describes the unique characteristics of RF circuits, the methods developed to simulate these circuits, and the application of these methods. Index Terms--- Circuit simulation, cyclostationary noise, envelope methods, harmonic balance, interchannel interference, intermodulation distortion, jitter, mixer noise, mixers, nonlinear oscillators, phase noise, quasi-periodic methods, shooting methods, SPICE. I. THE RF INTERFACE W IRELESS transmitters and receivers can be conceptually separated into baseband and radio-frequency (RF) sections. Baseband is the range of frequencies over which transmitters take their input and receivers produce their output. The width of the baseband determines the underlying rate at which data can flow through the system. There is a considerable amount of signal processing that occurs at baseband designed to improve the fidelity of the data stream being communicated and to reduce the load the transmitter places on the transmission medium for a particular data rate. The RF section of the transmitter is responsible for converting the processed baseband signal up to the assigned channel and injecting the signal into the medium. Conversely, the RF section of the receiver is responsible for taking the signal from the medium and converting it back down to baseband. With transmitters there are...

### Citations

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Citation Context ...ges, substrates and spirals can be modeled with large lumped networks,butsuchsystemscanbetoolargetobeefficiently incorporated into a time domain simulation, and so some form of reduction is necessary =-=[11,42]-=-. D. Semiconductor Models The semiconductor models used by RF simuators must accurately model the high-frequency small-signal behavior of the devices to accurately perdict the behavior of RF circuits.... |

145 |
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Citation Context ...e timestep is constrained to be uniform by the FFT, however there is new work that explores the possibility of using new FFT algorithms that do not require equally spaced points with harmonic balance =-=[10, 37]-=-. The strong convergence properties of shooting methods result from its implementation as a multilevel Newton method, and not from the fact it is a time-domain method. Indeed, it is possible to formul... |

124 | Phase noise in oscillators: a unifying theory and numerical methods for characterization
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Citation Context ...folding and cyclostationary noise sources can be implemented [40,52], which fills a critically important need for RF circuits. When applied to oscillators, it also accounts for oscillator phase noise =-=[8,9,21,22]-=-. C. Linear Passive Components At the high frequencies present in RF circuits, the passive components, such as transmission lines, spiral inductors, packages (including bond wires) and substrates, oft... |

113 |
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Citation Context ...s to allow the receiver to tune over a range of channels. Complex feedback schemes are often necessary to satisfy the often competing requirements of frequency tuning resolution and noise performance =-=[14]-=-. Predicting the phase noise and jitter of such circuits can be quite difficult. They rarely operate with simple periodic or quasiperiodic signals, and so the PNoise and QPNoise analyses cannot be dir... |

73 |
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Citation Context ...aminating the small input signal. For various reasons, the direct conversion or homodyne architecture is a candidate to replace the superheterodyne architecture in some wireless communication systems =-=[1,24,25]-=-. In this architecture the RF input signal is directly converted to baseband in one step. Thus, most of the gain will be at baseband and the LO will be at the same frequency as the input signal. In I ... |

68 |
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Citation Context ... is still a large number if the number of fundamentals is large. In practice, these methods are typically limited to a maximum of 3 or 4 fundamental frequencies. 1) Harmonic Balance: Harmonic balance =-=[27,30,36,47]-=- formulates the circuit equations and their solution in the frequency domain. The solution is written as a Fourier series that cannot represent transient behavior, and so harmonic balance directly fin... |

65 | A study of phase noise in CMOS oscillators
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Citation Context ...her baseband or at a sideband. The former is referred to as baseband to AM/PM conversion and the latter is SSB to AM/PM conversion. Both cases were demonstrated in the case of an oscillator by Razavi =-=[43]-=-. A PAC analysis directly computes the transfer function from some small input signal to the upper and lower sidebands components of a modulated carrier. It is also possible, using a change of basis, ... |

54 |
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Citation Context ...tion between noise at different frequencies. As can be seen from Figure 13, at frequencies separated by kf cyclostationary noise is correlated, where f is the modulation frequency and k is an integer =-=[15]-=-. 1) Noise Figure: Noise is a critical concern in receivers because of the small input signals. Typically designers characterize the noise of individual blocks using the noise figure of the block beca... |

41 | Efficient steady-state analysis based on matrix-free Krylov-subspace methods
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Citation Context ...to relatively small circuits. Recently, Krylov subspace methods have been applied to accelerate both harmonic balance and the shooting methods, which allows them to be applied to much larger circuits =-=[13,30,33,58,60,61]-=-. A. Periodic and Quasiperiodic Analysis Periodic and quasiperiodic analyses can be thought of as RF extensions of SPICE’s DC analysis. In DC analysis one applies constant signals to the circuit and i... |

38 | Efficient AC and Noise Analysis of Two-Tone RF Circuits
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Citation Context ...to relatively small circuits. Recently, Krylov subspace methods have been applied to accelerate both harmonic balance and the shooting methods, which allows them to be applied to much larger circuits =-=[13,30,33,58,60,61]-=-. A. Periodic and Quasiperiodic Analysis Periodic and quasiperiodic analyses can be thought of as RF extensions of SPICE’s DC analysis. In DC analysis one applies constant signals to the circuit and i... |

34 | Future Directions in Silicon ICs for RF Personal Communications
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(Show Context)
Citation Context ...aminating the small input signal. For various reasons, the direct conversion or homodyne architecture is a candidate to replace the superheterodyne architecture in some wireless communication systems =-=[1,16,44,45]-=-. In this architecture the RF input signal is directly converted to baseband in one step. Thus, most of the gain will be at baseband and the LO will be at the same frequency as the input signal. In th... |

32 |
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Citation Context ... final state v(T) with respect to changes in the initial state v(0). The sensitivity is used to determine how to correct the initial state to reduce the difference between the initial and final state =-=[2,58]-=-. 5) Autonomous Shooting Methods: As with harmonic balance, it is extremely important to be able to determine the steady-state behavior of oscillators. To do so it is necessary to modify shooting meth... |

32 | Cyclostationary noise analysis of large RF circuits with multitone excitations
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(Show Context)
Citation Context ...er, enabling a wide variety of applications (some of which are described in [59]). In particular, a noise analysis that accounts for noise folding and cyclostationary noise sources can be implemented =-=[40,52]-=-, which fills a critically important need for RF circuits. When applied to oscillators, it also accounts for oscillator phase noise [8,9,21,22]. C. Linear Passive Components At the high frequencies pr... |

32 |
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Citation Context ... Δ « f « fc ∞ = ∑ k = – ∞ X 2 1 f Δ Sv( f c + f m) = -----------------------------π f 2 Δ + f 2 m Jk = kaT † Other references report that L(fm)=Sφ(fm)/2, which is true when Sφ is the single-sided PSD =-=[50,63]-=-. Here, Sφ is the doubled-sided PSD. log S φ log S vsKUNDERT: INTRODUCTION TO RF SIMULATION AND ITS APPLICATION 1304 In the case where u represents flicker noise, S u ( f ) is generally pink or propor... |

31 |
Nonlinear Microwave Circuits
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Citation Context ...umber of transfer functions between any particular input and output. Each represents a different frequency translation. Versions of this type of small-signal analysis exists for both harmonic balance =-=[17,24,31]-=- and shooting methods [39,40,61]. There are two different ways of formulating a small-signal analysis that computes transfer functions [59,61]. The first is akin to traditional AC analysis, and is ref... |

29 |
Analysis and Simulation of Noise in Nonlinear Electronic Circuits and Systems
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(Show Context)
Citation Context ...folding and cyclostationary noise sources can be implemented [40,52], which fills a critically important need for RF circuits. When applied to oscillators, it also accounts for oscillator phase noise =-=[8,9,21,22]-=-. C. Linear Passive Components At the high frequencies present in RF circuits, the passive components, such as transmission lines, spiral inductors, packages (including bond wires) and substrates, oft... |

29 |
A piecewise harmonic balance technique for determination of periodic response of nonlinear systems
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Citation Context ... is still a large number if the number of fundamentals is large. In practice, these methods are typically limited to a maximum of 3 or 4 fundamental frequencies. 1) Harmonic Balance: Harmonic balance =-=[27,30,36,47]-=- formulates the circuit equations and their solution in the frequency domain. The solution is written as a Fourier series that cannot represent transient behavior, and so harmonic balance directly fin... |

29 |
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Citation Context ...eral methods have been developed to extend harmonic balance to handle quasiperiodic signals [14,37]. In addition, it has also be extended to allow the Fourier coefficients to be vary slowly with time =-=[6,15,18,29]-=-. These methods are referred to as envelope methods and are usedsKUNDERT: INTRODUCTION TO RF SIMULATION AND ITS APPLICATION 7 to simulate modulated carrier signals where the modulation is not a repeti... |

28 |
Efficient multi-tone distortion analysis of analog integrated circuits
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Citation Context ...to relatively small circuits. Recently, Krylov subspace methods have been applied to accelerate both harmonic balance and the shooting methods, which allows them to be applied to much larger circuits =-=[13,30,33,58,60,61]-=-. A. Periodic and Quasiperiodic Analysis Periodic and quasiperiodic analyses can be thought of as RF extensions of SPICE’s DC analysis. In DC analysis one applies constant signals to the circuit and i... |

24 |
E - cient full-wave electromagnetic analysis via modelorder reduction of fast integral transforms
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Citation Context ...ges, substrates and spirals can be modeled with large lumped networks,butsuchsystemscanbetoolargetobeefficiently incorporated into a time domain simulation, and so some form of reduction is necessary =-=[11,42]-=-. D. Semiconductor Models The semiconductor models used by RF simuators must accurately model the high-frequency small-signal behavior of the devices to accurately perdict the behavior of RF circuits.... |

21 |
State of the art and present trends in nonlinear microwave CAD techniques
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(Show Context)
Citation Context ... is still a large number if the number of fundamentals is large. In practice, these methods are typically limited to a maximum of 3 or 4 fundamental frequencies. 1) Harmonic Balance: Harmonic balance =-=[27,30,36,47]-=- formulates the circuit equations and their solution in the frequency domain. The solution is written as a Fourier series that cannot represent transient behavior, and so harmonic balance directly fin... |

21 | Efficient Methods for Simulating Highly Nonlinear Multi-Rate Circuits
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Citation Context ...tors. However, there other methods that have the potential to become significant to RF designers. In particular, two families of methods seem promising: the Volterra methods [31] and the MPDE methods =-=[4,51]-=-. The Volterra methods are similar in concept to the small-signal analyses in that they represent the circuit using a Taylor series expansion, except they take into account more than just the first te... |

20 |
Numerical steady state analysis of electronic circuits driven by multi-tone signals
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Citation Context ...tors. However, there other methods that have the potential to become significant to RF designers. In particular, two families of methods seem promising: the Volterra methods [31] and the MPDE methods =-=[4,51]-=-. The Volterra methods are similar in concept to the small-signal analyses in that they represent the circuit using a Taylor series expansion, except they take into account more than just the first te... |

20 | Fast simulation algorithms for rf circuits
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Citation Context |

19 | Simulation methods for RF integrated circuits
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Citation Context ...onents is infinite dimensional. The state vector must somehow be discretized before shooting methods can be applied. However, even then shooting methods will be expensive if the state vector is large =-=[27,28]-=-. This disadvantage explains why shooting method-based RF simulation techniques have mainly been applied to RFICs. Most RFICs can be modeled completely with lumped components. New approaches for gener... |

19 |
An Efficient Numerical Method for highly Oscillatory Ordinary Differential Equations
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Citation Context ...hod: With the sample envelope method the envelope is represented by slowly-varying sampled version of the waveform as shown in Figure 10. First developed by Petzold and referred to envelope following =-=[41]-=-, it is a transient-envelope method based on shooting methods. It approximates the sample envelope as a piecewise polynomial [25,41] in a manner that is analogous to conventional transient analysis. T... |

18 |
Scattering parameter transient analysis of transmission lines loaded with nonlinear terminations
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Citation Context ...A alternative approach is to compute the impulse response for a distributed component from a frequency domain description and use convolution to determine the response of the component in the circuit =-=[20,54,56]-=-. Evaluating lossy or dispersive transmission line models or tables of S-parameters with this approach is generally expensive and error-prone. Packages, substrates and spirals can be modeled with larg... |

17 |
New Method of Analysis of Communication Systems
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Citation Context ...ER 1999 1) Fourier-Envelope Method: With the Fourier-envelope method the envelope is represented by slowly-varying Fourier coefficients. First developed by Sharrit and referred to as circuit envelope =-=[55]-=-, the Fourier-envelope method is a transient-envelope method based on harmonic balance. In the Section IV-A, the concept of harmonic balance with timevarying Fourier coefficients was introduced. In th... |

17 | Receiver Characterization Using Periodic Small-Signal Analysis
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- 1996
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Citation Context ...results, which does exhibit frequency translation. All of the traditional small-signal analyses can be extended in this manner, enabling a wide variety of applications (some of which are described in =-=[59]-=-). In particular, a noise analysis that accounts for noise folding and cyclostationary noise sources can be implemented [40,52], which fills a critically important need for RF circuits. When applied t... |

16 | Robust Rational Function Approximation Algorithm for Model Generation
- Coelho
- 1999
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Citation Context ...ion lines, are described with partial-differential equations (PDE) and are problematic for time-domain simulators. Generally, the PDEs are converted to a set of ODEs using some form of discretization =-=[6,35]-=-. Such approaches suffer from bandwidth limits. A alternative approach is to compute the impulse response for a distributed component from a frequency domain description and use convolution to determi... |

16 | Full chip harmonic balance
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- 1997
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16 |
Numerical Noise Analysis for Nonlinear Circuits with a Periodic Large Signal Excitation Including Cyclostationary Noise Sources
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Citation Context ...er, enabling a wide variety of applications (some of which are described in [59]). In particular, a noise analysis that accounts for noise folding and cyclostationary noise sources can be implemented =-=[40,52]-=-, which fills a critically important need for RF circuits. When applied to oscillators, it also accounts for oscillator phase noise [8,9,21,22]. C. Linear Passive Components At the high frequencies pr... |

15 |
An Efficient Small Signal Frequency Analysis Method of Nonlinear Circuits with Two Frequency Excitations
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Citation Context ...n. Given that us() t Use j2πf st = the perturbation in steady-state response is given by vs() t Vsk e j2π f = ∞ ( s + kfL)t ∑ k = – ∞ (40) (41) where fL =1/TLis the large signal fundamental frequency =-=[39,61]-=-. Vsk represents the sideband for the kth harmonic of VL . In this situation, shown in Figure 11, there is only one sideband per harmonic because Us is a single frequency complex exponential and the c... |

14 |
An envelope-following method for the efficient transient simulation of switching power and filter circuits
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- 1988
(Show Context)
Citation Context ...gure 10. First developed by Petzold and referred to envelope following [41], it is a transient-envelope method based on shooting methods. It approximates the sample envelope as a piecewise polynomial =-=[25,41]-=- in a manner that is analogous to conventional transient analysis. This approach is efficient if the sequence formed by sampling the state at the beginning of each clock cycle, v(0), v(T), v(2T),..., ... |

12 |
General-purpose harmonic balance analysis of nonlinear microwave circuits under multitone excitation
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- 1988
(Show Context)
Citation Context ... When evaluating I and Q the multidimensional discrete Fourier transform is used. Using a multidimensional Fourier transform is just one way of formulating harmonic balance for quasiperiodic problems =-=[49,66]-=-. It is used here because of its simple derivation and because it introduces ideas that will be used later in Section IV-C. An alternate approach that is generally preferred in practice is the false f... |

12 | Efficient Frequency-Domain Modeling and Circuit Simulation of Transmission Lines
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(Show Context)
Citation Context ...A alternative approach is to compute the impulse response for a distributed component from a frequency domain description and use convolution to determine the response of the component in the circuit =-=[20,54,56]-=-. Evaluating lossy or dispersive transmission line models or tables of S-parameters with this approach is generally expensive and error-prone. Packages, substrates and spirals can be modeled with larg... |

11 |
A mixed frequency-time approach for distortion analysis of switching filter circuits
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(Show Context)
Citation Context ...recovered by knowing the waveform over 2K + 1 cycles of the carrier that are evenly distributed over the period of the modulation. This is the basic idea behind the Mixed Frequency-Time or MFT method =-=[13,26,27]-=-. Consider a circuit driven by two periodic signals that responds in steady-state by producing 2-fundamental quasiperiodic waveforms as in (15). Designate the fundamental frequencies as f1 and f2 and ... |

9 |
Determination of the Correlation Spectrum of Oscillators with Low Noise
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(Show Context)
Citation Context ...folding and cyclostationary noise sources can be implemented [40,52], which fills a critically important need for RF circuits. When applied to oscillators, it also accounts for oscillator phase noise =-=[8,9,21,22]-=-. C. Linear Passive Components At the high frequencies present in RF circuits, the passive components, such as transmission lines, spiral inductors, packages (including bond wires) and substrates, oft... |

9 |
Analysis of white and f –� noise in oscillators
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- 1990
(Show Context)
Citation Context |

9 | A substitution algorithm for solving nonlinear circuits with multi-frequency components - Usihda, Chua, et al. - 1987 |

8 |
Simulation and modeling of intermodulation distortion in communication circuits
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- 1999
(Show Context)
Citation Context ...lternative to transient-envelope simulations is to extract a behavioral baseband-equivalent table model of the transmitter using one of the large-signal steady-state methods described in Section IV-A =-=[5]-=-. Because the behavioral model abstracts away the carrier and unnecessary circuit details, the ACPR calculation step is fast regardless of circuit size or complexity. In transmitter circuits the input... |

8 | Efficient computation of quasiperiodic circuit operating conditions via a mixed frequency/time approach
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- 1998
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Citation Context |

8 |
Modeling and simulation of jitter in phase-locked loops
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- 1997
(Show Context)
Citation Context ... simulation of PLLs can be quite effective. If the behavioral models can be made to include the effect of component noise sources, they can be used to efficiently predict the noise and jitter of PLLs =-=[8,29]-=-. To do so, it is first necessary to individually characterize the noise behavior of the blocks that make up the PLL using transistor-level simulation. For each block, representative periodic signals ... |

8 |
Hagreraats “A 2.5-GHz BiCMOS transceiver for wireless LAN’s
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- 1997
(Show Context)
Citation Context ...LO alone, linearizing the circuit about the LO, and applying the PNoise analysis of Section IV-B. Actual measurements on bipolar mixers have shown that noise figure can be predicted to within 0.25 dB =-=[7,34]-=-. Results are not expected to be as accurate on CMOS mixers because the noise model for MOS transistors is not as accurate as the one for bipolar transistors. 2) Impact of Cyclostationarity on Subsequ... |

8 | Frequency-domain nonlinear circuit analysis using generalized power series - Rhyne, Steer, et al. - 1988 |

8 |
F.Zappa, “Spectrum folding and phase noise in LC tuned oscillators
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- 1998
(Show Context)
Citation Context ...he upper and lower sidebands components of a modulated carrier. It is also possible, using a change of basis, to recast these transfer functions in terms of the AM and PM components of the modulation =-=[53]-=-. To show this, consider a circuit that is generating a sinusoidal carrier. Assume that the carrier is both amplitude and phase modulated by small complex exponentials at the same frequency fm . The r... |

8 |
RF design guide systems, circuits and equations (Artech
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(Show Context)
Citation Context ... the noise of individual blocks using the noise figure of the block because it is relatively simple to combine the noise figure of cascaded blocks to determine the noise figure of the entire receiver =-=[44,65]-=-. The noise figure (NF) of a block is a measure of how much the signal-to-noise ratio (SNR) degrades as the signal passes through the block. It is defined as SNRin NF = 10 log---------------- (57) SNR... |

7 |
Computation of circuit waveform envelopes using an efficient, matrix-decomposed harmonic balance algorithm
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- 1996
(Show Context)
Citation Context ...g Fourier coefficients was introduced. In that case, the Fourier coefficients were assumed to be periodic, with the result that signals themselves were quasiperiodic. With the Fourier-envelope method =-=[12,38,48]-=-, the Fourier coefficients in (20) are time varying, but are not necessarily periodic. Instead, the Fourier coefficients X are taken to be slowly varying transient waveforms. Thus, signals take the fo... |

7 | Efficient time-domain simulation of frequency-dependent elements
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- 1996
(Show Context)
Citation Context ...A alternative approach is to compute the impulse response for a distributed component from a frequency domain description and use convolution to determine the response of the component in the circuit =-=[20,54,56]-=-. Evaluating lossy or dispersive transmission line models or tables of S-parameters with this approach is generally expensive and error-prone. Packages, substrates and spirals can be modeled with larg... |

7 |
Noise and loss in balanced and subharmonically pumped mixers: Part i—theory
- Kerr
(Show Context)
Citation Context ...umber of transfer functions between any particular input and output. Each represents a different frequency translation. Versions of this type of small-signal analysis exists for both harmonic balance =-=[17,24,31]-=- and shooting methods [39,40,61]. There are two different ways of formulating a small-signal analysis that computes transfer functions [59,61]. The first is akin to traditional AC analysis, and is ref... |