Combined Transfer Function Research Articles

Systematic Order Fitting Algorithm in Neuro-TF for Parametric Modeling of Microwave Components

This letter proposes a novel systematic order fitting technique for parametric modeling using a combined neural network and transfer function (neuro-TF) technique for electromagnetic (EM) behaviors of microwave components. The proposed technique can not only systematically determine the minimum order of the transfer function, but also systematically change the orders. By using the proposed technique, the problem of pole/residue discontinuity due to the improperly fitting can be solved, which results in a better neuro-TF modeling accuracy over the existing order-changing technique. The proposed method is demonstrated by two microwave examples.

Reconstructing sea-level change in the Falkland Islands (Islas Malvinas) using salt-marsh foraminifera, diatoms and testate amoebae

Proxy records of past sea-level change provide a means of extending sea-level histories from tide gauges into the pre-industrial period. This is especially valuable in the South Atlantic region where sea-level data are limited to only a few tide-gauge records. Multi-proxy approaches to sea-level reconstruction are relatively rare but have distinct benefits when groups of micro-organisms are sparse or under-represented in modern or fossil sediments. Here, we address this challenge by utilising surface foraminifera, testate amoebae and diatoms from a salt marsh at Swan Inlet, East Falkland. All three micro-organism groups occupied distinct vertical niches in the contemporary salt-marsh. We investigated the relative performance of each group of micro-organisms in providing a sea-level reconstruction using individual (group-specific) regression models and with a multi-proxy regression model that combined all three groups. Foraminifera alone were not a suitable proxy. Surveyed sample elevations were closely matched by estimated elevations using Weighted-Average (WA) and Weighted-Average Partial-Least-Squares (WA- PLS) regressions. Relative sea-level reconstructions were derived by applying each model to microfossil assemblages recovered from a core (SI-2) from the same site. The combined transfer function yielded reconstructive precision (± 0.08 m) comparable to our best single-proxy transfer function (± 0.06 m) but only 18% of palaeo-samples were identified as having “close” or “good” analogues in the combined training data set. We highlight the benefit of a pragmatic approach to sea-level reconstructions whereby additional proxies should be employed if the use of only one proxy performs poorly across the width of the elevation gradient.

Restoration and Calibration of Tilting Hyperspectral Super-Resolution Image

Tilting sampling is a novel sampling mode for achieving a higher resolution of hyperspectral imagery. However, most studies on the tilting image have only focused on a single band, which loses the features of hyperspectral imagery. This study focuses on the restoration of tilting hyperspectral imagery and the practicality of its results. First, we reduced the huge data of tilting hyperspectral imagery by the p-value sparse matrix band selection method (pSMBS). Then, we restored the reduced imagery by optimal reciprocal cell combined modulation transfer function (MTF) method. Next, we built the relationship between the restored tilting image and the original normal image. We employed the least square method to solve the calibration equation for each band. Finally, the calibrated tilting image and original normal image were both classified by the unsupervised classification method (K-means) to confirm the practicality of calibrated tilting images in remote sensing applications. The results of classification demonstrate the optimal reciprocal cell combined MTF method can effectively restore the tilting image and the calibrated tiling image can be used in remote sensing applications. The restored and calibrated tilting image has a higher resolution and better spectral fidelity.

Recent advances in parametric modeling of microwave components using combined neural network and transfer function

AbstractParametric modeling of electromagnetic (EM) behaviors has become important for EM design optimizations of microwave components. This paper provides an overview of recent advances in parametric modeling of microwave components using combined neural network and transfer function (neuro‐TF). Transfer functions are used to represent the EM responses of passive components vs frequency. With the help of the transfer function, the nonlinearity of the neural network structure can be significantly decreased. We first introduce the neuro‐TF modeling approach in rational format. We also review the pole‐residue‐based neuro‐TF modeling technique. The orders of the pole‐residue transfer functions may vary over different regions of geometrical parameters. A pole‐residue tracking technique can be used to solve this order‐changing problem. As a further advancement, we discuss the sensitivity analysis‐based neuro‐TF modeling technique. The purpose is to increase the model accuracy by utilizing EM sensitivity information and to speed up the model development process by reducing the number of training data required for developing the model. After the modeling process, the trained model can be used to provide accurate and fast prediction of the EM responses w.r.t. the geometrical variables and can be subsequently used in the high‐level circuit and system design.

Towards a Holarctic synthesis of peatland testate amoeba ecology: Development of a new continental-scale palaeohydrological transfer function for North America and comparison to European data

Fossil testate amoeba assemblages have been used to reconstruct peatland palaeohydrology for more than two decades. While transfer function training sets are typically of local-to regional-scale in extent, combining those data to cover broad ecohydrological gradients, from the regional-to continental- and hemispheric-scales, is useful to assess if ecological optima of species vary geographically and therefore may have also varied over time. Continental-scale transfer functions can also maximise modern analogue quality without losing reconstructive skill, providing the opportunity to contextualise understanding of purely statistical outputs with greater insight into the biogeography of organisms. Here, we compiled, at moderate taxonomic resolution, a dataset of nearly 2000 modern surface peatland testate amoeba samples from 137 peatlands throughout North America. We developed transfer functions using four model types, tested them statistically and applied them to independent palaeoenvironmental data. By subdividing the dataset into eco-regions, we examined biogeographical patterns of hydrological optima and species distribution across North America. We combined our new dataset with data from Europe to create a combined transfer function. The performance of our North-American transfer function was equivalent to published models and reconstructions were comparable to those developed using regional training sets. The new model can therefore be used as an effective tool to reconstruct peatland palaeohydrology throughout the North American continent. Some eco-regions exhibited lower taxonomic diversity and some key indicator taxa had restricted ranges. However, these patterns occurred against a background of general cosmopolitanism, at the moderate taxonomic resolution used. Likely biogeographical patterns at higher taxonomic resolution therefore do not affect transfer function performance. Output from the combined North American and European model suggested that any geographical limit of scale beyond which further compilation of peatland testate amoeba data would not be valid has not yet been reached, therefore advocating the potential for a Holarctic synthesis of peatland testate amoeba data. Extending data synthesis to the tropics and the Southern Hemisphere would be more challenging due to higher regional endemism in those areas.

Using Transfer Function Analysis to develop biologically and economically efficient restoration strategies

Rare species across taxonomic groups and biomes commonly suffer from multiple threats and require intensive restoration, including population reintroduction and threat control. Following reintroduction, it is necessary to identify what level of threat control is needed for species to persist over time. Population reintroduction and threat control are time intensive and costly. Thus, it is pragmatic to develop economically efficient restoration strategies. We combined transfer function analysis and economic cost analysis to evaluate the effects of biologically meaningful increases in demographic processes on the persistence of a reintroduced population of a Hawaii endemic long-lived shrub, Delissea waianaeensis. We show that an increase in fertility by 0.419 following the suppression of non-native rodents or an increase by 0.098 in seedling growth following the suppression of invasive molluscs would stabilize the population (i.e., λ = 1). Though a greater increase in fertility than seedling growth was needed for the reintroduced population to persist over time, increasing fertility by suppressing rodents was the most cost effective restoration strategy. Our study emphasizes the importance of considering the effects of large increases in plant vital rates in population projections and incorporating the economic cost of management actions in demographic models when developing restoration plans for endangered species.

Spontaneous baroreceptor reflex sensitivity for risk stratification of heart failure patients: optimal cut-off and age effects.

Baroreceptor reflex sensitivity (BRS) is an important prognostic factor because a reduced BRS has been associated with an adverse cardiovascular outcome. The threshold for a 'reduced' BRS was established by the ATRAMI study at BRS <3 ms/mmHg in patients with a previous myocardial infarction, and has been shown to improve risk assessment in many other cardiac dysfunctions. The successful application of this cut-off to other populations suggests that it may reflect an inherent property of baroreflex functioning, so our goal is to investigate whether it represents a 'natural' partition of BRS values. As reduced baroreflex responsiveness is also associated with ageing, we investigated whether a BRS estimate <3 ms/mmHg could be the result of a process of physiological senescence as well as a sign of BRS dysfunction. This study involved 228 chronic heart failure patients and 60 age-matched controls. Our novel method combined transfer function BRS estimation and automatic clustering of BRS probability distributions, to define indicative levels of different BRS activities. The analysis produced a fit clustering (cophenetic correlation coefficient 0.9 out of 1) and identified one group of homogeneous patients (well separated from the others by 3 ms/mmHg) with an increased BRS-based mortality risk [hazard ratio (HR): 3.19 (1.73, 5.89), P<0.001]. The age-dependent BRS cut-off, estimated by 5% quantile regression of log (BRS) with age (considering the age-matched controls), provides a similar mortality value [HR: 2.44 (1.37, 4.43), P=0.003]. In conclusion, the 3 ms/mmHg cut-off identifies two large clusters of homogeneous heart failure (HF) patients, thus supporting the hypothesis of a natural cut-off in the HF population. Furthermore, age was found to have no statistical impact on risk assessment, suggesting that there is no need to establish age-based cut-offs because 3 ms/mmHg optimally identifies patients at high mortality risk.

Comparison between chironomid-inferred mean-August temperature from varved Lake Żabińskie (Poland) and instrumental data since 1896 AD

Chironomids preserved in varved Lake Żabińskie (54°07′54.5″ N; 21°59′01.1″ E; 120 m a.s.l), northeastern Poland, were used to reconstruct mean-August air temperature since 1896 AD at annual (1949–2011 AD) and at 3–4 year resolutions (1896–1948 AD). This is one of very few studies using chironomids at such high temporal resolution, for a total of 130 sediment samples analyzed. To infer temperature a combined (Northeastern Canada and Poland) transfer function was developed. This transfer function had 112 lakes (50 from Poland, 72 from Canada) and 95 taxa. The mean-August-air temperature gradient was 23.5 °C. The coefficient of determination (r2jack) was 0.88, the root-mean-square error of prediction (RMSEP) was 1.30 °C and the maximum bias was 2.08 °C.The average mean August air temperature measured during the 1961–1990 AD reference period was 16.3 °C and the chironomid-inferred average temperature for the same period was 16.6 °C. For the 1981–2010 AD reference period, the temperatures were 1 °C warmer (i.e. measured = 17.3 °C; inferred = 17.6 °C). The differences of only 0.3 °C between the measured and the inferred temperatures during the reference periods were a first indication that the merged transfer function could provide accurate estimates of mean August air temperature.Chironomid-inferred mean August air temperatures since 1896 AD had strong relationships with instrumental data at near-annual resolution (rPearson = 0.74, pcorr < 0.01). The relationship increased on decadal scale (rPearson = 0.91, pcorr < 0.01) when the errors in varve chronology (−5 to 2 years) were considered.Although the RMSEP of the transfer function was higher than most changes recorded, the average difference between the instrumental and the inferred mean August air temperatures was 0.75 °C. The results of this study suggest that not only do chironomids reconstruct the right pattern in temperature changes (83% of the inferences recorded the same temperature increase or decrease than the instrumental data) they can also infer changes with the right amplitude (61% of the inferences had differences with the instrumental data below 0.7 °C). This study is one more example of the possible accuracy of chironomids to reconstruct climate, at least for the past 100 years and at this site.

Equivalent Load Identification Algorithm Based on Least-squares in Frequency Domain

Because the real load location of structure in working condition is unknown or cannot be reached, a new equivalent load identification algorithm in frequency domain is proposed. This new algorithm can guarantee the same vibration response signals for key examination points and can reach purposes of reliability test and environmental adaptability test then. Based on the assumptions of linear system and uncorrelated of each load, combined transfer functions, least-squares of generalized matrix inverse in frequency domain, this new algorithm identify multi-sources loads at the same time. In this new algorithm, the Identification load location, direction and form can be specify prior and not be the same with real load location, the boundary condition of structure in test cannot be the same with working condition. Formula derivation, application scope and steps of this method were summarized then. Load identification results of vibration simulations in cylindrical shell showed that this new method could basically meet the engineering precision requirement of ± 3db at last, which valid the effectiveness and reasonableness of this method.

Parametric modeling of microwave passive components using combined neural networks and transfer functions in the time and frequency

A novel parametric modeling technique is proposed to develop combined neural network and transfer function models for both time and frequency (TF) domain applications of passive components, where the neural network is trained to map geometrical variables to the coefficients of transfer functions. Built on our previous work, a new order-changing module is developed to enforce stability of transfer functions and simultaneously guarantee continuity of coefficients. A constrained optimization strategy is introduced to enforce passivity of transfer functions through a neural network training process. A general equivalent circuit for two-port passive components is generated directly from coefficients of arbitrary-order transfer functions. Once trained, the parametric model can provide accurate and fast prediction of the electromagnetic behavior of passive components with geometrical parameters as variables. Compared to our previous work, the proposed method enables models to work well in the time domain providing good accuracy in challenging modeling applications. Two parametric modeling examples of spiral inductors and interdigital capacitors, and their application in both time and frequency domain simulations of a power amplifier are examined to demonstrate the validity of the proposed technique. © 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE , 2013.

An Innovations-Based Noise Cancelling Technique on Inverse Kepstrum Whitening Filter and Adaptive FIR Filter in Beamforming Structure

This paper presents an acoustic noise cancelling technique using an inverse kepstrum system as an innovations-based whitening application for an adaptive finite impulse response (FIR) filter in beamforming structure. The inverse kepstrum method uses an innovations-whitened form from one acoustic path transfer function between a reference microphone sensor and a noise source so that the rear-end reference signal will then be a whitened sequence to a cascaded adaptive FIR filter in the beamforming structure. By using an inverse kepstrum filter as a whitening filter with the use of a delay filter, the cascaded adaptive FIR filter estimates only the numerator of the polynomial part from the ratio of overall combined transfer functions. The test results have shown that the adaptive FIR filter is more effective in beamforming structure than an adaptive noise cancelling (ANC) structure in terms of signal distortion in the desired signal and noise reduction in noise with nonminimum phase components. In addition, the inverse kepstrum method shows almost the same convergence level in estimate of noise statistics with the use of a smaller amount of adaptive FIR filter weights than the kepstrum method, hence it could provide better computational simplicity in processing. Furthermore, the rear-end inverse kepstrum method in beamforming structure has shown less signal distortion in the desired signal than the front-end kepstrum method and the front-end inverse kepstrum method in beamforming structure.

A New Training Approach for Parametric Modeling of Microwave Passive Components Using Combined Neural Networks and Transfer Functions

This paper presents a novel technique to develop combined neural network and transfer function models for parametric modeling of passive components. In this technique, the neural network is trained to map geometrical variables onto coefficients of transfer functions. A major advance is achieved in resolving the discontinuity problem of numerical solutions of the coefficients with respect to the geometrical variables. Minimum orders of transfer functions for different regions of geometrical parameter space are identified. Our investigations show that varied orders used for different regions result in the discontinuity of coefficients. The gaps between orders are bridged by a new order-changing module, which guarantees the continuity of coefficients and simultaneously maintains the modeling accuracy through a neural network optimization process. This technique is also expanded to include bilinear transfer functions. Once trained, the model provides accurate and fast prediction of the electromagnetic behavior of passive components with geometrical parameters as variables. Compared to conventional training methods, the proposed method allows better accuracy in challenging applications involving high-order transfer functions, wide frequency range, and large geometrical variations. Three examples including parametric modeling of slotted patch antennas, bandstop microstrip filters, and bandpass coupled-line filters are examined to demonstrate the validity of this technique.

Binaural Rendering in MPEG Surround

This paper describes novel methods for evoking a multichannel audio experience over stereo headphones. In contrast to the conventional convolution-based approach where, for example, five input channels are filtered using ten head-related transfer functions, the current approach is based on a parametric representation of the multichannel signal, along with either a parametric representation of the head-related transfer functions or a reduced set of head-related transfer functions. An audio scene with multiple virtual sound sources is represented by a mono or a stereo downmix signal of all sound source signals, accompanied by certain statistical (spatial) properties. These statistical properties of the sound sources are either combined with statistical properties of head-related transfer functions to estimate that represent the perceptually relevant aspects of the auditory scene or used to create a limited set of combined head-related transfer functions that can be applied directly on the downmix signal. Subsequently, a binaural rendering stage reinstates the statistical properties of the sound sources by applying the estimated binaural parameters or the reduced set of combined head-related transfer functions directly on the downmix. If combined with parametric multichannel audio coders such as MPEG Surround, the proposed methods are advantageous over conventional methods in terms of perceived quality and computational complexity.

Kinetic Aspects of Ion Exchange in KhFek[Fe(CN)6]l·mH2O Compounds: A Combined Electrical and Mass Transfer Functions Approach

The present paper quantifies and develops the kinetic aspects involved in the mechanism of interplay between electron and ions presented elsewhere(1) for K(h)Fe(k)[Fe(CN)(6)](l)*mH(2)O (Prussian Blue) host materials. Accordingly, there are three different electrochemical processes involved in the PB host materials: H(3)O(+), K(+), and H(+) insertion/extraction mechanisms which here were fully kinetically studied by means of the use of combined electronic and mass transfer functions as a tool to separate all the processes. The use of combined electronic and mass transfer functions was very important to validate and confirm the proposed mechanism. This mechanism allows the electrochemical and chemical processes involved in the K(h)Fe(k)[Fe(CN)(6)](l)*mH(2)O host and Prussian Blue derivatives to be understood. In addition, a formalism was also developed to consider superficial oxygen reduction. From the analysis of the kinetic processes involved in the model, it was possible to demonstrate that the processes associated with K(+) and H(+) exchanges are reversible whereas the H(3)O(+) insertion process was shown not to present a reversible pattern. This irreversible pattern is very peculiar and was shown to be related to the catalytic proton reduction reaction. Furthermore, from the model, it was possible to calculate the number density of available sites for each intercalation/deintercalation processes and infer that they are very similar for K(+) and H(+). Hence, the high prominence of the K(+) exchange observed in the voltammetric responses has a kinetic origin and is not related to the amount of sites available for intercalation/deintercalation of the ions.

Combined Transfer Function Analysis and Modelling of Cerebral Autoregulation

The clinical importance of cerebral autoregulation has resulted in a significant body of literature that attempts both to model the underlying physiological processes and to estimate the mathematical relationships between clinically measurable variables, the most common of which are Arterial Blood Pressure and Cerebral Blood Flow Velocity. These approaches have, however, rarely been used together to interpret clinical data. A simple model of cerebral autoregulation is thus proposed here, based on a flow dependent feedback mechanism with gain and time constant that adjusts arterial compliance. Analysis of this model shows that it closely approximates a second order system for typical values of physiological parameters. The model parameters can be optimally estimated from available experimental data for the Impulse Response (IR), yielding physiologically reasonable values, although there is one free parameter that must be fixed. The effects of changes in feedback gain and time constant are found to be significant on the predicted IR and can thus be estimated robustly from experimental data. The effects of elevated baseline Intracranial Pressure (ICP) are found to be exactly equivalent to a reduced feedback gain, although the solution is much less sensitive to the former effect. A transfer function approach can be used to estimate autoregulation status clinically using a physiologically-based model, thus providing greater insight into the processes that govern cerebral autoregulation.

Combined analyser-based and propagation-based phase-contrast imaging of weak objects

A new theoretical method combining analyser-based and propagation-based hard X-ray phase-contrast imaging is investigated. Unlike the previous theoretical model of the combined imaging method constructed under the assumption of slow variation of the individual transfer functions (large Fresnel numbers), a new model proposed in this paper uses the assumption of a weak scatterer (analogous to the first Born approximation). Consequently, the results are not limited to the case of short propagation distances or low-resolution imaging. An explicit expression for the combined transfer function is derived and analytical and numerical examples solving related inverse imaging problems are presented.

Transfer function measurements in a model combustor: Application to adaptive instability control

Due to its good performance in terms of low pollutant emissions, the lean premixed, prevaporized combustor design has received a special interest in the field of gas turbines. This article reports an experimental study of a laboratory-scale, premixed, prevaporized, swirl-stabilized combustor operated with preheated air and fed with liquid heptane as fuel. Under certain operating conditions, it exhibits well-established combustion oscillations at frequencies near 400 Hz. To understand and control these instabilities, it is useful to determine the burner response to external modulations. A novel actuator relying on secondary air modulation was designed and integrated in the injector to allow measurements of the combined transfer function of the burner and actuation system. Two different techniques were used for this determination: direct measurement between command and pressure on photo-multiplier signals based on spectral and cross-spectral densities, or identification through a finite impulse response numerical filter. The two techniques are in good agreement. This identification is then successfully applied to active control of combustion instabilities. The mean pressure oscillation amplitude is reduced by about 50% (from 650 Pa to 400 Pa, the noise level being 130 Pa). This modest reduction is due to the fact that the actuator operates at a frequency that is slightly greater than its cutoff frequency and that its effectiveness is therefore degraded. This highlights the difficulties associated with the bandwidth limitations of practical actuators.

Spectro-temporal response field characterization with dynamic ripples in ferret primary auditory cortex.

To understand the neural representation of broadband, dynamic sounds in primary auditory cortex (AI), we characterize responses using the spectro-temporal response field (STRF). The STRF describes, predicts, and fully characterizes the linear dynamics of neurons in response to sounds with rich spectro-temporal envelopes. It is computed from the responses to elementary "ripples," a family of sounds with drifting sinusoidal spectral envelopes. The collection of responses to all elementary ripples is the spectro-temporal transfer function. The complex spectro-temporal envelope of any broadband, dynamic sound can expressed as the linear sum of individual ripples. Previous experiments using ripples with downward drifting spectra suggested that the transfer function is separable, i.e., it is reducible into a product of purely temporal and purely spectral functions. Here we measure the responses to upward and downward drifting ripples, assuming reparability within each direction, to determine if the total bidirectional transfer function is fully separable. In general, the combined transfer function for two directions is not symmetric, and hence units in AI are not, in general, fully separable. Consequently, many AI units have complex response properties such as sensitivity to direction of motion, though most inseparable units are not strongly directionally selective. We show that for most neurons, the lack of full separability stems from differences between the upward and downward spectral cross-sections but not from the temporal cross-sections; this places strong constraints on the neural inputs of these AI units.

Generalised iterative method for solving 2D multiscattering problems using spectral techniques

An iterative method for two-dimensional problems with multiple scatterers is presented. Spectral techniques are used to characterise a scatterer by its transfer function. To obtain this transfer function physical optics and the method of moments have been used. An iterative method analysing the existence of a new object at each new iteration is outlined. The method produces combined transfer functions that give the response of the multiple scattering problem to several incident fields. A useful alternative iteration for clumped groups of scatterers is also presented. Results for a parabolic reflector, a Cassegrain structure and a horn for TM incidence are analysed and comparison with theoretical data is discussed. Finally, the extension of the method for the TE case and for 3D problems is outlined.

Analysis of dynamic spectra in ferret primary auditory cortex. I. Characteristics of single-unit responses to moving ripple spectra.

1. Auditory stimuli referred to as moving ripples are used to characterize the responses of both single and multiple units in the ferret primary auditory cortex. Moving ripples are broadband complex sounds with a sinusoidal spectral profile that drift along the logarithmic frequency axis at a constant velocity. 2. Neuronal responses to moving ripples are locked to the phase of the ripple, i.e., they exhibit the same periodicity as that of the moving ripple profile. Neural responses are characterized as a function of ripple velocity (temporal property) and ripple frequency (spectral property). Transfer functions describing the response to these temporal and spectral modulations are constructed. Temporal transfer functions are inverse Fourier transformed to obtain impulse response functions that reflect the cell's temporal characteristics. Ripple transfer functions are inverse Fourier transformed to obtain the response field, a measure analogous to the cell's response area. These operations assume linearity in the cell's response to moving ripples. 3. Transfer functions and other response functions are shown to be fairly independent on the overall level or depth of modulation of the ripple stimuli. Only downward moving ripples were used in this study. 4. The temporal and ripple transfer functions are found to be separable, in that their shapes remain unchanged for different test parameters. Thus ripple transfer functions and response fields remain statistically similar in shape (to within an overall scale factor) regardless of the ripple velocity or whether stationary or moving ripples are used in the measurement. The same stability in shape holds for the temporal transfer functions and the impulse response functions measured with different ripple frequencies. Separability implies that the combined spectrotemporal transfer function of a cell can be written as the product of a purely ripple and a purely temporal transfer functions, and thus that the neuron can be computationally modeled as processing spectral and temporal information in two separate and successive stages. 5. The ripple parameters that characterize cortical cells are distributed somewhat evenly, with the characteristic ripple frequencies ranging from 0.2 to > 2 cycles/octave and the characteristic angular frequency typically ranging from 2 to 20 Hz. 6. Many responses exhibit periodicities in the spectral envelope of the stimulus. These periodicities are of two types. Slow rebounds, not found in the spectral envelope, and with a period of approximately 150 ms, appear with various strengths in approximately 30% of the cells. Fast regular firings with interspike intervals of approximately 10 ms are much less common and appear to correspond to interactions between the component tones that make up a ripple.