Amplitude Estimation Research Articles

Phase-matching enhanced quantum phase and amplitude estimation of a two-level system in squeezed reservoir

Phase-matching enhanced quantum phase and amplitude estimation of a two-level system in squeezed reservoir

Comparison of synergy extrapolation and static optimization for estimating multiple unmeasured muscle activations during walking

BackgroundCalibrated electromyography (EMG)-driven musculoskeletal models can provide insight into internal quantities (e.g., muscle forces) that are difficult or impossible to measure experimentally. However, the need for EMG data from all involved muscles presents a significant barrier to the widespread application of EMG-driven modeling methods. Synergy extrapolation (SynX) is a computational method that can estimate a single missing EMG signal with reasonable accuracy during the EMG-driven model calibration process, yet its performance in estimating a larger number of missing EMG signals remains unknown.MethodsThis study assessed the accuracy with which SynX can use eight measured EMG signals to estimate muscle activations and forces associated with eight missing EMG signals in the same leg during walking while simultaneously performing EMG-driven model calibration. Experimental gait data collected from two individuals post-stroke, including 16 channels of EMG data per leg, were used to calibrate an EMG-driven musculoskeletal model, providing “gold standard” muscle activations and forces for evaluation purposes. SynX was then used to predict the muscle activations and forces associated with the eight missing EMG signals while simultaneously calibrating EMG-driven model parameter values. Due to its widespread use, static optimization (SO) applied to a scaled generic musculoskeletal model was also utilized to estimate the same muscle activations and forces. Estimation accuracy for SynX and SO was evaluated using root mean square errors (RMSE) to quantify amplitude errors and correlation coefficient r values to quantify shape similarity, each calculated with respect to “gold standard” muscle activations and forces.ResultsOn average, compared to SO, SynX with simultaneous model calibration produced significantly more accurate amplitude and shape estimates for unmeasured muscle activations (RMSE 0.08 vs. 0.15, r value 0.55 vs. 0.12) and forces (RMSE 101.3 N vs. 174.4 N, r value 0.53 vs. 0.07). SynX yielded calibrated Hill-type muscle–tendon model parameter values for all muscles and activation dynamics model parameter values for measured muscles that were similar to “gold standard” calibrated model parameter values.ConclusionsThese findings suggest that SynX could make it possible to calibrate EMG-driven musculoskeletal models for all important lower-extremity muscles with as few as eight carefully chosen EMG signals and eventually contribute to the design of personalized rehabilitation and surgical interventions for mobility impairments.

Impact on preoperative estimation of atrial mechanical amplitudes with vdd leadless pacemaker

Abstract Background It would be useful to determine whether the device can sense the mechanical atrial amplitude (A4) before the implantation of a VDD leadless pacemaker (Micra AV^RT, Medtronic, US). Purpose To evaluate whether identifiable A4 is associated with preoperative echocardiographic parameters. Methods Echocardiographic measurements were performed to quantify atrial motion of the interventricular septum in patients who received a leadless pacemaker implantation. We visually divided the septum into five segments from the base to the apex. As shown in Figure 1, the myocardial velocity of each segment measured on tissue doppler imaging (TDI) was defined as EV1, EV2, EV3, EV4, and EV5 (m/s), respectively. After the implantation, amplitude of A4 was compared to the local EV. Results We retrospectively analyzed 30 patients who had Micra AV implanted into the right ventricular septum by imaging examination and had measurable EV for each segment. The mean age was 82.3±7.8 years, 15 (50%) patients were female, 11 (37%) patients had organic heart disease, and 23 (77%) patients had atrial ventricular block. The mean LVEF and A4 amplitude were 60.4 ± 8.9% and 2.7 ± 1.8 m/s^2, respectively, and the mean atrial mechanical sensing rate (AMSR) was 79.9±24.0%. The A4 amplitude was positively correlated to the AMSR (r=0.48), while the AMSR was significantly better in patients with A4 amplitude over 1.0 m/s^2 (88.0 ± 10.4% vs. 41.0 ± 33.6%, p=0.04). The A4 amplitude was correlated to the EV of the implantation site estimated from fluoroscopy and echocardiography (r=0.62) in Figure 2. 24 patients had local EV over 3.0 cm/s, and among them, 22 (92%) patients had A4 amplitude over 1.0 m/s^2. Conclusion Local EV over 3.0 cm/s, as determined by preoperative echocardiography, might predict high-frequency AMSR.

Toward Utilizing Similarity in Hydrologic Data Assimilation

Similarity to reality is a necessary property of models in earth sciences. Similarity information can thus possess a large potential in advancing geophysical modeling and data assimilation. We present a formalism for utilizing similarity within the existing theoretical data assimilation framework. Two examples illustrate the usefulness of utilizing similarity in data assimilation. The first, theoretical example shows changes in the accuracy of the amplitude estimate in the presence of a phase error in a sine function, where correcting the phase error prior to the assimilation reduces the degree of ill-posedness of the assimilation problem. This signifies the importance of accounting for the phase error in order to reduce the error in the amplitude estimate of the sine function. The second, real-world example illustrates that timing errors in simulated flow degrade the data assimilation performance, and that the flow gradient-informed shifting of rainfall time series improved the assimilation results with less adjusting model states. This demonstrates the benefit of utilizing streamflow gradients in shifting rainfall time series in a way to improve streamflow timing—vital information for flood early warning and preparedness planning. Finally, we discuss the implications, potential issues, and future challenges associated with utilizing similarity in hydrologic data assimilation.

Option pricing under stochastic volatility on a quantum computer

We develop quantum algorithms for pricing Asian and barrier options under the Heston model, a popular stochastic volatility model, and estimate their costs, in terms of T-count, T-depth and number of logical qubits, on instances under typical market conditions. These algorithms are based on combining well-established numerical methods for stochastic differential equations and quantum amplitude estimation technique. In particular, we empirically show that, despite its simplicity, weak Euler method achieves the same level of accuracy as the better-known strong Euler method in this task. Furthermore, by eliminating the expensive procedure of preparing Gaussian states, the quantum algorithm based on weak Euler scheme achieves drastically better efficiency than the one based on strong Euler scheme. Our resource analysis suggests that option pricing under stochastic volatility is a promising application of quantum computers, and that our algorithms render the hardware requirement for reaching practical quantum advantage in financial applications less stringent than prior art.

Research on the fusion imaging method of sign coherence and time reversal for Lamb wave sparse array

Time-reversal imaging struggles to detect plate-like structures due to interference from Lamb wave mode conversion and the processing demands, leading to less effective outcomes. This paper proposes a sign coherence factor and time reversal fusion (SCF-TR) imaging method based on amplitude and phase estimation. This method removes the coherence of array signals during signal reversal and refocusing. It reintroduces the sign coherence component to reduce interference from non-target scattered waves and partially overcome the constraints imposed by the Rayleigh criterion. The method allows imaging at a resolution smaller than the wavelength of Lamb and enhances the quality of the resulting images. In addition, a sparse array design utilizing the White Shark Optimisation Algorithm (WSO) is proposed to streamline the SCF-TR calculation process. This design utilizes sparse full matrix data to improve imaging efficiency. The experimental results show that for single blind hole defects, the SCF-TR method improves the array performance metrics and signal-to-noise ratio by 22.46% and 42.50%, respectively, compared to the TR method. For multiple asymmetric blind hole defects, when the defect size exceeds the resolution threshold, SCF-TR accurately reflects the position and morphology of defects smaller than the wavelength. When the defect size is below the resolution threshold, SCF-TR achieves super-resolution imaging. The sparse array designed using the White Shark Optimization algorithm demonstrates good sidelobe characteristics, effectively reducing sidelobe noise without reducing the array aperture. Moreover, the SCF-TR imaging time is reduced by approximately half while maintaining imaging accuracy.

Mechanisms of multi-layered Rayleigh noise in Brillouin optical correlation-domain reflectometry

In Brillouin optical correlation-domain reflectometry (BOCDR), sinusoidal modulation is applied to the output frequency of a light source, with spatial resolution inversely related to the modulation amplitude. We have developed an effective method to estimate the modulation amplitude using the width of the noise spectrum caused by Rayleigh scattering, eliminating the need for an optical spectrum analyzer or modifications to existing equipment. However, the Rayleigh noise spectrum often displays a three-layered structure, complicating the identification of the appropriate spectral components for estimating the modulation amplitude. In this work, we investigate the origins of this three-layered Rayleigh noise spectrum and identify the directivity of an optical circulator as the source of the third noise component. As replacing the circulator with alternative optical components is not easy, it remains an essential part of the system. Our analysis shows that the third noise component exhibits significantly small variation in spectral width with changes in modulation frequency compared to the first and second components. This characteristic allows for the effective separation and identification of the third noise component, thereby enhancing the accuracy and convenience of modulation amplitude estimation in BOCDR.

Quantum Probabilistic Model Checking for Time-Bounded Properties

Probabilistic model checking (PMC) is a verification technique for analyzing the properties of probabilistic systems. However, existing techniques face challenges in verifying large systems with high accuracy. PMC struggles with state explosion, where the number of states grows exponentially with the size of the system, making large system verification infeasible. While statistical model checking (SMC) avoids PMC’s state explosion problem by using a simulation approach, it suffers from runtime explosion, requiring numerous samples for high accuracy. To address these limitations in verifying large systems with high accuracy, we present quantum probabilistic model checking (QPMC), the first method leveraging quantum computing for PMC with respect to time-bounded properties. QPMC addresses state explosion by encoding PMC problems into quantum circuits that superpose states within qubits. Additionally, QPMC resolves runtime explosion through Quantum Amplitude Estimation, efficiently estimating the probabilities of specified properties. We prove that QPMC correctly solves PMC problems and achieves a quadratic speedup in time complexity compared to SMC.

Dual-stream Noise and Speech Information Perception based Speech Enhancement

In real-world scenarios, dynamic ambient noise often degrades speech quality, highlighting the need for advanced speech enhancement techniques. Traditional methods, which rely on static embeddings as auxiliary features, struggle to address the complexities of varying noise conditions. To overcome this, we propose a Dual-stream Noise and Speech Information Perception (DNSIP) approach that dynamically detects and processes both noise and speech through innovative information extraction and suppression mechanisms. Initially, non-speech segments predominantly contain environmental noise, while speech segments carry information about the intended speaker. To handle this dynamic nature, real-time voice activity detection (VAD) is employed to accurately differentiate between speech and noise components. Building on VAD estimates, we propose an innovative information extraction framework that selectively extracts relevant noise and speech features from the noisy input, establishing a dual-stream network for concurrent noise and speech learning. To account for the temporal and spectral variability of noise and speech, a frequency-sequence attention mechanism is integrated, enhancing the model’s ability to learn contextual and spectral dependencies. Additionally, an information suppression module is introduced to minimize cross-stream interference by attenuating noise within the speech stream and suppressing speech content within the noise stream. The derived noise and speech spectrograms are then utilized to formulate a minimum mean square error log-spectral amplitude (MMSE-LSA) estimator for robust speech enhancement. Experimental evaluations on the WSJ0 and VCTK+DEMAND datasets demonstrate that our DNSIP approach surpasses existing state-of-the-art methods, underscoring its efficacy in challenging acoustic environments.

Accurate Measurement of the Lensing Magnification by BOSS CMASS Galaxies and Its Implications for Cosmology and Dark Matter

Magnification serves as an independent and complementary gravitational lensing measurement to shear. We develop a novel method to achieve an accurate and robust magnification measurement around BOSS CMASS galaxies across physical scales of 0.016h −1 Mpc < r p < 10h −1 Mpc. We first measure the excess total flux density δ M of the source galaxies in the deep DECaLS photometric catalog that are lensed by CMASS galaxies. We convert δ M to magnification μ by establishing the δ μ–δ M relation using a deeper photometric sample. By comparing magnification measurements in three optical bands (grz), we constrain the dust attenuation curve and its radial distribution, discovering a steep attenuation curve in the circumgalactic medium of CMASS galaxies. We further compare dust-corrected magnification measurements to model predictions from high-resolution dark matter-only (DMO) simulations in Wilkinson Microwave Anisotropy Probe (WMAP) and Planck cosmologies, as well as the hydrodynamic simulation TNG300-1, using precise galaxy–halo connections from the Photometric objects Around Cosmic webs method and the accurate ray-tracing algorithm P3MLens. For r p > 70h −1 kpc, our magnification measurements are in good agreement with both WMAP and Planck cosmologies, resulting in an estimation of the matter fluctuation amplitude of S 8 = 0.816 ± 0.024. However, at r p < 70h −1 kpc, we observe an excess magnification signal, which is higher than the DMO model in Planck cosmology at 2.8σ and would be exacerbated if significant baryon feedback is included. Implications of the potential small scale discrepancy for the nature of dark matter and for the processes governing galaxy formation are discussed.

Competition between transient oscillations and early stochasticity in exponentially growing populations

It has been recently shown that the exponential growth rate of a population of bacterial cells starting from a single cell shows transient oscillations due to early synchronized bursts of division. These oscillations are enhanced by cell size regulation and contain information about single-cell growth statistics. Here, we report a phase transition in these oscillations as a function of growth rate variability. Below the transition point, these oscillations become asymptotically deterministic and can be measured experimentally, while above the transition point, the stochasticity in population growth dominates the oscillations and masks all the information about single cell growth statistics. The analytically calculated transition point, which roughly corresponds to 13% variability in single-cell growth rate, falls within physiologically relevant parameters. Additionally, we show that the oscillations can stochastically emerge even when the initial state contains multiple cells with out-of-phase division cycles. We show that the amplitude and phase of these oscillations are stochastic and vary across repeated measurements with the same initial conditions. We provide analytic expressions as well as numerical estimates for the typical oscillation amplitude and the number of generations before the amplitude falls below a given measurement threshold for in multiple growth conditions. Published by the American Physical Society 2024

Deep neural network based distribution system state estimation using hyperparameter optimization

In the past decade, distribution system state estimation has become a crucial topic in power system research due to the increasing importance of distribution networks amidst the decline of centralized energy production. This paper addresses a gap in the literature regarding the application of modern hyperparameter optimization techniques in low-voltage distribution system state estimation using deep neural networks. In particular, it demonstrates the use of the Tree-structured Parzen Estimator algorithm, which is a Bayesian hyperparameter optimization method, for distribution system state estimation on real Hungarian low-voltage networks. The study uses data from four real-life low-voltage supply areas in Hungary, which were modeled to address the challenges in obtaining network information. Compared to traditional methods like the weighted least squares method, the Tree-structured Parzen Estimator algorithm significantly improves the accuracy of the voltage amplitude and angle estimations, reducing the relative error by 14–73%. Additionally, it is shown that TPE outperforms simpler methods like Random Search in hyperparameter optimization. The results also reveal connections between the distribution system size and optimal hyperparameters, such as batch size, learning rate, and hidden layer configuration. The proposed non-iterative algorithm, combined with the parallel computation capabilities of deep neural networks utilizing GPU, resulted in four orders of magnitude improvement in runtime. These advancements make the proposed approach a valuable tool for renewable energy integration planning and real-time monitoring, highlighting its potential for practical applications in the power industry.

Data-driven determination of 1H-MRS basis set composition.

Metabolite amplitude estimates derived from linear combination modeling of MR spectra depend upon the precise list of constituent metabolite basis functions used (the "basis set"). The absence of clear consensus on the "ideal" composition or objective criteria to determine the suitability of a particular basis set contributes to the poor reproducibility of MRS. In this proof-of-concept study, we demonstrate a novel, data-driven approach for deciding the basis-set composition using Bayesian information criteria (BIC). We have developed an algorithm that iteratively adds metabolites to the basis set using iterative modeling, informed by BIC scores. We investigated two quantitative "stopping conditions", referred to as max-BIC and zero-amplitude, and whether to optimize the selection of basis set on a per-spectrum basis or at the group level. The algorithm was tested using two groups of synthetic in-vivo-like spectra representing healthy brain and tumor spectra, respectively, and the derived basis sets (and metabolite amplitude estimates) were compared to the ground truth. All derived basis sets correctly identified high-concentration metabolites and provided reasonable fits of the spectra. At the single-spectrum level, the two stopping conditions derived the underlying basis set with 77-87% accuracy. When optimizing across a group, basis set determination accuracy improved to 84-92%. Data-driven determination of the basis set composition is feasible. With refinement, this approach could provide a valuable data-driven way to derive or refine basis sets, reducing the operator bias of MRS analyses, enhancing the objectivity of quantitative analyses, and increasing the clinical viability of MRS.

Estimation of cerebrovascular reactivity amplitude and lag using breath-holding fMRI and the global BOLD signal: Application in diabetes and hypertension.

In this work, we demonstrate a data-driven approach for estimating cerebrovascular reactivity (CVR) amplitude and lag from breathhold (BH) fMRI data alone. Our approach employs a frequency-domain approach that is independent of external recordings. CVR amplitude is estimated from the BOLD frequency spectrum and CVR lag is estimated from the Fourier phase using the global-mean BOLD signal as reference. Unlike referencing to external recordings, these lags are specific to the brain. We demonstrated our method in detecting regional CVR amplitude and lag differences across healthy (CTL), hypertensive (HT) and hypertension-plus-type-2-diabetes (HT + DM) groups of similar ages and sex ratios, with a total N of 49. We found CVR amplitude to be significantly higher in CTL compared to HT + DM, with minimal difference between CTL and HT. Also, voxelwise CVR lag estimated in the Fourier domain is a more sensitive marker of vascular dysfunction than CVR amplitude. CVR lag in HT is significantly shorter than in CTL, with minimal difference between CTL and HT + DM. Our results support the importance of joint CVR amplitude and lag assessments in clinical applications.

Power system reliability assessment technique and modeling approach based on quantum computing theory

Aiming at the reliability assessment of power system, the state sampling process as a critical step requires to generate power system states one by one using classical computer. Due to the quantum superposition, all system states can be generated at once using the quantum computing and then assess the reliability, and thus it will improve the assessment efficiency. However, how to sample system states and assess reliability using the quantum computing theory will be very important. Therefore, this paper proposes a power system reliability assessment technique and its modeling approach based on the quantum computing theory. Firstly, this paper develops a quantum model of power system reliability (including reliability models of electrical device and system) and proposes a new system state sampling method based on the quantum computing theory, which samples all system states at once and improves the sampling efficiency. Secondly, this paper proposes a quantum computing theory based on reliability assessment technique and its modeling approach for the reliability indices: Loss of Load Probability (LOLP) and Expected Energy Not Supplied (EENS). Thirdly, this paper uses an iterative quantum amplitude estimation (IQAE) algorithm to measure the reliability indices for improving the measurement accuracy. Finally, this paper investigates both designed Case and RBTS, and results shows the correctness and effectiveness of reliability assessment of power system based on the quantum computing.

A square-root speedup for finding the smallest eigenvalue

We describe a quantum algorithm for finding the smallest eigenvalue of a Hermitian matrix. This algorithm combines quantum phase estimation and quantum amplitude estimation to achieve a quadratic speedup with respect to the best classical algorithm in terms of matrix dimensionality, i.e. O~(N/ε) 9 9In this work O~ ignores terms that are polylogarithmic in N or 1/ε . black-box queries to an oracle encoding the matrix, where N is the matrix dimension and ɛ is the desired precision. In contrast, the best classical algorithm for the same task requires Ω(N)polylog(1/ε) queries. In addition, this algorithm allows the user to select any constant success probability. We also provide a similar algorithm with the same runtime that allows us to prepare a quantum state lying mostly in the matrix’s low-energy subspace. We implement simulations of both algorithms and demonstrate their application to problems in quantum chemistry and materials science.

Real-Time Milling Chatter Detection and Control with Axis Encoder Feedback and Spindle Speed Manipulation

This paper introduces a complete real-time algorithm, where the chatter is detected and eliminated by spindle speed manipulation via the chatter energy feedback calculated from the axis encoder measurement. The proposed method does not require profound knowledge of the machining dynamics; instead, the entire algorithm exploits the fact that milling vibrations consist of forced vibrations at spindle speed harmonics and chatter vibrations that are close to one of the natural modes, with sidebands which are spread at the multiples of spindle speed frequency above and below the chatter frequency. The developed algorithm is able to identify the amplitude, phase and frequency of all the harmonics constituting the periodic forced and chatter vibrations. The key challenge is to select dominant chatter frequencies for the calculation of a robust and accurate chatter energy ratio feedback; this is achieved by utilizing the frequency estimation variance of EKF as a novel chatter indicator. Based on the chatter energy ratio feedback, the controller overrides the spindle speed in order to suppress the chatter energy below a particular threshold value. The varying spindle speed challenge is handled by updating the state transition matrices of the Kalman filters and real-time calculation of the band-pass filter coefficients, based on the derived discrete time transfer functions. The developed algorithm is tested on a Deckel FP5cc CNC which is in-house retrofitted and has a PC-based controller for the real-time application of the proposed algorithm. It is shown that the real-time chatter frequency and amplitude estimates are compatible with off-line FFT analysis, and chatter can be successfully eliminated by energy feedback.

Carbon market risk estimation using quantum conditional generative adversarial network and amplitude estimation

AbstractAccurately and efficiently estimating the carbon market risk is paramount for ensuring financial stability, promoting environmental sustainability, and facilitating informed decision‐making. Although classical risk estimation methods are extensively utilized, the implicit pre‐assumptions regarding distribution are predominantly contained and challenging to balance accuracy and computational efficiency. A quantum computing‐based carbon market risk estimation framework is proposed to address this problem with the quantum conditional generative adversarial network‐quantum amplitude estimation (QCGAN‐QAE) algorithm. Specifically, quantum conditional generative adversarial network (QCGAN) is employed to simulate the future distribution of the generated return rate, whereas quantum amplitude estimation (QAE) is employed to measure the distribution. Moreover, the quantum circuit of the QCGAN improved by reordering the data interaction layer and data simulation layer is coupled with the introduction of the quantum fully connected layer. The binary search method is incorporated into the QAE to bolster the computational efficiency. The simulation results based on the European Union Emissions Trading System reveals that the proposed framework markedly enhances the efficiency and precision of Value‐at‐Risk and Conditional Value‐at‐Risk compared to original methods.

Global assessment of linear trend and seasonal variations of GNSS-IR sea level retrievals with nearby tide gauges

Global Navigation Satellite System Interferometric Reflectometry (GNSS-IR) sea level retrievals and tide gauges at 40 globally distributed stations spanning from about 4.5 to 18 years are compared on a site-by-site basis, in terms of noise background, rate and seasonal variations by using the weighted least squares estimation (LSE) along with the Maximum likelihood estimation (MLE). The result shows that monthly GNSS-IR data agree well with tide gauges for most stations except the site Tuktoyaktuk, Canada. The mean correlation is 0.95 and the mean root mean square difference is 2.9 cm, respectively. The discrepancies of rate and seasonal amplitude estimates are within ± 1 cm for most stations. We confirm both the two sea level data exhibit temporal correlation, which has a great effect on the rate uncertainty estimates. Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) favor Matérn and first-order autoregressive (AR1) the preferred stochastic model for the daily and monthly mean sea level time series, respectively. Owing to the data span dependence for the rate uncertainty estimates, to get an accuracy of sub-mm/yr in linear rate using the weighted LSE, at least 30 years of data (depends on data quality) is required. We recommend using long time series and a proper stochastic model for the rate estimation of sea level.

Model-based frequency-and-phase correction of 1H MRS data with 2D linear-combination modeling.

Retrospective frequency-and-phase correction (FPC) methods attempt to remove frequency-and-phase variations between transients to improve the quality of the averaged MR spectrum. However, traditional FPC methods like spectral registration struggle at low SNR. Here, we propose a method that directly integrates FPC into a 2D linear-combination model (2D-LCM) of individual transients ("model-based FPC"). We investigated how model-based FPC performs compared to the traditional approach, i.e., spectral registration followed by 1D-LCM in estimating frequency-and-phase drifts and, consequentially, metabolite level estimates. We created synthetic in-vivo-like 64-transient short-TE sLASER datasets with 100 noise realizations at 5 SNR levels and added randomly sampled frequency and phase variations. We then used this synthetic dataset to compare the performance of 2D-LCM with the traditional approach (spectral registration, averaging, then 1D-LCM). Outcome measures were the frequency/phase/amplitude errors, the SD of those ground-truth errors, and amplitude Cramér Rao lower bounds (CRLBs). We further tested the proposed method on publicly available in-vivo short-TE PRESS data. 2D-LCM estimates (and accounts for) frequency-and-phase variations directly from uncorrected data with equivalent or better fidelity than the conventional approach. Furthermore, 2D-LCM metabolite amplitude estimates were at least as accurate, precise, and certain as the conventionally derived estimates. 2D-LCM estimation of FPC and amplitudes performed substantially better at low-to-very-low SNR. Model-based FPC with 2D linear-combination modeling is feasible and has great potential to improve metabolite level estimation for conventional and dynamic MRS data, especially for low-SNR conditions, for example, long TEs or strong diffusion weighting.