the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Mar 2026
An Order of Magnitude Signal-to-Noise Improvement of Magnetic Resonance Spectra using a Segmented-Overlap Fourier-Filtering and Averaging (SOFFA) Approach
Abstract. Segmented-Overlap Fourier-Filtering and Averaging (SOFFA) data acquisition method is described in detail for magnetic resonance spectroscopy. In this work the four processes that encompass the SOFFA data acquisition method are detailed: (i) oversampling spectral segments, (ii) Fourier block-filtering, (iii) segment-overlap averaging, and (iv) decimation. Three experimental examples are shown. Conventional Continuous Wave (CW) Electron Paramagnetic Resonance (EPR) is compared to SOFFA-CW of a single reduced [4Fe-4S]+ (S=1/2) at concentrations of 1 mM, 100 µM, and 10 µM showing an average increase in concentration sensitivity by a factor of 5.6. Experimental comparison of CW and SOFFA nonadiabatic rapid scan (SOFFA-NARS) data with similar filter parameters and field-modulation amplitude demonstrates a factor of 10.3 in signal-to-noise improvement for a 150 µM sitedirected spin-labeled Hemoglobin in 82 % glycerol at 18 °C. The signal-to-noise improvements were made for the same data acquisition times on standard commercial instruments. This method can be implemented to perform real-time segmented processing and, combined with more sophisticated averaging methods, will push the state-of-the-art sensitivity in magnetic resonance spectroscopy.
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Status: open (until 03 Apr 2026)
- RC1: 'Comment on mr-2026-6', Anonymous Referee #1, 19 Mar 2026 reply
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RC2: 'Comment on mr-2026-6', Anonymous Referee #2, 27 Mar 2026
reply
This manuscript describes a segment-based block filtering and averaging approach that aims at improving signal-to-noise ratio (SNR). This is an intriguing paper, particularly given the strong claim in its title. Improving SNR is dear to every magnetic resonance spectroscopist!
The manuscript needs substantial work before it can be considered for publication. There are two primary concerns:
(1) It is very difficult to understand the method, since its description is long-winded, partially redundant and spread over three sections (Methods; Theory; Appendix C). The notation is confusing. Pseudo-code or actual code for the proposed processing method would be crystal clear, but it is missing (there is no supplementary material). Therefore, it was impossible for me to reproduce and test the claims in this manuscript. The method description should be consolidated, shortened and made more consistent to increase readability. A Python script should be provided that takes acquired segments and required processing parameters as inputs and produces a filtered spectrum as output. This would enhance the impact of this work considerably.(2) The work does not clearly explain the origin of the apparent SNR improvements. Therefore, it does not convince me that the claim of SNR improvement is actually valid. In general, any improvement in SNR during processing comes at the cost of a reduction in bandwidth. One can always increase SNR by low-pass filtering a signal, making sure the filter stopband and passband droop do not encroach on the signal band. Also, it appears that the SOFFA method acquires more points in the same amount of time (see Fig. 7 for instance; "oversampling" is mentioned a lot) - for a bandwidth-limited signal this would provide more data, and an improved SNR would be the trivial consequence. What is the true origin of the (apparent) SNR improvement?
If these two concerns are addressed, this would constitute a substantial and impactful contribution that merits publication in Magn.Reson.
Below are my detailed notes from working through the manuscript:
- It is unclear what t represents. In Figs.1 and B1, t appears to be a time or time index, defining the spectral axis. In Eq.(2), it is the total spectral width (which is indicated as g in Eq.(1)). What is the difference between t and g?
- It is unclear which of the parameters t, s, g, N, L, m are the independent ones and which are the dependent ones.
- Eq.(1): Why is this formulated as an estimate - there must be a precise relationship between integer n and the right-hand side quantities.
- Line 68: "Data is discretized". This is unclear. Data is usually discretized already during measurement (which is point 1 on line 67), since the spectral intensity is only measured at discrete field values. What is this second discretization under point 2?
- Eq.(2) appears not general enough. Assume the sweep width is 10 G, the step size is 1 G, and m is set to 5, then there are only 6 segments, but this equation gives K = 10. It appears the equation is only correct if m=1.
- Line 81: D is not defined. Is it equal to N-L, as inferred from Fig.1? This should be stated explicitly.
- Eq.(3) mentions frequency omega, whereas up to this point only field sweeps and their FT are discussed. Unclear how the frequency relates to the field.
- gamma in Eq.(3) is defined as the "central point in the Fourier transformed data". What does this mean? The expectation value of the frequency over the FT of the field sweep? Add mathematical definition.
- Line 87: Provide more details about the concatenation. Are all segments just appended? If m=5 and there are K=6 segments, this leads to a total length m*K = 30 segments, which exceeds the length of the original field sweep. This probably boils down to what a "segment" is. Fig.1 and Fib.B1 indicate that it includes the overlap regions - segments are 3*n long in both figures. Fig.1 caption says "segments x_k[j]".
- Line 93: How is the "mean value of the peak-to-peak signal" defined? Is the peak-to-peak height averaged over several points, or several repeat measurements?
- Line 98: Mathematica doesn't appear to have a RandReal function, but a RandomReal function.
- Line 100: The use of a uniform distribution for (white) noise is entirely non-physical. For instance, it has a non-zero mean, whereas physical noise (white or pink) has zero mean. A normal distribution should be used instead.
- Line 103: What is the reason that white and pink noise are combined? The noise model used should match the actual noise as it occurs in experimental spectra.
- Line 121: "either random (white) or frequency-dependent (pink) noise". This is inaccurate language. All noise, including pink, is random. White noise has a frequency-independent power spectrum.
- In the main text, Figure 3 appears to be mentioned before Figure 2. The figure numbering should be adjusted.
- Paragraph starting Line 140: It is not clear how the discussion of parallel processing and real-time processing is relevant here. This should be clarified.
- Line 158 says that there are k total segments, whereas Eq.(2) says it is K. Make notation consistent.
- Line 160 mentions a "zero-fill parameter". How is it defined?
- "Oversampling" is mentioned many times throughout the manuscript. This term should be defined. Does it relate to the Nyquist sampling criterion? The FT of a Gaussian line is a Gaussian, so theoretically the signal is not band limited, and there is no oversampling.
- Eq.(6) uses t_e. How does that relate to t in Fig. 1, in Eq.(2) and in Fig.B1.
- Line 175 "Since...": This sentence is unclear.
- Line 175: "Fig A1 (ii and iii)" - unclear which panel in the figure is referred to. Same in line 184.
- Line 188: How is the frequency center of the Gaussian determined? Is the center at zero frequency?
- Line 214: A moving-average decimation filter is possibly suboptimal for this step, since it has the worst sidelobe behavior. Are there better alternatives? What about polyphase filtering, see SciPy.signal.resample_poly.
- There is a formatting issue in the caption of Figure 4.
- For the analsis in Figure 4, whas the white noise uniformly or normal distributed?
- Line 258: Correct subscripts for BGO chemical formula.
- Section 4.2 and Figure 5: The FT of the CW and SOFF-CW spectra should be shown, so that the frequency composition of the noise remaining after filtering/averaging can be assessed.
- Figure 5B is not referenced anywhere in the text.
- Paragraph starting in line 288: issues with citations.
- Figure 7 caption: typo "On 13C"
- Figure 7 caption: "3mT sweep ..., with 100 steps of 0.1 mT each" - unclear how this applies to the total sweep width of 10 mT. Is the 0.1 mT the step size s from Fig.1? That would give 100 subdivisions of the 10 mT range. Does a 3 mT sweep then mean that m=3/0.1=30? Clarify.
- Figure 7 should give the SNRs for both spectra and both noise models.
- The paragraph starting line 302 appears to describe a method to generate 1/f noise. Is this consistent with the use of AudioGenerator["pink"] earlier on? Also, is there are reference for this 1/f generation algorithm?
- Line 316: muM -> µM
- Line 318: This is the first time phase noise is mentioned. If this is relevant, it should be introduced earlier. Its effect on the measured amplitude noise as well its behavior under the SOFFA method should be discussed. Experimentally, what is the source of this phase noise?
- Line 332: "be adjusted to optimize the oversampled segments". What aspect of the segments is optimized? Maximize the SNR? Minimize the acquisition time? Details would be helpful here.
- Line 335: "higher frequency-content" -> "higher-freuqency content"
- Line 355: Remove first names from citation
- Line 378: ProDEL code is mentioned, but not included.
- Line 381: A Mathematica script is mentioned, but not included. I suggest providing a Phython script instead, since access to Mathematica is rare, and Python is free.
- Line 398: Brackets around citationsCitation: https://doi.org/10.5194/mr-2026-6-RC2
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- 1
The author proposes a segmented strategy for acquiring and filtering magnetic resonance spectra that improves the signal-to-noise by up to an order of magnitude. Examples are given for simulated EPR data with white or ‘pink’ noise and experimental EPR data for an iron-sulfur cluster and for spin-labeled hemoglobin in the intermediate tumbling regime.
2. Specific Comments
The examples selected for the samples have relatively broad lines. Would the method work as well for spectra with well-resolved (narrow) hyperfine lines? If so, what data acquisition and signal processing parameters would need to be adjusted?
Line 178 - what tools are used for 'noise-shaping?
The caption for Fig. 4 should state that the spectral parameters are the same as for Fig. 3.
In Fig. 5 is the lowest-field feature that dominates the signal for the 10 mM sample a background signal? If so, that should be clarified. The caption does not define the significance of the asterisks. Also, the caption states that the 1 mM reference spectrum was acquired by averaging 9 times, although the description in the main text for this spectrum does not mention averaging. This need clarification.
The method was implemented on a Bruker spectrometer, which presumably means that sinusoidal modulation scans were used. If so, over what fraction of the sinusoidal scan was assumed to sufficiently linear to be used in reconstructing spectra?
How much software overhead is there in the implementation with ProDEL?
How sensitive are the S/N improvements to the extent of oversampling that is reduced in the final decimation step?
The ProDel code is available through github. However the Mathematica code and sample data are only available 'on request'. The effectiveness of the method may be strongly dependent on details of implementation. The paper should only be published if the code and sample data are made publicly available.
3. Technical corrections