Three concentrations of apo [FeFe]-hydrogenase from Chlamydomonas reinhardtii (single reduced [4Fe-4S]+, S=1/2) were prepared at 1 mM, 100 µM, and 10 µM and confirmed by UV-VIS absorption at 400 nm . All spectra were collected at 15 K with an incident microwave power of 0.63 mW and a field modulation amplitude of 0.5 mT at 100 kHz. The 1 mM reference spectrum was collected as a 4 min scan over 100 mT with 4096 points averaged 9 times. The 100 µM and 10 µM concentrations were each collected with 25 averages (4 min scan, 100 mT, 4096 points) for a total measurement time of 100 min.

For the SOFFA-CW experiments, the center field was stepped in 0.5 mT increments over a total sweep of 100 mT, yielding K=200 steps, with a segment sweep width g=25mT and 4096 points per segment. Each segment was collected over 30 s for a total measurement time of 100 min. Segments were acquired using a ProDEL script in Bruker Xepr (v.2.6b-160), which steps the center field and stores each segment in Bruker DSC/DTA format.

The ProDEL script and Python GUI used to collect and process SOFFA data are available at https://github.com/jsidabras/SOFFABruker (last access: 18 June 2026).

The single reduced [4Fe-4S]+ CW EPR spectra and complementary SOFFA-CW EPR spectra were collected on a Bruker E580 Elexsys X-band spectrometer (9.70 GHz) equipped with a bismuth germanate (Bi4(GeO4)3, BGO) dielectric resonator retrofitted in a Bruker MD5 housing .

The 10 µM TEMPO spectra were recorded on a Bruker E500 Elexsys II X-band spectrometer equipped with a Super High-Q cylindrical TE011 model (no. E4122011). The traditional CW EPR was measured with a 10 mT sweep of 1024 points over 41.94 s with 400 averages, 20.48 ms time constant at 3.2 mW, and 0.1 mT field modulation at 100 kHz. SOFFA-CW was performed with a 3 mT sweep of 8192 points over 10.39 s, with 100 steps of 0.1 mT each, a 1.28 ms time constant at 3.2 mW, and 0.1 mT field modulation at 100 kHz.

The conventional CW comparison and NARS spectra were collected on a custom-built X-band bridge dedicated to segmented-NARS spectroscopy . A 150 µM site-directed spin-labeled hemoglobin sample in 82 % glycerol was measured at 18 °C in a five-loop–four-gap loop–gap resonator at 9.81 GHz. For the conventional CW reference, 16 averages of 2 min scans were collected over 20 mT with 0.1 mT field modulation at 100 kHz, a 5 ms time constant, and 1 mW of incident microwave power. The 5 ms time constant was chosen to minimize lock-in pre-filtering within each scan, preserving the full signal bandwidth for subsequent filtering post-processing. For the SOFFA-NARS experiment, the static field was stepped in 0.025 mT increments with an overlap of m=40 using a trapezoidal field ramp at 11.33 kHz with a 50 µs flat region and an amplitude of 0.5 mT. Data were acquired over 95 % of the trapezoidal sweep in one direction; the resulting edge effects appear as periodic noise outside the signal power spectral density and are suppressed by segment-overlap averaging. Both experiments required approximately 32 min of measurement time.

The SNR is defined as 1SNR=Spkσnoise, where Spk is the peak-to-peak signal amplitude, and σnoise is the standard deviation of the noise, computed from an off-resonance region of the spectrum .

In the SOFFA data acquisition method, each field- and/or frequency-swept segment has the following properties: (i) the overlapping noise is non-coherent, (ii) field- or frequency-dependent features are considered to be the signal, and (iii) each segment is an oversampled section of the whole spectrum. Here, oversampled means that each SOFFA segment is acquired with substantially more points than what is required to represent the underlying spectral features at the target final resolution.

In the case of the SOFFA method, all segments are collected separately and can then be further windowed to the desired length to remove potential edge effects with the window size L. Stepping the field by s over the spectrum yields k individually collected segments. The step size s is represented by the discrete shift step size n, illustrated in Fig. . Since magnetic resonance produces a stationary and time-invariant signal, the corresponding output blocks can be zero-filled by a length of the total collection size to begin at the same absolute step size, aligning the field- or frequency-dependent features. The zero-fill parameter is then shift-stepped by n to align field-dependent features. To maximize the overlap averaging after filtering, the windowing function is chosen to be rectangular, and no aliasing issues exist if oversampling is used. Unlike windowing in typical block-filtering, which benefits greatly from window choice , a rectangular window will maximize the individual contribution of each segment during the concatenation process. Any higher-order frequencies generated by discontinuities are removed in the filtering process and result in proper concatenation and aliasing.

In the SOFFA data acquisition method, each segment is oversampled in order to reduce the noise spectral density while keeping the signal spectral density constant . One can estimate an effective time constant, 3τc=1fs=tenK, where fs is the effective sampling frequency, te is the total experiment scan time, n is the shift step size, and K is the total number of segments. If the noise is predominantly white and a perfect low-pass filter is used, the SNR is improved by the oversampling ratio, 4OSR=fs2fcw, where fcw is the inverse of the time constant τcw of the conventional CW data collection.

SNR is improved in oversampling by reducing the spectral bandwidth by the oversampling ratio (OSR) and effectively spreading noise power spectral density by the same amount; see Fig.  in the Appendix (ii and iii) . Since segmented magnetic resonance spectra are collected, a further decrease in the spectral bandwidth is exhibited since only a fragment of the spectrum is sampled. Ultimately, the bandwidth of the noise increases, but it is the reduction in the bandwidth of the signal (due to the collection of partial spectra) that is reduced, allowing a more aggressive filter without spectral broadening.

A key advantage of the segmented acquisition strategy is that it allows the whole spectrum to be oversampled far beyond what the spectrometer hardware permits in a single continuous sweep. Legacy instruments such as some Bruker E500 consoles are limited to 8192 points per continuous sweep. For example, with SOFFA, collecting a spectrum with a 2 mT segment sweep width, a 0.1 mT field step size, 4096 points per segment, and 100 steps produces 204 800 effective data points across the whole spectrum. Because the magnetic resonance signal is stationary and time-invariant, the signal contribution from every segment that covers a given field position is fully correlated, meaning the same EPR line appears at the same field value in each overlapping segment. The noise, however, is independent during each acquisition and is therefore uncorrelated between segments. The result is a whole-spectrum dataset with correlated signal and uncorrelated noise, which is precisely the condition that enables effective Fourier filtering and signal averaging to improve the SNR ratio.

Another advantage of oversampled segmented data collection has to do with noise that is loosely correlated to the signal or noise that is colored, such as 1/f noise. In conventional CW, the time constant acts as a low-pass filter on the data, while the sweep duration acts as a high-pass filter on the noise. Repeating full-spectrum sweeps for signal averaging introduces 1/f noise that cannot be directly filtered without broadening the EPR signal. Furthermore, because this noise is time-correlated, traditional signal averaging becomes less effective, improving the SNR ratio by less than the ideal ma factor. However, by reducing the sweep into segments and maintaining the sweep rate (e.g., 60 s for 10 mT and 12 s for 2 mT; 0.167 mTs-1), the shorter observation window inherently rejects ultra-low-frequency drift. Furthermore, due to the segment step timing and excessive overlap, any remaining correlated noise is randomized between segments. This decorrelation allows the signal averaging to approach the theoretical white-noise limit of m.

Once each of the segments is sufficiently oversampled, filtering can be employed to reduce the high-frequency content outside the spectral bandwidth by applying finite-impulse response (FIR) block-filtering techniques.

The SOFFA data acquisition method differs from typical Fourier block-filtering based on two important factors: (i) each segment is a new collection of data with field- or frequency-stepped correlated signal and uncorrelated noise, and (ii) each segment overlaps the next segments for a total of m overlapping segments; see Fig. . Due to oversampling, the spectral bandwidth of the spectrum in each segment is reduced; see Fig. , comparing spectra ii and iii.

Fourier block-filtering is illustrated in Fig. . Each segment is convolved with the filter H[ω], such that 5Gk[ω]=Yk[ω]Hk[ω], where Yk[ω] is the Fourier transform (Fourier space ω) of the discrete signal, and the convolution is Gk[ω]. In this work, a fixed FIR Gaussian filter is used, defined by 6Hk[ω]=exp⁡-(ω-γ)22σn2, where ω is the discretized frequency, and γ in the central point in the Fourier-transformed data and has a variance of σn2. The smallest variance σn2, which shows no spectral broadening, is chosen for each L. However, it should be noted that these filters are fixed in this work but, in principle, can vary at each k value or be more sophisticated, such as in adaptive averaging techniques .

Spectral leakage occurs when the chosen Gaussian standard deviation σn is more narrow than the spectral bandwidth of the desired signal. As the length of L is increased for each segment, the OSR must be increased to maintain the same spectral bandwidth. However, in practice, this is impractical, and, instead, the standard deviation of the filter must be modified to ensure no spectral leakage. This is visually evidenced by a broadening of the linewidth and the reduction in features of the EPR spectrum. By choosing to adjust σn, the spectral bandwidth will approach that of the whole spectrum as L approaches the total length of x̃[i] (i=0,1,…,N), and the SNR improvement will approach that of filtering after concatenation, as employed by Hyde et al. .

Since the optimum windowing method for the SOFFA algorithm is rectangular, abrupt discontinuities may exist in field space as the data are collected segmentally. Discontinuities cause Sinc-like features in Fourier space. However, Fourier filtering each segment with an FIR Gaussian filter removes these discontinuities and produces smooth overlap and concatenation. Filtering each segment separately removes the need for filtering the periodic noise associated with the step, as in , which is required if the spectrum is filtered after concatenation.

As m increases, the length of the tail further overlaps with the head of multiple segments, resulting in averaging.

If a signal input is infinitely long or for signals that need to be filtered in real time, the overlap-add method is used with short-time Fourier transform (STFT) block-filtering for efficient digital signal processing . Block-filtering breaks a long or continuous signal x(t) into a series of spectral windows with a fixed discrete length. Each window is filtered individually in Fourier space and concatenated back together, as illustrated in Fig. A. Finally, the output of block-filtering and recombining the data with an overlap-add or overlap-save method is equivalent to filtering the long-signal x(t) data directly. In the case of the SOFFA method, illustrated in Fig. B, each component is acquired separately.

For overlap-add processing in SOFFA, a high-resolution overlap-add grid is constructed at 4 times the target resolution, with 4P evenly spaced points spanning the full field range t. Each acquired segment xk[j] is mapped onto this grid by computing the field position fk,j=Ffirst+k⋅s+δj for each point j, where δj is the field offset of point j within the segment sweep. The value xk[j] is accumulated into the nearest grid bin, and a count is incremented. After all K segments have been processed, each bin is divided by its count to yield the averaged value, and bins with no contributions are filled by linear interpolation from neighboring bins. This accumulation and averaging step is where the segment-overlap averaging takes place on the fine grid.

Once each segment is successfully filtered, the segments must be concatenated together. Due to the use of a rectangular window, the segments can be concatenated in Fourier space to save processing time, and a single inverse Fourier transform (IFFT) is needed to obtain the discretized spectrum. Because the data are zero-filled and aligned, this is done simply by summation 7ytot[ω]=IFFT∑k=0K-1Yk[ω]Hk[ω]. Since each of the segments has a portion of the signal and non-coherent noise that lies within the spectral bandwidth (low frequency), an SNR improvement of m is expected by averaging the overlapped segments.

When a fixed filter (i.e., the same filter used on each segment) is employed and each block is recombined using the overlap-add or overlap-save algorithm, the SNR of the filtered signal is exactly equivalent as if the significantly oversampled whole signal was processed directly.

Finally, the first and last L points are called the head and tail, respectively. Since the head and tail do not have a complete segment-overlap dataset, they must be removed. As m increases, the head and tail regions may encroach on the magnetic resonance signal. In practice, for large values of m, the total sweep width should be increased to accommodate for the removal of the head and tail region.

The total number of points in the final concatenated spectrum ytot[i] is a function of the oversampling rate, the step size, and the number of steps. To generate a SOFFA spectrum with a conventional number of output points P (for example, 512, 1024, 2048, or 4096), the filtered grid is decimated to P output points using a moving-average window, where each output point is the mean of a fixed number of neighboring fine-grid points centered on the corresponding output field position. This produces the final output spectrum y[p], p=0,…,P-1. The decimation step conserves the SNR gains achieved during Fourier block-filtering and segment-overlap averaging. In addition, the moving-average window acts as a final low-pass filter on the fine grid while reducing the number of points to a standard (typically 1024).

The block-filtering approach employed by the SOFFA method provides flexibility in filter design. If the filter H[ω] is constant, as in this work, the output y[i] is equivalent whether block-filtering is used or the oversampled data are filtered after concatenation but before decimation. One must be careful to always filter before decimation to not reduce the filtering effectiveness. If the filter varies per segment Hk[ω] then filtering after concatenation is no longer equivalent . There is no direct analogue in conventional CW averaging to segment-specific filtering because a full sweep is processed as a single globally averaged trace, whereas SOFFA permits each independently acquired segment to be denoised with its own locally appropriate filter, for example by applying a Wiener filter separately to each segment before recombination .

In general, the SOFFA method splits data processing into two complementary operations: (i) high-frequency filtering and noise shaping and (ii) low-frequency averaging.

A series of CW EPR spectra of apo [FeFe]-hydrogenase from Chlamydomonas reinhardtii of varying concentrations were collected using a bismuth germanate (Bi4(GeO4)3, BGO) dielectric resonator retrofitted in a Bruker MD5 housing . The apo protein has one reduced [4Fe-4S]+ cluster with an effective S=1/2. Three concentrations of 1 mM, 100 µM, and 10 µM were prepared and confirmed with UV-VIS by measuring the characteristic 400 nm feature indicating relative amounts of [4Fe-4S] .

The 100 µM CW spectrum yielded a SNR of 181.8, while the SNR of the 10 µM CW spectrum was calculated to be 16.5. For SOFFA-CW, an SNR improvement of 6.1 (1114:181.8) is exhibited between the 100 µM CW experiment and the SOFFA-CW experiment. On the other hand, the SOFFA-CW experiment with 10 µM concentration had an SNR ratio that was calculated to be 83.8 and an SNR improvement of 5.1 (83.8:16.5). An average of 5.6 increase in concentration sensitivity is shown for the two concentrations using SOFFA-CW.

Conventional CW data are compared to SOFFA-NARS of a 150 µM site-directed spin-labeled hemoglobin sample in 82 % glycerol (18 °C) at X-band (9.81 GHz), shown in Fig. . Experimental parameters are stated in the figure caption. This sample is in the slow-tumbling regime of approximately 15×10-9s rotational correlation time τcorr. After 16 averages of 2 min scans, a CW SNR of 9.9 is realized. Adding a Gaussian filter increases the SNR by a factor of 4.7 (46.9:9.9). By processing the segmented-NARS data with the SOFFA filtering method, the collected pure-absorption signal has an SNR of 3890 (data omitted). In order to display the spectrum in a more conventional form, the first derivative is computed using an MDIFF pseudo-modulation of 0.1 mT. An SNR increase of 10.3 (483.3:46.9 compared to the filtered CW data) was achieved for the same measurement time of 32 min. The zoomed inset is the first 500 points of the NARS signal multiplied by a factor of 50. Since NARS uses a balanced mixer and an A/D converter, the dispersion signal was also collected for no increase in measurement time (data omitted). The NARS linear background was baseline-corrected in Xepr.

The SNR ratio of EPR is fundamental to the usefulness of the technique. Improvements in EPR fall into three categories: technical developments, such as resonator and spectrometer advances; multiharmonic data collection ; and advanced averaging and filtering techniques, such as, wavelet analysis and adaptive averaging techniques. One must take care that any new method does not over-filter and deliver a distorted EPR spectrum.

In addition, the evaluation of new technology must be done with both white noise and correlated 1/f noise under consideration. White noise has a flat power spectral density across all frequencies with statistically independent samples, plotted in Fig. B spectrum vi. Correlated noise (“pink” or 1/f noise), shown in Fig. B, spectrum v, concentrates power at low frequencies that directly overlap with the EPR signal bandwidth, making it impossible to filter without distorting the spectrum. The 1/f noise power spectral density is a function of scan time and is therefore a measure of instrument stability , and changing the sweep speed shifts the correlated noise frequency content relative to the EPR signal bandwidth.

For this experiment, the data are plotted in Fig. A; the 13C lines of 10 µM TEMPO have been highlighted by using traditional averaging, shown in black, by scanning 400 times over a 273 min time period for an SNR of 1146. In contrast, the same SNR ratio can be achieved in only 20 min (red) without loss of the subtle 13C lines by using the SOFFA-CW method for an SNR of 1218.

Since the power spectral density of 1/f noise is similar to the EPR signal, both correlated noise and EPR signal have a similar shape in the field domain, making fitting subtle spectral features difficult without significantly improving the SNR ratio of the EPR signal with hardware improvements since excessively long averaging will not average out 1/f noise.

To isolate performance under strictly controlled 1/f drift conditions, a hybrid empirical-simulated approach was utilized. Specifically, the empirically acquired EPR spectra shown in Fig. A were corrupted by mathematically injecting randomly generated 1/f noise into the signal. The noise was implemented in the frequency domain following a 1/f power law. The resulting spectrum for the continuous wave is shown in black in Fig. B.

For SOFFA, the randomly generated correlated noise was added to each of the 100 segments, and averaging was performed through the SOFFA algorithm. Since the continuous-wave EPR experiment and the SOFFA-CW experiment have different sweep times (41.49 and 10.39 s, respectively) the frequency content of the correlated noise is different, and this is taken into account in this approach. The resulting spectrum for SOFFA-CW is shown in red in Fig. B. From this, it is clear that the subtle 13C lines of the 10 µM TEMPO are lost in the continuous-wave EPR spectrum and remain in the SOFFA-CW due to the reduction in SNR from 1146 to 205 when 1/f noise was added. The SOFFA method demonstrates enhanced EPR spectral features by not only enhancing the SNR ratio by a factor of 3.7 compared to traditional continuous-wave measurements (767:205) but also effectively mitigating 1/f correlated noise, thereby preserving subtle spectral features such as 13C lines and enabling more reliable spectral fitting even in challenging low-concentration samples where 1/f starts to dominate due to averaging requirements.

Finally, the version of Bruker Xepr (v.2.6b-160) used to collect the SOFFA spectra is not optimized for segmented data acquisition. Future work will implement segmented-NARS using the Xepr Python API interface to decrease the latency associated with large datasets. An SNR improvement is expected by increasing the number of averages for the same total data collection time. Despite current software limitations, the spectra collections shown in Figs.  and  were performed on a commercial Bruker E500 instrument with no hardware modifications. Further improvement in segmented-NARS is expected by using trapezoidal sweeps and processing both the sweep-up and sweep-down data .