1 Introduction

Much of analysis of time-dependent data in interdisciplinary research relies decisively upon accurate signal processing. The most difficult problem in most spectra from measured time signals is unambiguous separation of overlapping structures. These structures are peaks associated with the characteristics of the constituents of the investigated system. Such constituents are molecules (metabolites) in specimens scanned by nuclear magnetic resonance (NMR) spectroscopy, which is synonymously called magnetic resonance spectroscopy (MRS) in medicine [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43].

Resonances in MRS spectra, assignable to metabolites contained in biofluids and tissues, are of key relevance to diagnostics in medicine. Clinical usefulness of resonances depends heavily on their quantification, which is hampered by abundant overlaps and unavoidable noise. The linear low-resolution fast Fourier transform (FFT) cannot separate overlapped peaks nor suppress noise. Noise (random and systematic alike) is always present in measured time signals that are in MRS alternatively referred to as the free induction decay (FID) data.

One of the promising avenues to tackle the resolution-noise bottleneck is offered by derivative signal processing. Derivative spectra in MRS are obtained by applying the \(m\,\)th order \((m=1,2,\ldots )\) derivative operator \(\textrm{D}_m=(\text{ d }/\text{d }\nu )^m\) to the given nonderivative spectrum \((m=0)\). Here, \(\nu \) denotes linear frequency. The principal reason for utilizing derivative lineshapes is in the possibility for simultaneous improvements of resolution and signal to noise ratio (SNR), an unachievable goal in the FFT for encoded FIDs. The optimized derivative fast Fourier transform (dFFT) [44,45,46] can enhance both resolution and SNR at the same time.

This is feasible since in the optimized dFFT the operator \(\textrm{D}_m\) narrows the peak widths, increases the peak heights and flattens the background baselines. Moreover, \(\textrm{D}_m\) can transform a shape estimator to a parameter estimator. Such a mapping occurs when a nonderivative total shape spectrum (envelope), subjected to \(\textrm{D}_m\), becomes resolved into its physical well-isolated and unambiguously quantifiable components.

In practice, e.g. the second derivative lineshapes are of insufficient quality and accuracy to represent the end-point of signal processing. On the other hand, for higher m, the number and intensity of sidelobes increase. In absorptive derivative spectra, this enhances the likelihood of confusing sidelobes with physical resonances. Such obstacles are largely absent from the magnitude mode due to an approximate compensation of the real and imaginary parts of each sidelobe. Moreover, unlike the absorptive lineshapes, their magnitude-mode counterparts are, by definition, phase-insensitive and positive-definite. Both features are very important for quantification.

Ordinarily, the nonderivative FFT \((m=0)\) magnitude spectra \(\left| \textrm{FFT}\right| \) are not used because of the uniformly widened peak widths by a factor of \(\sqrt{3}\) relative to the associated absorptive lineshapes Re(FFT). The situation changes with derivative estimations. Importantly, the peak widths are the same in the magnitude first derivative \((m=1)\) lineshapes \(\left| \mathrm{D_1FFT}\right| \) and Re(FFT) [47]. For \(m\ge 2\), peaks can be much narrower in \(\left| \textrm{D}_m\textrm{FFT}\right| \) than in Re(FFT) [45].

By design, the FFT is a shape estimator. It provides merely the lineshape profiles that are not autonomously quantified. The optimized dFFT does not solve explicitly any version of the quantification problem (eigenvalue equations, polynomial rootings,...). Therefore, this processor too is a shape estimator until all the components of the initial envelope are reconstructed at which point it becomes a parameter estimator. Isolating all the components with no remaining overlapping resonances effectively amounts to quantification (extraction of peak locations, widths, heights, area). The found peak parameters from the examined derivative and nonderivative lineshapes are connected by their scaling relationships [47].

Generally, higher-order derivative estimations are prone to instabilities. Therefore, it is of utmost practical interest to find out whether the optimized dFFT is resilient to derivative-induced lineshape distortions for higher values of m. Higher derivative orders are necessary to resolve the most closely packed resonances that are abundantly encountered in MRS spectra.

Therein, often singlet-appearing symmetric resonances could well be sub-structured. This would lead to ambiguities in quantification and could compromise vital decision making (benign versus malignant) in diagnostics within cancer medicine. We address this critically important issue by applying the optimized dFFT to the FIDs encoded in Ref. [16] with water suppression by in vitro proton MRS at a Bruker 600 MHz spectrometer. These encodings were made using the samples of ovarian cyst fluid, benign and malignant, from patients diagnosed histopathologically as serous cystadenoma and serous cystoadenocarcinoma, respectively.

2 Theory

2.1 Optimized derivative fast Fourier transform, dFFT

Data sets encoded by MRS are in the digitized form of the original analog time signals c(t). The waveforms of these FIDs, as a function of time t, do not reveal the metabolic content of the scanned human fluids or tissues. However, the representation in the domain of linear frequency \(\nu \) is by far more transparent. Thus, e.g. a spectrum in the FFT [48,49,50,51,52,53] can provide a number of discernible peaks. As is well-known, the FFT is the fast Cooley-Tukey computation [48] of the discrete Fourier transform (DFT).

The continuous Fourier integral \(F(\nu )\) over t, or equivalently, the continuous Fourier transform (CFT), is the result of a smoothing operator. In the CFT, the analog time signal c(t) is multiplied by the oscillating unattenuated exponential \(\textrm{e}^{-2\pi i\nu t}\) and integrated within a finite interval of t. After equidistantly discretizing t, the CFT becomes the Riemann sum, which is the simplest quadrature rule, the trapezoidal histogram. If in this Riemann sum, \(\nu \) is also equidistantly discretized, the DFT is obtained.

The stick spectrum of the spectral Fourier intensities exists only at a set of discrete FFT linear frequencies \(k/T\, (0\le k\le N-1)\). Here, N and T are the total length and the total duration of the digitized time signal \(c_n=c(n\tau )\), where \(\tau \) is the sampling rate \((\tau =T/N)\). In the DFT and FFT, resolution 1/T, as a separation between two adjacent peaks, is predetermined by the choice of the total acquisition time T.

Integration in the CFT and summing in the DFT are the smoothing operations. This is the case because at each Fourier frequency k/T, e.g. the Riemann sum in the DFT is actually the mean or the average value of the input N signal points \(\{c_n\}\, (0\le n\le N-1)\), multiplied by the harmonic function \(\textrm{e}^{-2\pi ink/N}\). An averaging procedure smooths out the finer details of the summand \(\textrm{e}^{-2\pi ink/N}c_n\) in the DFT or of the integrand \(\textrm{e}^{-2\pi i\nu t}c(t)\) in the CFT.

In an MRS envelope from the frequency domain, resonances can have very complicated lineshapes with smaller or larger departures from the symmetric Lorentzians, Gaussians or their products or sums or convolutions (Voigtians). Moreover, even a seemingly bell-shaped singlet resonance can still be comprised of one or more hidden peaks that may significantly alter the sought true concentration of the provisionally assigned metabolite [44,45,46]. Such complications can pose severe challenges to quantification and interpretation of the data in the frequency domain.

However, derivative estimation by the dFFT should be able to rescue the situation. The dFFT is the result of application of \(\textrm{D}_m\) to the FFT. In the derivative continuous Fourier transform (dCFT), operator \(\textrm{D}_m\) applied to the integrand \(\textrm{e}^{-2\pi i\nu t}c(t)\), generates the function \((-2\pi it)^m\textrm{e}^{-2\pi i\nu t}c(t)\). Thus, the unweighted c(t) from the CFT becomes weighted by the complex time power function \((-2\pi it)^m\) in the dCFT. The power or degree of monomial \((-2\pi it)^m\) is equal to the derivative order m. The dCFT can alternatively be conceived as a weighted version of the CFT, which uses the monomial \((-2\pi it)^m\) as a weight function. This function could be an anti-smoother. It might counter an intrinsic smoothing invoked by integration in the CFT. Smoothing from integrations blurs the finer details in a spectrum. Unsmoothing from derivatives can unravel the hidden spectral details.

In principle, derivative operators are naturally suitable for investigating the characteristics of spectral lineshapes. For instance, as per the elementary calculus, finding the extremal values of a real part \(R(\nu )=\textrm{Re}[F(\nu )]\) of a complex spectrum \(F(\nu )\) reduces to searching for the zeros of its first derivative, \(R^{(1)}(\nu )=0\), where \(R^{(1)}(\nu )=\textrm{D}_1R(\nu )\). Further, the negative or positive signs of the second derivative \(R^{(2)}(\nu )=\textrm{D}_2R(\nu )\) determine whether the found extremum is a maximum or a minimum, respectively. Moreover, the third derivative \(R^{(3)}(\nu )=\textrm{D}_3R(\nu )\) can be employed to locate the points of inflection in a lineshape. Also, coupled with \(\textrm{D}_2R(\nu )\) and \(\textrm{D}_3R(\nu )\), the fourth derivative \(R^{(4)}(\nu )=\textrm{D}_4R(\nu )\) contributes to the peak parametrization by solving a system of equations defined by the relations \(\textrm{D}_2R(\nu )>0\), \(\textrm{D}_3R(\nu )=0\) and \(\textrm{D}_4R(\nu )<0\).

A direct consequence of unsmoothing by \(\textrm{D}_m\) is observed by comparing some typical lineshapes for \(m=0\) and \(m\ge 1\). Thus, a Lorentzian lineshape \(L(\nu )\propto 1/(\nu -\nu _k)\) decreases slowly as \(1/\nu \) away from the resonance frequency \(\nu _k\), where \(\nu \) is real and \(\nu _k\) complex. This causes the adjacent peaks to overlap (possibly coalescing into a singlet) with no chance for peak quantification in the FFT.

However, within the same region of \(\nu \), the \(m\,\)th derivative \(L^{(m)}(\nu )=\textrm{D}_mL(\nu )\) of \(L(\nu )\) for \(m\ge 1\), given by \(L^{(m)}(\nu )\propto 1/(\nu -\nu _k)^{m+1}\), exhibits a faster decay as \(1/\nu ^{m+1}_k\) [47]. As a result, the usually notable elevations of the Lorentzian tails from the chemical shift axis are strongly diminished. The same occurs with the entire background baseline. This implies noise suppression by \(\textrm{D}_m\). Localization of resonances minimizes their overlaps, thus leading to increased resolution.

Unsmoothing by \(\textrm{D}_m\) decreases the peak widths and concomitantly increases the peak heights. The net gain is the ’excess information’ with derivative lineshapes. This signifies that information provided by the dFFT about the physical content of the examined specimen is enhanced relative to noise. The implication is simultaneously improved resolution and SNR. In the FFT, if the SNR were increased, resolution would be decreased and vice versa. Thus, if the FFT is filtered (e.g. by an exponential), SNR will be increased (truncation artifacts reduced), but at the expense of the peak widening (resolution worsening).

All these beneficial features of the dFFT are strictly preserved only for ideally clean time signals with no artifacts. Such FIDs do not exist in any measurement. For measured time signals, invariably contaminated by noise (systematic and random), the dFFT breaks down. In these cases, SNR is worsened because for larger t, the power function \(t^m\) emphasizes the tail of noisy time signals. At longer times, especially towards the end of encoding \((t=T)\), mainly random noise is sampled.

The unwieldy effect of the power function \(t^m\) for noisy FIDs is mitigated in the optimized dFFT [44,45,46]. In this processor, monomial \(t^m\) is tempered by a time-decaying weight function, which is either an exponential filter (EF) or a Gaussian filter (GF). They are jointly denoted by \(\textrm{e}^{-\lambda _p(m,T)t^p},\) where the attenuating or damping parameter \(\lambda _p(m,T)>0\) leads to the EF or GF for \(p=1\) or \(p=2\), respectively.

Dependence of \(\lambda _p\) on m and T is crucial to simultaneously mitigate two perturbations, noise enhancement by \(t^m\) and truncation distortions in measured time signals. Thus, the optimized dFFT processes c(t) multiplied by the combined weight function \((-2\pi it)^m\textrm{e}^{-\lambda _p(m,T)t^p}\) called the adaptive power-exponential filter (APEF) for \(p=1\) and the adaptive power-Gaussian filter (APEF) for \(p=2\). Here, the adjective ’adaptive’ implies that these attenuating filters are tailored to both the derivative order m and SNR of the original FID. An alternative is to view the optimized dFFT as the FFT applied to an encoded FID weighted by either the APEF or APGF.

The essence of this tailoring is understood by reference to the typical envelope \(A\textrm{e}^{-t/T^\star _2}\) of an FID transient from MRS encodings, where A is the intensity at \(t=0\) and \(T^\star _2\) is the effective spin-spin relaxation time. At the end of encoding \((t=T)\), the value of quotient \(T/T^\star _2\) informs about the extent of the FID decay to zero. For example, \(T/T^\star _2=3\) would mean that the FID has practically relaxed (i.e. approximately immersed in noise) since its intensity \(A\textrm{e}^{-3}\) at \(t=T\) dropped to about 5% \((e^{-3}\approx 0.0498)\) relative to the initial intensity A at \(t=0\), i.e. \([(A\textrm{e}^{-3})/A]\times 100\%\approx 4.98\%\).

To mimic \(T/T^\star _2\) within the damping \(\lambda _p(m,T)\), a parameter \(\alpha _p>0\) is introduced by the requirement \(t^m\textrm{e}^{-\lambda _p(m,T)t^p}=\textrm{e}^{-m\alpha _p}\) at \(t=T\). This condition specifies the explicit analytical form of the optimized damping as \(\lambda _p(m,T)=m(t/T)^p\textrm{ln}(T\textrm{e}^{\alpha _p})\). The corresponding optimized filter \((-2\pi it)^m\textrm{e}^{-m(t/T)^p\textrm{ln}(T\textrm{e}^{\alpha _p})}\), which multiplies measured c(t), has a twofold self-correcting property [46]. In the APEF or APGF, noise from \(t^m\) is damped by \(\textrm{e}^{-m(t/T)^p\textrm{ln}(T\textrm{e}^{\alpha _p})}\). Moreover, lineshape broadening by \(\textrm{e}^{-m(t/T)^p\textrm{ln}(T\textrm{e}^{\alpha _p})}\) is mitigated by \(t^m\) in both filters.

This is the mechanism by which resolution and SNR are simultaneously improved by the APEF or APGF in the optimized dFFT. Such a concept has been shown to work excellently through lower derivative orders \(m\le 5\) for FIDs encoded by in vitro and in vivo proton MRS with and without water suppression [44,45,46].

However, there is always a possibility that derivative estimations become quickly invalidated with the rising values of m. Therefore, it is of utmost practical interest to find out whether the optimized dFFT is resistant to derivative-induced lineshape distortions also for higher derivative orders m.

2.2 Working formulae of the optimized dFFT

From the preceding exposition, the optimized derivative discrete Fourier transform (dDFT) reads as:

$$\begin{aligned}{} & {} \textrm{dFFT}=\frac{1}{N}\sum _{n=0}^{N-1}\textrm{e}^{-2\pi ink/N}c_n(m,T;p),\quad 0\le k\le N-1, \end{aligned}$$
(1)
$$\begin{aligned}{} & {} c_n(m,T;p)=W_n(m,T;p)c_n\,,\,\, 0\le n\le N-1 \,\, (m\ge 1,\,\, p\ge 1), \end{aligned}$$
(2)
$$\begin{aligned}{} & {} W_n(m,T;p)=(-2\pi in\tau )^m\textrm{e}^{-m\mu _p(n\tau )^p},\quad \mu _p=\frac{1}{T^p}\textrm{ln}(T\textrm{e}^{\alpha _p}). \end{aligned}$$
(3)

Note that both functions \((-2\pi in\tau )^m\) and \(\textrm{e}^{-m\mu _p(n\tau )^p}\) in their product \(W_n(m,T;p)\) from (3) are raised to the same power m. In other words, to cope with the enhanced noise due to the increasing power m of the monomial \((-2\pi in\tau )^m\), a variation with the derivative order is also included in the attenuating filter viz \(\textrm{e}^{-m\mu _p(n\tau )^p}\).

The dDFT, as the derivative estimator from (1), can alternatively be viewed as the nonderivative estimator DFT applied to the weighted time signal \(\{c_n(m,T;p)\}\) from (2). The weight function \(\{W_n(m,T;p)\}\, (m=1,2,\ldots )\) is the APEF \((p=1)\) or APGF \((p=2)\). The optimized dFFT follows from computing the dDFT by the fast Cooley-Tukey algorithm [48].

2.2.1 Magnitude spectra in the normalized form

The magnitude mode of e.g. the complex FFT spectrum \((m=0)\), is related to its real \(\mathrm{Re(FFT)}\) and imaginary \(\mathrm{Im(FFT)}\) parts by \(\left| \textrm{FFT}\right| {=}\sqrt{[\mathrm{Re(FFT)}]^2{+}[\mathrm{Im(FFT)}]^2}\). Similarly, the magnitude derivative spectra are \(\left| \textrm{dFFT}\right| =\sqrt{[\mathrm{Re(dFFT)}]^2+[\mathrm{Im(dFFT)}]^2}\). To facilitate monitoring of the stability of the reconstructed lineshape profiles, normalized magnitude spectra are particularly suitable. A normalized magnitude spectrum \(\left| \textrm{D}_{m}\textrm{FFT}\right| _{\textrm{N}}\) \((m>0)\) is, in fact, a scaled version of the corresponding derivative spectrum in the magnitude mode, \(\left| \textrm{D}_{m}\textrm{FFT}\right| \, (m=1,2,\ldots )\). For each fixed \(m\ge 0\), we extract the maximum value \(\textrm{max}\left| \textrm{D}_{m}\textrm{FFT}\right| \equiv \left| \textrm{D}_{m}\textrm{FFT}\right| ^\textrm{max}\) of the computed assembly \(\left| \textrm{D}_{m}\textrm{FFT}\right| \) from its set at the N Fourier grid frequencies \(\{k/T\}\, (0\le k\le N-1)\). Note that \(m=0\) is included here as \(\left| \textrm{D}_{0}\textrm{FFT}\right| ^\textrm{max}=\left| \textrm{FFT}\right| ^\textrm{max}\). The nonderivative spectrum \(\left| \textrm{FFT}\right| ^\textrm{max}\) is not normalized. The normalized derivative spectrum is defined by:

$$\begin{aligned} \left| \textrm{D}_{m}\textrm{FFT}\right| _{\textrm{N}} ={\displaystyle { \left| \textrm{D}_{m}\textrm{FFT}\right| \frac{\left| \textrm{FFT}\right| ^\textrm{max}}{\left| \textrm{D}_{m}\textrm{FFT}\right| ^\textrm{max}} }} \quad (m=1,2,\ldots ). \end{aligned}$$
(4)

In other words, the normalized magnitude derivative spectrum \(\left| \textrm{D}_{m}\textrm{FFT}\right| _{\textrm{N}}\, (m=1,2,\ldots )\) is the product of the unnormalized spectrum \(\left| \textrm{D}_{m}\textrm{FFT}\right| \) and \(\left| \textrm{FFT}\right| ^\textrm{max}/\left| \textrm{D}_{m}\textrm{FFT}\right| ^\textrm{max}.\) The latter ratio contains the maximal nonderivative (\(m=0\)) and the maximal unnormalized derivative spectra \((m>0\)). For convenience, the same notation \(\left| \textrm{D}_{m}\textrm{FFT}\right| _{\textrm{N}}\) from (4) will be used for both the optimized and unoptimized dFFT. No confusion should arise in Sect. 3 since the acronym dFFT will consistently be used with the adjectives ’optimized’ and ’unoptimized’ dFFT in the illustrative figures and in the accompanying text.

3 Results and discussion

The digitized form of the time-domain data c(t) to be processed here have been encoded with water suppression by the authors of Ref. [16] using in vitro proton MRS at a Bruker 600 MHz spectrometer. These patient data refer to the \(\mathrm{D_2O}-\)bathed excised samples of ovarian cyst fluid with benign and malignant lesions, serous cystadenoma and serous cystadenocarcinoma, respectively. The illustrative spectra, reconstructed using the FIDs encoded from both samples, will be shown in five figures.

The acquisition parameters were: the Larmor frequency \(\nu _{\textrm{L}}=600\,\textrm{MHz}\) (\(B_0\approx 14.1\)T), repetition time TR=1200 ms, the bandwidth BW=6667 Hz, the full length of each of the FIDs, \(N=\)16384, the sampling time \(\tau =1/\textrm{BW}\approx 0.15\) ms, the number of excitations NEX=128 and the echo time T=30 ms.

After encodings, the usual arithmetic average value of the encoded 128 FIDs is employed in the spectrometer. For chemical shift calibration, trimethylsilyl-2-2-3-3 tetradeuteropropionic (TSP) acid (sodium salt) was added to the cyst fluid samples. The resonance frequency of the TSP singlet is \(\nu _{\textrm{TSP}}=0.00000\,\textrm{ppm}\) (parts per million). All the other resonance frequencies are counted relative to this location of the TSP peak used as the internal reference standard.

In the real part of the FFT complex spectrum, the \(\mathrm{H_2O}\) molecule is set to resonate at \(\nu _{H_2O}=4.70755\, \textrm{ppm}\). For all the other resonances, their dimensionless chemical shifts \(\nu \)(ppm) are related to \(\nu _{\mathrm{H_2O}}\)(ppm), \(\nu \)(Hz) and \(\nu _{\textrm{L}}\)(Hz) by the standard formula:

$$\begin{aligned} \nu (\textrm{ppm})=\nu _{\mathrm{H_2O}}(\textrm{ppm})-\frac{\nu (\textrm{Hz})}{\nu _{\textrm{L}}(\textrm{Hz})}. \end{aligned}$$
(5)

The results from Figs. 1–5 depict the nonderivative (\(m=0\)) as well as derivative \((m=1-20)\) spectra. The reconstructed spectra are graphed as the magnitudes of spectral intensities in arbitrary units (au) versus chemical shifts in ppm.

The lineshape profiles with \(m=0\) are from the FFT, whereas those with \(m\ge 1\) are from the dFFT (optimized, unoptimized). The FFT is unweighted as it uses the original FIDs whose sole modification is in one zero-padding. In the unoptimized and optimized dFFT, the time signal c(t) is multiplied by the power function \((-2\pi it)^m\) which stems from the derivative operator \(\textrm{D}_m=(\text{ d }/\text{d }\nu )^m\) applied to the FFT.

Additionally, in the optimized dFFT, the time monomial \((-2\pi it)^m\) is multiplied by either the exponential filter, EF, or the Gaussian filter, GF (both attenuating and adapted to the derivative order m). According to (3), the product of \((-2\pi it^m)\) with the EF or GF is the mentioned adaptive power-exponential filter, APEF, or the adaptive power-Gaussian filter, APGF, \([(-2\pi it)^m\textrm{e}^{-m\mu _1t}]\) or \([(-2\pi it)^m\textrm{e}^{-m\mu _2t^2}]\), respectively, where \(\mu _p=(1/T^p)\textrm{ln}(T\textrm{e}^{\alpha _p})\).

In the present analysis, both the APEF and APGF were employed. The ensuing reconstructions were very similar for the same parameters \(\alpha _1=\alpha _2=3\). Therefore, it suffices to give here the results for e.g. the APEF alone. For a convenient referencing in the discussion, this filter, deduced from (3) for \(p=1\), is now rewritten as the product of the power function (PF) and the exponential filter, EF:

$$\begin{aligned} \left. \begin{array}{c} {\displaystyle { \textrm{APEF}=\textrm{PF}\cdot \textrm{EF} }}\\ {\displaystyle { \textrm{PF}=(-2\pi it)^m }}\\ {\displaystyle { \textrm{EF}=\textrm{e}^{-m\mu _1t} }}\\ {\displaystyle { \mu _1=\frac{1}{T}\textrm{ln}(T\textrm{e}^{\alpha _1})\,>\,0 }} \end{array}\right\} . \end{aligned}$$
(6)

Here, as in Sect. 2, the continuous time variable t is digitized according to \(t=n\tau \, (0\le n\le N-1)\). As stated, all the computations are carried out by using the same numerical value of the smoothing parameter, \(\alpha _1=3\), in the attenuating term \(\mu _1\) from (6).

The exposition is arranged for three chemical shift bands covering the selected salient aspects of derivative signal processing. The main emphasis is on the larger values (10–20) of m. Within the frequency window 1.2\(-\)5.1 ppm of the aliphatic region, Figs. 1–5 deal with the three strongest resonances, the \(\mathrm{H_2O}\) residual, lactate quartet Lac:q and lactate doublet Lac:d. We begin with the water residual (\(\sim \)4.70 ppm) in Figs. 1–3 (4.6\(-\)5.1 ppm), continue with Lac:q (\(\sim \)4.36 ppm) in Fig. 4 (4.25\(-\)4.56 ppm) and finalize with Lac:d (\(\sim \)1.41 ppm) in Fig. 5 (1.2\(-\)1.56 ppm). Figures 1, 4 and 5 are on the malignant sample alone. Figure 2 is only on the benign sample, whereas Fig. 3 is on both the malignant and benign samples.

The analysis for the mentioned bands will not be limited to the said three main metabolites, \(\mathrm{H_2O}\), Lac:q and Lac:d. Their close surroundings are also addressed. Thus, within 4.6\(-\)5.1 ppm from the first three figures, the band on the \(\mathrm{H_2O}\) residual structure contains the unknown quartet U:q (Figs. 1, 3) as well as the unknown singlet U:s and the unknown doublet U:d (Figs. 2, 3). Moreover, Fig. 4 (4.25\(-\)4.56 ppm) on Lac:q includes the tyrosine doublet Tyr:d, threonine doublet Thr:d and creatinine singlet Crn:s. Further, in Fig. 5 (1.2\(-\)1.56 ppm), Lac:d is accompanied by the alanine doublet Ala:d, threonine doublet Thr:d and \(\beta -\)hydroxybutyrate doublet \(\beta -\)HB:d (alternatively denoted as 3-HB:d).

All the five figures share a similar configuration pattern. They have two columns with four rows. Thus, in Figs. 1, 2, 4 and 5, the FFT and the unoptimized dFFT are on the left columns, whereas the optimized dFFT is on the right columns. Figure 3 compares the spectra for the malignant and benign samples on the left and right columns, respectively. Figures 1, 2, 4 and 5 are on the FFT, unoptimized dFFT and optimized dFFT. Figure 3 is on the FFT and optimized dFFT.

There is another common feature in Figs. 1, 2, 4 and 5 consisting of at least two of their panels having more than one layouts (traces) as the bottom (lower) and top (upper) displays. Such are panels (a,e) in Figs. 1–4 and (a, e–h) in Fig. 5. Some of these multi-layouts are made to simply economize the panel spaces (Figs. 1–3). The other layouts are the special insets (Figs. 4, 5) extracted from the principal source profiles. Such insets are plotted to highlight certain sub-bands. This is done through multiplying by a factor and lifting the original lineshapes.

Water, the theme of Figs. 1–3, is the most abundant substance in the human body. Therefore, if unsuppressed, the water peaks completely mask all the other metabolites in conventional NMR spectra computed from encoded biomedical FIDs. Giant resonances, assigned to the \(\mathrm{H_2O}\) molecules, have partially been suppressed during the process of the FID encodings from both benign and malignant samples [16]. It is very important to appropriately treat the residual water structure so as to minimize the troublesome complications and ambiguities in quantification of the remainder of the spectrum.

Regarding Fig. 1 for the malignant sample, the lineshape of the FFT (a: \(m=0)\) is lifted by 180 au in the upper trace. The lower trace on (a) is the original unoptimized dFFT for \(m=5\). On the left column (a-d: \(m=5-20\)), the latter estimator gradually loses the entire physically interpretable information for the increased values of m. Therein, all the derivative lineshapes from this processor appear with some pronounced fluctuating deformations.

These distortions are caused by the detrimental impact of the noise-enhancing power function \((-2\pi it)^m\), which heavily weights the tail of the processed FID as m rises. As a consequence, both the \(\mathrm{H_2O}\) peak and the unknown quartet U:q (\(\sim \)5.05 ppm) are poorly noticeable due to their surroundings, which is inundated with the intense noisy structures. The U:q resonance is much better predicted by the FFT (a: \(m=0\)) than by the unoptimized dFFT (a-d: \(m=5-20\)).

This situation is to be contrasted to its counterpart associated with the optimized dFFT on the right column of Fig. 1. There are two derivative lineshapes on panel (e) of this figure. The top trace (lifted by 130 au) is for \(m=1\), whereas the bottom trace is the intact spectrum for \(m=5\). Throughout the right column (e–h), the optimized dFFT (\(m=5-20\)) is observed to efficiently narrow the residual water structure and, at the same time, clearly delineates the unknown quartet resonance, which is bell-shaped even for \(m=1\) (e).

Furthermore, the unknown quartet U:q is markedly stable at \(m=5-20\) (e–h). The stage for this crucial advancement is set already with \(m=1\) in the upper trace of (e). Resilience of U:q to the derivative-caused distortions is secured by the tempering effect of the adaptive exponential filter, EF, \(\textrm{e}^{-m\mu _1t}\) from (6). This stability is both qualitative (shape-wise) and quantitative (intensities-wise).

As such, with the optimized dFFT at \(m=1-20\) (e-h), the unknown quartet resonance remains unaltered (i.e. it ’stays put’) in every regard with the augmented m. The implication is that all the four individual peaks in this quartet are of the same height and width, as is noticeable on Fig. 1e–h by the naked eye. This is commendable given that the deformed U:q (with its two outermost peaks pointing downwards) was a weak resonance superimposed on the tail of the water residual for \(m=0\) on the upper trace of (a).

Also of importance is that the optimized dFFT in Fig. 1 largely diminishes the water residual structure to about 80 au (e: \(m=5\)) and 66 au (h: \(m=20\)) from about 400 au (a: \(m=0\)) in the FFT. The two wings (one thinner, the other broader) of the water residual on (a: \(m=0\)) completely disappear from the view on (f-h). Moreover, the former narrow dip in between these wings on (a: \(m=0\)) in the FFT is converted into a sharp upward-pointing peak on (e–h) in the optimized dFFT. Further, several smaller peaks on (f–h) are quite stable too. This can be spotted by following e.g. the arrow above the unknown triplet (to the right of the \(\mathrm{H_2O}\) peak, near 4.675 ppm), which persists throughout the right column.

Fig. 1
figure 1

In vitro proton MRS for samples of malignant ovarian cyst fluid from a patient. Signal processing is performed using the average of the 128 time signals encoded with water suppression at a Bruker 600 MHz (\(B_0\approx 14.1\)T) spectrometer [16]. The magnitude spectra are for the frequency band (4.6\(-\)5.1 ppm) around the water residual structure and the unknown quartet U:q. The lineshapes profiles are from the FFT with \(m=0\) (a), the unoptimized dFFT with \(m=5-20\) (a–d) and the optimized dFFT with \(m=1-20\) (e–h). Ordinates (spectral intensities) are in arbitrary units (au) and abscissae (chemical shifts) are in parts per million (ppm). For details, see the accompanying text (color online)

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Likewise, immediately to the left of the \(\mathrm{H_2O}\) peak, surviving even at \(m=20\), is an unmarked doublet which might be a part of the nitrogen acetyl aspartate (NAA) multiplet. A potential mechanism for the detected presence of the NAA molecules in samples from malignant ovarian cyst fluid has been addressed in Refs. [21, 27, 28, 32, 33]. If these immediate left and right neighborhoods of the water peak on (e-h) were noise-rooted, they would appreciably change some of their characteristics (locations, heights, shapes) with alteration of m. This does not happen, indicating that their origin is likely to be genuine rather than spurious (noisy).

Figure 2 for the benign sample exhibits a lineshape pattern of the water residual fairly similar to that just discussed. Here too, the upper trace is the FFT envelope (a: \(m=0\)) lifted by 180 au. The intact unoptimized dFFT for \(m=5\) is on the lower trace on (a). The \(\mathrm{H_2O}\) residual on (a: \(m=0\)) from the FFT is fractured into two broad wings separated by an extremely sharp peak. The derivative lineshapes (a-d: \(m=5-20\)) from the unoptimized dFFT are very noisy. Nevertheless, despite the reduced levels relative to the FFT (a: \(m=0\)), the water residuals still dominate the spectrum. As such, nothing else on the left column is meaningfully identifiable/interpretable from the noise-contaminated background baseline. In particular, U:q is absent.

The right column of Fig. 2 is only on the derivative spectra (e-f: \(m=1-20\)) from the optimized dFFT. On panels (e), the upper trace for \(m=1\) is lifted by 150 au, whereas the lower trace is the original spectrum for \(m=5\). Therein, already for \(m=1\), two well-delineated, unknown resonances, a singlet U:s and a doublet U:d are seen on the left and right of the reduced \(\mathrm{H_2O}\) residual, respectively.

Beneath the water residual on (e: \(m=1\)), the background is visibly elevated. It becomes completely flattened on (e: \(m=5\)) as well as on (f–h: \(m=10-20\)). With this notable progress, the \(\mathrm{H_2O}\) residual is narrowed on (e–h: \(m=5-20\)), so that U:s and U:d are distinctly propelled in the spectrum. Importantly, their stability is maintained throughout (e–h: \(m=5-20\)). On the other hand, the unknown quartet U:q, so prominent on (e–h) from Fig. 1 for the malignant sample, is not seen in the background on (e–h) from Fig. 2 for the benign sample.

Figure 3 visualizes more directly all the differences between the spectra within 4.6\(-\)5.1 ppm from the optimized dFFT for the two samples. The most striking differences in Fig. 3 are the presence and absence of U:q on the left and right column for the malignant and benign samples, respectively. The other marked difference is that U:s and U:d near the water residual are strongly elevated above the background on (e–h: \(m=5-20\)) for the benign sample, as opposed to the malignant sample (a-d: \(m=5-20\)).

Fig. 2
figure 2

In vitro proton MRS for samples of benign ovarian cyst fluid from a patient. Signal processing is performed using the average of the 128 time signals encoded with water suppression at a Bruker 600 MHz (\(B_0\approx 14.1\)T) spectrometer [16]. The magnitude spectra are for the frequency band (4.6\(-\)5.1 ppm) around the water residual structure, including the two unknown resonances, singlet U:s and doublet U:d. The lineshapes profiles are from the FFT with \(m=0\) (a), the unoptimized dFFT with \(m=5-20\) (a-d) and the optimized dFFT with \(m=1-20\) (e-h). Ordinates (spectral intensities) are in arbitrary units (au) and abscissae (chemical shifts) are in parts per million (ppm). For details, see the accompanying text (color online)

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Fig. 3
figure 3

In vitro proton MRS for ovarian cyst fluid samples (malignant: left column and benign: right column) from patients. Signal processing is performed using the average of the 128 time signals encoded with water suppression at a Bruker 600 MHz (\(B_0\approx 14.1\)T) spectrometer [16]. The magnitude spectra are for the frequency band (4.6\(-\)5.1 ppm) around the water residual structure. Threin, also present is the unknown quartet U:q for the malignant sample (a–d). For the benign sample (e–h), there are two unknown resonances, a singlet U:s and a doublet U:d. The lineshapes profiles are from the FFT with \(m=0\) (a, e) and the optimized dFFT with \(m=5-20\) (a–h). Ordinates (spectral intensities) are in arbitrary units (au) and abscissae (chemical shifts) are in parts per million (ppm). For details, see the accompanying text (color online)

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Overall, it can be stated that the optimized dFFT in Figs. 1–3 is capable of performing an extremely narrow localization of the residual water structure. The considerable reduction of spectral perturbations from the water residual permits the nearby hidden resonances to emerge from their opaque presence in the background baseline within the FFT (a: \(m=0\)). This can be beneficial in practice since some of these newly appearing resonances (e.g. U:q for the malignant sample as well as U:s and U:d for the benign sample) might well be of diagnostic relevance in the clinic.

In the MRS literature, the water residuals are treated in a rather dubious way. They are usually fitted with some 3–10 arbitrary and unphysical Lorentzians. Subsequently, the resulting model spectrum around the \(\mathrm{H_2O}\) structure is subtracted from the corresponding original lineshape. This yields some noisy leftovers in that part of the spectrum. However, such a procedure is bound to uncontrollably wash out also some of the genuine resonances from the zoomed frequency window around the \(\mathrm{H_2O}\) residual structure. In contradistinction, the optimized dFFT handles the residual water structure in a conceptually and fundamentally different manner. This estimator performs no subtraction of spectra at all. Moreover, it can unravel some of the genuine resonances in the tight vicinity of the water residual (Figs. 1–3).

Such an earlier inapproachable chemical shift band, adjacent to the center of the \(\mathrm{H_2O}\) resonance, widens the opportunity for detecting some additional diagnostically informative metabolites. Similar supplementary detection possibilities are offered by the optimized dFFT also farther away from the water residual. This is expected to abundantly occur because many hidden resonances can pop out from the flattened background baseline throughout the entire Nyquist range of resonance frequencies.

The next metabolites whose spectra are to be analyzed are the lactate quartet (Fig. 4) and lactate doublet (Fig. 5). Lactate molecules are among the recognized cancer biomarkers. Their abundance levels are often considerably higher in cancerous than in benign lesions across the vital human organs such as brain, breast, prostate and ovary. In the case of the presently studied FIDs from human ovarian cyst fluid (malignant, benign), the most intense peak heights ratios \(\left| \textrm{FFT}\right| _{\textrm{Malignant}}/\left| \textrm{FFT}\right| _{\textrm{Benign}}\) are about 18 for Lac:q and Lac:d. The widths of the individual 6 peaks in each of these two resonances are practically the same, so that the stated ratios are also the corresponding concentration quotients.

In Fig. 4 (4.25\(-\)4.56 ppm) for the lactate quartet centered at 4.36 ppm, all 8 panels are with the ordinates maximized at 2000 au. The first row of this figure with panels (a: \(m=0\)) for the FFT and (e: \(m=1,\, 5\)) for the optimized dFFT is multi-layered. The layers consist of the main full lineshapes \((m=0,\, 1)\) on the bottom and the insets on the top. The insets are within the windows 4.45\(-\)4.51 and 4.25\(-\)4.34 ppm. The first window (4.45\(-\)4.51 ppm) to the left of Lac:q contains 2 small peaks within the tyrosine doublet Tyr:d. The second window (4.25\(-\)4.34 ppm) to the right of Lac:q includes several small peaks, two of which refer to the threonine doublet Thr:d and creatinine singlet Crn:s. The insets on (a) are multiplied by 5 and lifted by 100. Panel (e) has two sets of insets, both multiplied by 20, but they are subjected to different liftings, 380 for \(m=1\) and 1250 for \(m=5\).

The individual four peaks in Lac:q from the FFT (a: \(m=0\)) have broad bases. On the lower trace of (a), the mentioned small resonances superimposed on the tails of Lac:q show up as the minuscule wiggles. Their magnified appearance in the insets on (a) points to the dominating dispersive part of the displayed spectral magnitude mode. Staying still within the first row of Fig. 4, but switching to the optimized dFFT (e: \(m=1\)) it appears that this processor narrows each of the four peaks of Lac:q on the bottom, unsplit spectrum. Simultaneously, most of the tail of Lac:q is flattened down to to the chemical shift axis.

The lower and upper insets on (e: \(m=1\)) highlight the clearly delineated seven peaks, two in 4.45\(-\)4.51 ppm for Tyr:d and five (four thinner, one broader) in 4.25\(-\)4.34 ppm with Thr:d and Crn:s in the central part of this tiny sub-band. Within 4.45\(-\)4.51 ppm on (e), the lineshape of Try:d in the upper inset \((m=5)\) is quite similar to that in the associated lower inset \((m=1)\). However, within 4.25\(-\)4.34 ppm on (e), there are changes that are notable in the upper inset \((m=5)\) relative to the corresponding lower inset \((m=1)\). Specifically, the four thinner peaks, including Thr:d and Crn:s, are better resolved for \(m=5\) (top inset) than for \(m=1\) (bottom inset). To the right of Crn:s, the outermost formerly wider peak for \(m=1\) in the lower inset becomes fragmented into a notably weaker multiplet for \(m=5\) in the upper inset.

The higher order derivative lineshapes \((m=10-20)\) from the dFFT in Fig. 4 on (b–d) and (f–h) without and with optimization, respectively, have no insets. Thus, within the full depicted range 4.25\(-\)4.56 ppm on (b–d, f–h), the focus is only on the lactate quartet. The left column (b–d) for the unoptimized dFFT proceeds in an ever enlarging disarray when m is increased from 10 through 15 to 20. This is manifested in the dip filling between the adjacent individual peaks of Lac:q by the impact of the ill-tempered power function \((-2\pi it)^m\).

Fig. 4
figure 4

In vitro proton MRS for samples of malignant ovarian cyst fluid from a patient. Signal processing is performed using the average of the 128 time signals encoded with water suppression at a Bruker 600 MHz (\(B_0\approx 14.1\)T) spectrometer [16]. The magnitude spectra are for the frequency band (4.28\(-\)4.51 ppm) around the lactate quartet Lac:q accompanied by the doublets of tyrosine Tyr:d and threonine Thr:d as well as the creatinine singlet Crn:s). The lineshape profiles are from the FFT with \(m=0\) (a), the unoptimized dFFT with \(m=5-20\) (a–d) and the optimized dFFT with \(m=1-20\) (e–h). Ordinates (spectral intensities) are in arbitrary units (au) and abscissae (chemical shifts) are in parts per million (ppm). For details, see the accompanying text (color online)

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Such a circumstance lifts (slowly, but persistently) the pertinent parts of the base of Lac:q. Eventually, for even larger m in the unoptimized dFFT (not shown, but verified), the lactate quartet collapses into a singlet and thus loses every physical meaning. Large spectral deteriorations also occur in the tails of Lac:q. In the end then, the unoptimized dFFT for \(m=10-20\) (b–d) is perceived as being startlingly inferior even to its seed spectrum, the nonderivative FFT (a: \(m=0\)).

This runs contrary to the associated predictions by the optimized dFFT on the right column (f–h: \(m=10-20\)) of Fig. 4. On these panels, the Lac:q profiles are significantly improved with respect to their counterpart from the first derivative on (e: \(m=1\)). Such an observation is supported by the bottom parts (feet) of the four component peaks in the Lac:q resonance. They are all baseline-resolved on (f–h: \(m=10-20\)), unlike the situation on (e: \(m=1\)).

In Fig. 4 too, the appropriateness of the optimized dFFT on (e-h: \(m=1-20\)) is evidenced in countering the offending power function, PF, \((-2\pi it)^m\) by the attenuated, derivative-adapted, tempering exponential filter, EF. As discussed, the ill-behavior of the PF at larger times t is manifested by noise amplification in the processed product \(\textrm{PF}c(t)\). This diverging pattern is forced into convergence through multiplication of the encoded FID by the adaptive power-exponential filter, APEF, given in (3) for \(p=1\) and explicitly recapitulated in (6).

For theoretically synthesized noiseless time signals, the PF itself would suffice for resolution improvement. The reason is that this function has an unsmoothing feature, which can unfold the components beneath the given total shape spectrum or envelope. In such cases (not encountered in practice from any measurement), the PF is well-behaved as it introduces no divergences in the product \(\textrm{PF}c(t)\). Hence, it is only this idealized situation (unrealistic in practice) which can successfully be covered by the unoptimized dFFT.

However, all measured time signals inevitably contain noise which makes the PF ill-tempered in the product \(\textrm{PF}c(t)\). That is why for encoded FIDs the unwanted influence of the PF must be mitigated. This is done via tempering the PF by the EF, as in the optimized dFFT, which then processes the product \(\textrm{PF}\times \textrm{EF}c(t)\equiv \textrm{APEF}c(t)\). With such an amelioration at hand, the optimized dFFT becomes a veritable ’spectral component explorer’. It can deconvolve the physical components invisibly folded within the envelopes. All told, Fig. 4 testifies to the capability of the optimized dFFT to fulfill the major promise of derivative estimations by delivering the Lac:q lineshapes of supreme quality for \(m=10-20\) (f-h).

Finally, we are guided to Fig. 5 (1.2\(-\)1.56 ppm) for the lactate doublet centered at 1.41 ppm. Herein, on the lower trace of (a) for the FFT (\(m=0\)), the base is broad in Lac:d. On the large span of the ordinate scale, which maximizes at 8000 au, the dip between the two lactate peaks is seen to descend all the way down to the abscissa, the chemical shift axis. On both sides of Lac:d, superimposed on its tail, the lower trace shows three small doublets consisting of Ala:d, Thr:d and \(\beta -\)HB:d. However, therein they become more visible on the right (1.2\(-\)1.393 ppm) and left (1.425\(-\)1.56 ppm) parts of the upper trace through multiplication by 10 and lifting by 1500 au. The lineshape of these doublets are distorted by the lactate tail, which rises sharply when the running frequency \(\nu (\textrm{ppm})\) approaches the center 1.41 ppm of Lac:d.

The asymmetry on (a) of the Ala:d, Thr:d and \(\beta -\)HB:d lineshapes is partially caused by mixing the constituents \(\mathrm{Re(FFT)}\) and \(\mathrm{Im(FFT)}\) in the magnitude spectrum \(\left| \textrm{FFT}\right| =\sqrt{[\mathrm{Re(FFT)}]^2+[\mathrm{Im(FFT)}]^2}\). Herein, whenever e.g. \([\mathrm{Im(FFT)}]^2\) is appreciably larger than \([\mathrm{Re(FFT)}]^2\), the shape of \(\left| \textrm{FFT}\right| \) acquires a form reminiscent of that from the imaginary part of the complex FFT spectrum. On panel (a) of Fig. 5, this lineshape dispersiveness is hinted already on the bottom trace and is more pronounced with the enlargement and upscaling in the corresponding top trace.

Fig. 5
figure 5

In vitro proton MRS for samples of malignant ovarian cyst fluid from a patient. Signal processing is performed using the average of the 128 time signals encoded with water suppression at a Bruker 600 MHz (\(B_0\approx 14.1\)T) spectrometer [16]. The magnitude spectra are for the frequency band (1.2\(-\)1.56 ppm) around the lactate doublet Lac:d alongside the doublets of alanine Ala:d, threonine Thr:d and \(\beta -\)hydroxybutyrate \(\beta -\)HB:d. The lineshape profiles are from the FFT with \(m=0\) (a), the unoptimized dFFT with \(m=10-20\) (b–d) and the optimized dFFT with \(m=1-20\) (e-h). Ordinates (spectral intensities) are in arbitrary units (au) and abscissae (chemical shifts) are in parts per million (ppm). For details, see the accompanying text (color online)

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The derivative spectra in the unoptimized dFFT (\(m=10-20\)) are on (b–d) from the left column of Fig. 5. As pointed out earlier, in this processor, noise from the encoded FIDs is magnified by the unattenuated power function \((-2\pi it)^m\). Therefore, some spectral distortions in derivative lineshapes are expected to ensue. Such distortions are seen on panel (b–d) to be very severe indeed. The derivative-amplified noise lifts the dips between the two lactate peaks for \(m=10\) (b), 15 (c) and 20 (d) all the way up to about 1000, 2000 and 3000 au, respectively.

Moreover, the outer part of the base and tail of Lac:d on (b–d) are respectively broadened and lifted from the abscissae. As a result, the Ala:d, Thr:d and \(\beta -\)HB:d resonances on these panels become unrecognizably immersed into the heavily noise-loaded background baselines. Taken together, due to the severity of the lineshape deformations for \(m=10-20\) on (b–d) relative to \(m=0\) (a), the unoptimized dFFT is found to exploit no theoretically anticipated advantages of derivative signal processing.

An alternative prospect is provided by the optimized dFFT on the right column of Fig. 5. The derivative lineshapes for \(m=1-20\) from this processor have two traces throughout (e–h). On these panels, the bottom traces within 1.2\(-\)1.56 ppm represent the unmodified spectra. However, the top two traces (1.2\(-\)1.393, 1.425\(-\)1.56 ppm) are multiplied by 15 and lifted by 2000 au to monitor the derivative lineshapes of the Ala:d, Thr:d and \(\beta -\)HB:d resonances. They are straightened and symmetrized already for the minuscule profiles in the bottom trace on (e: \(m=1\)). The most striking is to compare the upper traces for \(m=0\) (a) and \(m=1\) (e) with a substantial improvement in favor of the latter case. On these upper traces, the highly climbing parts of the lineshapes on (a: \(m=0\)) to the left of Thr:d and to the right of Ala:d are strongly lowered on (e: \(m=1\)).

Another important improvement in the bottom trace of (e: \(m=1\)) relative to (a: \(m=0\)) is in a marked localization of the base of the Lac:d resonance. Moreover, the tail of Lac:d is also diminished on (e: \(m=1\)) compared to (a: \(m=0\)). Such a twofold benefit translates directly into the simultaneous enhancement of resolution and SNR on (e: \(m=1\)) with respect to (a: \(m=0\)).

Scrolling down from (f) to (h) on the right column of Fig. 5, the Lac:d resonance is observed to remain robustly stable regarding its shape and the peak parameters. The same holds true also for the Ala:d, Thr:d and \(\beta -\)HB:d resonances that, despite their relative weakness compared to Lac:d, still retain excellent visibility on the bottom traces at \(m=10-20\) (f–h). Stability of these small doublets is by far more manifest in the top traces on (f–h).

Further, zooming into the Ala:d, Thr:d and \(\beta -\)HB:d resonances on the top traces can also give a rough hint about the onset of noise contamination (even the slightest) which, after certain derivative order m, is likely to settle in for the optimized dFFT, as well. Derivatives are not the panacea for all the spectral obstacles with no limitations.

Indeed, an astute reader would notice some small irregularities in the insets on the base of the Ala:d resonance near 1.51 ppm. Herein, interlaced with Ala:d resides a minuscule doublet of an unassigned metabolite. On the upper trace for \(m=1\) (e), this doublet is not resolved since it appears only as some two shoulders around the base of Ala:d. By comparison, they are baseline-resolved for some of the higher derivative orders, including the shown spectrum for \(m=10\) (f). Thereafter, however, first one and subsequently both peaks of this small doublet on (g: \(m=15\)) and (h: \(m=20\)), respectively, partially overlap (and, hence, obstruct) the base of Ala:d. Within the optimized dFFT, this implies that the various derivative lineshapes for e.g. Ala:d are best estimated from their fully stabilized profiles if m is not pushed beyond \(m=10\) (f).

The circumstances of this and similar or more marked types can inform about the answer to a practical question: where does one stop with derivative estimations by the optimized dFFT when m is gradually augmented? In this processor, the stopping criterion vis-\(\grave{\textrm{a}}\)-vis m is similar to that in any conventional iterative procedure: one stops when the preassigned accuracy threshold has been attained.

In the case of derivative estimations, one would stop at any derivative order m from its range of several values within which the lineshapes of the resonances of interest are stabilized. Here, stabilization refers to the ’baseline-resolved’ resonances. The latter term, under the quotation marks, signifies that the given multiplet resonance, no matter how close its constituents might be, should be split apart into the individual peaks, down to the background baseline, which is presently near the abscissae.

This kind of stringent stabilization concept within the optimized dFFT has also been achieved in Ref. [45]. The latter study used the FIDs encoded from the same malignant sample addressed in the present work. In Ref. [45], the analyzed spectra were from a wider window 0.875\(-\)1.6 ppm with the additional peaks assigned to the other branched amino acids. These are the two doublets of valine Val:dd, the second (downfield) of which appearing as interlaced with the smaller doublet of isoleucine Iso:d.

Sufficiently good resolution and SNR, especially with the baseline-resolved resonances (whenever feasible in practice), are the ultimate prerequisite for unequivocal quantification, ’la raison d’\(\hat{\textrm{e}}\)tre’ for NMR spectroscopy. One of the first aspects the clinician would be in need to grasp from the perused spectra is the relative strength of the main resonances of diagnostic relevance, most notably cancer biomarkers (lactate, choline,...).

If the widths of a number of resonances of interest in the given chemical shift band could be estimated with certainty to be practically the same, their relative strengths would be equal to the peak area ratios. These ratios are proportional to the concentration quotients of the assigned metabolites, accounting for the number of the resonating nuclei (protons for \({}^1\textrm{H}\) MRS).

In the optimized dFFT for the present spectra throughout the whole Nyquist range (aromatic and aliphatic region alike), the peak area ratios of many (but not all) resonances remain almost the same even for larger values of m. The implication is that such resonances have the same widths. Consequently, the dimensionless metabolite concentrations can be extracted directly from the displayed peak heights by allowance for the number of protons contributing to each stabilized resonance. Such concentrations can be expressed in the physical units (e.g. micromol per gram, mM/gr), as the percentage of the concentration of the calibrating reference resonance, which is the TSP peak for the FID analyzed here.

4 Conclusion

Customarily, most processings of encoded time signals in magnetic resonance spectroscopy (MRS) are performed by the fast Fourier transform (FFT). The applications of this linear low-resolution estimator are heavily obstructed by noise, which is always present in encoded time signals, alternatively called free induction decay (FID) data. Besides low resolution and poor signal-to-noise ratio (SNR), yet another stumbling block to the FFT is the abundant presence of overlapping resonances in spectra from encoded FIDs. In this estimator, resolution is predetermined by total acquisition time. Added to these limitations of the FFT is the lack of autonomous quantification, which is the main task of MRS.

The optimized derivative fast Fourier transform (dFFT) simultaneously solves all these problems. It processes the given FID weighted by the adaptive power-exponential filter (APEF) or the adaptive power-Gaussian filter (APGF). Both filters attenuate the analyzed FID and thus improve SNR by reducing the truncation artifacts from the necessarily finite durations of any encoded time signal. Resolution is enhanced by the smoothing feature of a time power function (monomial). However, both the time monomial and the filter also possess their disadvantages, if they would multiply separately an encoded FID, since the former and the latter function amplifies noise and lowers resolution, respectively.

Such intrinsic disadvantages of these two functions are rooted in the following circumstance. A time power function itself worsens SNR because of weighting heavily the noisy tails of encoded FIDs. Moreover, resolution is lowered by an attenuating filter because of introducing a line broadening, which increases overlaps of adjacent resonances.

On the other hand, the optimized dFFT does not process an FID multiplied separately by either a power function or a decaying filter. Rather, in this processor, an FID is multiplied by both functions (monomial, filter) at the same time. It is this combined product of a monomial and an attenuating filter that compensates the said individual drawbacks of the two functions multiplying the processed FID. As a result, resolution and SNR are simultaneously improved by the optimized dFFT, the ultimate goal of signal processing as the prerequisite for achieving unequivocal quantification.

The adeptness of the optimized dFFT is born out from raising the attenuating filter to the same power as that in the monomial. In other words, the time decaying parameters in the attenuating filters are not constant, but rather they vary by gradually adapting themselves to the order of the derivative spectra. Overall, the optimization comes in an attractive form of the joint analytical expression for the APEF and APGF. This closed formula contains only one optimizing parameter, the time decaying constant multiplied by the given derivative order. Hence, it is a matching filter, which can still accurately describe all the resonances in the computed spectrum.

Recent experience with the optimized dFFT has provided a high level of fidelity to this estimator for lower derivative orders (1–5). Nevertheless, the common knowledge teaches us that the pitfalls of derivative estimations usually arise for increased derivative orders. Therefore, it is necessary to find out whether also the optimized dFFT could be prone to develop some worrisome derivative-induced lineshape distortions for higher derivative orders.

The present study addresses this issue and uses high derivative orders (up to 20). The optimized dFFT, applied to in vitro MRS, employs the FIDs encoded with water suppression at a Bruker 600 MHz spectrometer from human ovarian cyst fluid (benign, malignant) dissolved in deuterium dioxide. The obtained spectral lineshapes are demonstrated to be robustly resilient to derivative-induced deformations. The reconstructions illustrate three chemical shift bands containing the water residual structures, the lactate quartet and the lactate doublet resonances alongside their close surroundings.

The conclusions of practical usefulness were drawn for each of the bands. The resonance peak of water, as the most abundant substance in human biofluids, totally masks the signatures of all the other molecules (metabolites). Water suppression during FID encoding mitigates this problem, but nevertheless leaves some spectral residuals that deform the neighboring resonances. Therefore, a reliable handling of the water residual is essential and it is shown that derivative signal processing is highly suitable for this purpose.

The optimized dFFT completely localizes the water residual to allow detection of even its closest surrounding of potential diagnostic relevance. This processor does not resort to any modeling in either the time or frequency domains. Nor does it make any subtraction of spectra. By contrast, the conventional subtraction of the modeled water residual from the whole spectrum can wash out some of the genuine resonances, as well.

In the FFT spectra, the long-extending tails of strong peaks, such as the intense lactate quartet and doublet resonances, lift all the other lineshape profiles and prevent their analysis and interpretation. The ensuing background baseline elevation, coupled with peak overlaps, are insurmountable by the FFT. Such obstacles are overcome in the optimized dFFT by an extreme localization of resonances through the linewidth narrowing and concomitant peak height intensifying. As a result, even the smallest resonances can emerge from their formerly hidden configurations. This processor is shown to be capable of extracting even the tiniest peaks beneath much larger resonances.

The outcome from the optimized dFFT is the possibility of major practical benefit for autonomous quantification without any fitting. The reason is that the mentioned simultaneous localizations, background flattening, linewidth narrowing and peak height increasing effectively amount to unambiguously deconvolving the given total shape spectrum or envelope into its quantifiable physical (genuine) components. Robust stability of the reconstructed lineshapes, even for high-order derivative operators, is presently demonstrated for each of the analyzed frequency bands (and beyond, throughout the entire Nyquist range).

This is all the more remarkable since without optimization by the APEF or APGF, we prove that signal processing dramatically breaks down even for lower orders, let alone the high orders of the conventional derivative upgrade of the unweighted FFT. It is high time now to widely employ the automatic and fast Cooley–Tukey-computations of the optimized dFFT in MRS. Such an advance is foreseen to enlarge the diagnostic window of MRS and equip it with a tool of utmost reliability for quantitative interpretation of spectra by the clinician, while the patient is still in the scanner. The enumerated beneficial features of the optimized dFFT are by no means restricted to MRS alone. Quite the contrary, they are deemed to be of essential usefulness to the entire wide field of signal processing across interdisciplinary branches of research and development, from the academic arena to technology and industry.