Automatic adaptive gain for magnetic resonance sensitivity

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Automatic adaptive gain for magnetic resonance sensitivity enhancement Mazin Jouda, Erwin Fuhrer, Pedro Silva, Jan G. Korvink, and Neil MacKinnon Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.8b05148 • Publication Date (Web): 04 Jan 2019 Downloaded from http://pubs.acs.org on January 9, 2019

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Analytical Chemistry

Automatic adaptive gain for magnetic resonance sensitivity enhancement Mazin Jouda,∗ Erwin Fuhrer, Pedro Silva, Jan G. Korvink, and Neil MacKinnon∗ Institute of Microstructure Technology (IMT), Karlsruhe Institute of Technology (KIT), Karlsruhe, Baden-Württemberg 76131, Germany E-mail: [email protected]; [email protected]

1

ACS Paragon Plus Environment

Analytical Chemistry 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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract The decaying nature of magnetic resonance (MR) signals results in a decreasing signal-to-quantization noise ratio (SQNR) over the acquisition time. Here we describe a method to enhance the SQNR, and thus the overall signal-to-noise ratio (SNR), by dynamically adapting the gain of the receiver before analog-to-digital conversion (ADC). This is in contrast to a standard experiment in which the gain is fixed for a single data acquisition, and is thus adjusted only for the first points of the signal. The gain adjustment in our method is done automatically in a closed loop fashion by using the envelope of the MR signal as the control signal. Moreover, the method incorporates a robust mechanism that runs along with signal acquisition to monitor the gain modulation, enabling precise recovery of the signals. The automatic adaptive gain (AGAIN) method requires minimal additional hardware and is thus general and can be implemented in the signal path of any commercial spectrometer system. We demonstrate an SNR enhancement factor of 2.64 when applied to a custom spectrometer, while a factor of 1.4 was observed when applied to a commercial spectrometer.

Introduction Nuclear magnetic resonance (NMR) is a powerful method to determine molecular structure and dynamics with atomic resolution. It is a highly reproducible, non-invasive, and non-destructive method, and has found applications ranging from material surface characterization to functional imaging of the human brain. A well-known limitation to NMR is the rather poor sensitivity of the technique - typical limit of detection values are at the level of low µM concentrations. There has therefore been significant effort dedicated to improving the signal-to-noise ratio (SNR) so that ever decreasing quantities of analyte can be measured. While there are non-Faraday magnetic resonance detection methods that reduce the sensitivity to the level of single electron 1 or nuclear spins, 2 these methods have yet to translate to routine measurement and in any case would have the same dynamic range challenge. 2


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Improvements to SNR have been accomplished by various, diverse methods. SNR can be improved by increasing the magnetic field strength, decreasing the temperature of the RF electronics, improving the sample filling factor, and improving the detector induction efficiency. 3,4 Practically, this has resulted in developments of sophisticated magnet technologies attempting to move beyond the current 23.5 T field ceiling. 5 Developments in cryo-probe technology have offered an SNR gain on the order of a factor of 3-4. 6 Micro-coils of various geometries address the filling factor and detector efficiency, and offer signal enhancements on the order of a factor 10 for mass limited samples. 7–14 Moving beyond equilibrium polarization, methods that overcome Boltzmann statistics have been extremely successful in enhancing SNR. These hyperpolarization methods, offering orders of magnitude improvement, include dynamic nuclear polarization (DNP), 15 para-hydrogen induced polarization (PHiP), 16 and spin-exchange by optical pumping (SEOP). 17 There have been advances in SNR improvement by software approaches as well. Nonuniform sampling of the indirect dimensions of multidimensional experiments is a prominent example of such software-based methodology, requiring selection of both a sampling scheme and data reconstruction algorithm to maximize the benefit in SNR. 18–20 Given the goal of maximizing SNR, less attention hase been devoted to improving the dynamic range of signal digitization. NMR free induction decay (FID) signals have a large dynamic range, and currently the ADC is not often used to its full capability during the length of the acquired signal. In other words, the gain is set based on the amplitude of the first data points of the FID (often the strongest signals), and the full digital resolution of the ADC is therefore not used during the entire length of the FID. Efforts to enhance the dynamic range of the receiver in magnetic resonance imaging (MRI) systems were reported in. 21–23 For instance, Kose et al. 21 showed the possibility of achieving a significant improvement (a numerical simulation showed a noise reduction by a factor of 10 to 20) in the quantization noise of the ADC by applying nonlinear amplitude compression where the low-frequency signals are attenuated prior to the ADC while the 3



high-frequency signals remain unchanged. In a quite similar approach, Elliott et al. 22 and Oh et al. 23 demonstrated independently a step-wise gain change method, in which case the small-amplitude signals that correspond to large phase encoding steps are largely amplified while the large-amplitude signals are slightly amplified prior to the ADC, thereby improving the quantization noise in an MRI experiment. In 2011, Takeda et al. described APRICOT, 24 a method to reduce the quantization noise in NMR spectroscopy experiments by dynamic signal modulation. The method relies on multiplying the NMR signal by an external modulating waveform before the analog to digital conversion. When tested on a 14-bit home built spectrometer, 25 the method showed a remarkable reduction in the quantization noise. However, one of the major drawbacks of this method is that it necessitates prior knowledge of the NMR signal in order to generate the appropriate modulating waveform, which restricts the flexibility and generality of the approach, as well as its applicability to commercial systems. Moreover, the method lacks a mechanism to precisely monitor the change in amplitude of the NMR signal after modulation. It assumes that the knowledge of the modulating waveform is sufficient to recover the NMR signal, which is true only under the assumption that the multiplier is perfectly linear and no phase distortion occurs. This implies that a pre-calibration is necessary before any experiment. In this contribution, we describe a method to enhance the SNR of the NMR signal by taking full advantage of the ADC. The concept is to dynamically adapt the gain during the acquisition of the free induction decay (FID) under full automation. While APRICOT is an example of an open-loop control system, 24 our method is superior in the sense that it acts as a closed-loop system, thus it is more stable and requires no prior knowledge of the FID. Furthermore, our method features a robust mechanism to track the gain alteration, which enables a highly accurate recovery of the NMR signal. These advantages make our method more practical and thus applicable to every NMR spectrometer.

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Experimental section For a signal m(t) that exists over the time duration from 0 to T , the power of m(t) is calculated as Z

1 Sm = T

T

| m(t) |2 dt

(1)

0

The power of the quantization noise of a uniform quantizer is calculated, under the assumption that m(t) is much larger than the quantization level and thus the quantization noise is not largely correlated with m(t), from the following equation 26 Z

∆/2

Nq = −∆/2

∆2 1 2 e de = , ∆ 12

(2)

where ∆ is the quantization level, 1/∆ is the probability density function of the quantization noise, and e is the quantization noise voltage. The last equation shows that the quantization noise depends only on the quantization level and is constant over time. This leads us to the fact that the signal-to-quantization noise ratio of the NMR signal, SQNR = Sm /Nq , is decaying over time as the signal level decays. This means that, if we can stop the decay of the NMR signal envelope such that it appears as a constant at the input of the analogto-digital converter ADC, then we can obtain a higher SQNR and thus an enhanced ADC performance. Assume that the function, c(t), compensates for the decay of the NMR signal so that the new function mc (t) = m(t) · c(t)

(3)

is constant. Then the power of mc (t)

Sm,c

1 = T

Z

T

| m(t) · c(t) |2 dt

(4)

0

is larger than that of m(t), and therefore when digitized, the SQNR of mc (t) will be larger than that of m(t). This can be mathematically illustrated by assuming a typical NMR FID

5



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such as: (5)

m(t) = sin(ωl t)e−t/T2 ,

where ωl is the Larmor frequency and T2 is the spin-spin relaxation time. To compensate the decay of the NMR signal, we multiply it with c(t) = et/T2 , thus the resulting compensated signal is (6)

mc (t) = sin(ωl t).

Both the original NMR signal and the compensated one have the same maximum amplitude, therefore we can assume that the quantization noise power over the acquisition duration, T , is equal for both signals. The ratio of the SQNR of mc (t) to that of m(t) can be calculated as

SQNRm,c = SQNRm

1 Ti R Ti 1 Ti 0

R Ti 0

| sin(ωt) |2 dt

| sin(ωt) · e−t/T2 |2 dt

.

(7)

After some mathematical manipulation and assuming that the acquisition duration, T , is an integer number of periods, Equation 7 reduces to SQNRm,c 2T = . SQNRm T2 (1 − e−2T /T2 )

(8)

This clearly shows that for any value of T , the signal-to-quantization noise ratio (SQNR) of the modified signal, mc (t), is larger than that of the original signal m(t). Practically, c(t) is the adjusted gain profile. NMR spectroscopy experiments were performed on a Bruker AVANCE III 500 MHz widebore spectrometer operating at a 1 H frequency of 500.13 MHz. Single-pulse experiments were measured using the 5 mm saddle coil insert for the Micro5 microimaging probe (Bruker, Rheinstetten). Data were acquired with a spectral width of 20 ppm with 24 K data points. The FIDs were multiplied with an exponential function equivalent to 0.30 Hz line broadening prior to Fourier transform. All chemicals were purchased from Sigma Aldrich and used as received. 6


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The UHFLI lock-in amplifier was operated via the official software (LabOne 17.06) provided by Zurich Instruments® , and Matlab R2014a® was employed to accomplish all the signal post-processing. The UHFLI was configured such that all signal acquisitions were triggered by the Bruker spectrometer.

Results and discussion Clk Trig

RF out Sample

CM

CT

Coil

Switch

Demodulators x, y, r, Θ

x

ADC1

Power amplifier

fL

Spectrometer console

Low-noise amplifier

xAG, yAG, rAG, ΘAG

x

RF in

fL

ADC2

xP , y P , r P , Θ P

x

memory

fL+ Δf Position 2

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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


RF splitter

DAC1

fL+ Δf

Frequency synthesizer

Position 1 RF combiner

VGA

Gain control

r K

DAC2

Pilot

Commercial spectrometer

Digital lock-in amplifier

External components

Figure 1: The experimental setup of the Automatic Adaptive Gain (AGAIN) NMR experiment. The setup allows two scenarios for acquiring the NMR signals: (Position 1) NMR acquisition via the UHFLI lock-in amplifier and (Position 2) NMR acquisition via the commercial spectrometer. The first RF input of the UHFLI is routed to one demodulator and they are dedicated to extracting the FID envelope and controlling the VGA. The second RF input is routed to two demodulators where one is dedicated to monitoring the pilot, while the other is utilized to acquire the AGAIN-modulated NMR signal. Figure 1 depicts the setup of the automatic adaptive gain (AGAIN) NMR experiment. It shows the three main components of the experiment highlighted using gray, blue, and green boxes. These components are: 1. (Gray) The commercial spectrometer, which is an 11.7 T (500 MHz for 1 H) AVANCE III Bruker® spectrometer. 7



2. (Blue) The ultra-high frequency lock-in (UHFLI) amplifier from Zurich Instruments® . The UHFLI features two RF input channels that sample their input signals with 12-bit ADCs at a fixed rate of 1.8 GHz. In addition, the digital signals can be quadraturedemodulated with eight different-frequency demodulators, and then filtered with digital lowpass filters which can be set down to sub-Hertz bandwidth. Besides that, the UHFLI has a number of auxiliary inputs, outputs, and trigger ports that serve as general purpose signals for various control tasks. As the sample rate of the ADCs is fixed, the effective oversampling ratio (OSR) cannot be directly controlled. However, one can enhance the measurement resolution by reducing the input range of the ADCs and the bandwidth of the digital filters. Nevertheless, these two parameters are mainly restricted by the input signals. 3. (Green) The third part of the experimental setup represents all external discrete components, including an RF splitter/combiner (ZFSC-2-4-S+), and a variable-gain amplifier VGA (ZFL-1200GH+). The VGA was particularly chosen for its inverse gain-control voltage profile (Figure 2), which makes it naturally appropriate to compensate for the FID decay while being controlled by the FID itself. Moreover, if the FID of interest exhibits a beating due to having a pair of dominant peaks, then as the envelope of the FID decreases the gain will automatically increase, and once the FID starts increasing again due to the beating, then the gain will decrease automatically and so on. This guarantees the most efficient use of the ADC’s dynamic range and ensures, at the same time, that no ADC saturation occurs, and thus makes the AGAIN technique applicable to any FID profile. The connection diagram of the experiment can be explained as follows: the sample excitation together with the pre-adjustments such as the shim and the frequency search are carried out by the Bruker® spectrometer. The NMR signal is taken from the Bruker® system after the preamplifier. This is not only for ease of access, but also to ensure that the noise contribution of the externally added components is negligible. 27 This NMR signal is then 8


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5 V

3 0

G a in

c o n tro l

1 5 2

0 1

-1 5 0

-3 0

c o n tro l

[V ]

3

0 .0 2

0 .0 4

0 .0 6

0 .0 8

G a in [ d B ]

4

V

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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 .1 0

T im e [s ] Figure 2: The VGA response to an input signal of 500 MHz and a time-varying control voltage. This response was measured via the UHFLI where one RF output was used as VGA input, and the VGA output was recorded by one of the RF input channels. The control voltage was generated via the internal arbitrary wave generator (AWG) through one of the auxiliary outputs. split into two branches. The first branch is routed to the first RF input of the UHFLI (ADC1). The signal is quadrature-demodulated at the Larmor frequency by one of the inp ternal demodulators, and the magnitude, r = x2 + y 2 , is scaled by an internal digital gain, K, and is output on one of the auxiliary ports in order to control the VGA. The second branch of the NMR signal at the RF splitter is combined with a "pilot" and fed to the VGA. The "pilot" signal is generated by the UHFLI’s internal frequency synthesizer at a frequency slightly above the Larmor frequency, fpilot = fL + ∆f , to monitor the change in the gain over the FID decay. This is necessary to reconstruct the desired NMR signal when postprocessing the adaptive-gain-modulated FID. The connection diagram in Figure 1 allows two acquisition scenarios, such that the output of the VGA can be processed and recorded either by the UHFLI (Position 1) or the Bruker® spectrometer (Position 2). Both the Bruker 9



spectrometer and the UHFLI amplifier perform the same typical signal processing tasks, including quadrature-demodulation and analog-to-digital conversion; however, they conduct these tasks in opposite order. Specifically, the Bruker spectrometer does most of the signal processing in the analog domain and digitizes the signals only when they are at base band frequency, which allows the use of high resolution (≥ 18 bits) analog-to-digital converters (ADCs). On the other hand, the UHFLI performs all the signal processing in the digital domain, therefore it starts by sampling the input signals at a high (1.8 GHz) sample rate to cover a very wide band, and as a result, its ADC’s resolution is limited to 12 bits. To ensure the synchronous operation of the various components, the clock signal from the Bruker system is used as a master timing reference for the lock-in amplifier. In fact, almost all commercial spectrometers allow access to their clock signal so that the user can synchronize multiple instruments. Therefore, the AGAIN technique is applicable to any commercial spectrometer. Furthermore, a TTL trigger signal programmed into the pulse sequence is used to trigger the UHFLI to start acquisition. This maintains a correct phase of the acquired signals and thus allows error-free reconstruction of the NMR signals from the adaptive-gain-modulated FIDs. Throughout the rest of the paper we will use x, y, r, and θ to refer to the real, imaginary, magnitude, and phase of the NMR signal recorded without adaptive gain. On the other hand, xAGAIN , yAGAIN , rAGAIN , and θAGAIN refer to the NMR signal recorded with adaptive gain, while xP , yP , rP , and θP refer to the pilot signal. The concept of automatic adaptive gain (AGAIN) is demonstrated in Figure 3 which shows the experimental results of applying it on a simple water FID. The signal acquisition in this experiment was carried out by the UHFLI (Position 1) such that the NMR signal without AGAIN is digitized by ADC1 and demodulated at the Larmor frequency fL (the red curve , r, in Figure 3a). On the other hand, the VGA’s output is digitized by ADC2 and demodulated simultaneously by two demodulators; the first at fL to extract the NMR signal with AGAIN (the blue curve , rAGAIN ), whereas the second at fL + ∆f to extract 10


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3 0 r

2 5

A G A IN

r r

0 .0 2

2 0 P

1 5 1 0

G a in [ v / v ]

A m p lit u d e [ V ]

0 .0 4

5

0 .0 0 0 0 .1

(a )

0 .2

0 .3

w ith o u t A G A IN

F ID

0 .5

w ith A G A IN

(P ro c e s s e d )

1 .0

0 .5 4 9 8 .0 m

5 0 0 .0 m

0 .0 -0 .5 x

A m p lit u d e [ a . u . ]

1 .0

y

-1 .0 0 .0

(b )

0 .4

T im e [s ] F ID

A m p lit u d e [ a . u . ]

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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 .5 4 9 8 .0 m

-0 .5 x y

-1 .0

0 .2

0 .4 T im e [s ]

5 0 0 .0 m

0 .0

0 .0

(c )

0 .2

0 .4 T im e [s ]

Figure 3: Quantization noise enhancement via applying adaptive gain method. (a) The NMR signal enhancement without (red), and with (blue) adaptive gain, and the VGA response to the NMR signal (green). (b) Real and imaginary component of the NMR signal without AGAIN. (c) Real and imaginary of the post-processed NMR signal after AGAIN. the "pilot" signal (the green curve , rP ) in Figure 3a. In this and the other experiments, unless otherwise stated, we set the UHFLI to automatically adjust the input range of the ADCs based on the signal level so as to ensure full ADC resolution (ENOB = 12) in all acquisitions. Figure 3a shows that as r decreases the gain of the VGA, rP , increases and so does rAGAIN . This continues until the rate of the FID decay becomes larger than the rate of the gain increase which eventually becomes zero when the VGA reaches its maximum gain. The curves in Figure 3a are related through

rAGAIN (t) = r(t) · rP (t),

(9)

which corresponds in the frequency domain to the convolution

RAGAIN (f ) = R(f ) ∗ RP (f ), 11


(10)


thus the true NMR signal can be obtained from the adaptive-gain-modulated FID either via a division in the time domain, or a deconvolution in the frequency domain. The real and imaginary parts of the FID without AGAIN are shown in Figure 3b, while Figure 3c displays the real and imaginary parts of the AGAIN-modulated FID after post-processing. Comparing the two sub-figures, a clear reduction in quantization noise due to the adaptive gain method can be immediately observed. For a quantitative assessment of the quantization noise enhancement, we conducted a number of NMR experiments using both the UHFLI and the Bruker spectrometer, in order to explore the efficiency of the adaptive gain method on both low-resolution (high quantization noise) and high-resolution (low quantization noise) ADCs.

AGAIN-NMR via UHFLI An NMR spectroscopy experiment of a model test solution (45 mM alanine, 10 mM citric acid, 5 mM TSP in 90:10 H2 O:D2 O) was performed according to the experimental setup in Figure 1 using Position 1. In this single pulse experiment, the sample excitation was done by the Bruker system, whereas the signal readout was done by the UHFLI. A single acquisition with a bandwidth of 10 kHz and acquisition duration of 900 ms was measured. Figure 4 displays the spectrum of the signal acquired via ADC1, i.e., the signal without AGAIN (red), and the spectrum of the post-processed signal acquired with AGAIN via ADC2 (blue). In this figure, both spectra are normalized to the water peak and plotted with the same scale in order to facilitate the direct comparison of their quality. Comparing the two spectra in Figure 4, one can immediately notice the obvious improvement of the noise level due to the adaptive gain, as was also observed with APRICOT. 24 With a calculated SNR enhancement factor, β, of 2.64 (Equation 11) it is possible to resolve the multiplet structure of the signal at 3.7 ppm (alanine, α-1H, quartet) with a 12-bit ADC, which is not possible without AGAIN. Furthermore, figure 4 demonstrates clearly how the AGAIN technique, when combined with the oversampling of the UHFLI, can push the quantization 12


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noise enhancement beyond the limits of oversampling. The signal-to-noise ratio (SNR) enhancement factor, β, was calculated according to the following formula β=

SNRAGAIN SNRw/o AGAIN

(11)

and the SNR is calculated as SNR =

S 2·N

(12)

where S is the highest intensity in the signal region, and N is the noise which, in turn, can be calculated for a noise region, XN , of length, M , as 28 v  u u  u P  u n (XN (i))2 −  u i=−n   t

n P

!2

XN (i)

i=−n

M

N=

3· +

n P

2 i(XN (i)−XN (−i))

i=1

M ·(M 2 −1)

M −1

      

(13)

with n = (M − 1)/2. According to the previous equations, the SNR enhancement factor, calculated for a noise bandwidth of 2 kHz, was found to be 2.64, which corresponds to a 8.43 dB improvement in the dynamic range of channel 2. According to Equation 4, the SNR enhancement due to the adaptive gain depends on the decay-compensating function, c(t), which is, according to the experimental setup, a function of the digitally scaled-NMR signal, and hence can be written as

c(t) = G(K· | r(t) |)

(14)

where G is the response (gain) of the VGA. This means that the SNR enhancement factor, β, depends, for a certain VGA, on the digital scale factor, K, and the NMR signal, r(t). Therefore, the value of K must be optimized for different NMR samples. The dependence on K is demonstrated in Figure 5a. Interestingly, it was observed that the SNR enhancement factor was virtually independent of the digital resolution used to read the raw FID r(t) (Figure 5b). This leads to the conclusion that the VGA is not following the exact envelope 13



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of the FID but rather the general decay of its envelope, and therefore one can use a moderate ADC (with lower ENOB) for monitoring r(t) and still retain an SNR improvement after AGAIN.

AGAIN-NMR via commercial spectrometer The noise associated with the NMR signal, as can be calculated from Equation 13, contains noise contributions of various types including thermal noise, the ADC’s quantization noise, and the jitter of the sampling clock. Thereby, the effective SNR of the ADC can be described by 29

SNR = −20 log (2π · fa · tj,rms )2 +

1+ε 2

2 3

2N

+

2 √ 2 2Vn,rms N 2

1 2

(15)

where fa is the analog input frequency, tj,rms is root mean square jitter of the external clock, ε is the differential non-linearity of the ADC, N is the number of bits of the ADC, and Vn,rms is the effective thermal noise of the ADC. The jitter noise can be neglected due to the extremely accurate (better that 0.5 ppb) reference clock used in both commercial spectrometers and the UHFLI. Thus, the effective SNR of the ADC reduces to 1

  SNR = −20 log 

2

2

1+ε 2 N

2 |3 {z }

Quantization

+

|

2 √  2 2Vn,rms  N 2 {z

Thermal

,

(16)

}

where the first term represents the quantization noise while the second term stands for the thermal noise. From the previous equation, one can immediately conclude that the efficacy of the adaptive gain, as a technique to boost the signal to quantization noise ratio, drops as the quantization noise gets smaller. In other words, the adaptive gain technique is more efficient for low resolution ADCs for which the quantization noise has a larger contribution. To demonstrate this, we conducted a number of NMR experiments in which we combined

14


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the adaptive gain technique with the traditional commercial signal acquisition (Position 2 in the connection diagram of Figure 1). In all experiments, it was of key importance to ensure that the full ADC resolution was exploited, and therefore an automatic receiver gain adjustment routine was run prior to each experiment to ensure the full range of the ADC was used without overflow. Figure 6 shows an example of a spectrum acquired via the commercial system while applying AGAIN. The sample was nearly the same as the model solution above, only with [alanine] = 0.1 mM, in order to explore the limits of the AGAIN technique. The figure demonstrates that after 32 averages one can recognize a signal at 1.45 ppm in the spectrum with AGAIN, while in the spectrum that is recorded without AGAIN, it is not feasible to distinguish a signal from the noise. Moreover, one can clearly notice the improved noise floor in the case of AGAIN. Unsurprisingly however, the measured SNR enhancement (β = 1.4) confirms the aforementioned conclusion that the AGAIN technique is more efficient with low resolution ADCs. Another example is depicted in Figure 7, and shows the spectrum of a cell culture medium solution (taken as an example of a complex mixture). The measured SNR enhancement factor due to applying the adaptive gain technique was found to be approximately 1.45, which suffices to resolve many signal peaks that are otherwise unresolvable.

Conclusion In this paper we introduced a new technique with which to enhance the NMR sensitivity via boosting the signal to quantization noise ratio (SQNR) of the analog-to-digital converter (ADC) used, and thus improve the overall measured signal to noise ratio (SNR). The technique is based on the automatic adaptation of the in-loop gain of the NMR receiver before the ADC such that the FID decay is fully or partially compensated, and consequently a higher SQNR is obtained over the acquisition time.

15



The concept of automatic adaptive gain was validated experimentally using 2 signal pathways: first in a nonstandard signal acquisition scheme using a digital lock-in amplifier with 12-bit resolution, and second using the commercial system. In the former scheme, the AGAIN technique showed a 2.64 enhancement factor in the SNR, while the maximum enhancement factor in the case of using commercial acquisition was 1.45. This difference in SNR enhancement comes as no surprise since the AGAIN technique effectively reduces the quantization noise and is thus most effective when using ADC’s with lower digital resolution. This makes the technique particularly valuable for micro-coils, cryogenic-coils, 30 and cryogenic electronics, 31 where the thermal noise is usually reduced by several folds and thus the relative contribution of quantization noise is increased. Furthermore, as the AGAIN technique provides an enlarged dynamic range of the receiver, it will be perfectly suitable for experiments that exhibit signals with large dynamic range such as the simultaneous multinuclear acquisition via broadband detectors using a single ADC. Another example is the acquisition of a frequency division multiplexed signal from an array of detectors through a single ADC. 32 In both examples, in the frequency domain the signal appears as a group of sub-signals occupying different frequency bands. However, in the time domain it appears as a single signal with a large dynamic range depending on the number of incorporated sub-signals. Further potential applications that can benefit largely from AGAIN are (1) high field (above 22 T) NMR where the noise level is similar to that of low fields while the signal level is much increased and thus the dynamic range is much larger, (2) hyperpolarized NMR where a similar argument holds. We additionally anticipate that MRI would stand to benefit from the AGAIN method to improve signal quality away from the center of k-space. We believe that the SNR enhancement values achieved here can be improved further, provided that a VGA with larger gain range is available. Unfortunately, cascading multiple VGAs does not help in this case, as the combination very quickly becomes nonlinear. In summary, the following key features of the technique make it a very promising candi16


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date to enhance the NMR receiver’s dynamic range and hence its sensitivity. • The technique is fully automatic thus does not necessitate prior knowledge of the NMR signal to generate the VGA control waveform. In full automation, the only requirement is a pre-scan adjustment to discover the optimal value of K, similar to the standard automatic receiver gain adjustment. • It is also fully automatic in the sense that it tracks the gain modulation simultaneously along with the NMR signal acquisition through the pilot signal. Therefore, no extra runs or dedicated acquisitions are required for post-processing and reconstructing the NMR data. • It can be readily adapted to operate in any commercial spectrometer system with minimal hardware changes. • The experiments showed that the requirements of the VGA control signal are not stringent. Thus the AGAIN technique can, upon integration with commercial systems, be realized in a much more cost-efficient way. For instance, the UHFLI can be entirely replaced with a low-cost high performance envelope detector, such as ADL6010 from Analog Devices® or the one reported in, 33 as only the magnitude of the FID is of interest. The only additional modification then is the acquisition of the pilot signal, potentially by one of the X-channels of the spectrometer after down-conversion.

Acknowledgement M.J., E.F., and J.G.K. acknowledge funding from the European Union under the Horizon 2020-FET framework for the project TISuMR (#737043). P.S. and J.G.K. acknowledge support from Bürkert Fluid Control Systems. N.M. acknowledges funding from the Deutsche Forschungsgemeinschaft for the project Bio-PRICE (DFG MA 6653/1-1). J.G.K. acknowledges funding from the Deutsche Forschungsgemeinschaft for the project ScreeMR (DFG 17



KO 1883/29-1).

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Graphical TOC Entry

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W ith A G A IN W ith o u t A G A IN

2 .8

3 .8

5

4

2 .6

2 .4

1

3

2

1

0

H c h e m ic a l s h ift [p p m ]

Figure 4: 1 H NMR spectra of a model solution (45 mM alanine, 10 mM citric acid, 5 mM TSP in 90:10 H2 O:D2 O) with (blue) and without (red) applying the automatic adaptive gain. Each spectrum was obtained from a single scan. The signal acquisition was done via the UHFLI (Position 1, Figure 1) with a digital scale K = 500. The SNR enhancement factor, β, was 2.64.

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A m p litu d e [V ]

0 .1 0 O r K = K = K = K =

0 .0 5

ig 5 5 5 5

in a l F ID 0 0 , β= 2 0 0 , β= 2 0 0 , β= 2 0 0 , β= 2

.6 4 .3 7 .4 9 .4 7

0 .0 0 0 .0 5

0 .1 0

(a )

0 .1 5

0 .2 0

T im e [s ] 3 .2

β

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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 26

2 .4

1 .6 5

(b )

6

7

8

9

1 0

1 1

1 2

A D C 1 E N O B [b its ]

Figure 5: (a) Effect of the internal digital gain of the lock-in amplifier, K, on the SNR enhancement due to the adaptive gain. (b) Effect of ADC1 resolution on the SNR enhancement of the AGAIN-modulated signals acquired by ADC2 (ENOB = 12).

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2 .0 W W

3

1 .8

1 .6

1 .4

1 .2

1 .0

0 .2

0 .0

-0 .2

ith A G A IN ith o u t A G A IN

2 1

1 H

0

c h e m ic a l s h ift [p p m ]

Figure 6: A 1 H NMR spectrum of a model solution with [alanine] = 0.1 mM, [citrate] = 10 mM, and [TSP] = 5 mM without (red) and with (blue) AGAIN. Each spectrum contained 32 averages and was acquired via the Bruker spectrometer (Position 2, Figure 1). The digital gain of the UHFLI, K, was set to 300. The SNR enhancement factor, β, due to AGAIN was 1.4.

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W W

7 .2

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ith A G A IN ith o u t A G A IN

7

6 .9

6

1 .2

1

5 H

4

3

2

1

0

c h e m ic a l s h ift [p p m ]

Figure 7: 1 H NMR spectra of a cell culture medium as an example complex mixture. Each spectrum was a single scan, and was acquired via the Bruker spectrometer (Position 2, Figure 1). The digital gain of the UHFLI, K, was set to 300 and the measured SNR enhancement factor, β, was 1.45.

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Automatic adaptive gain for magnetic resonance sensitivity

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