SVD-Based Technique for Interference Cancellation and Noise ... - MDPI

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SVD-Based Technique for Interference Cancellation and Noise Reduction in NMR Measurement of Time-Dependent Magnetic Fields Wenjun Chen 1,† , Hong Ma 2,† , De Yu 1,† and Hua Zhang 2, * 1 2

* †

School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China; [email protected] (W.C.); [email protected] (D.Y.) School of Electronics Information and Communication, Huazhong University of Science and Technology, Wuhan 430074, China; [email protected] Correspondence: [email protected]; Tel.: +86-139-7107-9509 These authors contributed equally to this work.

Academic Editor: Andreas Hütten Received: 3 December 2015; Accepted: 27 January 2016; Published: 4 March 2016

Abstract: A nuclear magnetic resonance (NMR) experiment for measurement of time-dependent magnetic fields was introduced. To improve the signal-to-interference-plus-noise ratio (SINR) of NMR data, a new method for interference cancellation and noise reduction (ICNR) based on singular value decomposition (SVD) was proposed. The singular values corresponding to the radio frequency interference (RFI) signal were identified in terms of the correlation between the FID data and the reference data, and then the RFI and noise were suppressed by setting the corresponding singular values to zero. The validity of the algorithm was verified by processing the measured NMR data. The results indicated that, this method has a significantly suppression of RFI and random noise, and can well preserve the FID signal. At present, the major limitation of the proposed SVD-based ICNR technique is that the threshold value for interference cancellation needs to be manually selected. Finally, the inversion waveform of the applied alternating magnetic field was given by fitting the processed experimental data. Keywords: singular value decomposition; magnetic field measurement; time-dependent magnetic fields; interference cancellation; noise reduction

1. Introduction Nuclear magnetic resonance (NMR) is usually performed in a stable magnetic field. However, in many cases, we need to pursue NMR in a time-dependent magnetic field; for instance, the field induced normal state in high Tc superconductors lies above 100 T, but such fields can only be produced temporarily by a pulsed magnet [1–3]. To study the NMR experimental technique in time-dependent magnetic fields as well as the related data processing method, a 1 H-NMR experiment was conducted in an alternating magnetic field in this work. However, affected by some home-made devices in our setup and the experimental environment, there is non-negligible radio frequency interference (RFI) in the measured NMR data. As a result, the signal to interference plus noise ratio (SINR), which defined as the power of the free induction decay (FID) signal divided by the sum of the RFI power and the power of background noise, is rather low; in certain data sets, the spectral peak amplitude of RFI is even higher than that of FID signal, which will lead to confusion between the interference peaks and the FID signal spectral peak, and cause a deviation in the magnetic field fitting. Therefore, we have to employ an appropriate interference cancellation algorithm to identify and remove the RFI. There is no “universal” interference cancellation algorithm. For a specific RFI, various interference cancellation algorithms have different effects. If the RFI is a narrow-band and stationary signal, a simple Sensors 2016, 16, 323; doi:10.3390/s16030323

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notch filtering method can adequately remove the interference [4]. However, the frequency-domain notch filtering algorithm requires a priori information or high accurate frequency estimation of RFI. Furthermore, this method does not suffice in more complex RFI environments, for example, where the frequency of RFI changes over time. A more common approach of performing interference cancellation is the adaptive cancelling scheme. The representative adaptive algorithms (e.g., LMS, NLMS, and RLS) have been applied in the time domain [5,6] as well as various transform domains [7–9]. In most cases, however, these adaptive algorithms require an iterative process to obtain the optimal filtering weight coefficient, resulting in a high computational complexity and a large amount of calculation time. In addition, some interference cancellation algorithms based on singular value decomposition (SVD) method are reported [10,11]. These algorithms are mainly applied in processing synthetic aperture radar (SAR) data. In this case, after SVD, the SAR signal has small and similar order of magnitude singular values as the characteristic of white noise, whereas the RFI merely corresponds to several dominant singular values. The RFI can be estimated using these largest singular values, and subtracted from the measured data. The SVD-based algorithm described in these articles can effectively suppress the time-variant RFI. However, they are only applicable to remove the narrow band interference (NBI) from a wideband signal and suitable for the situation where the interference power is much higher than the interested signal. In this article, the NMR experimental setup and scheme in a sinusoidal alternating magnetic field are introduced first. Then, in order to rapidly suppress the RFI, whose power level is comparable with that of the FID signal, a new interference cancellation and noise reduction (ICNR) algorithm based on singular value decomposition is proposed. After the ICNR processing, the SINR of NMR data was improved significantly. At the end, the waveform of the magnetic field was given by data fitting. 2. NMR in Time-Dependent Magnetic Fields In this work, we employ a routine pulsed NMR technique to detect NMR signal in a sinusoidal alternating magnetic field. Our experimental setup comprises a magnet, an NMR probe and associated electronics devices. The alternating magnetic field Bz is generated by a hybrid magnet, which consists of a permanent magnet and an electromagnet. The field strength of the permanent magnet Bz0 , measured by a gauss meter based on a Hall effect sensor, is about 0.579 T. The gyromagnetic ratio γ of the 1 H nucleus is 2.6752 ˆ 108 rad¨ s´1 ¨ T´1 , the corresponding Larmor frequency f is about 24.652 MHz. 0 rz . The operating voltage of the The electromagnet produces an additional alternating magnetic field B electromagnet can be adjusted in the range of 0–250 V. When operating at the maximum voltage 250 V, the amplitude of the additional alternating magnetic field Bzp is about 0.012 T. Thus, the bandwidth ∆ f (“ γ ¨ 2Bzp {2π) of the RF excitation signal, NMR probe and receiver should be at least 1.02 MHz. Assuming ωz and ϕz0 are the actual angular frequency and initial phase angle of the alternating magnetic field, the strength of the alternating magnetic field at time t is: Bz ptq “ Bz0 ` Bzp sinpωz t ` ϕz0 q

(1)

The basic principle of NMR measurement is the measurement of Larmor precession frequency. Nuclear magnetic resonance will occur when the frequency f 1 of the RF excitation field is exactly equal to the Larmor frequency f 0 [12] (p. 11). By measuring the spectral peak frequency of each FID signal, we can calculate the corresponding magnetic field strength Bz ptq at the moment of pulsed excitation. Because of the rate of change of the alternating magnetic field is slow (nearly 50 Hz), the frequency and phase of FID signal modulated by the change of the magnetic field have be ignored. Then, all parameters of the alternating magnetic field can be estimated by fitting the measured data points according to Equation (1) using a least squares method. The accuracy of the magnetic field measurement is mainly determined by the precision of the measurement frequency of each FID signal.

Sensors 2016, 16, 323 Sensors 2016, 16, 323 to Equation (1) using a least squares method. The accuracy of the magnetic3field of 12 points according measurement is mainly determined by the precision of the measurement frequency of each FID signal. The experiment is home-made single The NMR NMR probe probe used used in in our our experiment is aa home-made single coil coil probe. probe. A A solenoid solenoid coil coil of of 10 turns is formed by wrapping a 30-gauge enameled copper wire upon a glass tube (5-mm 10 turns is formed by wrapping a 30-gauge enameled copper wire upon a glass tube (5-mm outer outer diameter, 8-mm length). length).The Thecoil coilpitch pitchisisadjusted adjusted approximately equal to the diameter diameter, 8-mm to to bebe approximately equal to the wirewire diameter and and then fixed with epoxy resin. The resistance and inductance of this solenoid coil is about 0.95 then fixed with epoxy resin. The resistance and inductance of this solenoid coil is about 0.95 OhmOhm and and 0.75 respectively. µH, respectively. In to order to transmit RFtopower to the NMR probe efficiently, the probe 0.75 µH, In order transmit RF power the NMR probe efficiently, the probe impedance impedance has been to the characteristic of cable the coaxial ( Z 0  50 Ohm). has been matched to matched the characteristic impedance impedance of the coaxial (Z “cable 50 Ohm). There are 0

There several types of impedance matching[13,14]. network [13,14]. L-type A routine L-typenetwork couplingis network severalare types of impedance matching network A routine coupling used for is used for this purpose due to its simple structure and high efficiency. The quality factor Q, with this purpose due to its simple structure and high efficiency. The quality factor Q, with the samplethe in sample place, about and the bandwidth of is the probe2 is nearly 2 MHz. place, isinabout 35,isand the35, bandwidth of the probe nearly MHz. The The sketch sketch of of our our NMR NMR system system is is shown shown in in Figure Figure 1. 1. The The experimental experimental setup setup is is built built around around aa National Instruments (NI) PXI system [15]. The PXI system is controlled by a high-bandwidth, National Instruments (NI) PXI system [15]. The PXI system is controlled by a high-bandwidth, 2.3 2.3 GHz GHz Dual-Core Embedded Controller (NI PXIe-8130, National Instruments, Austin, TX, USA). High Dual-Core Embedded Controller (NI PXIe-8130, National Instruments, Austin, TX, USA). High speed speed output (NI PXI-6733, National Instruments, is used to control analog analog output (NI PXI-6733, National Instruments, Austin, TX,Austin, USA) isTX, usedUSA) to control various single various single pole double throw (SPDT) switches (2ASWA-2-50DR+, Mini-circuits, New York, NY, pole double throw (SPDT) switches (2ASWA-2-50DR+, Mini-circuits, New York, NY, USA), whose USA), whose RF input is only 24 dBm, a home-made power SPDT (the maximum RFmaximum input power is onlypower 24 dBm, a home-made high powerhigh SPDT switch (theswitch maximum maximum RF input power reaches up to 100 W for a continuous wave signal, and the switching time RF input power reaches up to 100 W for a continuous wave signal, and the switching time is about is 12 µs), and the unblanking of the power amplifier. The RF signal generator (NI PXI-5422, 12about µs), and the unblanking of the power amplifier. The RF signal generator (NI PXI-5422, National National Instruments, Austin, TX, USA)a produces a wave continuous signal. RF signal is Instruments, Austin, TX, USA) produces continuous signal.wave This RF signalThis is shaped by the shaped by the SPDT 1 to form an RF pulse sequence and amplified by a power amplifier SPDT 1 to form an RF pulse sequence and amplified by a power amplifier (TwinPulse400, Tomco, (TwinPulse400, Tomco, Australia), the the high power switch intoeach the Stepney, Australia), thenStepney, passes through the then high passes power through switch into NMR probe. When NMR probe. When each pulseofisthe over, otherswitch channel of the high switch is excitation pulse is over, theexcitation other channel highthe power is unblanked forpower signal reception. unblanked for signal reception. The generated NMR signal is amplified by a preamplifier The generated NMR signal is amplified by a preamplifier (ZFL-1000LN +, Mini-circuits, New York, NY, (ZFL-1000LN +, Mini-circuits, New York, NY, USA),(NI then digitizedNational by a high-resolution USA), then digitized by a high-resolution digitizer PXI-5122, Instruments,digitizer Austin, (NI TX, PXI-5122, National Instruments, Austin, TX, USA) with 100 MS/s and recorded by a disk array USA) with 100 MS/s and recorded by a disk array (HDD-8264 RAID, National Instruments, Austin, (HDD-8264 RAID, National The SPDTthe 2 and 3 areAll used to protect the TX, USA). The SPDT 2 and Instruments, 3 are used to Austin, protect TX, the USA). receiver during pulse. components are receiver theMHz pulse.reference All components by the 10 MHz reference clock of the PXI system. clocked during by the 10 clock ofare theclocked PXI system.

Figure Figure 1. 1. Schematic Schematic diagram diagram of of the the NMR NMR system, system, all all gray gray parts are controlled by PXI-6773.

In In pulsed pulsed NMR NMR experiments, experiments, the the nutation nutation angle angle of of the the macroscopic macroscopic magnetic magnetic moment moment is is mainly mainly determined by the RF excitation field strength B and the pulse width  . When the nutation 1 determined by the RF excitation field strength B and the pulse width τ. When the nutation angle angle 1

is is π/2, π/2, the the macroscopic macroscopic magnetic magnetic moment moment is is flipped flipped to to the the xy-plane, xy-plane, meanwhile, meanwhile, the the NMR NMR signal signal strength reaches a maximum. Assuming an uniform magnetic field occupying the entire coil volume strength reaches a maximum. Assuming an uniform magnetic field occupying the entire coil volume V, V, width a π/2 pulse is approximately given [12] (pp. 38–39): thethe width of aofπ/2 pulse is approximately given byby [12] (pp. 38–39):  d 0V  / 2  π ω0 V τπ{2 “  2 0 QP γ 2µ0 QP

(2) (2)

Under the following conditions: the resonance angular frequency 0  2  24.652 MHz, the Under the following conditions: the resonance angular frequency ω0 “ 2π7 ˆ 24.652 MHz, the volume of the coil V  157  10´9 9 m33, the vacuum permeability  0  4  10´ H/m, the quality volume of the coil V “ 157 ˆ 10 m , the vacuum permeability µ0 “ 4π ˆ 10 7 H/m, the quality factor Q “ 35, and the RF power P “ 100 W, the π/2 RF pulse width calculated using Equation (2) is 3/12

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factor Q  35 , and the RF power P  100 W, the π/2 RF pulse width calculated using Equation (2) Sensors 2016, 16, 323 4 of 12 isfactor aboutQ0.62 Considering the inhomogeneity of π/2 RF excitation field, the calculated value is always  35µs. , and the RF power P  100 W, the RF pulse width calculated using Equation (2) smaller than the actual value. Therefore, this value can only provide a reference for the selection of is about 0.62 µs. Considering the inhomogeneity of RF excitation field, the calculated value is always about 0.62 µs. Considering the inhomogeneity of RF excitation field, the calculated value is always π/2 pulsethan width. results showed that the of 1a µs is an appropriate choice.of smaller the Experimental actual value. Therefore, this value canpulse onlywidth provide reference for the selection smaller than the actual value. Therefore, thisofvalue can only provideRF a reference for thefrom selection of The pulse sequence used here is a train identical equidistant pulses derived the RF π/2 pulse width. Experimental results showed that the pulse width of 1 µs is an appropriate choice. π/2 pulse width. Experimental results showed that the pulse of 1pulse µs is repetition an appropriate choice. carrier running 24.65 MHz. RF pulse is equidistant 1 µs,width and the 1 ms. The pulse at sequence usedThe hereduration is a trainofof identical RF pulses derivedtime fromisthe RF The pulse sequence used here is a train of identical equidistant RF pulses derived from the RF NMR data with a temporal length of 30 ms is recorded, which contains 29 integral FID signals. carrier running at 24.65 MHz. The duration of RF pulse is 1 µs, and the pulse repetition time is 1 ms. carrier running at 24.65 MHz. The duration of RF pulse is 1 RF µs, pulse and the repetition time is The dead thelength interval from end of the to pulse theintegral start ofFID the signals. signal NMR datatime with a during temporal of 30 ms the is recorded, which contains 29 1 ms. NMRisdata withIna order temporal length of a30high ms is recorded, which contains 29 integral FID signals. acquisition 15 µs. to interval guarantee thepulse interferences between raw FID The dead time  during the from thecorrelation end of theofRF to the start of the signal The dead time δ during the interval from the end of the RF pulse to the start of the signal acquisition data and reference data, the selected reference signal is an adjacent data set after the corresponding acquisition is 15 µs. In order to guarantee a high correlation of the interferences between raw FID is 15data µs. In order to guarantee a high correlation of the interferences between raw FID data and FID as illustrated in Figure data andset, reference data, the selected2.reference signal is an adjacent data set after the corresponding reference data, the selected reference signal is an adjacent data set after the corresponding FID data set, FID data set, as illustrated in Figure 2. as illustrated in Figure 2.

Figure 2. Sketch of the pulse sequence and the data selection. Figure 2. Sketch of the pulse sequence and the data selection. Figure 2. Sketch of the pulse sequence and the data selection.

The NMR sample is a saturated CuSO4 aqueous solution (mass fraction about 20% at 300 K). The addition ofsample paramagnetic salt canCuSO effectively accelerate the(mass relaxation rates of 1H nuclei, thus The NMR is aqueous solution (mass fraction about sample is aa saturated saturated solution fraction about 20% at 300 K). 4 aqueous 1 avoiding the saturation of NMRsalt sample in the process of pulse sequence excitation. spin-lattice 1H nuclei, The of can accelerate the relaxation rates thus The addition addition of paramagnetic paramagnetic salt can effectively effectively accelerate the relaxation rates of of The nuclei, thus 1H are approximately 0.78 ms and 0.60 ms relaxation time T1 and the spin-spin relaxation T2 of ofpulse avoiding the saturation of NMR in process sequence saturation NMR sample sample in the the time process of pulse sequence excitation. excitation. The The spin-lattice spin-lattice

1 1 inhomogeneity relaxation the spin-spin relaxation time 0.78 ms 0.60 at 300 K, respectively [16]. into account of theTTspatial of the applied magnetic time TT11 and and theTaking spin-spin relaxation of H Hare areapproximately approximately 0.78 ms and and 0.60 ms ms 22 of * at 300 K, respectively [16]. Taking into account of the spatial inhomogeneity of the applied magnetic field, the apparent relaxation time T [17] (pp. 50–52) will be shorter. The selected temporal length 2 at 300 K, respectively [16]. Taking into account of the spatial inhomogeneity of the applied magnetic ˚ *[17] (pp. 50–52) will be shorter. The selected temporal length of field, the apparent relaxation time T of the FID data set and the reference data set(pp. for signal are both ms. temporal length 2 field, the apparent relaxation time T2 [17] 50–52)processing will be shorter. The0.45 selected the FID data set and the reference data set for signal processing are both 0.45 ms. AllFID data processing is reference done on PC a signal self-written Matlab of the data set and the datausing set for processing areprogram. both 0.45The ms. data processing Allshown data processing is done on PC using areference self-written Matlab program. The data processing flow is flow is in Figure 3. The FID data and data are firstly demodulated All data processing is done on PC using a self-written Matlab program. Thewith dataquadrature processing shown in Figure 3. and The FID data andsignals reference data are firstly demodulated withthese quadrature reference reference signals, then, filtered andare extracted. After pre-processing flow is shown in Figure 3. Thethese FID data andare reference data firstly demodulated with quadrature signals, and then, these signals are filtered and extracted. After these pre-processing procedures, the procedures, the number of sample is dramatically from 45,000 900, which provides reference signals, and then, thesepoints signals are filtered reduced and extracted. Aftertothese pre-processing number of sample points is dramatically reduced from 45,000 to 900, which provides a convenient for aprocedures, convenient the for number subsequent matrixpoints operations. of sample is dramatically reduced from 45,000 to 900, which provides subsequent matrix operations. a convenient for subsequent matrix operations.

Figure 3. NMR data processing flow chart. The demodulation frequency is 24.65 MHz. The pass band Figure 3. NMR flow chart. demodulation of the digital lowdata passprocessing filter is 0–600 kHz, andThe the cutoff frequencyfrequency is 1 MHz.is 24.65 MHz. The pass band of 3. theNMR digital lowprocessing pass filter is 0–600 kHz,The anddemodulation the cutoff frequency is 1 MHz. Figure data flow chart. frequency is 24.65 MHz. The pass

band of the digital passin filter is 0–600 kHz, are andmany the cutoff frequency is 1 MHz. Unfortunately, aslow shown Figure 4, there large frequency-discrete comb interference Unfortunately, as shown in Figure 4, there are many large frequency-discrete comb interference peaks with a frequency interval nearly 284 kHz in the 4–38 MHz range. Moreover, this RFI is not a peaksUnfortunately, with a frequency interval nearly kHz in the 4–38 MHz range. Moreover,comb this RFI is not a as shown Figure 284 4, there many large frequency-discrete interference strictly stationary signal, as itsinfrequency variesare slowly over time. After testing, we found that this strictly stationary signal,interval as its frequency varies over MHz time. range. After testing, we this found thisa peaks with a frequency nearly 284 kHz slowly in the of 4–38 Moreover, RFIthat is not RFI mainly derives from the power supply component our home-made high power SPDT switch. RFI mainly derives from the power supply component of our home-made high power SPDT switch. strictly stationary signal, as its frequency varies slowly over time.from Afterthe testing, wepeak found that this Because the time-dependent magnetic field strength is calculated spectral frequency Because the derives time-dependent magnetic field component strength is calculated from thehigh spectral peak frequency RFI mainly from the power supply of our home-made power SPDT switch. of FID signal, a deviation will occur in final magnetic field fitting if the interference peak amplitude of FID signal, a deviation will occur in final magnetic field fitting if the interference peak amplitude Because the time-dependent magnetic field strengthlow is calculated is comparable with that of NMR signal in certain SINR datafrom sets.the In spectral order topeak deal frequency with this isofcomparable with that ofwill NMR signal in certain low SINR dataif sets. In order topeak deal amplitude with this FID signal, a deviation occur in final magnetic field fitting the interference problem conveniently and effectively, a new interference cancellation algorithm based on singular is comparable with that of NMR signal in certain low SINR data sets. In order to deal with this value decomposition is proposed.

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problem conveniently and effectively, a new interference cancellation algorithm based on singular Sensors 2016, 16, 323 5 of 12 value decomposition is proposed.

Figure Figure 4. 4. Spectrum Spectrum of of one one FID FID data data set set of of aa low low SIR. SIR.The Thedata datahas hasbeen beenpre-processed pre-processed (demodulation, (demodulation, low pass filtering and extraction). The FID signal peak is marked; others peaks. Obviously, low pass filtering and extraction). The FID signal peak is marked; others areare RFIRFI peaks. Obviously, the the spectral width of the FID signal is broader than that of RFI. spectral width of the FID signal is broader than that of RFI.

3. SVD-Based Interference Cancellation and Noise Reduction Method 3. SVD-Based Interference Cancellation and Noise Reduction Method reference data data set set isisy y“ty Supposed the FID FID data data set setisisx x“tx  x1, ,xx2,,¨¨ ¨, x, lx u and  y1,,yy2,,¨¨ ¨, ,yyl u.. Supposed the and the the reference 1

2

1

l

2

The vectors xxand and y can be used to construct Hankel matrices of dimension  follows: n as follows: y can be used to construct Hankel matrices of dimension mˆ m n as x 1x1 xx22 — x — x 2 2 xx33 1 A1 “A— .. .  — – .. . x x m m  xm xm`11

 ¨¨¨  ¨¨¨ ..  .  ¨¨¨

» y y y1 1 y22 —  y2 yy33  y2 A2— — A2 “ — .  .. – .. . ym ym 1  y m y m `1

 yn  ¨¨¨ yn  y n  1 ¨ ¨ ¨ yn1`   . .. . . .  yl  ¨¨¨ yl

»

xn xn xnx1n`1  .. . xl x l

l

fi ffi ffi ffi ffi fl

(3) (3)

fi ffi ffi ffi ffi fl

(4) (4)

where, which is is usually taken half thethe total length of where, m m is is the thenumber numberofofrows rowsofofthe theHankel Hankelmatrix, matrix, which usually taken half total length . The floor function gives the largest integer less than or equal to l / 2 . the data, i.e., m  l / 2 l / 2     of the data, i.e., m “ rl{2s. The floor function rl{2s gives the largest integer less than or equal to l{2. In addition, m and n subject to the the constraint constraint mm`nn´  11“l .l.Applying Applyingsingular singularvalue valuedecomposition decomposition to andAA , andwe weget: get: matrices AA1 and 2 ,2 and (5) A1 “ U1 Σ1 VT1 T A1  U11V1 (5) A2 “ U2 Σ2 V2 T (6) A2  U 2  2V2T (6) In Equations (5) and (6), the left eigenvector matrix U and the transposition of the right eigenvector T are the orthogonal In (5) and (6), the left U nand transposition of the right matrix VEquations matrices of eigenvector dimension mmatrix ˆ m and ˆ n, the respectively; the main diagonal T eigenvector V are the orthogonal matrices of dimension m andare n non-negative  n , respectively; elements of matrix the diagonal matrix Σ are called the singular values ofmA, which and the main diagonal elements arranged in descending order.of the diagonal matrix  are called the singular values of A , which are non-negative descending The FID dataand set arranged x and the in reference data order. set y both contain RFI. However, the actual correlation both contain RFI. However, the actual The RFI FIDindata setRFI x in and reference data setof yhardware between x and y isthe affected by a variety factors. Therfore, we plan to use the correlation between of U RFI from the FID signal. linear correlation between RFI the in columns in1 and y isUaffected by a variety of hardware factors.The Therfore, x and RFI 2 to distinguish

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we plan to use the correlation between the columns of U1 and U 2 to distinguish RFI from the FID signal. The linear correlation coefficient  ij between the i-th column vector u1i of left singular matrix U1 and the j-th column vector u2 j of matrix U 2 can be defined by the Pearson correlation Sensors 2016, 16, 323 coefficient [18] (pp. 37–40):

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cov(u1i , u2 j )  ij  (7) correlation coefficient γij between the i-th column u1vector u2 j u1i of left singular matrix U1 and the j-th i column vector u2j of matrix U2 can be defined by the Pearson correlation coefficient [18] (pp. 37–40): where the value range of subscript i and j is from 1 to m. cov(u1i , u2 j ) is the covariance of u1i and covpu1i , u2j q u2 j . u1i and u2 j are the standard deviation a γij “ of u1i and u2 j , respectively. Thus, we can obtain(7) σu1i σu2j correlation matrix C: where the value range of subscript i and jis from 1 to  m. covpu 1m 1i , u2j q is the covariance of u1i and u2j . 11 12  σu1i and σu2j are the standard deviation of u1i and u2j , respectively. Thus, we can obtain a correlation  21  22   2 m   C matrix C: (8) » fi  |   | |γ 11| |γ12 ¨ ¨ ¨ |γ 1m  — ffi  m|1 |γ 22 ¨ ¨ ¨  |γ m2|  mm 2m — |γ 21  | ffi ffi C“— (8) .. . .. — .. ffi . set.. x .flIf the maximum value in the k-th . Assuming the FID signal mainly–presents in. the data |γm1 | than γm2a given ¨ ¨ ¨ |γthreshold mm | row of the correlation matrix C is greater value kin , it means the k-th

U1 belongs the interference themaximum corresponding value column vector of matrix Assuming the FID signal mainly to presents in the datasubspace; set x. If the value singular in the k-th row of matrix C isgreater than a given threshold value k , it means the k-th column vector  k the of correlation the diagonal matrix can be set to zero. Thus, the original diagonal matrix  is replaced in 1 1 of matrix U1 belongs 1 . to the interference subspace; the corresponding singular value λk of the diagonal by a new matrix matrix Σ1 can be set to zero. Thus, the original diagonal matrix Σ is replaced by a new matrix Σ1 . In the diagonal matrix 11 , several previous large singular1 values corresponding to RFI 1has In the diagonal matrix Σ1 , several previous large singular values corresponding to RFI has been been removed, while the smaller singular values, mainly represent the noise component, still exist. removed, while the smaller singular values, mainly represent the noise component, still exist. The The SINR of NMR data can be further improved by removing the small singular values. Assuming SINR of NMR data can be further improved by removing the small singular values. Assuming the the FID signal is a single frequency signal, the random noise can be reduced with the greatest degree FID signal is a single frequency signal, the random noise can be reduced with the greatest degree by preserving only the largest one of the remaining singular values in 11 , and we can obtain a new by preserving only the largest one of the remaining singular values in Σ1 , and we can obtain a new 2 diagonal diagonal matrix matrix Σ11. . be reconstituted according to After the ICNR ICNR processing, processing, the theestimated estimatedmatrix matrixA1A1ofof After the A1 Acan be reconstituted according to the 1 can 1 the following formula: following formula: 2 A1 “ U1 Σ1 VT1 T (9) A  U11V1 (9) Generally, the reconstruction matrix A11 is no longer a Hankel matrix. The estimated signal Generally, the( reconstruction matrix A1 is no longer a Hankel matrix. The estimated signal 1 x “ x11 , x21 , ¨ ¨ ¨ , xl1 can be obtained by averaging the diagonal elements of the reconstruction matrix: x    x1, x2 , , xl can be obtained by averaging the diagonal elements of the reconstruction matrix: s ÿ 1 s 1 xi1 “ A1 j`1,j (10) xi s ´ p ` 1  ii´ (10) j 1, j 1 s  p  1 jj“ 1

min(n, iiq ) and and p p“maxp1, max(1, ii´mm` 1) 1q. . where, ss“ minpn, In summary, the architecture of the proposed SVD-based ICNR algorithm in thisinarticle is shown In summary, the architecture of the proposed SVD-based ICNR algorithm this article is in Figure 5. shown in Figure 5.

Figure 5. 5. Schematic Schematic overview overview of of the the ICNR ICNR algorithm algorithm based based on on singular singular value value decomposition. decomposition. Figure

4. Results and Discussion

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A plot of the interference cancellation effect of different threshold values is shown in Figure 6. The upper two parts of Figure 6 are the Fourier amplitude spectra of one of the 29 FID data sets and the

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4. Results and Discussion A plot of the interference cancellation effect of different threshold values is shown in Figure 6. 7 of 12 of Figure 6 are the Fourier amplitude spectra of one of the 29 FID data sets and the corresponding reference data set, respectively. It shows that the RFI in the two data sets are correlated. The lower four parts show various spectra after interference cancellation with a threshold corresponding reference data set, respectively. It shows that the RFI in the two data sets are correlated. value kin = 0.4, 0.5, 0.6, 0.7, respectively. This figure clearly demonstrates that, the FID signal is The lower four parts show various spectra after interference cancellation with a threshold value cancelled simultaneously when theThis selected threshold value is too low, 0.4. It is is cancelled unable to in =signal k in = 0.4, 0.5, 0.6, 0.7, respectively. figure clearly demonstrates that,e.g., the kFID k = 0.6 or 0.7, which suppress all of the interference peaks with a too high threshold value, such as simultaneously when the selected threshold value is too low, e.g., k in = 0.4. It isin unable to suppress implies the actualpeaks correlation between RFI in FID data such set and that in or reference dataimplies set is all of thethat interference with a too high threshold value, as k in = 0.6 0.7, which smaller threshold value.RFI When thedata threshold can be that the than actualthese correlation between in FID set and value that inkreference datainterference set is smaller than in = 0.5, the these threshold value. When the threshold value k = 0.5, the interference can be effectively cancelled effectively cancelled and the FID signal is better preserved. It should be noted that, the first RFI peak in andthe theleft FID is better preserved. be noted that, the first peak left is not on is signal not suppressed even whenIt kshould = 0.4, probably because the RFI power of on thisthe interference in suppressed even when k = 0.4, probably because the power of this interference component after in component after filtering is comparable with that of the noise in the pass band. filtering is comparable with that of the noise in the pass band. This means that, to achieve a good result using the above SVD-based ICNR algorithm, the key to achieve athreshold good result using above SVD-based ICNR algorithm, key factorThis is tomeans choosethat, an appropriate value kinthe . The optimal threshold value should bethe bigger factor is to choose an appropriate threshold value k in . The optimal threshold value should be bigger than the correlation between FID signal in the FID data set and RFI near the Larmor frequency in than the correlation between FID signal in the FID data set and RFI near the Larmor frequency in reference data set, meanwhile, smaller than the correlation between the RFI components need to be reference data set, meanwhile, smaller than the correlation between the RFI components need to be cancelled in these two data sets. However, we have not found a suitable way to quantitatively assess cancelled in these two data sets. However, we have not found a suitable way to quantitatively assess their correlations. At the present stage, the selection of the threshold value is primarily determined their correlations. At the present stage, the selection of the threshold value is primarily determined by by the interference cancellation effect. the interference cancellation effect. Sensors 2016, 16, 323parts The upper two

Figure 6. Fourier amplitude amplitude spectra spectra of of (A) (A) the the FID FID data; data; (B) (B) the Figure 6. Fourier the corresponding corresponding reference reference data; data; and and various spectra after after interference interference cancellation cancellationwith withdifferent differentthreshold thresholdvalue value(C) (C) kkin == 0.4; 0.4; (D) (D) kkin = 0.5; various spectra = 0.5; in in (E) k = 0.6; (F) k = 0.7, respectively. in in (E) k = 0.6; (F) k = 0.7, respectively. in

in

In Figure Figure 7, 7, the thespectra spectraofofthe theFID FIDdata, data, the reference data, and corresponding spectra of In the reference data, and thethe corresponding spectra of the the FID interference cancellation noise suppression shown. Figure 7C shows FID datadata afterafter interference cancellation andand noise suppression are are shown. Figure 7C shows that,that, an an interference cancellation effect of approximately 20–30 dB can be reached across the interesting interference cancellation effect of approximately 20–30 dB can be reached across the interesting frequency band using our our SVD-based SVD-based interference interference cancellation cancellation algorithm, algorithm, meanwhile, FID signal signal frequency band using meanwhile, the the FID peak peak at at the the Larmor Larmor frequency frequency is is left left undisturbed undisturbed after after interference interference cancellation. cancellation. Thus, Thus, the the spectral spectral peak amplitude of FID signal is significantly higher than those of RFI. It effectively avoids a wrong 7/12 in the subsequent data processing procedure. selection of the spectral peak frequency of FID signal After interference cancellation, the FID signal is still corrupted by random noise. To further improve

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peak amplitude of FID signal is significantly higher than those of RFI. It effectively avoids a wrong peak amplitude of FID signal is significantly higher than those of RFI. It effectively avoids a wrong selection of the spectral peak frequency of FID signal in the subsequent data processing procedure. selection of16,the Sensors 2016, 323spectral peak frequency of FID signal in the subsequent data processing procedure. 8 of 12 After interference cancellation, the FID signal is still corrupted by random noise. To further improve After interference cancellation, the FID signal is still corrupted by random noise. To further improve the SINR, some of the small singular values are removed to reduce the noise. The spectrum of the SINR, some of the small singular values are removed to reduce the noise. The spectrum of preserving only the largest one of the remaining singular values after interference cancellation is the SINR, some of the small singular are singular removed to reduce noise. The spectrum of preserving only the largest one of the values remaining values afterthe interference cancellation is given in Figure 7D. preserving only the largest one of the remaining singular values after interference cancellation is given given in Figure 7D. in Figure 7D.

The effect of interference cancellation and noise suppression. The spectra of (A) the raw FID Figure 7. 7. The Figure effect of interference cancellation and noise suppression. The spectra of (A) the raw FID data; (B) the reference data; (C) the FID data after interference cancellation with a threshold value of data; (B) the reference data; (C) the FID data after interference cancellation with a threshold value of preserving only only the the largest largest one of of the remaining remaining singularvalues. values. 0.5; (D) (D) noise noise suppression suppression by by preserving preserving 0.5; only the largest one one of the the remaining singular singular values.

Figure 8 8 gives gives the the singular singular value value curves of the FID data, data, the the reference reference data, data, and and the the data data after after Figure curves of of the the FID FID Figure 8 gives the singular value curves data, the reference data, and the data after interference cancellation. All of the first fifteen largest singular values, except the 7th singular value interference cancellation. cancellation. All All of of the the first first fifteen fifteen largest largest singular singular values, values, except except the the 7th 7th singular singular value value interference (marked inFigure Figure 8),are are identified and set to zero by the interference cancellation algorithm. And setset to to zero by by the the interference cancellation algorithm. And And then, (marked in in Figure8), 8), areidentified identifiedand and zero interference cancellation algorithm. then, only this marked singular value is preserved to reconstitute the FID signal. The corresponding only only this marked singular value is preserved to to reconstitute the corresponding then, this marked singular value is preserved reconstitute theFID FIDsignal. signal. The The corresponding spectrum, whose peak frequency is consistent with the Larmor frequency, is shown in Figure 7D, spectrum, whose peak Larmor frequency, is shown in Figure 7D, and spectrum, peak frequency frequencyisisconsistent consistentwith withthe the Larmor frequency, is shown in Figure 7D, and the waveform is illustrated in Figure 9 (red curve). the waveform is illustrated in Figure 9 (red curve). and the waveform is illustrated in Figure 9 (red curve).

Figure 8. Singular Singular value plots of Figure 7A–C. The marked singular value corresponds to Figure 7D. Figure 8. Singular value plots of Figure 7A–C. The marked singular value corresponds to Figure 7D.

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Time-series plots of Figure 7 are shown in Figure 9. In this figure, it is obviously easier to observe a FID signal after data processing, preliminarily verifying the validity of our SVD-based * ICNR 2016, algorithm. Sensors 16, 323 The rapid decay of FID signal illustrates that the apparent relaxation time T2 is Sensors 2016, 16, 323

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shorter than 0.15 ms, indicating a rather inhomogeneous applied magnetic field. Time-series plots of Figure 7 are shown in Figure 9. In this figure, it is obviously easier to observe a FID signal after data processing, preliminarily verifying the validity of our SVD-based ICNR algorithm. The rapid decay of FID signal illustrates that the apparent relaxation time T2* is shorter than 0.15 ms, indicating a rather inhomogeneous applied magnetic field.

Figure 9. The The waveforms waveformsof ofraw rawFID FIDdata data(cyan), (cyan), and data after interference cancellation (blue) Figure 9. and thethe data after interference cancellation (blue) and and noise suppression (red). noise suppression (red).

The NMR spectra of all 29 FID signals after ICNR are shown in Figure 10. It can be seen clearly Time-series plots of Figure 7 are shown in Figure 9. In this figure, it is obviously easier to observe a that the spectral peak frequency ofFID each FID(cyan), signaland variesdata sinusoidally, confirming the feasibility of Figure 9. The waveforms of raw data after of interference cancellation FID signal after data processing, preliminarily verifyingthe the validity our SVD-based ICNR(blue) algorithm. the magnetic field measurement scheme. and noise suppression (red). illustrates that the apparent relaxation time T ˚ is shorter than 0.15 ms, The rapid decay of FID signal 2 indicating a rather inhomogeneous applied magnetic field. The The NMR NMR spectra spectra of of all all 29 29 FID FID signals signals after after ICNR ICNR are are shown shown in in Figure Figure 10. 10. ItIt can can be be seen seen clearly clearly that the spectral peak frequency of each FID signal varies sinusoidally, confirming the feasibility of that the spectral peak frequency of each FID signal varies sinusoidally, confirming the feasibility of the the magnetic measurement scheme. magnetic fieldfield measurement scheme.

Figure 10. The waterfall graph of all 29 NMR spectra.

Figure 10. 10. The The waterfall waterfall 9/12 graph of of all all 29 29 NMR NMR spectra. spectra. Figure graph

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The magnetic field strength at each pulse excitation moment can be calculated from the spectral magnetic strength at eachFID pulse excitation can be calculated from the peakThe frequency of field the corresponding signal set. Allmoment of the parameters in Equation (1)spectral can be peak frequency of the corresponding FID signal set. All of the parameters in Equation (1) estimated by fitting the 29 data points of magnetic field strength using a least square method. Figurecan 11 be estimated by curve fittingof the data points of magnetic fieldthe strength using a residual least square shows the fitting the29alternating magnetic field and corresponding error.method. Fitting Figureindicates 11 showsthat, the the fitting curve ofmagnetic the alternating field and the corresponding residual result permanent field Bmagnetic z 0 is 0.5798 T; the amplitude of the alternating error. Fitting result indicates that, the permanent magnetic field is 0.5798 T; the frequency amplitude of 250 V; angular z the is magnetic field Bzp is 0.0117 T when the operating voltage isBz0 alternating magnetic field Bzp is 0.0117 T when the operating voltage is 250 V; angular frequency ωz is 2  50.0676 rad/s, which is close to the nominal alternating current frequency 50 Hz; the initial 2π ˆ 50.0676 rad/s, which is close to the nominal alternating current frequency 50 Hz; the initial phase phase  z 0 is 0.0066 rad. ϕz0 is 0.0066 rad.

Fitting curve curve of of the alternating alternating magnetic magnetic field field when when the the operating operating voltage voltage is is 250 V and Figure 11. Fitting residual error.

5. Conclusions Conclusions In this thispaper, paper, experiment in time-dependent magnetic fields,theincluding the the the NMRNMR experiment in time-dependent magnetic fields, including experimental experimental and the field measurement was The introduced. The NMR setup and thesetup magnetic fieldmagnetic measurement scheme, was scheme, introduced. NMR method can method achieve can achieve higher measurement accuracy, larger measurement well as has a speed faster higher measurement accuracy, larger measurement range, as well range, as has as a faster response response speed than other used inof measurement of time-dependent magnetic such than other techniques usedtechniques in measurement time-dependent magnetic fields, such asfields, flux meter, as flux meter, magneto-resistance, magneto-optical, Hall effect, etc. [19]. In addition, to improve the magneto-resistance, magneto-optical, Hall effect, etc. [19]. In addition, to improve the SINR, a new SINR, a new SVD-based ICNR algorithm and wasused proposed, and used in processing measured NMR SVD-based ICNR algorithm was proposed, in processing measured NMR data. The raw FID data. The corrupted raw FID data was corrupted bynoise; additive and noise; the data, selected reference data, which data was by additive RFI and the RFI selected reference which is close to the FID is close to the FID data, comprised only correlated interference and uncorrelated Applying data, comprised only correlated interference and uncorrelated noise. Applying SVD tonoise. the constructed SVD to the constructed Hankel matrices of these two signals, then removing the singular values Hankel matrices of these two signals, and then removing theand singular values corresponding to the corresponding to noise the interference and noiseour components using our SVD-based the interference and components using SVD-based ICNR algorithm, theICNR SINRalgorithm, of raw NMR SINR of raw NMR data was significantly improved. The interference cancellation result showed that data was significantly improved. The interference cancellation result showed that the amplitude of the amplitude of can interference peaksaround can be around to 30 dB. Based on that our interference peaks be suppressed 20 suppressed to 30 dB. Based on our20investigation, we found investigation, we found that the choice of the optimal threshold value for interference cancellation the choice of the optimal threshold value for interference cancellation depends on the correlation of depends on the correlation the interference components between the two data sets. the interference componentsofbetween the two data sets. Compared with other interference cancellation methods, the proposed SVD-based interference cancellation applications, such as for cancelling cancellation algorithm algorithm in in this this article article has has aa wider range of applications, non-stationary interference, interference,wideband wideband interference, or interference a comparable even interference, or interference with with a comparable or evenor smaller smaller power, and does not require an iterative process and is suitable for real-time processing. However, there are also some limitations of our algorithm. 10/12

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power, and does not require an iterative process and is suitable for real-time processing. However, there are also some limitations of our algorithm. The main problem is that there is no mature and effective way to determine the optimal threshold value. The current version of our SVD-based ICNR technique needs a user-defined threshold for retaining or eliminating the singular values. This is an important limitation of the technique at this stage of development. In this article, the selection of the optimal threshold value mainly relies on the actual cancelling effect. In addition, in the period of rapid change of an applied time-dependent magnetic field, the FID signal frequency is dramatically modulated by the magnetic field, resulting in spectral line broadening [20]. Preserving only the first largest singular value in the noise reduction procedure can naturally achieve a highest improvement in SINR. However, the reconstituted waveform of FID signal using this strategy may produce distortion if the NMR signal has a broad spectrum, although such a distortion does not change the spectral peak frequency and have no effect on the final magnetic field fitting result. To avoid distortion of the FID waveform in some special application environments, e.g., a pulsed magnetic field of a high field strength and a short duration, some other criteria of determining the optimal de-noising order can be used as a reference, such as the increment of the singular entropy [21], the difference spectrum of the singular values [22], the curvature spectrum of the increment of singular entropy [23], etc. Finally, in order to get a high correlated reference signal, besides sampling the raw FID data, we also need to record a reference data set of the same length during two excitation pulses. This strategy makes the available number of FID signal sets is reduced by half in the same data acquisition time, which may result in an accuracy reduction of data fitting. At the end, we fitted the 29 processed experimental data points using a least square method and estimated all parameters of the alternating magnetic field. The results are consistent with the measured values using a gauss meter, indicating the validity of the ICNR algorithm and magnetic field measurement scheme. Acknowledgments: This work was supported by the National Natural Science Foundation of China (Grant No. 11475067), the Innovative Research Program of Huazhong University of Science and Technology (Grant No. 2015ZDTD017), and the Experimental Apparatus Research Project of Wuhan Pulsed High Magnetic Field Center (Grant No. 2015KF17). Author Contributions: Hong Ma and Wenjun Chen conceived and designed the experiments; Wenjun Chen performed the experiments; De Yu and Hua Zhang analyzed the data; all authors took part in the evaluation and discussion of results, as well as manuscript editing. Conflicts of Interest: The authors declare no conflict of interest.

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