Research ArticleAnalysis of NMR Spectrometer Receiver Noise Figure

Peter Andris 1 Earl F Emery2 and Ivan Frollo1

1 Institute of Measurement Science Slovak Academy of Sciences Dubravska Cesta 9 841 04 Bratislava Slovakia2Tecmag Inc 10161 Harwin Dr Suite 150 Houston TX 77036 USA

Correspondence should be addressed to Peter Andris umerandrsavbask

Received 13 February 2019 Accepted 16 May 2019 Published 26 June 2019

Academic Editor Giorgio Besagni

Copyright copy 2019 Peter Andris et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This article describes the measurement and evaluation of the system noise figure of a nuclear magnetic resonance spectrometerA method was used which involved the console of the spectrometer calibrated for real voltage (in volts) and used for noisesignal digitisation and measurement The resulting digitised signal was exported and processed and the root mean square valuecalculated This value was utilised for the noise power calculation which was compared with the theoretical value and the noisefigure calculatedThe method presented enables analysis of the bandwidth of the noise Many of the equations that are commonlyused for signal processing have been derived for the specific task and require verification We have verified the technique using acommercial console with two preamplifier pairs The experimental data agree well with the theoretical values confirming that thepresented method is a valid simple and fast tool for the inspection of the spectrometer receiver

1 Introduction

Many parameters of complex instrumentation must bechecked regularly in order to ensure that high quality resultsare reported The noise figure is an important parameterof the receiver in a nuclear magnetic resonance (NMR)spectrometer Regular inspection of the receiver can revealdefects of the hardware or sources of interference It ispossible tomeasure the noise voltage only but the noise figureis a widely reported parameter and so use of this value allowsfor easy comparison Professional instruments for noisefigure measurement are expensive and only provide accurateresults for frequencies above tens of MHz At frequenciesof MHz the noise figure must be measured using differentinstruments such as the console of the NMR instrument [1]The noise figure can be calculated by processing themeasurednoise voltage however some researchers have calculated thenoise figure from continuous signals rather than discretesignals [2ndash4] The noise level is always evaluated as part ofa basic NMR experiment [5 6] Connecting the receivercoils of the NMR using inductive coupling can provide someadvantages [7 8] (higher comfort for operators balancing thereceive coil) and the noise figure of the entire inductivelycoupled system can be calculated using an equation [7]

Signal processing and noise matching have been used tooptimise NMR results [9ndash15] although optimisation must becarried out in amanner specific to the particular experiment

The system noise figure at frequencies of MHz can bedetermined using the ldquohotcold resistorrdquo method howeverthis method has the disadvantage of requiring liquid nitrogento cool the cold resistor Not every working place may beequipped for such operations The technique that we presenthere for noise figure determination utilises the NMR consolebut in this case it is calibrated to the real values of voltage andthe noise voltage is measured and subsequently processedOur technique provides a fast and accurate approach whichonly requires standard laboratory instruments and materialsThe measured data is processed in two ways and the resultsare compared First the noise figure has been determined ata narrow noise bandwidth where the spectra may be con-sidered constant and second at wider noise bandwidth withnonconstant spectra using the calculation of the equivalentnoise bandwidth The processing of the measured data isdiscussed elsewhere [16]The theory and results described inthe present article are useful for technical and experimentalpurposes and provide a platform for theoretical studiesMoreover our research offers options for techniques whichutilise the NMR spectroscopic console which is available at

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1083706 7 pageshttpsdoiorg10115520191083706

2 Mathematical Problems in Engineering

Vg V1

Rg

VlRlRin=Rg

(a)

P1_n

Rg

radic4kTRgΔf

Rin

(b)

Figure 1 A simplified circuit diagram of the NMR receiver (a) Signal and noise voltages at the two-port (b) incoming noise power at thetwo-port 1198811 group of incoming signal or noise voltages eg 1198811 119904 1198811 119899 1198811 119904 119903119898119904 or 1198811 119899 119898119890119886119899 119881119897 group of outgoing signal or noise voltageseg 119881119897 119904 119881119897 119899 119881119897 119904 119903119898119904 or 119881119897 119899 119894 (i-th term of the samples series) 119891 is the noise bandwidth of the receiver The noise spectra within the noisebandwidth should be constant119881119892 is signal voltage from source119877119892 119877119894119899 119877119897 are resistances of source input resistance of the two-port and loadresistance of the two-port119879 is the absolute temperature of the resistance119877119892radic4119896119879119877119892119891 is RMS value of the thermal noise voltage generatedin the resistance 119877119892 (Nyquist equation) 1198751 119899 is the noise power from 119877119892 incoming into the two-port

many NMR laboratories The technique has been tested at thehardware for measurements at lowmagnetic field but there isno reason why it could not be used also for higher spectrafrequencies

2 Theory and Methods

A simplified circuit diagram of the receiver is presented inFigure 1(a) All impedances and voltages indicated are realvalues (not complex) In practice these values are rarelyaccurate but considerable research is dedicated to achievingaccuracy including the application of transmission linesmatching

The power gain of the receiver including the preamplifiercan be calculated using (1)

119866 = 1198812119897 11990411988121 119904 ∙119877119894119899119877119897 (1)

Consider the spectra frequencies and impedance tempera-tures where ℎ119891119896119879 ≪ 1 (ℎ = 662 times 10minus34 Js is the Planckconstant 119896 = 138 times 10minus23 JK is the Boltzmann constant) Thecondition limits the region where the noise is approximatedby the Nyquist equation sufficiently accuratelyThe signal-to-noise ratio (SNR) of the signal incoming into the input of thetwo-port is given by (2)

1198781198731198771 = 1198751 1199041198751 119899 = 11988121 119904 119903119898119904119877119894119899119896119879119891 (2)

The calculation behind the relationship 1198751 119899 = 119896119879119891 [2ndash4] can be explained using Figure 1(b) The power of thethermal noise 1198751 119899 coming into the receiver from the sourceimpedance 119877119892 is split into two equal resistances 119877g and 119877inand is defined by (3)

1198751 119899 = 119877119894119899119877119892 + 119877119894119899 ∙(radic4119896119879119877119892119891)2119877119892 + 119877119894119899

= 12 ∙ (radic4119896119879119877119892119891)22119877119892 = 119896119879119891(3)

119879 is the absolute temperature of both 119877in and the matchedsource resistance 119877g and 119891 is the noise bandwidth of thecircuit For impedance matching 119877119892 = 119877119894119899 the value of 1198751 119899from the noise source (119877g) into the input resistance (119877in) ismaximal and equal to the value of 119896119879119891 The SNR of theoutgoing signal is given by (4)

119878119873119877119897 = 119875119897 119904119875119897 119899 = 1198812119897 119904 1199031198981199041198771198971198812119897 119899 119903119898119904

119877119897 = 1198812119897 119904 1199031198981199041198812119897 119899 119903119898119904

(4)

Calculation of the noise factor using the incoming andoutgoing signal SNRs can be achieved using (5)

119865 = 1198781198731198771119878119873119877119897 = 11988121 119904 119903119898119904 ∙ 1198812119897 119899 1199031198981199041198812119897 119904 119903119898119904

∙ 119877119894119899119896119879119891 = 119875119897 119899119866 ∙ 119896119879119891= 1198702 ∙ 1198812119897 119899 119903119898119904119877119894119899119896119879119891

(5)

Here always 119865 gt 1 The noise figure in decibels is given by(6)

119873119865 = 10 ∙ log119865 (6)

In practice the noise factor and the noise figure should bemeasured at the room temperature 119879 = 1198790 = 290119870

Calibration of the receiver is carried out using thecoefficient 119870 (7)

119870 = 1198811 119904119881119897 119904 (7)

The coefficient can be measured and calculated using aharmonic signal voltage (RMS mean or amplitude) froman external signal generator or using the signal from thetransceiver of the console provided that signal measurementis accurate If 119870 is in voltsADC units the receiver iscalibrated and 119865 is dimensionless as it should be The sizeof 1198811 119904 must not exceed the limit for the 119881119897 119904 truncationThe noise bandwidth of the receiver is not equivalent to thefrequency bandwidth of the receiver The spectrum related tothe noise bandwidth should be constant and can be calculated

Mathematical Problems in Engineering 3

d [samples]

Vl [ADC units]

800 000

600 000

400 000

200 000

100000 200000 300 000 400 000 500 0000(a)

d [samples]

Vl [ADC units]

800 000

600 000

400 000

200 000

50 000 100000 150 000 200 000 250 000 300 0000(b)

Figure 2Thenoise voltage from thematched receiver (a) All measuredmodules of frequency samples (b) cropped frequency samples seriesThe new series can be used for the noise bandwidth calculation as the spectra are approximately constant

from the frequency bandwidth (equivalent noise bandwidth)provided that the parameters of the receiver filter are known(gain versus frequency)

Given the shape of the noise spectrum we presenta simplified approach to the calculation The measurednoise voltage output of the matched receiver is shown inFigure 2(a) By cropping both margins of the spectra wegenerated modified spectra with a rectangular-shaped noisebandwidth indicating a more constant value of the spectra(Figure 2(b)) This processing should be acceptable for mostmeasurements Equation (5) requires the route mean square(RMS) value of the output voltage In order to calculatethe RMS voltage frequency samples must be calculated Bymeasuring the noise of the matched receiver we obtaineda series of 119873 time domain samples To convert these intofrequency spectra a discrete Fourier transform (FT) wascarried out The FT can be performed directly using theconsole or later using computer processing of the exporteddata The RMS voltage can then be calculated from thefrequency samples (119881119897 119899 119894) using (8) Spectra in Figure 2consist of absolute values of such frequency samples

119881l n 119903119898119904 = radic 11198732119873sum119894=1

1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162 = radic 119873sum119894=1

10038161003816100381610038161003816100381610038161003816119881119897 119899 119894119873100381610038161003816100381610038161003816100381610038162

(8)

Equation (8) was derived from definition of RMS for timesamples using the information in [17] (the Parseval theorem)119881119897 119899 119894 is i-th term of the outgoing noise frequency samplesseries

A computer simulation was used for the verification Atime function with known RMS value was selected and givenby (9)

119881119894 = 1198601 ∙ cos 21205871198911119905 (119894 minus 1) + 1198602 ∙ cos 21205871198912119905 (119894 minus 1)+ 1198603 ∙ cos 21205871198913119905 (119894 minus 1) (9)

For 119894 = 1 divide 119873 a vector of time samples was calculated underthe following conditions

number of samples 119873=10000

12

10

08

06

04

02

0 2000 4000

d [samples]

6000 8000 10000

Vi N [V]

Figure 3 RMS voltages of the spectral components in the verifica-tion computer simulation

frequency 1198911 = 100 Hzfrequency 1198912 = 250 Hzfrequency 1198913 = 500 Hztime interval 119905 = 10minus4 samplitude 1198601 = 05 Vamplitude 1198602 = 10 Vamplitude 1198603 = 20 V

The vector was processed using the discrete Fourier trans-form resulting in119873 frequency samples (10)

119881 = 1198811 divide 119881119873 (10)

At the first sight (8) is similar to an average but Figure 3depicts absolute value of spectra (10) divided by 119873 Asassumed all terms except those corresponding to the har-monic voltages are of zero values The RMS voltage accordingto (8) is given by

119881119903119898119904 = 162019 V (11)

4 Mathematical Problems in Engineering

Calculate the 119881119903119898119904 only for nonzero terms according toFigure 3 It is a known fact that a resulting RMS voltage mustbe calculated using the quadrates sum of the participatingRMS components

119881119903119898119904 = radic102 + 052 + 0252 + 0252 + 052 + 102= 162019 V

(12)

A transform to single sided spectrum yields

119881119903119898119904 = radic0522 + 1022 + 2022 = 162019 V (13)

It indicates the following

1198811 119903119898119904 = 1198601radic21198812 119903119898119904 = 1198602radic21198813 119903119898119904 = 1198603radic2

(14)

This are generally known RMS values of a real (not complex)harmonic signal

It is evident that the RMS voltage of the i-th frequencysample is given by (15)

119881119894 119903119898119904 = 10038161003816100381610038161198811198941003816100381610038161003816119873 (15)

The RMS voltage of all 119873 samples is given by (8)For cropping of noise bandwidth (8) can be modified to

119881l n 119903119898119904 = radic( 1119873)2 ∙ 1198992sum119894=1198991

1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162 (16)

where 1 le 1198991 lt 1198992 le 119873Ultimately the first noise factor is calculated using

1198651 = 1198702 (11198732)sum119889119894=1 1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162119896119879((119889 minus 1) 119873119879119904) 119877119894119899 (17)

where 119889 is the number of samples after cropping and 119879119904 is thesampling time interval of the samples

For verification we compared results using our simplifiedapproach with those obtained using the conventional calcu-lation (equivalent noise bandwidth) with (16)-(17) describedpreviouslyThenoise spectra in Figure 2(a) are thewhite noisecreated by resistance matching of the receiver input includinga preamplifier amplified by the receiver Thus the spectralenvelope is modulated by the gain of the receiver and thereceiver filter The equivalent number of samples is given by(18)

119889119889 = sum119873119894=1 1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162100381610038161003816100381610038161198812119897 119899 11989410038161003816100381610038161003816 (1198910) (18)

RF Generator

Attenuator

Preamplifier

NMR Console

PC

Filter

Rg

Figure 4 Block diagram of the verification experiment

The second powers of all sample voltages are summed anddivided by the mean value of |119881119897 119899 119894|2 round the operationfrequency1198910The second noise factor for the entire frequencybandwidth of the receiver is given by (19)

1198652 = 1198702 (11198732)sum119873119894=1 1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162119896119879((119889119889 minus 1) 119873119879119904) 119877119894119899 (19)

Sample noise can be eliminated in two ways by filtering themeasured voltages of samples using a software low-pass filteror by averaging the noisy samples Following the latter optionwe calculated |1198812

119897 119899 119894|(1198910) of 10000 frequency samples and

compared the results of both approaches (17) and (19) fornoise factors calculation The values of 1198651 and 1198652 (1198731198651 and1198731198652) should be similar but do not have to be equal

The measured noise signal was compared to the theo-retical values of the noise power which ensures excellentcomparability with other systems

The processed data have beenmeasured under the follow-ing conditions

operation frequency of the scanner 1198910=445 MHzsampling interval 119879119904 = 100 120583sfrequency bandwidth of the receiver 119861 = 10 kHztemperature of 119877119892 119879=290 Koriginal number of samples 119873=500000number of samples after cropping d=300000

Verification experiments were performed in Bratislava andHouston on similar but not the same instrumentation There-fore only the results measured in Bratislava are publishedin this article Nevertheless the results from Houston wereimportant for verification of the experimental results carriedout in Bratislava mainly in the beginning of the work

Block diagram of the verification experiment is depictedin Figure 4 An external RF generator is assumed for thereceiver calibration

3 Results

Verification experiments were performed on an experimentalNMR scanner equipped with a home-made resistive mag-net of 01 T and the Apollo spectroscopic NMR console

Mathematical Problems in Engineering 5

Table 1 Results of verification

No 1198910[MHz] Preamplifier119873119865 1198731198651[dB] 1198731198652[dB] Filter 119889[samples] 119873[samples]1 445 broadband-1 1 dB 44 442 No 300000 5000002 445 broadband-2 1 dB 445 442 No 300000 5000003 445 broadband-1 1 dB 145 137 L-P 300000 5000004 445 broadband-2 1 dB 148 151 L-P 300000 5000005 445 tuned-1 1 dB 134 127 No 300000 5000006 445 tuned-2 1 dB 153 166 No 300000 5000007 445 tuned-1 1 dB 136 129 L-P 300000 5000008 445 tuned-2 1 dB 144 148 L-P 300000 500000

(Tecmag Inc Houston TX) The console was utilised forimaging operated at a frequency of 445 MHz Consolecalibration was performed using the GFG-3015 functiongenerator (Good Will Instrument Co Ltd New Taipei CityTaiwan) Calibration using the console transceiver was alsotested in Houston An external attenuator was necessaryin both cases The broadband preamplifier pair of AU-1579 (MITEQ Hauppauge NY) with 50-Ohm input andoutput impedances and 119873119865 = 1 dB (a catalogue datum) wasused (selected randomly from three pieces) Another tunedpreamplifier pair P445VD (Advanced Receiver ResearchBurlington CT) with 50-Ohm input and output impedancesand 119873119865 = 1 dB (a catalogue datum) was also used selectedrandomly from three pieces We verified our theory throughseveral experiments the results of which are presented inTable 1 The receiver with MITEQ broadband preamplifiersexhibited significant noise which was improved by insertinga low-pass filter between the preamplifier and the consoleThe low-pass filter applied had a cut-off frequency of 107MHz SLP-107+ (Mini-Circuits Brooklyn NY)Using a filterwith a cut-off frequency closer to the operation frequencyof the spectrometer may further improve the noise figurevalue however the noise figure obtained with broadbandpreamplifiers and filters is similar to that observed usingtuned preamplifiers The data obtained using tuned ARRpreamplifiers did not require a filter as the measured noisefigure values were in good agreement with the catalogue dataOnly small differences were seen between the results of 119889(cropped) samples compared with those from 119873 samplesThe difference did not exceed 013 dB (experiment no 6tuned-2 without a filter) All measurements were performedin the shielding cage of the scanner and all connectionsused shielded cables Measurements without a shieldingcage resulted in much higher noise which varied betweenmeasurements depending on the actual interference in themeasurement space Shielded cables alone are therefore notsufficient for an accurate measurement and all instrumentsshould be warmed up for some time before measurementsare carried out Also changes in the inputs and outputs ofthe preamplifiers should be performed some time before themeasurement

The noise figures of the receiver with each of the fourpreamplifiers depending on the noise bandwidth are shownin Figure 5 It can be seen that for frequencies close to 1198910 themeasured noise figure is inaccurate possibly due to the low

number of frequency samples Between 50000 and 300000samples the value is approximately constant falling rapidlybeyond 300000 samples

The frequency or noise bandwidth in Hz can be esti-mated using (20) taken from the graph of Figure 6

119861 = 119889119873119879119904 (20)

The results of the noise figure measurements reflect ourexpectations and it can therefore be concluded that the ver-ification was successful Furthermore equations and derivedformulae provided acceptable results

4 Discussion and Conclusions

The purpose of this study was to develop and to test atechnique for the determination of the NMR spectrometerreceiver noise figure We endeavoured to use only widelyavailable or cheap instruments The noise figure of thereceiver obtained using a broadband preamplifier dependson the filter between the preamplifier and the console Thefilter type and characteristic frequencies are the subject offurther studies It is possible that a suitable selection offrequencies could improve the noise figure of the entirereceiver which cannot be achieved by replacing the hardwarelow-pass filter with a suitable software low-pass filter Thissuggests that the point of the receiver chain at which thefilter is connected is an important consideration Measureddata was processed using calculations from the cited sourcesbut also from own derivations The final results have proventhat all the equations and derived formulae are correct Inthe present study methods were tested using a spectroscopicNMR console However the console must be able to calibratethe voltage in order to carry out the calculations and so thismay not be possible on all consoles

The primary application of the technique is to inspectthe receiver of the spectrometer but in some cases it can beapplied for analysis of the noise figure of a tested preamplifiermeasurement The method can be also applied to a magneticresonance imaging (MRI) scanner provided that the consoleis suitable If the signal for the console is amplified by abroadband preamplifier it is recommended that a suitablefilter is connected between the preamplifier and the consolefor significant improvements in the noise figure

6 Mathematical Problems in Engineering

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(a)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(b)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(c)

18

16

14

12

1

08

NF1 [dB]

0 100000 200000 300000 400000 500000

d [samples]

(d)

Figure 5 Influence of the noise bandwidth on themeasured receiver noise figure using (a) the broadband-1 preamplifier (b) the broadband-2preamplifier (c) the tuned-1 preamplifier and (d) the tuned-2 preamplifier The noise bandwidth d is symmetrical to the operation frequency1198910

10000B [Hz]

8000

6000

4000

2000

0100000 200000 300000 400000 500000

d [samples]

Figure 6 Calculation of the noise or frequency bandwidth (B) fromfrequency samples

Data Availability

The measured digital data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Grant AgencyVEGA 2000117 and within the project of the Research andDevelopment Agency No APVV-15-0029

References

[1] P Andris V Jacko T Dermek and I Frollo ldquoNoise measure-ment of a preamplifier with high input impedance using anNMR consolerdquoMeasurement vol 55 pp 408ndash412 2014

[2] V Zalud and V N Kulesov Semiconductor Circuits with LowNoise SNTL Prague Czechia 1980 (in Czech)

[3] H Bittel and L Storm Noises Springer-Verlag Berlin Ger-many 1971 (in German)

[4] B Schiek and H J Siweris Noises in RF Circuits HuthigHeidelberg Germany 1990 (in German)

[5] D Hoult and R Richards ldquoThe signal-to-noise ratio of thenuclear magnetic resonance experimentrdquo Journal of MagneticResonance vol 24 no 1 pp 71ndash85 1976

[6] D Hoult and P C Lauterbur ldquoThe sensitivity of the zeug-matographic experiment involving human samplesrdquo Journal ofMagnetic Resonance vol 34 no 2 pp 425ndash433 1979

[7] A Raad and L Darrasse ldquoOptimization of NMR receiver band-width by inductive couplingrdquoMagnetic Resonance Imaging vol10 no 1 pp 55ndash65 1992

[8] MDecorps P BlondetH Reutenauer J Albrand andC RemyldquoAn inductively coupled series-tuned NMR proberdquo Journal ofMagnetic Resonance vol 65 no 1 pp 100ndash109 1985

[9] J Weis A Ericsson and A Hemmingsson ldquoChemical shiftartifact-free microscopy spectroscopic microimaging of thehuman skinrdquoMagnetic Resonance in Medicine vol 41 no 5 pp904ndash908 1999

[10] P Marcon K Bartusek Z Dokoupil and E GescheidtovaldquoDiffusionMRI mitigation of magnetic field inhomogeneitiesrdquoMeasurement Science Review vol 12 no 5 pp 205ndash212 2012

[11] K Bartusek Z Dokoupil and E Gescheidtova ldquoMapping ofmagnetic field around small coils using the magnetic resonancemethodrdquoMeasurement Science and Technology vol 18 no 7 pp2223ndash2230 2007

Mathematical Problems in Engineering 7

[12] D Nespor K Bartusek and Z Dokoupil ldquoComparing SaddleSlotted-tube and Parallel-plate Coils for Magnetic ResonanceImagingrdquo Measurement Science Review vol 14 no 3 pp 171ndash176 2014

[13] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSimple phase method for measurement of magneticfield gradient waveformsrdquoMagnetic Resonance Imaging vol 25no 9 pp 1272ndash1276 2007

[14] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSingle-point imagingwith a variable phase encodingintervalrdquoMagnetic Resonance Imaging vol 26 no 1 pp 109ndash1162008

[15] D Gogola P Szomolanyi M Skratek and I Frollo ldquoDesignand construction of novel instrumentation for low-field MRtomographyrdquo Measurement Science Review vol 18 no 3 pp107ndash112 2018

[16] G Wimmer V Witkovsky and T Duby ldquoProper roundingof the measurement results under normality assumptionsrdquoMeasurement Science and Technology vol 11 no 12 pp 1659ndash1665 2000

[17] D W Kammler A First Course in Fourier Analysis CambridgeUniversity Press New York NY USA 2007

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Mathematical Problems in Engineering 3

d [samples]

Vl [ADC units]

800 000

600 000

400 000

200 000

100000 200000 300 000 400 000 500 0000(a)

d [samples]

Vl [ADC units]

800 000

600 000

400 000

200 000

50 000 100000 150 000 200 000 250 000 300 0000(b)

Figure 2Thenoise voltage from thematched receiver (a) All measuredmodules of frequency samples (b) cropped frequency samples seriesThe new series can be used for the noise bandwidth calculation as the spectra are approximately constant

from the frequency bandwidth (equivalent noise bandwidth)provided that the parameters of the receiver filter are known(gain versus frequency)

Given the shape of the noise spectrum we presenta simplified approach to the calculation The measurednoise voltage output of the matched receiver is shown inFigure 2(a) By cropping both margins of the spectra wegenerated modified spectra with a rectangular-shaped noisebandwidth indicating a more constant value of the spectra(Figure 2(b)) This processing should be acceptable for mostmeasurements Equation (5) requires the route mean square(RMS) value of the output voltage In order to calculatethe RMS voltage frequency samples must be calculated Bymeasuring the noise of the matched receiver we obtaineda series of 119873 time domain samples To convert these intofrequency spectra a discrete Fourier transform (FT) wascarried out The FT can be performed directly using theconsole or later using computer processing of the exporteddata The RMS voltage can then be calculated from thefrequency samples (119881119897 119899 119894) using (8) Spectra in Figure 2consist of absolute values of such frequency samples

119881l n 119903119898119904 = radic 11198732119873sum119894=1

1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162 = radic 119873sum119894=1

10038161003816100381610038161003816100381610038161003816119881119897 119899 119894119873100381610038161003816100381610038161003816100381610038162

(8)

Equation (8) was derived from definition of RMS for timesamples using the information in [17] (the Parseval theorem)119881119897 119899 119894 is i-th term of the outgoing noise frequency samplesseries

A computer simulation was used for the verification Atime function with known RMS value was selected and givenby (9)

119881119894 = 1198601 ∙ cos 21205871198911119905 (119894 minus 1) + 1198602 ∙ cos 21205871198912119905 (119894 minus 1)+ 1198603 ∙ cos 21205871198913119905 (119894 minus 1) (9)

For 119894 = 1 divide 119873 a vector of time samples was calculated underthe following conditions

number of samples 119873=10000

12

10

08

06

04

02

0 2000 4000

d [samples]

6000 8000 10000

Vi N [V]

Figure 3 RMS voltages of the spectral components in the verifica-tion computer simulation

frequency 1198911 = 100 Hzfrequency 1198912 = 250 Hzfrequency 1198913 = 500 Hztime interval 119905 = 10minus4 samplitude 1198601 = 05 Vamplitude 1198602 = 10 Vamplitude 1198603 = 20 V

The vector was processed using the discrete Fourier trans-form resulting in119873 frequency samples (10)

119881 = 1198811 divide 119881119873 (10)

At the first sight (8) is similar to an average but Figure 3depicts absolute value of spectra (10) divided by 119873 Asassumed all terms except those corresponding to the har-monic voltages are of zero values The RMS voltage accordingto (8) is given by

119881119903119898119904 = 162019 V (11)

4 Mathematical Problems in Engineering

Calculate the 119881119903119898119904 only for nonzero terms according toFigure 3 It is a known fact that a resulting RMS voltage mustbe calculated using the quadrates sum of the participatingRMS components

119881119903119898119904 = radic102 + 052 + 0252 + 0252 + 052 + 102= 162019 V

(12)

A transform to single sided spectrum yields

119881119903119898119904 = radic0522 + 1022 + 2022 = 162019 V (13)

It indicates the following

1198811 119903119898119904 = 1198601radic21198812 119903119898119904 = 1198602radic21198813 119903119898119904 = 1198603radic2

(14)

This are generally known RMS values of a real (not complex)harmonic signal

It is evident that the RMS voltage of the i-th frequencysample is given by (15)

119881119894 119903119898119904 = 10038161003816100381610038161198811198941003816100381610038161003816119873 (15)

The RMS voltage of all 119873 samples is given by (8)For cropping of noise bandwidth (8) can be modified to

119881l n 119903119898119904 = radic( 1119873)2 ∙ 1198992sum119894=1198991

1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162 (16)

where 1 le 1198991 lt 1198992 le 119873Ultimately the first noise factor is calculated using

1198651 = 1198702 (11198732)sum119889119894=1 1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162119896119879((119889 minus 1) 119873119879119904) 119877119894119899 (17)

where 119889 is the number of samples after cropping and 119879119904 is thesampling time interval of the samples

For verification we compared results using our simplifiedapproach with those obtained using the conventional calcu-lation (equivalent noise bandwidth) with (16)-(17) describedpreviouslyThenoise spectra in Figure 2(a) are thewhite noisecreated by resistance matching of the receiver input includinga preamplifier amplified by the receiver Thus the spectralenvelope is modulated by the gain of the receiver and thereceiver filter The equivalent number of samples is given by(18)

119889119889 = sum119873119894=1 1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162100381610038161003816100381610038161198812119897 119899 11989410038161003816100381610038161003816 (1198910) (18)

RF Generator

Attenuator

Preamplifier

NMR Console

PC

Filter

Rg

Figure 4 Block diagram of the verification experiment

The second powers of all sample voltages are summed anddivided by the mean value of |119881119897 119899 119894|2 round the operationfrequency1198910The second noise factor for the entire frequencybandwidth of the receiver is given by (19)

1198652 = 1198702 (11198732)sum119873119894=1 1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162119896119879((119889119889 minus 1) 119873119879119904) 119877119894119899 (19)

Sample noise can be eliminated in two ways by filtering themeasured voltages of samples using a software low-pass filteror by averaging the noisy samples Following the latter optionwe calculated |1198812

119897 119899 119894|(1198910) of 10000 frequency samples and

compared the results of both approaches (17) and (19) fornoise factors calculation The values of 1198651 and 1198652 (1198731198651 and1198731198652) should be similar but do not have to be equal

The measured noise signal was compared to the theo-retical values of the noise power which ensures excellentcomparability with other systems

The processed data have beenmeasured under the follow-ing conditions

operation frequency of the scanner 1198910=445 MHzsampling interval 119879119904 = 100 120583sfrequency bandwidth of the receiver 119861 = 10 kHztemperature of 119877119892 119879=290 Koriginal number of samples 119873=500000number of samples after cropping d=300000

Verification experiments were performed in Bratislava andHouston on similar but not the same instrumentation There-fore only the results measured in Bratislava are publishedin this article Nevertheless the results from Houston wereimportant for verification of the experimental results carriedout in Bratislava mainly in the beginning of the work

Block diagram of the verification experiment is depictedin Figure 4 An external RF generator is assumed for thereceiver calibration

3 Results

Verification experiments were performed on an experimentalNMR scanner equipped with a home-made resistive mag-net of 01 T and the Apollo spectroscopic NMR console

Mathematical Problems in Engineering 5

Table 1 Results of verification

No 1198910[MHz] Preamplifier119873119865 1198731198651[dB] 1198731198652[dB] Filter 119889[samples] 119873[samples]1 445 broadband-1 1 dB 44 442 No 300000 5000002 445 broadband-2 1 dB 445 442 No 300000 5000003 445 broadband-1 1 dB 145 137 L-P 300000 5000004 445 broadband-2 1 dB 148 151 L-P 300000 5000005 445 tuned-1 1 dB 134 127 No 300000 5000006 445 tuned-2 1 dB 153 166 No 300000 5000007 445 tuned-1 1 dB 136 129 L-P 300000 5000008 445 tuned-2 1 dB 144 148 L-P 300000 500000

(Tecmag Inc Houston TX) The console was utilised forimaging operated at a frequency of 445 MHz Consolecalibration was performed using the GFG-3015 functiongenerator (Good Will Instrument Co Ltd New Taipei CityTaiwan) Calibration using the console transceiver was alsotested in Houston An external attenuator was necessaryin both cases The broadband preamplifier pair of AU-1579 (MITEQ Hauppauge NY) with 50-Ohm input andoutput impedances and 119873119865 = 1 dB (a catalogue datum) wasused (selected randomly from three pieces) Another tunedpreamplifier pair P445VD (Advanced Receiver ResearchBurlington CT) with 50-Ohm input and output impedancesand 119873119865 = 1 dB (a catalogue datum) was also used selectedrandomly from three pieces We verified our theory throughseveral experiments the results of which are presented inTable 1 The receiver with MITEQ broadband preamplifiersexhibited significant noise which was improved by insertinga low-pass filter between the preamplifier and the consoleThe low-pass filter applied had a cut-off frequency of 107MHz SLP-107+ (Mini-Circuits Brooklyn NY)Using a filterwith a cut-off frequency closer to the operation frequencyof the spectrometer may further improve the noise figurevalue however the noise figure obtained with broadbandpreamplifiers and filters is similar to that observed usingtuned preamplifiers The data obtained using tuned ARRpreamplifiers did not require a filter as the measured noisefigure values were in good agreement with the catalogue dataOnly small differences were seen between the results of 119889(cropped) samples compared with those from 119873 samplesThe difference did not exceed 013 dB (experiment no 6tuned-2 without a filter) All measurements were performedin the shielding cage of the scanner and all connectionsused shielded cables Measurements without a shieldingcage resulted in much higher noise which varied betweenmeasurements depending on the actual interference in themeasurement space Shielded cables alone are therefore notsufficient for an accurate measurement and all instrumentsshould be warmed up for some time before measurementsare carried out Also changes in the inputs and outputs ofthe preamplifiers should be performed some time before themeasurement

The noise figures of the receiver with each of the fourpreamplifiers depending on the noise bandwidth are shownin Figure 5 It can be seen that for frequencies close to 1198910 themeasured noise figure is inaccurate possibly due to the low

number of frequency samples Between 50000 and 300000samples the value is approximately constant falling rapidlybeyond 300000 samples

The frequency or noise bandwidth in Hz can be esti-mated using (20) taken from the graph of Figure 6

119861 = 119889119873119879119904 (20)

The results of the noise figure measurements reflect ourexpectations and it can therefore be concluded that the ver-ification was successful Furthermore equations and derivedformulae provided acceptable results

4 Discussion and Conclusions

The purpose of this study was to develop and to test atechnique for the determination of the NMR spectrometerreceiver noise figure We endeavoured to use only widelyavailable or cheap instruments The noise figure of thereceiver obtained using a broadband preamplifier dependson the filter between the preamplifier and the console Thefilter type and characteristic frequencies are the subject offurther studies It is possible that a suitable selection offrequencies could improve the noise figure of the entirereceiver which cannot be achieved by replacing the hardwarelow-pass filter with a suitable software low-pass filter Thissuggests that the point of the receiver chain at which thefilter is connected is an important consideration Measureddata was processed using calculations from the cited sourcesbut also from own derivations The final results have proventhat all the equations and derived formulae are correct Inthe present study methods were tested using a spectroscopicNMR console However the console must be able to calibratethe voltage in order to carry out the calculations and so thismay not be possible on all consoles

The primary application of the technique is to inspectthe receiver of the spectrometer but in some cases it can beapplied for analysis of the noise figure of a tested preamplifiermeasurement The method can be also applied to a magneticresonance imaging (MRI) scanner provided that the consoleis suitable If the signal for the console is amplified by abroadband preamplifier it is recommended that a suitablefilter is connected between the preamplifier and the consolefor significant improvements in the noise figure

6 Mathematical Problems in Engineering

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(a)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(b)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(c)

18

16

14

12

1

08

NF1 [dB]

0 100000 200000 300000 400000 500000

d [samples]

(d)

Figure 5 Influence of the noise bandwidth on themeasured receiver noise figure using (a) the broadband-1 preamplifier (b) the broadband-2preamplifier (c) the tuned-1 preamplifier and (d) the tuned-2 preamplifier The noise bandwidth d is symmetrical to the operation frequency1198910

10000B [Hz]

8000

6000

4000

2000

0100000 200000 300000 400000 500000

d [samples]

Figure 6 Calculation of the noise or frequency bandwidth (B) fromfrequency samples

Data Availability

The measured digital data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Grant AgencyVEGA 2000117 and within the project of the Research andDevelopment Agency No APVV-15-0029

References

[1] P Andris V Jacko T Dermek and I Frollo ldquoNoise measure-ment of a preamplifier with high input impedance using anNMR consolerdquoMeasurement vol 55 pp 408ndash412 2014

[2] V Zalud and V N Kulesov Semiconductor Circuits with LowNoise SNTL Prague Czechia 1980 (in Czech)

[3] H Bittel and L Storm Noises Springer-Verlag Berlin Ger-many 1971 (in German)

[4] B Schiek and H J Siweris Noises in RF Circuits HuthigHeidelberg Germany 1990 (in German)

[5] D Hoult and R Richards ldquoThe signal-to-noise ratio of thenuclear magnetic resonance experimentrdquo Journal of MagneticResonance vol 24 no 1 pp 71ndash85 1976

[6] D Hoult and P C Lauterbur ldquoThe sensitivity of the zeug-matographic experiment involving human samplesrdquo Journal ofMagnetic Resonance vol 34 no 2 pp 425ndash433 1979

[7] A Raad and L Darrasse ldquoOptimization of NMR receiver band-width by inductive couplingrdquoMagnetic Resonance Imaging vol10 no 1 pp 55ndash65 1992

[8] MDecorps P BlondetH Reutenauer J Albrand andC RemyldquoAn inductively coupled series-tuned NMR proberdquo Journal ofMagnetic Resonance vol 65 no 1 pp 100ndash109 1985

[9] J Weis A Ericsson and A Hemmingsson ldquoChemical shiftartifact-free microscopy spectroscopic microimaging of thehuman skinrdquoMagnetic Resonance in Medicine vol 41 no 5 pp904ndash908 1999

[10] P Marcon K Bartusek Z Dokoupil and E GescheidtovaldquoDiffusionMRI mitigation of magnetic field inhomogeneitiesrdquoMeasurement Science Review vol 12 no 5 pp 205ndash212 2012

[11] K Bartusek Z Dokoupil and E Gescheidtova ldquoMapping ofmagnetic field around small coils using the magnetic resonancemethodrdquoMeasurement Science and Technology vol 18 no 7 pp2223ndash2230 2007

Mathematical Problems in Engineering 7

[12] D Nespor K Bartusek and Z Dokoupil ldquoComparing SaddleSlotted-tube and Parallel-plate Coils for Magnetic ResonanceImagingrdquo Measurement Science Review vol 14 no 3 pp 171ndash176 2014

[13] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSimple phase method for measurement of magneticfield gradient waveformsrdquoMagnetic Resonance Imaging vol 25no 9 pp 1272ndash1276 2007

[14] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSingle-point imagingwith a variable phase encodingintervalrdquoMagnetic Resonance Imaging vol 26 no 1 pp 109ndash1162008

[15] D Gogola P Szomolanyi M Skratek and I Frollo ldquoDesignand construction of novel instrumentation for low-field MRtomographyrdquo Measurement Science Review vol 18 no 3 pp107ndash112 2018

[16] G Wimmer V Witkovsky and T Duby ldquoProper roundingof the measurement results under normality assumptionsrdquoMeasurement Science and Technology vol 11 no 12 pp 1659ndash1665 2000

[17] D W Kammler A First Course in Fourier Analysis CambridgeUniversity Press New York NY USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

Calculate the 119881119903119898119904 only for nonzero terms according toFigure 3 It is a known fact that a resulting RMS voltage mustbe calculated using the quadrates sum of the participatingRMS components

119881119903119898119904 = radic102 + 052 + 0252 + 0252 + 052 + 102= 162019 V

(12)

A transform to single sided spectrum yields

119881119903119898119904 = radic0522 + 1022 + 2022 = 162019 V (13)

It indicates the following

1198811 119903119898119904 = 1198601radic21198812 119903119898119904 = 1198602radic21198813 119903119898119904 = 1198603radic2

(14)

This are generally known RMS values of a real (not complex)harmonic signal

It is evident that the RMS voltage of the i-th frequencysample is given by (15)

119881119894 119903119898119904 = 10038161003816100381610038161198811198941003816100381610038161003816119873 (15)

The RMS voltage of all 119873 samples is given by (8)For cropping of noise bandwidth (8) can be modified to

119881l n 119903119898119904 = radic( 1119873)2 ∙ 1198992sum119894=1198991

1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162 (16)

where 1 le 1198991 lt 1198992 le 119873Ultimately the first noise factor is calculated using

1198651 = 1198702 (11198732)sum119889119894=1 1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162119896119879((119889 minus 1) 119873119879119904) 119877119894119899 (17)

where 119889 is the number of samples after cropping and 119879119904 is thesampling time interval of the samples

For verification we compared results using our simplifiedapproach with those obtained using the conventional calcu-lation (equivalent noise bandwidth) with (16)-(17) describedpreviouslyThenoise spectra in Figure 2(a) are thewhite noisecreated by resistance matching of the receiver input includinga preamplifier amplified by the receiver Thus the spectralenvelope is modulated by the gain of the receiver and thereceiver filter The equivalent number of samples is given by(18)

119889119889 = sum119873119894=1 1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162100381610038161003816100381610038161198812119897 119899 11989410038161003816100381610038161003816 (1198910) (18)

RF Generator

Attenuator

Preamplifier

NMR Console

PC

Filter

Rg

Figure 4 Block diagram of the verification experiment

The second powers of all sample voltages are summed anddivided by the mean value of |119881119897 119899 119894|2 round the operationfrequency1198910The second noise factor for the entire frequencybandwidth of the receiver is given by (19)

1198652 = 1198702 (11198732)sum119873119894=1 1003816100381610038161003816119881119897 119899 11989410038161003816100381610038162119896119879((119889119889 minus 1) 119873119879119904) 119877119894119899 (19)

Sample noise can be eliminated in two ways by filtering themeasured voltages of samples using a software low-pass filteror by averaging the noisy samples Following the latter optionwe calculated |1198812

119897 119899 119894|(1198910) of 10000 frequency samples and

compared the results of both approaches (17) and (19) fornoise factors calculation The values of 1198651 and 1198652 (1198731198651 and1198731198652) should be similar but do not have to be equal

The measured noise signal was compared to the theo-retical values of the noise power which ensures excellentcomparability with other systems

The processed data have beenmeasured under the follow-ing conditions

operation frequency of the scanner 1198910=445 MHzsampling interval 119879119904 = 100 120583sfrequency bandwidth of the receiver 119861 = 10 kHztemperature of 119877119892 119879=290 Koriginal number of samples 119873=500000number of samples after cropping d=300000

Verification experiments were performed in Bratislava andHouston on similar but not the same instrumentation There-fore only the results measured in Bratislava are publishedin this article Nevertheless the results from Houston wereimportant for verification of the experimental results carriedout in Bratislava mainly in the beginning of the work

Block diagram of the verification experiment is depictedin Figure 4 An external RF generator is assumed for thereceiver calibration

3 Results

Verification experiments were performed on an experimentalNMR scanner equipped with a home-made resistive mag-net of 01 T and the Apollo spectroscopic NMR console

Mathematical Problems in Engineering 5

Table 1 Results of verification

No 1198910[MHz] Preamplifier119873119865 1198731198651[dB] 1198731198652[dB] Filter 119889[samples] 119873[samples]1 445 broadband-1 1 dB 44 442 No 300000 5000002 445 broadband-2 1 dB 445 442 No 300000 5000003 445 broadband-1 1 dB 145 137 L-P 300000 5000004 445 broadband-2 1 dB 148 151 L-P 300000 5000005 445 tuned-1 1 dB 134 127 No 300000 5000006 445 tuned-2 1 dB 153 166 No 300000 5000007 445 tuned-1 1 dB 136 129 L-P 300000 5000008 445 tuned-2 1 dB 144 148 L-P 300000 500000

(Tecmag Inc Houston TX) The console was utilised forimaging operated at a frequency of 445 MHz Consolecalibration was performed using the GFG-3015 functiongenerator (Good Will Instrument Co Ltd New Taipei CityTaiwan) Calibration using the console transceiver was alsotested in Houston An external attenuator was necessaryin both cases The broadband preamplifier pair of AU-1579 (MITEQ Hauppauge NY) with 50-Ohm input andoutput impedances and 119873119865 = 1 dB (a catalogue datum) wasused (selected randomly from three pieces) Another tunedpreamplifier pair P445VD (Advanced Receiver ResearchBurlington CT) with 50-Ohm input and output impedancesand 119873119865 = 1 dB (a catalogue datum) was also used selectedrandomly from three pieces We verified our theory throughseveral experiments the results of which are presented inTable 1 The receiver with MITEQ broadband preamplifiersexhibited significant noise which was improved by insertinga low-pass filter between the preamplifier and the consoleThe low-pass filter applied had a cut-off frequency of 107MHz SLP-107+ (Mini-Circuits Brooklyn NY)Using a filterwith a cut-off frequency closer to the operation frequencyof the spectrometer may further improve the noise figurevalue however the noise figure obtained with broadbandpreamplifiers and filters is similar to that observed usingtuned preamplifiers The data obtained using tuned ARRpreamplifiers did not require a filter as the measured noisefigure values were in good agreement with the catalogue dataOnly small differences were seen between the results of 119889(cropped) samples compared with those from 119873 samplesThe difference did not exceed 013 dB (experiment no 6tuned-2 without a filter) All measurements were performedin the shielding cage of the scanner and all connectionsused shielded cables Measurements without a shieldingcage resulted in much higher noise which varied betweenmeasurements depending on the actual interference in themeasurement space Shielded cables alone are therefore notsufficient for an accurate measurement and all instrumentsshould be warmed up for some time before measurementsare carried out Also changes in the inputs and outputs ofthe preamplifiers should be performed some time before themeasurement

The noise figures of the receiver with each of the fourpreamplifiers depending on the noise bandwidth are shownin Figure 5 It can be seen that for frequencies close to 1198910 themeasured noise figure is inaccurate possibly due to the low

number of frequency samples Between 50000 and 300000samples the value is approximately constant falling rapidlybeyond 300000 samples

The frequency or noise bandwidth in Hz can be esti-mated using (20) taken from the graph of Figure 6

119861 = 119889119873119879119904 (20)

The results of the noise figure measurements reflect ourexpectations and it can therefore be concluded that the ver-ification was successful Furthermore equations and derivedformulae provided acceptable results

4 Discussion and Conclusions

The purpose of this study was to develop and to test atechnique for the determination of the NMR spectrometerreceiver noise figure We endeavoured to use only widelyavailable or cheap instruments The noise figure of thereceiver obtained using a broadband preamplifier dependson the filter between the preamplifier and the console Thefilter type and characteristic frequencies are the subject offurther studies It is possible that a suitable selection offrequencies could improve the noise figure of the entirereceiver which cannot be achieved by replacing the hardwarelow-pass filter with a suitable software low-pass filter Thissuggests that the point of the receiver chain at which thefilter is connected is an important consideration Measureddata was processed using calculations from the cited sourcesbut also from own derivations The final results have proventhat all the equations and derived formulae are correct Inthe present study methods were tested using a spectroscopicNMR console However the console must be able to calibratethe voltage in order to carry out the calculations and so thismay not be possible on all consoles

The primary application of the technique is to inspectthe receiver of the spectrometer but in some cases it can beapplied for analysis of the noise figure of a tested preamplifiermeasurement The method can be also applied to a magneticresonance imaging (MRI) scanner provided that the consoleis suitable If the signal for the console is amplified by abroadband preamplifier it is recommended that a suitablefilter is connected between the preamplifier and the consolefor significant improvements in the noise figure

6 Mathematical Problems in Engineering

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(a)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(b)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(c)

18

16

14

12

1

08

NF1 [dB]

0 100000 200000 300000 400000 500000

d [samples]

(d)

Figure 5 Influence of the noise bandwidth on themeasured receiver noise figure using (a) the broadband-1 preamplifier (b) the broadband-2preamplifier (c) the tuned-1 preamplifier and (d) the tuned-2 preamplifier The noise bandwidth d is symmetrical to the operation frequency1198910

10000B [Hz]

8000

6000

4000

2000

0100000 200000 300000 400000 500000

d [samples]

Figure 6 Calculation of the noise or frequency bandwidth (B) fromfrequency samples

Data Availability

The measured digital data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Grant AgencyVEGA 2000117 and within the project of the Research andDevelopment Agency No APVV-15-0029

References

[1] P Andris V Jacko T Dermek and I Frollo ldquoNoise measure-ment of a preamplifier with high input impedance using anNMR consolerdquoMeasurement vol 55 pp 408ndash412 2014

[2] V Zalud and V N Kulesov Semiconductor Circuits with LowNoise SNTL Prague Czechia 1980 (in Czech)

[3] H Bittel and L Storm Noises Springer-Verlag Berlin Ger-many 1971 (in German)

[4] B Schiek and H J Siweris Noises in RF Circuits HuthigHeidelberg Germany 1990 (in German)

[5] D Hoult and R Richards ldquoThe signal-to-noise ratio of thenuclear magnetic resonance experimentrdquo Journal of MagneticResonance vol 24 no 1 pp 71ndash85 1976

[6] D Hoult and P C Lauterbur ldquoThe sensitivity of the zeug-matographic experiment involving human samplesrdquo Journal ofMagnetic Resonance vol 34 no 2 pp 425ndash433 1979

[7] A Raad and L Darrasse ldquoOptimization of NMR receiver band-width by inductive couplingrdquoMagnetic Resonance Imaging vol10 no 1 pp 55ndash65 1992

[8] MDecorps P BlondetH Reutenauer J Albrand andC RemyldquoAn inductively coupled series-tuned NMR proberdquo Journal ofMagnetic Resonance vol 65 no 1 pp 100ndash109 1985

[9] J Weis A Ericsson and A Hemmingsson ldquoChemical shiftartifact-free microscopy spectroscopic microimaging of thehuman skinrdquoMagnetic Resonance in Medicine vol 41 no 5 pp904ndash908 1999

[10] P Marcon K Bartusek Z Dokoupil and E GescheidtovaldquoDiffusionMRI mitigation of magnetic field inhomogeneitiesrdquoMeasurement Science Review vol 12 no 5 pp 205ndash212 2012

[11] K Bartusek Z Dokoupil and E Gescheidtova ldquoMapping ofmagnetic field around small coils using the magnetic resonancemethodrdquoMeasurement Science and Technology vol 18 no 7 pp2223ndash2230 2007

Mathematical Problems in Engineering 7

[12] D Nespor K Bartusek and Z Dokoupil ldquoComparing SaddleSlotted-tube and Parallel-plate Coils for Magnetic ResonanceImagingrdquo Measurement Science Review vol 14 no 3 pp 171ndash176 2014

[13] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSimple phase method for measurement of magneticfield gradient waveformsrdquoMagnetic Resonance Imaging vol 25no 9 pp 1272ndash1276 2007

[14] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSingle-point imagingwith a variable phase encodingintervalrdquoMagnetic Resonance Imaging vol 26 no 1 pp 109ndash1162008

[15] D Gogola P Szomolanyi M Skratek and I Frollo ldquoDesignand construction of novel instrumentation for low-field MRtomographyrdquo Measurement Science Review vol 18 no 3 pp107ndash112 2018

[16] G Wimmer V Witkovsky and T Duby ldquoProper roundingof the measurement results under normality assumptionsrdquoMeasurement Science and Technology vol 11 no 12 pp 1659ndash1665 2000

[17] D W Kammler A First Course in Fourier Analysis CambridgeUniversity Press New York NY USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

Table 1 Results of verification

No 1198910[MHz] Preamplifier119873119865 1198731198651[dB] 1198731198652[dB] Filter 119889[samples] 119873[samples]1 445 broadband-1 1 dB 44 442 No 300000 5000002 445 broadband-2 1 dB 445 442 No 300000 5000003 445 broadband-1 1 dB 145 137 L-P 300000 5000004 445 broadband-2 1 dB 148 151 L-P 300000 5000005 445 tuned-1 1 dB 134 127 No 300000 5000006 445 tuned-2 1 dB 153 166 No 300000 5000007 445 tuned-1 1 dB 136 129 L-P 300000 5000008 445 tuned-2 1 dB 144 148 L-P 300000 500000

(Tecmag Inc Houston TX) The console was utilised forimaging operated at a frequency of 445 MHz Consolecalibration was performed using the GFG-3015 functiongenerator (Good Will Instrument Co Ltd New Taipei CityTaiwan) Calibration using the console transceiver was alsotested in Houston An external attenuator was necessaryin both cases The broadband preamplifier pair of AU-1579 (MITEQ Hauppauge NY) with 50-Ohm input andoutput impedances and 119873119865 = 1 dB (a catalogue datum) wasused (selected randomly from three pieces) Another tunedpreamplifier pair P445VD (Advanced Receiver ResearchBurlington CT) with 50-Ohm input and output impedancesand 119873119865 = 1 dB (a catalogue datum) was also used selectedrandomly from three pieces We verified our theory throughseveral experiments the results of which are presented inTable 1 The receiver with MITEQ broadband preamplifiersexhibited significant noise which was improved by insertinga low-pass filter between the preamplifier and the consoleThe low-pass filter applied had a cut-off frequency of 107MHz SLP-107+ (Mini-Circuits Brooklyn NY)Using a filterwith a cut-off frequency closer to the operation frequencyof the spectrometer may further improve the noise figurevalue however the noise figure obtained with broadbandpreamplifiers and filters is similar to that observed usingtuned preamplifiers The data obtained using tuned ARRpreamplifiers did not require a filter as the measured noisefigure values were in good agreement with the catalogue dataOnly small differences were seen between the results of 119889(cropped) samples compared with those from 119873 samplesThe difference did not exceed 013 dB (experiment no 6tuned-2 without a filter) All measurements were performedin the shielding cage of the scanner and all connectionsused shielded cables Measurements without a shieldingcage resulted in much higher noise which varied betweenmeasurements depending on the actual interference in themeasurement space Shielded cables alone are therefore notsufficient for an accurate measurement and all instrumentsshould be warmed up for some time before measurementsare carried out Also changes in the inputs and outputs ofthe preamplifiers should be performed some time before themeasurement

The noise figures of the receiver with each of the fourpreamplifiers depending on the noise bandwidth are shownin Figure 5 It can be seen that for frequencies close to 1198910 themeasured noise figure is inaccurate possibly due to the low

number of frequency samples Between 50000 and 300000samples the value is approximately constant falling rapidlybeyond 300000 samples

The frequency or noise bandwidth in Hz can be esti-mated using (20) taken from the graph of Figure 6

119861 = 119889119873119879119904 (20)

The results of the noise figure measurements reflect ourexpectations and it can therefore be concluded that the ver-ification was successful Furthermore equations and derivedformulae provided acceptable results

4 Discussion and Conclusions

The purpose of this study was to develop and to test atechnique for the determination of the NMR spectrometerreceiver noise figure We endeavoured to use only widelyavailable or cheap instruments The noise figure of thereceiver obtained using a broadband preamplifier dependson the filter between the preamplifier and the console Thefilter type and characteristic frequencies are the subject offurther studies It is possible that a suitable selection offrequencies could improve the noise figure of the entirereceiver which cannot be achieved by replacing the hardwarelow-pass filter with a suitable software low-pass filter Thissuggests that the point of the receiver chain at which thefilter is connected is an important consideration Measureddata was processed using calculations from the cited sourcesbut also from own derivations The final results have proventhat all the equations and derived formulae are correct Inthe present study methods were tested using a spectroscopicNMR console However the console must be able to calibratethe voltage in order to carry out the calculations and so thismay not be possible on all consoles

The primary application of the technique is to inspectthe receiver of the spectrometer but in some cases it can beapplied for analysis of the noise figure of a tested preamplifiermeasurement The method can be also applied to a magneticresonance imaging (MRI) scanner provided that the consoleis suitable If the signal for the console is amplified by abroadband preamplifier it is recommended that a suitablefilter is connected between the preamplifier and the consolefor significant improvements in the noise figure

6 Mathematical Problems in Engineering

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(a)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(b)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(c)

18

16

14

12

1

08

NF1 [dB]

0 100000 200000 300000 400000 500000

d [samples]

(d)

Figure 5 Influence of the noise bandwidth on themeasured receiver noise figure using (a) the broadband-1 preamplifier (b) the broadband-2preamplifier (c) the tuned-1 preamplifier and (d) the tuned-2 preamplifier The noise bandwidth d is symmetrical to the operation frequency1198910

10000B [Hz]

8000

6000

4000

2000

0100000 200000 300000 400000 500000

d [samples]

Figure 6 Calculation of the noise or frequency bandwidth (B) fromfrequency samples

Data Availability

The measured digital data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Grant AgencyVEGA 2000117 and within the project of the Research andDevelopment Agency No APVV-15-0029

References

[1] P Andris V Jacko T Dermek and I Frollo ldquoNoise measure-ment of a preamplifier with high input impedance using anNMR consolerdquoMeasurement vol 55 pp 408ndash412 2014

[2] V Zalud and V N Kulesov Semiconductor Circuits with LowNoise SNTL Prague Czechia 1980 (in Czech)

[3] H Bittel and L Storm Noises Springer-Verlag Berlin Ger-many 1971 (in German)

[4] B Schiek and H J Siweris Noises in RF Circuits HuthigHeidelberg Germany 1990 (in German)

[5] D Hoult and R Richards ldquoThe signal-to-noise ratio of thenuclear magnetic resonance experimentrdquo Journal of MagneticResonance vol 24 no 1 pp 71ndash85 1976

[6] D Hoult and P C Lauterbur ldquoThe sensitivity of the zeug-matographic experiment involving human samplesrdquo Journal ofMagnetic Resonance vol 34 no 2 pp 425ndash433 1979

[7] A Raad and L Darrasse ldquoOptimization of NMR receiver band-width by inductive couplingrdquoMagnetic Resonance Imaging vol10 no 1 pp 55ndash65 1992

[8] MDecorps P BlondetH Reutenauer J Albrand andC RemyldquoAn inductively coupled series-tuned NMR proberdquo Journal ofMagnetic Resonance vol 65 no 1 pp 100ndash109 1985

[9] J Weis A Ericsson and A Hemmingsson ldquoChemical shiftartifact-free microscopy spectroscopic microimaging of thehuman skinrdquoMagnetic Resonance in Medicine vol 41 no 5 pp904ndash908 1999

[10] P Marcon K Bartusek Z Dokoupil and E GescheidtovaldquoDiffusionMRI mitigation of magnetic field inhomogeneitiesrdquoMeasurement Science Review vol 12 no 5 pp 205ndash212 2012

[11] K Bartusek Z Dokoupil and E Gescheidtova ldquoMapping ofmagnetic field around small coils using the magnetic resonancemethodrdquoMeasurement Science and Technology vol 18 no 7 pp2223ndash2230 2007

Mathematical Problems in Engineering 7

[12] D Nespor K Bartusek and Z Dokoupil ldquoComparing SaddleSlotted-tube and Parallel-plate Coils for Magnetic ResonanceImagingrdquo Measurement Science Review vol 14 no 3 pp 171ndash176 2014

[13] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSimple phase method for measurement of magneticfield gradient waveformsrdquoMagnetic Resonance Imaging vol 25no 9 pp 1272ndash1276 2007

[14] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSingle-point imagingwith a variable phase encodingintervalrdquoMagnetic Resonance Imaging vol 26 no 1 pp 109ndash1162008

[15] D Gogola P Szomolanyi M Skratek and I Frollo ldquoDesignand construction of novel instrumentation for low-field MRtomographyrdquo Measurement Science Review vol 18 no 3 pp107ndash112 2018

[16] G Wimmer V Witkovsky and T Duby ldquoProper roundingof the measurement results under normality assumptionsrdquoMeasurement Science and Technology vol 11 no 12 pp 1659ndash1665 2000

[17] D W Kammler A First Course in Fourier Analysis CambridgeUniversity Press New York NY USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(a)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(b)

18

16

14

12

1

080 100000 200000 300000 400000 500000

d [samples]

NF1 [dB]

(c)

18

16

14

12

1

08

NF1 [dB]

0 100000 200000 300000 400000 500000

d [samples]

(d)

Figure 5 Influence of the noise bandwidth on themeasured receiver noise figure using (a) the broadband-1 preamplifier (b) the broadband-2preamplifier (c) the tuned-1 preamplifier and (d) the tuned-2 preamplifier The noise bandwidth d is symmetrical to the operation frequency1198910

10000B [Hz]

8000

6000

4000

2000

0100000 200000 300000 400000 500000

d [samples]

Figure 6 Calculation of the noise or frequency bandwidth (B) fromfrequency samples

Data Availability

The measured digital data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Grant AgencyVEGA 2000117 and within the project of the Research andDevelopment Agency No APVV-15-0029

References

[1] P Andris V Jacko T Dermek and I Frollo ldquoNoise measure-ment of a preamplifier with high input impedance using anNMR consolerdquoMeasurement vol 55 pp 408ndash412 2014

[2] V Zalud and V N Kulesov Semiconductor Circuits with LowNoise SNTL Prague Czechia 1980 (in Czech)

[3] H Bittel and L Storm Noises Springer-Verlag Berlin Ger-many 1971 (in German)

[4] B Schiek and H J Siweris Noises in RF Circuits HuthigHeidelberg Germany 1990 (in German)

[5] D Hoult and R Richards ldquoThe signal-to-noise ratio of thenuclear magnetic resonance experimentrdquo Journal of MagneticResonance vol 24 no 1 pp 71ndash85 1976

[6] D Hoult and P C Lauterbur ldquoThe sensitivity of the zeug-matographic experiment involving human samplesrdquo Journal ofMagnetic Resonance vol 34 no 2 pp 425ndash433 1979

[7] A Raad and L Darrasse ldquoOptimization of NMR receiver band-width by inductive couplingrdquoMagnetic Resonance Imaging vol10 no 1 pp 55ndash65 1992

[8] MDecorps P BlondetH Reutenauer J Albrand andC RemyldquoAn inductively coupled series-tuned NMR proberdquo Journal ofMagnetic Resonance vol 65 no 1 pp 100ndash109 1985

[9] J Weis A Ericsson and A Hemmingsson ldquoChemical shiftartifact-free microscopy spectroscopic microimaging of thehuman skinrdquoMagnetic Resonance in Medicine vol 41 no 5 pp904ndash908 1999

[10] P Marcon K Bartusek Z Dokoupil and E GescheidtovaldquoDiffusionMRI mitigation of magnetic field inhomogeneitiesrdquoMeasurement Science Review vol 12 no 5 pp 205ndash212 2012

[11] K Bartusek Z Dokoupil and E Gescheidtova ldquoMapping ofmagnetic field around small coils using the magnetic resonancemethodrdquoMeasurement Science and Technology vol 18 no 7 pp2223ndash2230 2007

Mathematical Problems in Engineering 7

[12] D Nespor K Bartusek and Z Dokoupil ldquoComparing SaddleSlotted-tube and Parallel-plate Coils for Magnetic ResonanceImagingrdquo Measurement Science Review vol 14 no 3 pp 171ndash176 2014

[13] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSimple phase method for measurement of magneticfield gradient waveformsrdquoMagnetic Resonance Imaging vol 25no 9 pp 1272ndash1276 2007

[14] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSingle-point imagingwith a variable phase encodingintervalrdquoMagnetic Resonance Imaging vol 26 no 1 pp 109ndash1162008

[15] D Gogola P Szomolanyi M Skratek and I Frollo ldquoDesignand construction of novel instrumentation for low-field MRtomographyrdquo Measurement Science Review vol 18 no 3 pp107ndash112 2018

[16] G Wimmer V Witkovsky and T Duby ldquoProper roundingof the measurement results under normality assumptionsrdquoMeasurement Science and Technology vol 11 no 12 pp 1659ndash1665 2000

[17] D W Kammler A First Course in Fourier Analysis CambridgeUniversity Press New York NY USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

[12] D Nespor K Bartusek and Z Dokoupil ldquoComparing SaddleSlotted-tube and Parallel-plate Coils for Magnetic ResonanceImagingrdquo Measurement Science Review vol 14 no 3 pp 171ndash176 2014

[13] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSimple phase method for measurement of magneticfield gradient waveformsrdquoMagnetic Resonance Imaging vol 25no 9 pp 1272ndash1276 2007

[14] P Latta M L Gruwel V Volotovskyy M H Weber and BTomanek ldquoSingle-point imagingwith a variable phase encodingintervalrdquoMagnetic Resonance Imaging vol 26 no 1 pp 109ndash1162008

[15] D Gogola P Szomolanyi M Skratek and I Frollo ldquoDesignand construction of novel instrumentation for low-field MRtomographyrdquo Measurement Science Review vol 18 no 3 pp107ndash112 2018

[16] G Wimmer V Witkovsky and T Duby ldquoProper roundingof the measurement results under normality assumptionsrdquoMeasurement Science and Technology vol 11 no 12 pp 1659ndash1665 2000

[17] D W Kammler A First Course in Fourier Analysis CambridgeUniversity Press New York NY USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom