Sampling. David Rovnyak 4 th Biomolecular SSNMR Winter School Stowe, VT, 2016

Transcription

1 Sampling David Rovnyak 4 th Biomolecular SSNMR Winter School Stowe, VT, 2016

2 Acknowledgements! Bucknell Levi Craft, Melissa Palmer, Riju Gupta, Mark Sarcone, Ze Jiang Delaware Tatyana Polenova, Chris Suiter UCHC Jeffrey C. Hoch UVA James Rovnyak, Virginia Rovnyak Susquehanna Geneive Henry! Funding NSF-RUI, NIH-AREA! Thank you s Adam Schuyler (UCHC), Frank Delaglio (NIST/UMD)!! Big thank you to Mei Hong and Chris Jaroniec and Sponsors.!

3 Thank You CCSA NIH-AREA ACS-PRF-B NSF-MRI NSF-RUI Bucknell-Geisinger Research Initiative Dean Fellow Startup Matching

4 Goals/Outcomes! I. Interesting foundations of sampling/ Classic Fourier Calcs! II. III. IV. NUS fundamentals; good NUS schedules! How to calculate a NUS sensitivity enhancement! Speculations on frontiers in sampling!

5 I. Interesting Foundations! Aiming for key aspects of sampling which are assumed/skimmed in other texts.!! The Fourier Transform and Fourier Pairs!! Parseval s Theorem : the FT is power conserving!! Noise in the FID and the frequency spectrum!! Signal in the FID and the frequency spectrum!! Signal to Noise Ratio in the time domain!

6 Signal-to-Noise and Sensitivity! " rms = $ # 1 N N i=1 % s(ω i ) 2 ' & 1 2 s(ω 0 ) = height S(ω 0 ) rms noise SNR time ω 0 SNR : ratio of the peak maximum to the rms noise. Sensitivity: ratio of the SNR to the square root of total time (Ernst) delay acquisition time P-Set 1.6

7 Noise is white gaussian!! Thermal noise in a copper wire theoretically white gaussian!!white: all frequencies occur; no holes!!gaussian: intensity of a noise peak follows a Gaussian distribution! 1.00 Relative Probability Frequency (a.u.) Spectrometer noise Simulated white gaussian noise Varian RNMRTK is this signal or a Gaussian excursion? Noise magnitude

8 Noise and temperature! Noise measured with temperature of a 50 Ohm load (Glenn Facey)

9 Complex Fourier Transform! S(ω ) = S(t)e i2πωt (1) S(t) = S(ω )e i2πωt (2) P-Set 1.1

10 The FT is Linear! S(ω ) = ( f (t) + g(t))e i2πωt = f (t)e i2πωt + g(t)e i2πωt = f (ω ) + g(ω ) ω 1 1 ω 2 0 = time FT FT FT ω 1 ω 2 frequency ω 1 ω 2

11 Discrete Fourier Transform! f ω k N 1 j=0 ( ) = s j (t)e i N jk f(w) s(t) Fourier transform V coil time ν L frequency P-Set 1.7

12 Discrete Fourier Transform - Nyquist! Nyquist frequency, v max : let v max be the highest frequency you wish to detect in units of Hz (s -1 ). Then you must digitize 2v max times per second. This requires collecting two points per wavelength max. dwell time, Δt : time interval between points; Δt=1/(2v max ); v max =1/(2Δt)" no sign discrim. " sw=v max " +v max " -v rf " max " noise is reflected as well as signal!" NMR spectral width, part 2 : sign discrimination detects frequencies from - v max to +v max and so the NMR spectral width is 2v max =1/( t). with sign discrim." sw=2v max " +v max " -v rf " max " noise is reduced by a factor sqrt(2)" Example : Δt = 20 μs, then sw NMR = 50 khz"

13 The Discrete Fourier Transform is Cyclic F.Delaglio

14 The Discrete Fourier Transform is Cyclic As Signals Get Faster Than The Sampling Frequency F.Delaglio

15 Folding! ν max! Δν" ν rf! 0.0" NMR spectrum, with all lines shown at their correct frequencies" ν max! ν rf! Δν" 0.0" shift the center to lower frequency by Δν, and use a smaller ν max." Folding is circular. " " How far out can you fold? Filters remove noise (and signal) typically above *v max." " Coming up: NUS and spectral width."

16 DFT and Resolution! acquisition time, t a : this is NΔt, where N is the number of complex points" " digital resolution, R d : the frequency separation between points in the spectrum R d =1/t a " R d >> LW" R d ~ LW" R d << LW" Zerofilling improves R d without distorting the spectrum:" Why is this legal? Roughly : appending 0 s is adding no data. Formally: Parseval." Keeler Notes

17 Let s see the DFT in action.!! Credits and thanks: Frank Delaglio!

18 X( f ) = x( t ) [ cos( 2 ft / N ) - i sin( 2 ft / N ) ] Each Fourier term corresponds to a point in the spectrum.

19 First, Let s View the Fourier Transform with the Imaginary Parts Hidden

20

21 + - When the Fourier term does not match any frequency in the data, the product has balanced amounts of positive and negative intensity, and sums to zero.

22 + -

23 Now, Let s View the Fourier Transform with the Both the Real and Imaginary Parts Shown

24 + -

25 + -

26 + -

27 Combined, the Real and Imaginary Parts Distinguish Between Positive and Negative Frequencies + =

28 In Biomolecular NMR, Indirect Dimensions are Often Truncated and Have Limited Numbers of Points

29 + - Since the Fourier product is truncated, there is no longer ideal cancelation of positive and negative intensities, giving broad lines and artifacts

30 FT : Theorems!! Convolution!! Parseval!! Nyquist! F( f ) = f (ω ) = f (t)e i2πωt F(g) = g(ω ) = g(t)e i2πωt f (t)g(t)e i2πωt = f (τ )g(t τ )dτ F( f g) = f (ω ) g(ω ) = F( f ) F(g) (note: horrible typo in original notes) The FT of the product of two functions is related to the FT s of the individual functions by convolution: the overlap of scanning one function across the other.

31 FT : Theorems!! Convolution!! Parseval!! Nyquist! F( f g) = f (ω ) g(ω ) f(t) x g(t) f(t) g(t) = X FT ω f(ω) x g(ω) = X ω 0 f(ω) g(ω)

32 FT of an Exponential Decay! S(ν)! An exponential decay transforms to a Lorentzian! T 2 = s LW = FWHH = 1 πτ 2 (Hz) S(ω ) = = = = = e t/t 2 e i2πνt dt = e t(1/t +i2πν ) 2 dt 1 (1/ T 2 + i2πν) e t(1/t +i2πν ) (1/ T 2 + i2πν) (1/ T 2 + i2πν) 1/ T 2 1/ T π 2 ν 2 0 (1/ T 2 i2πν) (1/ T 2 i2πν) ( ) + i2πν ( 1/ T π 2 ν 2 ) frequency (Hz) = 1 2π 1 2 LW LW 2 +ν 2 ( ) + i2πν ( 1/ T π 2 ν 2 ) Real Part Imaginary Part P-Set 1.2

33 Other Fourier Pairs!! A Gaussian transforms to a Gaussian (not the same one)! S(ω ) = e t2 /2σ 2 e i2πνt dt = e (t2 /2σ 2 +iωt) dt Lorentzian Gaussian From tables e ax2 +bx dt = e b 2 4a π a S(ν) There a number of ways to tackle this integral. A fun alternative is to notice that you can turn this in to a more conventional Gaussian integral. Consider the exponent: t 2 ν 2 1 S(ω ) = α 2π e 2α 2, α = 1 2πσ 2σ 2 iωt Complete the square with some b frequency (Hz) t 2 2σ 2 iωt b 2 + b 2 Figure out b to lead you to 1 (t + iωσ 2 ) 2 ω 2 σ 2 2σ 2 2 P-Set 1.2

34 Parseval : the FT is Power Conserving! s(t) 2 = 1 2π s(ω ) 2 Practical consequence: the FT is like life, what you get out of it, is exactly what you put in to it. P-Set 1.3

35 Noise! 10*T 2! 3*T 2! noise time 1*T 2! Analogy to signal averaging. See for example: Spencer RG Equivalence of the time-domain matched filter and the spectral-domain matched filter in one-dimensional NMR spectroscopy. Concepts Magn Reson A 36A:

36 Noise! 0.2! 0.4! 0.6! Evolution time (sec) JBNMR 2004, 30, 1-10

37 Signal! Peak Height Area ppm time / T 2 peak height = integral of signal envelope t S(ω = ω 0 ) = e iω 0 t e R 2 t e iωt dt 0 t = e R 2 t dt 0 ( ) = T 2 1 e t acq /T 2

38 SNR in time domain ( ) integral of signal T 2 1 envelope e t/t 2 S N (t) = time t C N = SNR in frequency domain peak height r.m.s. noise SNR this solution strictly true for power conserving transform, such as FT

39 SNR in the Time Domain! Matson, G Signal Integration and the Signal-to-Noise Ratio in Pulsed NMR Relaxation Measurements. J Magn Reson 25,

40 Regarding 1.26 T 2! " Matson, G Signal Integration and the Signal-to-Noise Ratio in Pulsed NMR Relaxation Measurements. J Magn Reson 25, " Becker, E.D., Ferretti. J.A., Gambhir, P.N., Selection of Optimum Parameters for Pulse Fourier Transform Nuclear Magnetic Resonance. Analytical Chemistry 51(9), " D. Rovnyak, J.C. Hoch, A.S. Stern, G. Wagner, Resolution and Sensitivity of High Field Nuclear Magnetic Resonance. J. Biomol. NMR. 30, " T. Vosegaard, N. Chr. Nielsen, Defining the sampling space in multidimensional NMR experiments: What should the maximum sampling time be? J. Magn. Reson. 199, " Eriks Kup e, Lewis E. Kay, and Ray Freeman, Detecting the Afterglow of 13C NMR in Proteins Using Multiple Receivers. J. Am. Chem. Soc. 132 (51), pp " Spencer RG Equivalence of the time-domain matched filter and the spectral-domain matched filter in one-dimensional NMR spectroscopy. Concepts Magn Reson A 36A: P-Set-1.5

41 Signal Envelope Determines Sensitivity! 1984 J Magn Reson 58:

42 NUS and Sensitivity! Barna JCJ, Laue ED, Mayger MR, Skilling J, Worrall SJP (1986) Reconstruction of phase-sensitive two-dimensional NMR-spectra using maximum-entropy. Biochem Soc Trans 14(6): Barna JCJ, Laue ED, Mayger MRS, Skilling J, Worrall SJP (1987) Exponential sampling, an alternative method for sampling in two-dimensional NMR experiments. J Magn Reson 73(1):69 77

43 Definitions! Some misconceptions have arisen in the literature we ll talk about more of those throughout, but a great deal can be clarified by these definitions, which are suggested for broader adoption. Intrinsic SNR (isnr) SNR of raw data, prior to any manipulation Once receiver is off, the isnr cannot be changed May always be computed in time domain Equivalent to frequency spectrum for power conserving transform. Apparent SNR (SNR) SNR after signal manipulation: post-apodization post-linear prediction May be computed in time domain or frequency domain

44 II. NUS Fundamentals!! On grid NUS!! Weighted NUS!! Point Spread Function and Convolution!! Partial Component; Nonuniform Weighting!! Spectral window of NUS!! What is a good NUS schedule?!! What is good spectral reconstruction?!

45 Preaching to converted, but why NUS?! Uniform: 1024 samples Splitting=10 Hz FT Uniform: 128 samples LP-FT Non-Uniform: 128 samples (matched exponential density) MaxEnt Full Resolution in 1/8 Experimental Time Comment: building trust with non-fourier spectral estimation Stern AS, Li KB, Hoch JC. Modern spectrum analysis in multidimensional NMR spectroscopy: comparison of linear-prediction extrapolation and maximum-entropy reconstruction. J Am Chem Soc Mar 6; 124(9):

46 On Grid NUS! Most common approach Samples are a subset of the evenly (uniformly) spaced Nyquist grid. Equal number of transients for each sample.

47 Weighted Sampling! NUS and NUWS have same density of samples. Weighted Sampling (NUS) Sampling Density Weighted averaging (NUWS) (Kumar et al., JMR 95, 1-9, 1991)

48 Aside: Processing Weighted Averaging! FT-NUWS lb = 2 Hz FT-US lb = 2 Hz FT-US lb = 22 Hz (Kumar et al., JMR 95, 1-9, 1991) NUWS (weighted averaging) is on-grid so the FFT may be used. But NUWS introduces the apparent line broadening of the sampling density. The FFT will be power conserving, fast, and easy, but introduces this broadening. Non-Fourier techniques avoid the broadening of the sampling density. (just like NUS) Perhaps applicable in imaging where lines are very broad. Some other NUWS works. Christopher A. Waudby, John Christodoulou, An Analysis Of NMR Sensitivity Enhancements Obtained Using non-uniform Weighted sampling, and the application To Protein NMR, JMR V219, (2012) Qiang W (2011) Signal enhancement for the sensitivity-limited solid state NMR expderiments using a continuous non-uniform acquisition scheme. J Magn Reson 213:

49 Exponential Weighted Sampling! ZF Sample Number Exponential NUS BIAS 4.0 Signal Intensity (a.u.) (match) Evolution time / T 2

50 Aside: Random Sampling, Random Order! Conventional Uniformly Sampled (US) Schedule Non-Uniform Sampling Schedule Non-Uniform Sampling Schedule in Random Order Random order: A high res spectrum develops in real time. Allows for time resolved NMR, for example:

51 Aside: Random Sampling, Random Order! (Mayzel, Rosenlow, Isaksson, Orekhov, JBNMR 2014, 58: )

52 Convolution and Point Spread Function! PSF the Fourier transform of the sampling schedule Mobli, Hoch, Concepts in Magn. Reson. A, 32A, (2008)

53 Point Spread Function! What is a good PSF? Larger gaps Smaller gaps between samples (aside ignore stray horizontal line) Low noise around central feature corresponds to more robust reconstructions P-Set

54 Partial Component NUS! (t 1,t 2 ) t 3 t 1 t 2 (t 1,t 2 ) = RR, RI, IR, II Example NUS sampling schedule for 3D NMR; NUS in both t 1 and t 2. t 1 = indirect t 2 = indirect t 3 = direct Proposal: select RR,RI,IR,II spectra nonuniformly as well. Schuyler, Maciejewski, Stern, Hoch, JMR (2013) 227,

55 Partial Component NUS! Schuyler, Maciejewski, Stern, Hoch, JMR (2015) 254, Summary: PC is a route to introducing randomness, may help improve PSF s with sparsity.

56 Spectral Window of NUS! On grid NUS breaks Nyquist. Main concern is to avoid aliasing within the spectral window. General rule: (on or off grid) the upper limit of the spectral window is determined by the greatest common divisor of the samples. (in other words, the smallest gap) In noiseless spectra, the GCD translates in principle in to the exact spectral window. Manassen, Y.; Navon, G. Nonuniform Sampling in NMR Experiments. J. Magn. Reson. 1988, 79, On Grid: as long as sampling not too sparse, the effective spectral window corresponds closely to the Nyquist frequency of the original grid; Proposal choose NUS from oversampled Nyquist grid guideline - BURSTY samples (some uniform tracts) Maciejewski, M. W.; Qui, H. Z.; Rujan, I.; Mobli, M.; Hoch, J. C. Nonuniform sampling and spectral aliasing. J. Magn. Reson. 2009, 199, P-Set-2.2

57 Spectral Window of NUS! Example of selecting NUS from an oversampled grid. Maciejewski, M. W.; Qui, H. Z.; Rujan, I.; Mobli, M.; Hoch, J. C. Nonuniform sampling and spectral aliasing. J. Magn. Reson. 2009, 199, Don t run for the hills notice the cross-sections small effect but worth knowing.

58 Good schedule? Processor?! Random Coverage Weighted Reproducibility Gridding for SW PSF s, Gapping, Partial Component Non-sparse: 25-50% reduction Sparse: < 25% reduction Sensitivity, Resolution Reporting, validating, regulatory issues Minimizing GCD of samples In order to focus on fundamentals of sampling, processing is not the focus here. To make a long story short: Processing: CS, L1, L2, hmsist, IST, MaxEnt (MINT, FM, more), MDD, more. While improvements remain, reasonably modern implementations of ALL of these are normally robust for non-sparse NUS. In contrast, best processing for sparse NUS remains a going concern.

59 Calculate a NUS Enhancement!! Working through an example!! Exact Theory for several cases!! Two Theorems for NUS!! Fundamentals: Parseval

60 Example: NUS Enhancement!! Let s see a practical example before looking at the theory.!

61 An FID! Evolution time / T 2

62 Uniform Sampling! uniform samples Evolution time / T 2

63 Sum = 9.97! Evolution time / T 2

64 NUS 8/32 2X bias exponential NUS 8/32 1.5X bias exponential NUS 8/32 matched exponential Uniform 32 samples Sample Index

65 Equal Experiment Times!! Suppose the uniform experiment used 4 transients for each of 32 increments, for!!4 * 32 = 128 transients (uniform)!! Then NUS must use fourfold more, or 16 transients for each of 8 increments, for!!16 * 8 = 128 transients!!(nus)!

66 NUS 8/32 2X bias exponential NUS 8/32 1.5X bias exponential NUS 8/32 matched exponential Uniform 32 samples Sample Index

67 Matched NUS! x x x x 4 Sum = x x x x

68 Noise is a spectator!! Recall that the uniform experiment and the NUS experiment each used 128 transients!! Noise is the same in each experiment.!

69 Enhancement: Match NUS!! Enhancement for acquisition spanning 3.14 T 2 and with NUS probability density matched to T 2 : Signal Sum by NUS!! !=! = 1.78 Signal Sum by Uniform!! 9.97!! Computed by exact theory:*!!enhance = 1.71! *MRC, V49, , 2011

70 NUS 8/32 2X bias exponential NUS 8/32 1.5X bias exponential NUS 8/32 matched exponential Uniform 32 samples Sample Index

71 1.5X Bias NUS! x x x x x x 4 Sum = Enhance = 19.57/9.97 = 1.96 Theory = x x

72 Slight discrepancy?! match match match compute theory

73 Sample Index

74 (MRC, V49, , 2011) Exact Theory How does this work? Double Dipping: Enhance SNR by NUS 1. Eliminate samples > 1.26T 2 Decrease Noise. 2. Use time savings to increase transients for all remaining samples. Increase Signal.

75 Back to NUWS for a moment! corrected S/N enhance (Kumar et al., JMR 95, 1-9, 1991)

76 Step 1: Enforce Equal Experiment Times! Sample Density (a.u.) χ exp (match) h(t) Force area of h(t) equal to area of uniform with a scaling factor χ gauss (match) h(t) 1.0 uniform

77 Step 2: Form Ratio Of US to NUS Signal! For any nonuniform sampling density h(t) applied to an exponentially decaying signal: η = 0 t max 0 χh( t)e t /T 2 dt t max e t /T 2 dt t max χ h( t)e t /T 2 dt = 0 T 1 e t max /T 2 2 ( )

78 Cartoon Proof of Enhancement s(t) e t /T 2 = = Signal Uniform! 1!!"# $ $"# % %"# & s(t) e t /T 2 χ e t /T SMP = = Signal NUS!!!"# $ $"# % %"# &

79 Example Enhancements!

80 Exponential NUS-based enhancement Verified with MINT* * Power Conserving Regime of MaxEnt * Non sparse * large lambda (Paramasivam, Suiter, Sun, Palmer, Hoch, Rovnyak, Polenova, JPC B, V116, , 2012.)

81 Gaussian NUS! Sampling Density Free Induction Decay Scaled SIgnal Sample Density x bias Gaussian 1.5x biasgaussian matched Gaussian 1.0 x = Theoritical Sensitivity Gain Relative to Uniform Gaussian Matched x Bias x Bias NUS signal intensities uniform signal Some Gaussian NUS: Eddy MT, Ruben D, Griffin RG, Herzfeld J (2012) Deterministic schedules for robust and reproducible non-uniform sampling in multidimensional NMR. J Magn Reson 214(1): Qiang W (2011) Signal enhancement for the sensitivity-limited solid state NMR expderiments using a continuous nonuniform acquisition scheme. J Magn Reson 213:

82 See also: Arbogast, Brinson, Marino, Mapping Monoclonal Antibody Structure by 2D 13C NMR at Natural Abundance, Anal. Chem., 2015, 87 (7), pp (JBNMR 58, 303, 2014)

83 Trade-Off? 1.0 Signal (a.u.) ZF Sample Number Exponential NUS BIAS Uniform,FFT Evolution time / T N (ppm) 1.0 (match) (JBNMR 58, 303, 2014)

84 NUS Density and Line Shapes Sampling Density Matched Exp: enhance = 1.7 SIN (1.702) EXP (1.714) Sinusoid: enhance = time/t2 More samples by sinusoid density (JBNMR 58, 303, 2014)

85 Improved Peak Base: Sine vs. Matched Exp Uniform Sine Match Exp C (ppm) (JBNMR 58, 303, 2014)

86 Compound SNR! 3D biosolids NMR: measured enhancements 2.7 to 3.3 (Paramasivam, Suiter, Sun, Palmer, Hoch, Rovnyak, Polenova, JPC B, V116, , 2012.)

87 BioSolids NMR! Key: already operating at long evolution times in multiple indirect dimensions. Best positioned to take advantage of compounded NUS enhancements since CT periods less common. (Paramasivam, Suiter, Sun, Palmer, Hoch, Rovnyak, Polenova, JPC B, V116, , 2012.)

88 Towards stronger statements about NUS sensitivity SNR Absolute Uniform SNR x Enhancement Possible NUS Enhancement x Bias Matched = SNR Absolute (NUS) Sample Sensitivity x Bias Matched time / T 2 (Palmer et al, J. Phys. Chem. 2015)

89 FFT of Simulated NUWS data a) NUS: match b) NUS: 2x bias uniform t/t 2 t/t 2 (Palmer et al, J. Phys. Chem. 2015)

90 EXPERIMENTS Very little improvement due to NUS at 1.26 T a) NUS 2x Bias * 2.0 SNR x Bias Matched b) NUS Matched * Uniform 98.0 c) Uniform 0 time / T * H / ppm 1 H / ppm 13 C / ppm (Palmer et al, J. Phys. Chem. 2015)

91 Experiments Improvement due to NUS after 1.26 T a) NUS 2x Bias 2.0 SNR x Bias Matched Uniform b) NUS Matched time / T c) Uniform H / ppm 1 H / ppm 13 C / ppm (Palmer et al, J. Phys. Chem. 2015)

92 Validation with MINT NUS35 to 6.4 ms (2T2,*average) * US to 6.4 ms (2T2,average ) 13 C (ppm) US to 34.1 ms (πt2,*max) * ) NUS35 to 34.1 ms (πt2,max C Chemical Shift (ppm) (MINT: Paramasivum et al, J. Phys. Chem. 2012) (Palmer et al, J. Phys. Chem. 2016)

93 Is the enhancement always greater than 1? 2.5 4x Bias 3x Bias Enhancement x Bias Matched (1x) time/t 2

94 Yes. Theorem I (NUS Sensitivity Theorem). Assume h(t) is a positive, nonincreasing function on some interval 0 t < A, where A. For any positive T 2 and 0 < t max < A, the time domain isnr enhancement satisfies η = α ( 1 e α ) t max 0 0 h(t)e t T 2 dt t max h(t) dt 1, (X) where α = t max /T 2, and equality holds if and only if h(t) is constant for 0 < t < t max. Proving the equality statement settles something important: unweighted NUS (sometimes called random NUS) has equal isnr to uniform sampling. (Palmer et al, J. Phys. Chem. 2015)

95 Positive SNR slope with time Theorem 2 (Matched NUS SNR Theorem): The isnr of matched exponential NUS always has a positive slope for an arbitrary, positive T 2 and positive evolution time t max. 2.0 SNR x Bias Matched Uniform time / T 2 Narrower scope, but means: Exponential NUS simultaneously improves resolution and isnr with additional experimental time. 1 hr 3 hr P-Set-3.1

96 Examples of Theorem 1 a) normalized probability densities SINE GAUSS η b) SNR enhancement by NUS

97 Examples of Theorem 1 a) normalized probability densities LINEAR COSINE η b) SNR enhancement by NUS (Palmer et al, J. Phys. Chem. 2015) P-Set-3.2

98 Examples of Theorem 2 Sine Gauss Separate cases of Thm 2 could be analytically proven, but notice that we can see the positive slopes and can see visually that these densities have the properties of Thm. 2 SNR c) SNR time dependence time/t 2 time/t 2

99 Examples of Theorem 2 Sine c) SNR time dependence Gauss SNR Hmmm. Is isnr increasing? time/t 2 time/t 2

100 The Scope of Theorem 2 Exponential Sampling Densities Gaussian Sampling Densities 1.8 LW SMP /LW sig LW SMP /LW sig SNR increasing SNR match increasing SNR match uniform uniform time/t 2 time/t 2 (Palmer et al, J. Phys. Chem. 2015)

101 Speculations on Frontiers!! Building consensus on good NUS scheduling!! Enhancement and line shape are now metrics in identifying good schedules!! Need more criteria adherence to weighting, accessibility to broader user base!! What constitutes the ability to detect a peak?!! Constrained by the raw data prior isnr plays a role!! More to the story!! Vendor Implementations!! Other signal envelopes!! Public Relations.!

102 Consensus on Scheduling! Many recent approaches to schedules (us included) seem to be converging -Random - Reproducible / Deterministic - Minimize Gaps - Sensitivity and Lineshape - Ease of Implementation - Adherence to Weighting - Generalizable to multiple NUS dimensions - Generalizable to Partial Component Propose that the efforts of a number of folks may be converging mathematically.

103 How to select samples via a density? Quadratic decay for sampling density Random Reproducible / Deterministic Minimize Gaps Sensitivity and Lineshape Ease of Implementation Adherence to Weighting And more time/t2 (ms in preparation; if you use please reference Private Communication with DSR and VGR)

104 How to select samples via a density? (ms in preparation; if you use please reference Private Communication with DSR and VGR)

105 Vendor implementations* Uniformly organize NUS data in memory; NUS schedule always in header. More transparent/user-enabled approach to weighting. Automate NUS from oversampled grid. Automate real-time and time-resolved NUS. Calculators to estimate T 2 * and to estimate enhancements. Need a suite of standardized test data sets so that schedules/ processing can be certified. *Some of these may already be addressed.

106 Other Signal Envelopes Kumar A, Brown CB, Donlan ME, Meier BU, Jeffs PW (1991), Optimization of two-dimensional NMR by matched accumulation. J Magn Reson 95(1):1 9 Peter Schmieder, Alan S. Stern, Gerhard Wagner, Jeffrey C. Hoch, Application of nonlinear sampling schemes to COSY-type spectra, P-Set-3.3

107 PR Starting to see more formal assurances for NUS from several avenues. Important to increase breadth of understanding of these fundamentals. Improved understanding/reassurance of spectral windows. Improved/expanded metrics for schedules. More routine reports on NUS in two or more indirect dimensions. Improved understanding of sensitivity, with strong theorems. Need to be cautious to distinguish bleeding edge applications from established, robust NUS applications, and cautious about overuse of the straw man.

108 Thank you. 0 NUS time domain NON-U UNIF Uniform t/t 2 Exact numeric enhancement MaxEnt / MINT IST/hmsIST NESTA / L1 / L2 MDD CLEAN Compound in multiple dimensions. power conserving Compound with any hardware-derived enhancements. FORM? Guesswork eliminated - allows for rational design at detection limits M 0 processed peaks t/t 2