David Rovnyak 4’th Biomolecular SSNMR Winter School Stowe ...winterschool.mit.edu/sites/default/files/documents/... · Bucknell Levi Craft, Melissa Palmer, Riju Gupta, Mark Sarcone,

Sampling

David Rovnyak 4’th Biomolecular SSNMR Winter School

Stowe, VT, 2016

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.!

NSF-MRI NSF-RUI

NIH-AREA

Bucknell-Geisinger Research Initiative

CCSA ACS-PRF-B

Dean’ Fellow Startup Matching

Thank You

Goals/Outcomes!I.  Interesting foundations of sampling/ Classic Fourier Calcs!

II. NUS fundamentals; “good” NUS schedules!

III. How to calculate a NUS sensitivity enhancement!

IV. Speculations on frontiers in sampling!

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!

Signal-to-Noise and Sensitivity!

SNR : ratio of the peak maximum to the rms noise.

rms = 1N

s(ωi )2

i=1

N

∑"

#$

%

&'

12

s(ω0 ) = height

ω0

Sensitivity: ratio of the SNR to the square root of total time (Ernst)

S(ω0 )rmsnoise

SNRtime

P-Set 1.6

delay acquisition

time

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!

RNMRTKVarian

-1.5 -1.0 -0.5 0 0.5 1.0 1.50.00

0.33

0.67

1.00

Freq

uenc

y (a

.u.)

Spectrometer noise Simulated white gaussian noise is this signal or a

Gaussian excursion? Relative Probability

Noise magnitude

Noise and temperature!

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

Complex Fourier Transform!

S(ω ) = S(t)e−i2πωt

−∞

∞

∫

S(t) = S(ω )ei2πωt

−∞

∞

∫

(1)

(2)

P-Set 1.1

The FT is Linear!S(ω ) = ( f (t) + g(t))e−i2πωt

−∞

∞

∫ = f (t)e−i2πωt

−∞

∞

∫ + g(t)e−i2πωt

−∞

∞

∫ = f (ω ) + g(ω )

-1

0

1

-1

0

1

-2

-1

0

1

2

=

time

frequency

FT FT FT

ω1

+

ω2ω1

ω2ω1 ω2

Discrete Fourier Transform!B0

M

x

z

y

Vcoil

time

time

νL frequency

Precession of M induces current in coil

Fourier transform

signal digitized and stored in a computer

Vcoil

free induction decay (FID)

s(t) f(w)

f ωk( ) = s j (t)e−iNjk

j=0

N−1

∑

P-Set 1.7

Discrete Fourier Transform - Nyquist!Nyquist frequency, vmax: let vmax be the highest frequency you wish to detect in units of Hz (s-1). Then you must digitize 2vmax times per second. This requires collecting two points per wavelength �max.

dwell time, Δt : time interval between points; Δt=1/(2vmax); vmax=1/(2Δt)" NMR spectral width, part 2 : sign discrimination detects frequencies from - vmax to +vmax and so the NMR spectral width is 2vmax=1/(�t) .

Example : Δt = 20 μs, then swNMR = 50 kHz"

no sign discrim. "sw=vmax"

with sign discrim."sw=2vmax"

�rf"+vmax" -vmax"

�rf"+vmax" -vmax"

noise is reflected as well as signal!"

noise is reduced by a factor sqrt(2)"

The Discrete Fourier Transform is Cyclic …

F.Delaglio

The Discrete Fourier Transform is Cyclic … As Signals Get Faster Than The Sampling Frequency …

F.Delaglio

Folding!

0.0"νmax!

NMR spectrum, with all lines shown at their correct frequencies"

0.0"ν�max!

νrf!

ν�rf!

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 1.1-1.3*vmax.""Coming up: NUS and spectral width."

Δν"

Δν"

DFT and Resolution!acquisition time, ta : this is NΔt, where N is the number of complex points""digital resolution, Rd: the frequency separation between points in the spectrum Rd=1/ta"

Rd ~ LW"Rd >> LW" Rd << LW"

Zerofilling improves Rd without distorting the spectrum:"

Why is this legal? Roughly : appending 0�s is adding no data. Formally: Parseval."

Keeler Notes

Let’s see the DFT in action.!

!  Credits and thanks: Frank Delaglio!

X( f ) = � x( t ) [ cos( 2�ft / N ) - i sin( 2�ft / N ) ]

Each Fourier term corresponds to a point in the spectrum.

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

�+ �-

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.

�+ �-

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

�+ �-

�+ �-

�+ �-

+

=

Combined, the Real and Imaginary Parts Distinguish Between Positive and Negative Frequencies …

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

�+ �-

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

!  Convolution!

!  Parseval!

!  Nyquist!

FT : Theorems!

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)

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.

(note: horrible typo in original notes)

!  Convolution!

!  Parseval!

!  Nyquist!

FT : Theorems!

F ( f × g) = f (ω )⊗ g(ω )

X=

f(t) x g(t) f(t) g(t)

f(ω) x g(ω)

=ω 0

Xω

f(ω) g(ω)

FT

FT of an Exponential Decay!

!  An exponential decay transforms to a Lorentzian!

S(ω ) = e−t /T2e−i2πνt−∞

∞∫ dt = e−t(1/T2+i2πν )

0

∞∫ dt

= −1(1/T2 + i2πν )

e−t(1/T2+i2πν )⎡⎣⎢

⎤⎦⎥0

∞

= −1(1/T2 + i2πν )

0 −1⎡⎣ ⎤⎦

= 1(1/T2 + i2πν )

(1/T2 − i2πν )(1/T2 − i2πν )

=1/T2

1/T22 + 4π 2ν 2( ) +

−i2πν1/T2

2 + 4π 2ν 2( )= 12π

12 LW

LW 2 +ν 2( ) +−i2πν

1/T22 + 4π 2ν 2( )

Real Part Imaginary Part

P-Set 1.2

0.002

0.004

0.006

0.008

0.010

-200 -150 -100 -50 0 50 100 150 200

S(ν)

frequency (Hz)

T2 = 0.010 s

LW = FWHH = 1πΤ2

(Hz)

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 )

0

∞∫ dt

−t2 2σ 2 − iω t

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:

Complete the square with some b

−t2 2σ 2 − iω t − b2 + b2

Figure out b to lead you to

−1

2σ 2 (t + iωσ 2 )2 − ω 2σ 2

2

e−ax2+bx

−∞

∞∫ dt = e

b24a

πaFrom tables

S(ω ) = 1

α 2πe−ν 2

2α 2 , α = 12πσ

P-Set 1.2

-200 -150 -100 -50 0 50 100 150 200 frequency (Hz)

0.002

0.004

0.006

0.008

0.010

S(ν)

LorentzianGaussian

Parseval : the FT is Power Conserving!

s(t) 2

−∞

∞

∫ = 12π

s(ω ) 2

−∞

∞

∫

P-Set 1.3

Practical consequence: the FT is like life, what you get out of it, is exactly what you put in to it.

7006005004003002001000-100-200-300Chemical Shift (Hz)

noise ∝ time

See for example: Spencer RG. 2010. Equivalence of the time-domain matched filter and the spectral-domain matched filter in one-dimensional NMR spectroscopy. Concepts Magn Reson A 36A:255–265.

Noise!

1*T2!

3*T2!

10*T2!

Analogy to signal averaging.

Noise!

Evolution time (sec) 0.2! 0.4! 0.6!

JBNMR 2004, 30, 1-10

Signal!

S(ω =ω0 ) = eiω0te−R2te−iωt dt0

t∫

= e−R2t dt0

t∫

= T2 1− e−tacq /T2( )

0.5 1.0 1.5 2.0 2.5 3.0time / T2 ppm

-0.3-0.2-0.10.00.10.20.3

Area

Peak Height

peak height = integral of signal envelope

0.5 1.0 1.5 2.0 2.5 3.0time / T2 ppm

-0.3-0.2-0.10.00.10.20.3

Area

Peak Height

integral of signal envelope

time= peak height

r.m.s. noise

SNR in time domain SNR in frequency domain

SN (t) =

T2 1− e−t /T2( )CN t

SNR

this solution strictly true for power conserving transform, such as FT

•  Matson, G. 1977. Signal Integration and the Signal-to-Noise Ratio in Pulsed NMR Relaxation Measurements. J Magn Reson 25, 477-480.

SNR in the Time Domain!

Regarding 1.26 T2!"  Matson, G. 1977. Signal Integration and the Signal-to-Noise Ratio in Pulsed NMR Relaxation Measurements. J Magn Reson 25,

477-480.

"  Becker, E.D., Ferretti. J.A., Gambhir, P.N., 1979. Selection of Optimum Parameters for Pulse Fourier Transform Nuclear Magnetic Resonance. Analytical Chemistry 51(9), 1413-1420.

"  D. Rovnyak, J.C. Hoch, A.S. Stern, G. Wagner, 2004. Resolution and Sensitivity of High Field Nuclear Magnetic Resonance. J. Biomol. NMR. 30, 1-10.

"  T. Vosegaard, N. Chr. Nielsen, 2009. Defining the sampling space in multidimensional NMR experiments: What should the maximum sampling time be? J. Magn. Reson. 199, 147-158.

"  Eriks Kup�e, Lewis E. Kay, and Ray Freeman, 2010. Detecting the “Afterglow” of 13C NMR in Proteins Using Multiple Receivers. J. Am. Chem. Soc. 132 (51), pp 18008–18011.

"  Spencer RG. 2010. Equivalence of the time-domain matched filter and the spectral-domain matched filter in one-dimensional NMR spectroscopy. Concepts Magn Reson A 36A:255–265.

P-Set-1.5

Signal Envelope Determines Sensitivity!

1984  J Magn Reson 58:462–472

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):1262–1263 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

NUS and Sensitivity!

Definitions!

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

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.

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?!

Preaching to converted, but…why NUS?!

LP-FT

FT

MaxEnt

Splitting=10 HzUniform: 1024 samples

Uniform: 128 samples

Non-Uniform: 128 samples(matched exponential density)

Full Resolution in 1/8 Experimental Time

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. 2002 Mar 6; 124(9):1982-93.

Comment: building trust with non-Fourier spectral estimation

On Grid NUS!

Most common approach Samples are a subset of the evenly (uniformly) spaced Nyquist grid.

Equal number of transients for each sample.

Weighted Sampling!

NUS and NUWS have same density of samples.

(Kumar et al., JMR 95, 1-9, 1991)

Weighted Sampling (NUS)

Weighted averaging (NUWS)

Sampling Density

Aside: Processing Weighted Averaging!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.

FT-NUWS lb = 2 Hz

FT-US lb = 2 Hz

FT-US lb = 22 Hz

(Kumar et al., JMR 95, 1-9, 1991)

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) 46-52. Qiang W (2011) Signal enhancement for the sensitivity-limited solid state NMR expderiments using a continuous non-uniform acquisition scheme. J Magn Reson 213:171–175

Exponential Weighted Sampling!

0.5 1.0 1.5 2.0 2.5 3.0

20 40 60 80 100 120

1.0 (match)

1.5

2.0

2.5

3.0

3.5

4.0

ZF1.0

0.8

0.6

0.4

0.2

Evolution time / T2

Sign

al In

tens

ity (a

.u.)

ExponentialNUS BIAS

Sample Number

Aside: Random Sampling, Random Order!

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 3 5 8 10 11 15

0 1 11 5 8 3 15 10

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:

10 5 9 23 3 31 7 17 11 2 28 16 4 32 24 1 19 12 28 8 21 3 32 2 16 …..

Aside: Random Sampling, Random Order!

(Mayzel, Rosenlow, Isaksson, Orekhov, JBNMR 2014, 58: 129-139 )

Convolution and Point Spread Function!PSF – the Fourier transform of the sampling schedule

Mobli, Hoch, Concepts in Magn. Reson. A, 32A, 436-448 (2008)

Point Spread Function!What is a good PSF?

(aside – ignore stray horizontal line)

Smaller gaps between samples Larger gaps

Low noise around central feature corresponds to more robust reconstructions

P-Set 2.1 2.2

Partial Component NUS!

Schuyler, Maciejewski, Stern, Hoch, JMR (2013) 227, 20-24.

t2

t1

t3 (t1,t2)

(t1,t2) = RR, RI, IR, II t1= indirect t2 = indirect t3 = direct

Example NUS sampling schedule for 3D NMR; NUS in both t1 and t2.

Proposal: select RR,RI,IR,II spectra nonuniformly as well.

Partial Component NUS!

Schuyler, Maciejewski, Stern, Hoch, JMR (2015) 254, 121-130.

Summary: PC is a route to introducing randomness, may help improve PSF’s with sparsity.

Spectral Window of NUS!

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. 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, 88−93.

Manassen, Y.; Navon, G. Nonuniform Sampling in NMR Experiments. J. Magn. Reson. 1988, 79, 291−298.

On grid NUS breaks Nyquist.

Main concern is to avoid aliasing within the spectral window.

P-Set-2.2

Spectral Window of NUS!

Maciejewski, M. W.; Qui, H. Z.; Rujan, I.; Mobli, M.; Hoch, J. C. Nonuniform sampling and spectral aliasing. J. Magn. Reson. 2009, 199, 88−93.

Example of selecting NUS from an oversampled grid.

Don’t run for the hills – notice the cross-sections – small effect but worth knowing.

Good schedule? Processor?!Random PSF’s, Gapping, Partial Component

Coverage Non-sparse: 25-50% reduction Sparse: < 25% reduction

Weighted Sensitivity, Resolution

Reproducibility Reporting, validating, regulatory issues

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.

Gridding for SW Minimizing GCD of samples

Calculate a NUS Enhancement!

!  Working through an example!

!  Exact Theory for several cases!

!  Two Theorems for NUS !

!

Fundamentals: Parseval

Example: NUS Enhancement!

!  Let’s see a practical example before looking at the theory.!

An FID!

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5

Evolution time / T2

Uniform Sampling!

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5

32 uniform samples

Evolution time / T2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5

1.00

0.904

0.817 0.738

0.667 0.602

0.544 0.492

0.445 0.402

0.363 0.328

0.296 0.268

0.242

.043 .048 .053

.059 .065 .072

.079 .088 .097 .108 .119

0.132

0.219 0.198

0.179 0.161 0.146

Evolution time / T2

Sum = 9.97!

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Uniform 32 samples

NUS 8/32 matched exponential

NUS 8/32 1.5X bias exponential

NUS 8/32 2X bias exponential

Sample Index

!  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)!

Equal Experiment Times!

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Uniform 32 samples




Sample Index

1 2 3 5 8 12 17 32

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.00 x 4

0.904 x 4

0.817 x 4

0.667 x 4

0.492 x 4

0.328 x 4

.043 x 4

0.198 x 4

Matched NUS!

Sum = 17.79

!  Recall that the uniform experiment and the NUS experiment each used 128 transients !

!  Noise is the same in each experiment.!

Noise is a ‘spectator’!

!  Enhancement for acquisition spanning 3.14 T2 and with NUS probability density matched to T2:Signal Sum by NUS! ! 17.79------------------------------ != !--------- = 1.78 Signal Sum by Uniform ! ! 9.97!

!  Computed by exact theory:*

! !enhance = 1.71!

*MRC, V49, 483-491, 2011

Enhancement: Match NUS!

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Uniform 32 samples




Sample Index

1 2 3 4 8 6 13 32

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.00 x 4

0.904 x 4

0.817 x 4

0.738 x 4

0.492 x 4

0.602 x 4

.043 x 4

0.296 x 4

1.5X Bias NUS!

Sum = 19.57 Enhance = 19.57/9.97

= 1.96 Theory = 2.00

Slight discrepancy?!

compute 1.78 1.96 2.07

theory 1.71 2.00 2.20

match match match

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Sample Index

1 2 3 4

1 2 3

1 2 3 4 5 6

Exact Theory – How does this work? Double Dipping: Enhance SNR by NUS

(MRC, V49, 483-491, 2011)

1.  Eliminate samples > 1.26T2 Decrease Noise.

2.  Use time savings to increase transients for all remaining samples. Increase Signal.

Back to NUWS for a moment!

corrected S/N enhance

1.66

1.08 1.31

(Kumar et al., JMR 95, 1-9, 1991)

Step 1: Enforce Equal Experiment Times!

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

2.01.51.00.5 2.5 3.00.0

uniform

gauss (match)

exp (match)

Sample Density (a.u.)

Force area of h(t) equal to area of uniform with a scaling factor χ.

χ

h(t)

h(t)

Step 2: Form Ratio Of US to NUS Signal!

�

η =χh t( )e−t /T2dt

0

tmax∫e− t /T2dt

0

tmax∫=χ h t( )e− t /T2dt

0

tmax∫T2 1− e

−tmax /T2( )

For any nonuniform sampling density h(t) applied to an exponentially decaying signal:

Cartoon Proof of Enhancement

! !"# $ $"# % %"# &

1

s(t)

�

× =

�

e−t /T2 = SignalUniform!

! !"# $ $"# % %"# &

=

�

e−t /TSMPχ

s(t)

�

×

�

e−t /T2= SignalNUS!

Example Enhancements!

(Paramasivam, Suiter, Sun, Palmer, Hoch, Rovnyak, Polenova, JPC B, V116, 7416-7427, 2012.)

Exponential NUS-based enhancement

Verified with MINT* * Power Conserving Regime of MaxEnt * Non sparse * large lambda

Gaussian NUS!Sa

mpl

e D

ensi

ty

Theoritical Sensitivity Gain Relative to Uniform

Gaussian Matched..........1.59 1.5x Bias.........1.94 2x Bias............2.18

matched Gaussian

2x bias Gaussian

1.5x biasGaussian

x =

0.01.0 2.0 3.0

1.0

0.0

1.0 2.0 3.00.0

1.0

2.0

3.0

4.0

1.0 2.0 3.00.0

1.0

2.0

3.0

4.0

Scaled SIgnalSampling Density Free Induction Decay

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):296–301 Qiang W (2011) Signal enhancement for the sensitivity-limited solid state NMR expderiments using a continuous non-uniform acquisition scheme. J Magn Reson 213:171–175

(JBNMR 58, 303, 2014)

See also: Arbogast, Brinson, Marino, “Mapping Monoclonal Antibody Structure by 2D 13C NMR at Natural Abundance”, Anal. Chem., 2015, 87 (7), pp 3556–3561.

0.5 1.0 1.5 2.0 2.5 3.0

1.0

0.8

0.6

0.4

0.2

Evolution time / T2

Signal (a.u.)

20 40 60 80 100 120

1.0 (match)

1.5

2.0

2.5

3.0

3.5

4.0

ZF ExponentialNUS BIAS

Sample NumberTrade-Off ?

Uniform,FFT

126 125 124 12315N (ppm)

(JBNMR 58, 303, 2014)

NUS Density and Line Shapes

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0

SIN (1.702)

EXP (1.714)

time/T2

Sam

plin

g D

ensi

ty

(JBNMR 58, 303, 2014)

Sinusoid: enhance = 1.7

Matched Exp: enhance = 1.7

More samples by sinusoid density

Improved Peak Base: Sine vs. Matched Exp

(JBNMR 58, 303, 2014)

115.0

115.5

116.0

7.2 7.0 6.8

114.5 114.5114.5115.5 115.5115.5

Uniform Sine Match Exp.

115.0 115.0115.013C (ppm)

Compound SNR!


3D – biosolids NMR: measured enhancements 2.7 to 3.3

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.

00.20.40.60.81

1.2

0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3

1.41.61.8

22.22.4

time / T2

00.20.40.60.81

1.21.41.61.8

22.22.4

00.20.40.60.81

1.21.41.61.8

22.22.4

=xSN

R

En

ha

nce

me

nt

SN

R

Possible NUS EnhancementAbsolute Uniform SNR Absolute (NUS) Sample Sensitivity

Matched

2x Bias

Matched

2x Bias

(Palmer et al, J. Phys. Chem. 2015)

Towards stronger statements about NUS sensitivity

NUS: 2x bias

NUS: match

1.0 2.0 3.00.01.0 2.0 3.00.0t/T2t/T2

b)a)

uniform

FFT of Simulated NUWS data


98.0

99.0

98.0

99.0

98.0

99.0

0.4

0.8

1.2

1.6

2.0SNR

time / T20 1.0 2.0 3.0

Matched

2x Bias

Uniform

6.66.87.07.2

c)

b)

a)

95.100.

*

*

*

6.57.07.5

NUS 2x Bias

NUS Matched

Uniform

1H / ppm 1H / ppm 13C / ppm

EXPERIMENTS Very little improvement due to NUS at 1.26 T2


98.0

99.0

98.0

99.0

98.0

99.0

0.4

0.8

1.2

1.6

2.0SNR

time / T20 1.0 2.0 3.0

Matched

2x Bias

Uniform

6.66.87.07.2

c)

b)

a)

95.100.

*

*

*

6.57.07.5

NUS 2x Bias

NUS Matched

Uniform

1H / ppm 1H / ppm 13C / ppm

6.87.07.2

98.0

99.0

98.0

99.0

98.0

99.0

0.4

0.8

1.2

1.6

2.0SNR

0 1.0 2.0 3.0

Matched

2x Bias

Uniform

6.57.07.56.61H / ppm 1H / ppm 13C / ppm

95.100.

time / T2

c)

b)

a) NUS 2x Bias

NUS Matched

Uniform

Experiments Improvement due to NUS after 1.26 T2


0.4

0.8

1.2

1.6

2.0SNR

0 1.0 2.0 3.0

Matched

2x Bias

Uniform

time / T2

Validation with MINT

(Palmer et al, J. Phys. Chem. 2016) (MINT: Paramasivum et al, J. Phys. Chem. 2012)

13C Chemical Shift (ppm)

13C (ppm)

20

30

40

50

US to 6.4 ms (2T2,average)

NUS35 to 34.1 ms (πT2,max)US to 34.1 ms (πT2, max)

NUS35 to 6.4 ms (2T2, average)

60 50 40 30 20 10

* *

* *

60 50 40 30 20 10

60 50 40 30 20 10 60 50 40 30 20 10

10

20

30

40

50

1.0

1.5

2.0

2.5

0 3.00.5 1.0 1.5 2.0 2.5

4x� Bias

3x� Bias

2x� Bias

Matched(1x)

Enhancement

time/T2

Is the enhancement always greater than 1?

Theorem I (NUS Sensitivity Theorem). Assume h(t) is a positive, nonincreas-

ing function on some interval 0 ≤ t < A , where A≤∞ . For any positive T2 and 0 <

tmax < A, the time domain iSNR enhancement satisfies

η = α1− e−α( )

h(t)e−t T2 dt0

tmax∫h(t)dt

0

tmax∫≥ 1 , (X)

where α = tmax/T2 , and equality holds if and only if h(t) is constant for 0 < t < tmax.

Yes.


Proving the equality statement settles something important: unweighted NUS (sometimes called random NUS) has equal iSNR to uniform sampling.

Theorem 2 (Matched NUS SNR Theorem): The iSNR of matched exponential NUS always has a positive slope for an arbitrary, positive T2 and positive evolution time tmax.

Positive SNR slope with time

Narrower scope, but means:

Exponential NUS simultaneously improves resolution and iSNR with additional experimental time.

0.4

0.8

1.2

1.6

2.0SNR

0 1.0 2.0 3.0

Matched

2x Bias

Uniform

time / T2

1 hr 3 hr

P-Set-3.1

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

GAUSS SINE

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

a) normalized probability densities

b) SNR enhancement by NUS

c) SNR time dependence

0 1.0 2.0 3.0 0 1.0 2.0 3.0

0 1.0 2.0 3.0 0 1.0 2.0 3.0 time/T2 time/T2

ηSN

R

Examples of Theorem 1



COSINE

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

LINEAR

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 1.0 2.0 3.0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6




0 1.0 2.0 3.0

0 1.0 2.0 3.0 0 1.0 2.0 3.0

0 1.0 2.0 3.0 0 1.0 2.0 3.0 time/T2 time/T2

ηSN

R

P-Set-3.2

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

GAUSS SINE

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6




0 1.0 2.0 3.0 0 1.0 2.0 3.0

0 1.0 2.0 3.0 0 1.0 2.0 3.0 time/T2 time/T2

η

SNR

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

COSINE

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

LINEAR

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 1.0 2.0 3.0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6




0 1.0 2.0 3.0

0 1.0 2.0 3.0 0 1.0 2.0 3.0

0 1.0 2.0 3.0 0 1.0 2.0 3.0 time/T2 time/T2

η

SNR

Hmmm. Is iSNR increasing?

Sine Gauss


0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.5 1.0 1.5 2.0 2.5 3.0

LWSMP/LWsig

0.350.25

0.50

0.75

match

1.25

1.50

0.15uniform

SNR

increasing SNR

0 0.5 1.0 1.5 2.0 2.5 3.0

match

1.25

1.50

0.60

0.75

uniform

0.500.30

LWSMP/LWsig

Exponential Sampling Densities Gaussian Sampling Densities

increasing SNR

time/T2 time/T2

The Scope of Theorem 2


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.!

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.

How to select samples via a density?

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 time/T22.00.5 1.0 1.5 2.5 3.0

(ms in preparation; if you use please reference Private Communication with DSR and VGR)

Quadratic decay for sampling density

Random Reproducible / Deterministic Minimize Gaps Sensitivity and Lineshape Ease of Implementation Adherence to Weighting And more…

How to select samples via a density?

(ms in preparation; if you use please reference Private Communication with DSR and VGR)

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 T2* 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.

Other Signal Envelopes

P-Set-3.3

Peter Schmieder, Alan S. Stern, Gerhard Wagner, Jeffrey C. Hoch, ““Application of nonlinear sampling schemes to COSY-type spectra”,

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

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.

•  Exact numeric enhancement

•  Compound in multiple dimensions.

•  Compound with any hardware-derived enhancements.

•  Guesswork eliminated - allows for rational design at detection limits

MaxEnt / MINT IST/hmsIST NESTA / L1 / L2 MDD CLEAN …

1 2 30t/T2

1 2 30t/T2

NON-UNIFORMUNIFORM

time domain

processed peaks

NUS

Uniform ? 1 2 30t/T2

1 2 30t/T2

NON-UNIFORMUNIFORM

time domain

processed peaks

power conserving

Thank you.

Documents

David Rovnyak 4’th Biomolecular SSNMR Winter School Stowe ...winterschool.mit.edu/sites/default/files/documents/... · Bucknell Levi Craft, Melissa Palmer, Riju Gupta, Mark Sarcone,