Discrete decoding based ultrafast multidimensional nuclear
magnetic resonance spectroscopy
Zhiliang Wei, Liangjie Lin, Qimiao Ye, Jing Li, Shuhui Cai & Zhong Chen
Supplementary Materials
1. Comparisons between different 3D decoding strategies
Figure S1 Schematic diagrams of 3D CONECSY sequence developed with different decoding
strategies; (a) is developed from previously reported nested continuous decoding and (b) is
constructed based on discrete decoding and continuous decoding. All mixing periods are removed
to focus on differences among different decoding strategies. The open rectangles in the RF axis
represent 90o hard pulses; eclipses with an arrow chirp pulses; GEξ (ξ = X, Y, Z) encoding
gradients along the ξ axis; NDξ, GDξ and TEξ (ξ = X, Y, Z) repeating numbers of the basic decoding
unit, intensities of decoding gradients, and total encoding durations along the ξ axis, respectively;
TD duration of basic repetition module in nested decoding; and GPZ and GPX compensative
gradients for adjusting echo centers.
For focusing on differences between the continuous decoding and the discrete
decoding, all mixing periods are removed from the pulse sequences. The sequence
based on nested continuous decoding, as shown in Figure S1(a), is constructed
1
according to the previous reports1,2 by Frydman et al. Phase evolutions of the two
sequences will be carried out sequentially to demonstrate principles of different
decoding strategies. Compensative gradients (GPZ and GPX) prior to the acquisition
period merely serve to adjust the echo center and thus are omitted in the following
analyses. For descriptive convenience, some symbols are defined as follows: Rξ (ξ =
X, Y, Z) denotes the frequency-swept ratio of chirp pulse, Ω the chemical shift, J the
scalar-coupling constant, and Octr central excitation frequency of chirp pulses.
Regarding the sequence in Figure S1(a), the decoding gradient along the X axis
(denoted as X-decoding gradient) is numerically labelled from 1 to 2NDX. Within each
X-decoding gradient, the decoding gradient along the Z axis (denote as Z-decoding
gradient) is numerically labelled from 1 to 2NDZ. Durations of each X-decoding and
Z-decoding gradient are
TD
2
and TD
 4 N DZ 
, respectively. Regarding an
acquisition moment in the (2m+1)-th (m = 0, 1,…, NDZ-1) Z-decoding gradient within
the (2n+1)-th (n = 0, 1,…, NDX-1) X-decoding gradient, the phase can be given as
1 
4 GEZ Z
4 GEX X
   Octr    GDZ Zta 
   Octr 
RZ
RX
(S1)
 mTD



mTD
  GDX X 
 ta      J   nTD 
 ta  ,
2 N DZ
 2 N DZ



where ta the duration from the beginning of (2m+1)-th Z-decoding gradient to the
acquisition moment. Echoes arise when

 4    Octr  GDZ 

ta   0,
and
 GEZ Z 
RZ
GEZ 



 4    Octr  GDX  mTD



 ta    0.

 GEX X 
RX
GEX  2 N DZ



(S2)
Denoting dimensions encoded by Z and X gradients as the F1 and F2 dimensions, we
obtain the spectral width (SW) as3-5
GDZ RZ TD

 SWF 1  16G N ,

EZ
DZ

G
R
T
 SW  DX X D .
 F 2
8GEX
2
(S3)
The bipolar X-decoding gradients are repeated by NDX times, constructing a Fourier
transformation based dimension (denoted as the F3 dimension) with the dwell time of
TD. In other words, the spectral width in the F3 dimension is
SWF 3 
1
.
TD
(S4)
In spatiotemporal encoding experiments, the dynamic frequency range of chirp pulse
is set as the dispersed frequency range induced by the encoding gradient, namely,
R 
 GE L
TE
  X , Y , Z  ,
(S5)
where Lξ represents the detective sample length along the ξ axis. Therefore, the
spectral widths in different dimensions can be associated as
 LZ GDZ

 SWF 1 SWF 3  8T N ,

EZ
DZ

 SW SW   LX GDX .
F3
 F 2
4TEX
(S6)
From Eq. (S6), it can be seen that trade-offs arise among spectral widths in different
dimensions.
In sequence shown as Figure S1(b), discrete decoding is utilized to unravel the
encoded information in both X and Y axes. The discrete decoding gradients in the X
and Y axes are denoted as X-decoding and Y-decoding gradients, respectively.
Regarding an acquisition moment within the Z-decoding gradient following the
(2n+1)-th (n = 0, 1,…, NDY/2-1) Y-decoding gradient after the (2m+1)-th (m = 0, 1,…,
NDX/2-1) X-decoding gradient, the phase can be given as
1' 
2 m 1
4 GEX X
4 GEY Y
   Octr     GDXk X  DXk 
   Octr 
RX
RY
k 1
4 GEZ Z
   GDYl X  DYl 
   Octr    GDZ Zta ,
RZ
l 1
2 n 1
(S7)
where GDXk and GDYl represent the intensities of X- and Y-decoding gradients with
3
corresponding durations of  DXk and  DYl , respectively; ta the duration from the
beginning of the Z-decoding gradient to the acquisition moment. From Eq. (S7), the
spectral widths are

 LX N DX
SW

GDXk DXk ,

 F1
2
T
k

1
EX


 LY N DY
SW

GDYk DYk ,
 F2

2
T
k

1
EY


 LZ
GDZ DZ ,
 SWF 3 
2TEZ

(S8)
where  DZ denotes the duration of Z-decoding gradient. It can be observed from Eq.
(S8) that spectral widths in different dimensions are no longer associated with one
another, liberating the spatiotemporal encoding technique from trade-offs among
spectral-width performances in different dimensions.
In the case of nested continuous decoding (Figure S2(a)), the positive and
negative Z-decoding gradients decode mirror information. However, with the
utilization of discrete decoding, the positive and negative Z-decoding gradients
unravel different information (Figure S2(b)). From Figure S2(a), the nested decoding
performs uniform decoding without specificity. By utilizing a prior knowledge from
1D NMR measurement, the discrete decoding (Figure S2(b)) focuses the decoding on
several signal-bearing sites. Note that the data acquired by nested decoding should be
divided into four sets (interlaced Fourier transformation may be used). By contrast,
the data acquired with the module shown as Figure S2(d) can be directly constructed
into one single data set. This difference primarily results from the employment of
discrete decoding strategy.
4
Figure S2 Illustrative demonstrations of different decoding strategies; (a) and (b) represent the
scanning of acquisition space with decoding modules schematically shown in (c) and (d),
respectively.
2. 2D cross sections of the 3D JCOSY and CONECSY spectra
Figure S3 2D cross sections of the 3D JCOSY spectrum at F1 = 4.19 (a), 3.69 (b), 2.98 (c), 2.48
(d), 2.15 (e), 1.30 (f) and 0.96 (g) ppm, respectively.
5
Figure S4 2D cross sections of the 3D CONECSY spectrum at F1 = 0.92 (a), 1.50 (b), 1.94 (c)
and 3.69 (d) ppm, respectively.
3. JCOSY spectrum by the ultrafast technique with averages
Figure S5 JCOSY spectra by the ultrafast technique with 3840 averages; (a) represents original
JCOSY spectrum; (b) 2D projection onto the F1F2 plane; (c) 2D projection onto the F2F3 plane
along with 1D projection onto the F2 dimension.
6
As measured, the SNRs of conventional JCOSY, ultrafast JCOSY with and without
averages are 1589.7, 468.5 and 27.9, respectively.
As described in previous report6, the ultrafast experiments suffer from sensitivity loss
due to the enlargement of receiver bandwidth:
SNRCONV

SNRUF
fbwUF
,
fbwCONV
(S9)
where SNRUF and SNRCONV denote SNRs of the ultrafast experiment and conventional
2D experiment per scan; fbwUF and fbwCONV receiver bandwidths for ultrafast and
conventional experiments. Equation (S9) can also be written as
SNRCONV
 2 N1
SNRUF
(where N1 is the number of acquisitions along the indirect domain) in cases of
ultrafast 2D experiments. In this study, we deal with ultrafast 3D experiments and
thereby retain the original form as Eq. (S9). From Eq. (S9), we can infer that ultrafast
fbwUF
fbwCONV
experiments suffer from a ~
SNR penalty in comparison with
conventional counterparts. In JCOSY experiments, we set the parameters as
fbwUF  100.0 kHz and
experiment should be
fbwCONV  2.0 kHz , indicating that the SNR of ultrafast
1
that of conventional 3D experiment per scan.
50
The above analysis depicts the SNR comparison between ultrafast experiment
and conventional counterpart per scan. In order to analyze the multi-scan case, we
first explore the SNR difference between 2D and 1D experiments. The pulse sequence
for 2D experiment is 90o ~ t1 ~ t2 (mixing period is removed to focus on SNR analysis),
and for 1D experiment is 90o ~ t2. The digitized signal from the 2D experiment carried
out on a sample possessing a single chemical shift Ω and a magnetization per unit
length M0 is:
s(m, n)   dzM 0e
im1 
m1
n
in 2  2
T2
T2
e
,
(S10)
where m ranges from 0 to N1-1 and n from 0 to N2-1; τ1 and τ2 denotes dwell time for
the indirect and direct dimensions, respectively. For the purpose of calculating SNR, it
7
can be assumed without loss of generality that Ω = 0. Performing Fourier
transformation, we obtain
N 11 N 2 1
1
  s(m, n)e
N1N 2 m 0 n 0
F (k1 , k2 ) 
  2 ik1 
N 1  1 

N1 
 T2
LM 0 1  e

 2 ik1
 1
N1N 2
T2
N1
1 e
2 imk1
N1
1 e
e
2 ink2
N2
  2 ik2 
N 2  2 

N2 
 T2
1 e

 2 2 ik2
T2

(S11)
,
N2
where k1 ranges from 0 to N1-1 and k2 from 0 to N2-1; L the detective sample length;
The maximum value of F(k1, k2), corresponding to the peak’s maximum intensity,
arises when k1 = k2 = 0, namely,
 N1
Fmax
1
N 2
2
LM 0 1  e T2 1  e T2

.


 1
 2
N1N 2
T2
T2
1 e
1 e
(S12)
Similarly as the analyses in Reference 6, we assume that the total acquisition time in
the direct dimension and largest evolution duration in the indirect dimension are set
on the order of T2. Thereafter, Eq. (S12) can be rewritten as
Fmax 
1  e 
N1N 2
LM 0
1 2
T2 T2
1  2
 LM 0 N1N 2 1  e1  .
2
(S13)
Denoting the noise as nrms , we obtain SNR for the 2D experiment under analyses as
SNR2 D 
LM 0 N1N 2 1  e 1 
nrms
2
(S14)
.
After performing similar analyses to the 1D experiment, we obtain
SNR1D 
LM 0 N 2 1  e 1 
nrms
(S15)
.
Obviously, the 2D experiment involves N1 scans while the 1D experiment single scan
merely. Recording the 1D spectrum with N1 averages, we obtain
SNRAve1D 
LM 0 N1N 2 1  e 1 
nrms
.
(S16)
It can be observed from Eq. (S14) and Eq. (S16) the 2D experiment and 1D
experiment within identical time differ in SNR:
8
SNR2 D
 1  e 1 .
SNRAve1D
(S17)
From a macroscopic point of view, the 2D experiment suffers from more transversal
relaxations prior to the acquisition period than the 1D experiment does, thereafter
introducing SNR loss. Given identical time, averages are likely to generate better
SNR than scans, which experience progressive delays to construct the indirect
dimension. This principle also applies to conventional 3D JCOSY and ultrafast
JCOSY with averages. Since the conventional 3D JCOSY involves two indirect
dimensions, therefore, it can be concluded that
SNRAve UF  JCOSY
1

 35.4%.
2
SNRCONV  JCOSY
50 1  e 1 
From our measured results,
SNRAveUF  JCOSY
SNRCONV  JCOSY

mea .
(S18)
468.5
 29.5% , being smaller
1589.7
than the theoretical ratio. This difference may result from the sinc-wiggle in ultrafast
experiments.
4. Theoretical calculations of the continuous decoding based ultrafast
experiment
Figure S6 Schematic diagram of the ultrafast 3D JCOSY based on continuous decoding. Open
rectangles in the RF axis represent 90o hard pulses; eclipses with an arrow chirp pulses; GEξ (ξ = X,
Z) encoding gradients along the ξ axis; NDξ, GDξ and TEξ (ξ = X, Z) repeating numbers of the basic
decoding unit, intensities of decoding gradients, and total encoding durations along the ξ axis,
respectively; TD duration of basic repetition module in nested decoding; and GPZ and GPX
compensative gradients for adjusting echo centers.
9
The following calculations are performed in the precondition that continuous
decoding based JCOSY and discrete decoding based JCOSY share identical encoding
parameters. Regarding the system we study here, the spectral width is set as 4.0 ppm
(2.0 kHz), namely, SWF2 = SWF1 = 2 kHz. TD is set as 20 ms to provide a spectral
width of 50.0 Hz in the F3 dimension. The detective sample length along the X (LX)
and Z (LZ) axes are 5.0 and 16.0 mm, respectively. According to Eq. S6
 LZ GDZ

SW
SW

,
F
1
F
3

8TEZ N DZ

(
), it can be calculated that
 SW SW   LX GDX .
 F 2 F 3
4TEX
GDX 
4TEX SWF 2 SWF 3
 3.75(G/cm).
 LX
(S9)
Note that the digital resolution should be better than real spectral resolution, namely,
SWF 2
RX

.
N DZ
4 GEX LX
(S10)
It can be deduced that
N DZ 
4 GEX LX SWF 2
 36.
RX
(S11)
Then, it can be calculated that
GDZ 
8TEZ N DZ SWF 1SWF 3
 84.38(G/cm).
 LZ
(S12)
As described in the manuscript, the maximum available gradient intensity along the Z
axis is 64 G/cm for the spectrometer in this study. In other words, the required
gradient is beyond the hardware of our current spectrometer. It can also be inferred
that the nested continuous decoding scheme set higher requirements on the hardware.
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11