Clean Absorption Mode NMR Data Acquisition Based on Time-Proportional Phase Incrementation
Supporting Information
Clean absorption Mode NMR data acquisition based on time-proportional phase incrementation
Yibing Wu‡•Arindam Ghosh‡•Thomas Szyperski
Yibing Wu‡•Arindam Ghosh‡•Thomas Szyperski ()
Department of Chemistry, The StateUniversity of New York at Buffalo, Buffalo, NY 14260,
E-mail: .
‡: These authors made equal contributions to this work and should both be considered as first authors.
- c+- sampling(Forward TPPI)
- c- sampling (Backward TPPI)
IIII.Dual TPPI sampling
III.1 cADD- sampling (Sum offorward and backward TPPI interferograms)
III.2 cSUB-sampling (Difference offorward and backward TPPI interferograms)
III.3 Measurement of phase shift
III.4 Simulations to verify theory of dual TPPI data acquisition
IV.Generalization to multiple chemical shift evolution periods
IV.1 Multidimensional NMR
(a) c+c+-sampling for K = 1
(b) cc-sampling for K = 1
(c) cADDcADD-sampling for K = 1
IV.2 RD projection NMR
(a) c+c+-sampling for K = 1
(b) cc-sampling for K = 1
(c) c+c+ + cc-sampling for K = 1
(d) cADDcADD-sampling for K = 1
V.Comparative contour plots and cross sections
V.1 RD 2D CC(CON)HN recorded with forward TPPI in RD dimension
V.2 RD 2D CC(CON)HN recorded with ‘double’ dual TPPI in RD dimension
In this Supporting Information, we describein detail the theory of TPPI sampling schemes alluded to in the paper, using the following definition to characterize the schemes: the time evolution of chemical shift canbe sampled with N time domain data point (indexed by q = 1,2,3…,N)employing Time Proportional Phase Incrementation (TPPI) as or, equivalently, as , with representing a chemical shift, ‘±’ indicating forward (‘+’) and backward (‘’) sampling, denoting the TPPI phase increment and representing aphase shift which gives rise to a dispersive frequency domain peak component.In the following, equations are derived for the realtime domain signal c±(t).
The real part of complex Fourier transformation (FT) of c±(t) (or equivalently a real FT)yields a frequency domain peak located at /t)(see below) and itsquadrature image peak (‘quad peak’) located at /t), which is discarded(see below), both consisting of an absorptive peak component and a dispersive peak component. Here,represents the‘TPPI frequency shift’. The four peak components can be written as elements of a vector F. With
A+denoting theabsorptive peak component of the peak at +,
D+denoting the dispersive peak component of the peak at +
Adenoting the absorptive peak component of the peak at , and
Ddenoting the dispersive peak component of the peak at ,
the frequency domain signal is then proportional to
Re([c±(t)]) F= (S1).
denotes the complex FT and = represents a ‘coefficient vector’ which provides the relative intensities of absorptive (A+) and dispersive (D+) peak components located at frequency +, as well as the relative intensities of absorptive (A) and dispersive (D) peak components located at the quad peak frequency –.
I. c+ -sampling (Forward TPPI)
The forward TPPI sampled time domain signal is proportional to
(S2),
where c+(t) is the discretely forward TPPI sampled interferogram, t = (q1)t =0,t,2t,3t,…,Ntand = /t denotes the “TPPI frequency shift”. With 1/t = 2•SW, where SW is the spectral width in Hz and = /2, the TPPI shift for the chemical shift is given by = •SW rad s-1. Eq. S2 yields, after discarding the modulations giving rise to the discarded quad peak:
(S3).
After FT, one obtains for c+-sampling:
(S4),
yielding peak components A+ and D+ scaled with cos and sin, respectively.
Phase correction of the frequency domain spectrum requires both the real and the imaginary part. Complex FT of the time domain signal given in Eq. S2 is given by
(S5),
whereR and I are the real and the complex parts of the spectrum. The well known phase correction requires combining the real and imaginary parts according to cosR+sinI, which yields:
(S6).
Hence, the dispersive peak component of the desired frequency domain peak at is eliminated while thedispersive peak component of the discarded quad peak is doubled. A first order phase correction requires that the ‘pivot point’[2] is located at .
II. c-sampling (Backward TPPI)
The backward TPPI sampled time domain signalis proportional to
(S7).
which yields, after discarding the modulations giving rise to the discarded quad peak,
(S8).
After FT, one obtains for c-sampling:
(S9),
that is, the sign of the dispersive peak component D+ is inverted when compared with c+-sampling.
III. Dual TPPI sampling
III.1cADD-sampling (Sum of forward and backward TPPI interferograms)
Addition of St)in Eq. S3and St)in Eq. S8 yields for such dual TPPI sampling
(S10).
After FT, one obtains for -sampling:
(S11),
that is, the dispersive peak componentsD+ cancel and the absorptive peak component A+ is scaled by cos.
III.2cSUB-sampling (Difference of forward and backward TPPI interferograms)
Subtraction of St)in Eq. S8 from St)in Eq. S3 yields for such dual TPPI sampling
(S12).
After FT, one obtains for -sampling:
(S13),
that is, the absorptive peak componentsA+ cancel and the dispersive signal at D+ is scaled by sin. A zero order phase correction of /2, or a Hilbert transformation [1], converts D+ into A+, so that
(S14).
III.3Measurement of phase shift
Inspection of Eqs. S11 and S14 reveals that measurement of the ratio ofand , which can readily be measured by frequency domain signal integration, enables one to measure the phase shift according to
(S15).
For experimentsdesigned to encode an NMR parameter inthis enables one to measure .
III.4Simulations to verify theory ofdual TPPI data acquisition
Numerical simulations were performed using the program package MATLAB to verify equations provided in sections I-III. Constant amplitude time domain cosine modulations with oscillation frequency of 2,000 Hz and phase shift = 300weregenerated. The TPPI frequency shift () was chosen to be 6,000 Hz. Prior to performing complex FT, the time domain data was subjected to apodization using cosine bell window function.
Figure S1 shows the real part of the frequency domain spectrum obtained by complex FT of the simulated time domain signal. With a TPPI frequency shift of 6,000 Hz, the peak corresponding to frequency Hzappears at 8,000 Hz,while the quad peak appears at 8,000 Hz.
FigureS1.Simulated spectra obtained by employingTPPI sampling with = 6,000 Hz. The desired and the quad peaks(on the left, indicated with ‘Quad peak’) appear at 8,000 and 8,000 Hz, respectively(see text).
In Figs, S1a and S1b, the spectra corresponding to forward TPPI sampling (Eq. S2) are shown without and with phase correction, respectively.As predicted by Eq. S6, the phase correction eliminates the phase of the desired peak while that of the quad peak is doubled.
Fig. S1c shows the spectrumobtained when using backwardTPPI sampling (Eq. S7), where the phases of dispersive components of both the desired and the quad peak are reversed when compared to the forward TPPI sampled spectrum (Fig. S1a).
Fig. S1d shows the spectrum obtained afteraddingthe forward and backward TPPI sampling spectra shown in Figs. S1a and S1c, where the dispersive peak components are eliminated (Eq.S10).
Fig. S1e shows the spectrum obtained after forming the difference subtraction of the forward and backward TPPI sampling spectra shown in Figs. S1a and S1c, where the absorptive peak components are eliminated (Eq.S12).
Fig. S1f shows the spectrum obtained after a 90o zero-order phase correction of the spectrum shown in Fig. S1e. The ratio of peak volumes of the desired (or quad) peak in Fig. S1e and those in this spectrumis given by tan (Eq.S15) and enables measurement of Ф.
IV. Generalization to multiple chemical shift evolution periods
IV.1MultidimensionalNMR
TPPI sampling of indirect dimension can readily be generalized to K+1 indirect dimensions of a multi-dimensional NMR experiment, where K+1 chemical shifts 0, 1,…,K, that are associated with phase shifts 0, 1,…K, are sampled using either forward, backward or dual TPPI manner. For c±-sampling, the corresponding interferogramisobtained by K-fold tensor product formation:
(S16).
In the following, the generalization of TPPI based samplingof multidimensional NMR spectra is derived for two indirect dimensions (K = 1), considering that the majority of practical applications will be for K = 1. Generalization to higher dimensionality (K > 1) is straightforward.
(a)c+c+–sampling for K = 1:
Starting with Eq. S3, the time domain signal for two indirect dimensions is proportional to
(S17).
After FT, one obtains
(S18).
The intensity of the desired peak component (A1+)(A0+), which is absorptive in both dimensions and is located at (+1+,+), is modulated with cos(cos(. In general, for K+1 dimensions the relative intensity of the desired peak component is modulated with , where j represents the phase shift in the jth dimension. The other terms in Eq.S18 represent peak componentswhich are dispersive in either one or in both dimensions.
(b)cc–sampling for K = 1:
Starting with Eq. S8, the time domain signal for two indirect dimensions is proportional to
(S19).
After FT, one obtains
(S20).
(c)cADDcADD–sampling for K = 1:
Dual TPPI for both indirect dimensions requires all possible combinations of forward and backward TPPI sampling, that is,starting with Eq. S10, the time domain signal is proportional to
(S21).
The first and the second terms in Eq.S21 are represented, respectively, by Eqs.S17 and S19. It is then straightforward to show that the time domain signalcorresponding to the third term in Eq.S21 is proportional to
(S22).
After FT, one obtains
(S23).
Thetime domain signal represented by the fourth term in Eq.S21 is proportional to
(S24).
After FT, one obtains
(S25).
Addition of the spectra represented by Eqs. S18, S20, S23 and S25 give rise to aspectrum exhibitingonly the absorptive peak componentscaled bycos1cos0.
In general, for K+1 dimensions the relative intensity of the absorptive peak componentobtained using multiple dual TPPI is proportional to.
IV.2RD projectionNMR
TPPI can be applied to an arbitrary sub-set of K+1 chemical shift evolution periods which are jointly sampled in RD NMR, whereK+1 chemical shifts 0, 1,…K , which are associated with phase shifts 0, 1,…K, are measured as linear combinations (0+±(11+±…±KK+.The scaling factors j enable one to achieve different maximal evolution times for the different jointly sampled shifts, while proper adjustment of the TPPI frequency shiftsenablesone to shift peaks originating from different linear combinations of chemical shifts to different spectral regions,thereby avoiding additional spectral overlap arising from the projection.
Choosingc--or c+-sampling for each of the shift evolution periods, the resulting interferogramcan beobtained by K-fold tensor product formation according to:
(S26),
where. In the following, the generalization of TPPI based sampling ofRD NMR spectra is derived for two jointly sampled chemical shifts, where 0 is detected in quadrature and 1is encode in an in-phase splitting (K = 1), considering that the majority of practical applications will be for K = 1. Generalization to more than two jointly sampled chemical shifts (K > 1) is straightforward.
(a)c+c+ –sampling for K = 1:
Starting with Eq. S3, the time domain signal is proportional to
(S27).
FT reveals that the first two termsgive rise to the chemical shift doublet with components located at 0++ and +, with phase shifts 1+0 and 1+0, respectively.The locations of these peak components provide the desired linear combinations0 and 0of the jointly sampled chemical shifts 0 and The third and the fourth terms give rise to discarded quadrature image peaks. Thephaseassociated withagiven linear combinationof chemical shifts is then given by an analogous linearcombination of the associated phase shifts.
(b)cc –sampling for K = 1:
Starting with Eq. S8, the time domain signal is proportional to
(S28).
FT revealsthat the sameshift doublet components are obtained as with forward sampling (Eq. S27), while thephases are of opposite signs.
(c)cc+ cc –sampling for K = 1:
Addition ofEqs. S27 and S28 yields
(S29).
Inspection of Eq. S29 reveals that for obtaininga clean absorption mode RD NMR spectrum using dual TPPI, it is sufficient to addtwo spectra, i.e., one recorded with forward sampling of both chemical shift evolution periods and one recorded with backward sampling for both periods. However, the intensities of the peaks forming the peak doublet at 0++ and +, are then scaled differently by cos(1+0) and cos(1+0), respectively.
Eq. S29 can be extended for an arbitrary number of projected chemical shifts, where with associated phase shifts N,N-1,…,1yield
(S30).
Inspection of Eq. S30 reveals that, in contrast toclean absorption mode GFTNMR spectra (devoid of quadrature peaks) acquired using the hypercomplex method [7], clean absorption mode RD experiments can be obtained by recording only two RD NMR spectra, independent of the number of jointly sampled chemical shift evolution periods.
(d)cADDcADD–sampling for K = 1:
The imbalance of intensities of the peak pairs forming the chemical shift doublet (Eq. S29) can be eliminated by using dual TPPI for both evolution periods. This requires recoding of all possible combinations of forward and backward sampling for the two periods. Starting with Eq. S10, the resulting time domain signal is proportional to
(S31).
The first and the second terms in Eq.S31 are represented, respectively, by Eqs.S27 and S28.
The time domain signal corresponding to the third term in Eq.S31 is proportional to
(S32),
andthe time domain signal corresponding to the fourth term in Eq.S31 is proportional to
(S33).
Addition of two RD NMR spectra represented by Eqs. S32 and S33 yields aclean absorption mode spectrum. However, theintensity imbalance of the two components of the shift doublet is ‘swapped’ when compared to Eq. S29:
(S34).
Addition of the spectra represented by Eqs. S29 and S32yields:
(S35).
FT reveals that using ‘double’ dual TPPI sampling yields a clean absorption mode RD NMR spectrum in which the components of a given chemical shift doublet have the same intensity and are scaled bycos1cos0. In general, for K+1 jointly sampled chemical shifts, the relative intensity is proportional to.
If the quadrature detection of 0 is accomplished by using the hypercomplex method, the same results are obtained.
V. Comparative contour plots and cross sections
Contour plots ofthe RD 2D CC(CON)HNspectraalluded to in the paper are shown in this section.
V.1RD 2D CC(CON)HN recorded with forward TPPI in RD dimension
V.2RD 2D CC(CON)HN recorded with ‘double’ dual TPPI in RD dimension
S1