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A new scheme for resolution enhancement in fourier transform NMR
JOURNAL OF MAGNETIC RESONANCE 33, 457-463 (1979)
A New Scheme for Resolution Enhancement in Fourier Transform N M R B . CLIN,* J. DE BONY, P. LALANNE, J. BIAIS, AND B. L E M A N C E A U
Centre de Recherche Paul Pascal, Domaine Universitaire 33405 Talence Cedex France Received June 27, 1977; revised June 9, 1978 A new method for resolution enhancement in FT NMR is described, the physical principle of which is that of digital filtering by a sinusoidal window. The practical aspect of the method is developed, showing that the result can be obtained on a classical spectrometer without additional programming and with great versatility.
INTRODUCTION
A number of new methods have recently appeared for enhancing resolution or improving the signal to noise ratio in NMR spectroscopy (1, 2, 3). These techniques distort the shape of the peaks in order to achieve an enhanced resolution. For the "non-specialist" who wishes routinely to improve his measurements without having to cope with difficult mathematical or electronic problems, techniques with a minimum of adjustable parameters offer some advantage. From that point of view, digital filtering by a sinusoidal window, proposed by De Marco and Wiithrich (4), is particularly promising since its application does not require any delicate adjustment of parameters. The aim of this paper is to show that one can get the same results, without complicated mathematical manipulation of the signal, and that in the process of applying our technique, one can obtain a useful insight into the mechanism of digital filtering. MATHEMATICAL ANALYSIS
In their paper De Marco and Wiithrich (4) propose to multiply the free induction decay (FID) by a sinusoidal function with zero phase and a period of twice the acquisition time AT. If T* is the characteristic time for the decay, the lineshape after this procedure becomes, neglecting terms in exp(-AT/T*) (formula [2] of (4)). [~',trT22" 1 + (~'T2/AT) , 2 - Ato2T2,2 L(sin) a t - ~ - ) (1 + (zrT*/AT) 2 - AO92,~-2,2 12 ) +4Ato2T22"
[1]
What we want to point out is that this result can be writte 0 in another way:
~/AT-zato
1( ~r/AT + &a L(sin) a ~ 1 + T2*2 (w/AT + ato) 2 + 1 + T22 (~r/AT- &o)2J"
[2]
* To whom correspondance should be addressed 457
0022-2364/79/020457-07502~00/0 Copyright© 1979by AcademicPre~s,Inc. All rightsof reproductionin any formreserved. Printedin GreatBritain
458
CLIN
ET AL.
b C \
\ I | | I | |
V F~G. 1. (b) is obtained by adding (a) and (c). The new line shape is in fact obtained by addition of two dispersion Lorentzian functions of two different variables (zr/AT+A¢o) and (zr/AT-A¢o). This can be represented (Fig. 1) as two Lorentzian dispersion functions of the variable Aoj, shifted with respect to the origin by an amount of (+~r/AT), and 180 degrees out of phase. This result is quite general and using the theorem on Fourier transform and convolution products (5), the Fourier transform of an FID multiplied by a sinuso'/dal function can be written: FT(FID x sin
at) = FT(FID) ®
FT(sin at)
I-3]
where ® means "convolution product": 4-00
f(x)~) g(x)= f f(xt)g(x-xt) dxl
[4]
--oo
The Fourier transform of the function (sin at) represents the difference between the two delta functions 8(A~o + a ) and 8 ( A ~ 0 - a ) (5). Consequently the Fourier transform of the FID, filtered by a sinusoidal window of period 2rr/a will be the result of subtraction between the transform of initial FID, shifted in the frequency domain by an amount of a/2rr and the same, shifted by an amount of -a/27r. Note that the convolution of any function by a delta function leads to a change in the value
RESOLUTION ENHANCEMENT IN FOURIER TRANSFQRM
459
of the origin of the abscissa: +OO
f f(x)6(x-Xo) dx = f(xo).
[5]
--o0
The number of data points acquired for an FID is 2N. There is a relationship between the spectra width (SW), the acquisition time (AT) and this total number of samples taken (2N): 2 N = ~ × SW × A T
[6]
i.e., N = SW x AT. As, after Fourier transformation, the real part of the spectrum occupies only N data points, the frequency difference between two successive addresses is
kv = SW/N
= 1/AT
(=&o/2rr).
[7]
This value corresponds to the shift between the two Lorentzian curves (formula [2]) in the special case studied by De Marco and W/ithrich, where the period of the sinusoidal window is twice the acquisition time. APPLICATION The preceding remarks lead to a new method for obtaining the results of (4). From a practical point of view, one carries out the following procedure: (1) Calculate the FT of the original FID, and then perform the classical phase correction. It is the dispersion part of the signal that is of interest, because the sinusoidal function introduces a 90 ° phase shift. (2) Duplicate the signal in another block of memory. Then perform a left shift of this block by one address. (3) Subtract from this signal the original dispersion signal. (For users of Nicolet computers with any of a variety of programs, the procedure is to set DC-- - 1, define D F and D T blocks and issue the commands MV, LS and AT). DISCUSSION The method we have described, easy and fast to perform, also permits one to understand the non-universality of the "sine-bell" routine of De Marco and Wfithrich. For a signal with well-defined physical characteristics (i.e., resolution), the result of the "sine-bell" routine will depend upon the size of the block reserved for the acquisition, i.e., upon the acquisition time, for a given spectral width (formula [6]). As an example, we show in Fig. 2a a Lorentzian line obtained after Fourier transformation of a given FID. In Fig. 2b we show the result of the "sine-bell' routine in this case: it's clear that the loss of signal to noise is too severe. (In fact, the "sine-bell" routine has been simulated using the routine described in the preceeding paragraph). On Fig. 2c we show the result of the "sine-bell" routine applied to an
b
c
d
e
FIG. 2. (a) Conventional spectrum. (b) Resolution enhancement obtained by difference of two dispersed lorentzian curves shifted by one address, the acquisition having been performed on 16 K points (same result as "sine-bell" routine). (c) Resolution enhancement obtained by difference of two dispersed lorentzian curves shifted by one address, the acquisition having been performed on 4 K points (same result as "sine-bell" routine). (d) Resolution enhancement obtained by difference of two dispersed lorentzian curves shifted by four addresses, the acquisition having been performed on 16 K points. (e) Same results as (d) or (c) after addition of spectrum (a) multiplied by a factor 0.5.
a
RESOLUTION ENHANCEMENT IN FOURIER TRANSFORM
461
42,
fHOD 8
b
t
FIG. 3. Part of the spectrum of a polypeptide. (a) Conventional spectrum. (b) Resolution enhanced spectrum, following a classical sequence (two shifts, and add the real part, multiplied by a factor 0.3).
462
CLIN E T AL.
FID corresponding to the same sample, with the same characteristics for the acquisition, but for an acquisition time that is one fourth as long. On Fig. 2d we show the result of a more general ~ipplication of our routine to the same FID as the one used for the case of Fig. 2b. To obtain a result comparable with the one of Fig. 2c, we had only to perform four successive shifts before subtracting a Lorentzian curve from the other. It was obvious that, the only difference between the two cases being the two acquisition times ATa = 4ATe), the same result could be obtained by performing in one case a shift of four addresses, and in the other, a shift of one address. The technique presented here is in fact more versatile than one might infer from the paper of De Marco and Wiithrich. The influence of the resolution enhancement on the decrease of the signal to noise ratio and on the distortion of the lineshape has already been discussed (4) by De Marco and Wfithrich. It appears from Fig. 3 of their manuscript that the "sine-bell" routine fails for A T / T * < 5, a condition that corresponds to numerous cases of well resolved spectra. In fact, with the procedure presented here, one can adjust the shift between the lorentzian curves in order to compensate for lower values of A T / T ~ . We wish to point out that if the loss of signal to noise is too drastic, one can always add some suitable proportion of the absorption signal to the results obtained with the simple routine. Figure 2e can be compared to the original lineshape of Fig. 2a obtained from the same FID. We chose not to develop in this paper the detailed mathematical analysis of the influence of the adjustable parameter in our treatment, that is the number of one-address shifts. We believe in fact that this influence is quite easy to visualise. Nor do we develop in this paper a detailed discussion of the different precautions one has to take when resolution enhancement is used in NMR. One has to keep in mind that, for any routine leading to an improvement in resolution, there will be (i) a loss of signal to noise ratio or (ii) a lineshape distortion, or (iii) both of them. This has to be pointed out because, if some lines of a spectrum have not the same shape (same T~), they will undergo different corrections. In fact, this last point should be kept in mind by anyone using a resolution enhancement routine, as well a simple exponential multiplication as a somewhat more sophisticated routine. Last, it should be noted that the method can be used, without any modification, for quadrature detected data, since it only requires the knowledge of the imaginary (or 90 ° out of phase) part of the spectrum. The authors have written a program, overlay to the B~8-51218 from Bruker, for BNC-12 computers, that permits one to visualise the result of the correction without using supplementary memory; this allows one to fit the data before using additional memory or destroying the original data. Copies of the program are available upon request. REFERENCES I. I . D . CAMPBELL,C. M. DOBSON, R. J. P. WILLIAMS,AND A. V. XAVIER, jr. Magn. Reson. 11, 172 (1973). 2. W. B. MONIZ, C. F. PORANSKI, JR, AND S. A. SOJKA, J. Magn. Reson. 13, 110 (1974).
RESOLUTION ENHANCEMENT IN F O U R I E R TRANSFORM
463
3. N. N. SEMENDYAEV, Stud. Biophys. 47, 151 (1974). 4. A. DE MARCO AND K. WOTHRICH, J. Magn. Reson. 24, 201 (1976). 5. D. C. CHAMPENEY, "Fourier Transforms and Their Physical Applications," p. 29, Academic Press, New York, 1973.
A New Scheme for Resolution Enhancement in Fourier Transform N M R B . CLIN,* J. DE BONY, P. LALANNE, J. BIAIS, AND B. L E M A N C E A U
Centre de Recherche Paul Pascal, Domaine Universitaire 33405 Talence Cedex France Received June 27, 1977; revised June 9, 1978 A new method for resolution enhancement in FT NMR is described, the physical principle of which is that of digital filtering by a sinusoidal window. The practical aspect of the method is developed, showing that the result can be obtained on a classical spectrometer without additional programming and with great versatility.
INTRODUCTION
A number of new methods have recently appeared for enhancing resolution or improving the signal to noise ratio in NMR spectroscopy (1, 2, 3). These techniques distort the shape of the peaks in order to achieve an enhanced resolution. For the "non-specialist" who wishes routinely to improve his measurements without having to cope with difficult mathematical or electronic problems, techniques with a minimum of adjustable parameters offer some advantage. From that point of view, digital filtering by a sinusoidal window, proposed by De Marco and Wiithrich (4), is particularly promising since its application does not require any delicate adjustment of parameters. The aim of this paper is to show that one can get the same results, without complicated mathematical manipulation of the signal, and that in the process of applying our technique, one can obtain a useful insight into the mechanism of digital filtering. MATHEMATICAL ANALYSIS
In their paper De Marco and Wiithrich (4) propose to multiply the free induction decay (FID) by a sinusoidal function with zero phase and a period of twice the acquisition time AT. If T* is the characteristic time for the decay, the lineshape after this procedure becomes, neglecting terms in exp(-AT/T*) (formula [2] of (4)). [~',trT22" 1 + (~'T2/AT) , 2 - Ato2T2,2 L(sin) a t - ~ - ) (1 + (zrT*/AT) 2 - AO92,~-2,2 12 ) +4Ato2T22"
[1]
What we want to point out is that this result can be writte 0 in another way:
~/AT-zato
1( ~r/AT + &a L(sin) a ~ 1 + T2*2 (w/AT + ato) 2 + 1 + T22 (~r/AT- &o)2J"
[2]
* To whom correspondance should be addressed 457
0022-2364/79/020457-07502~00/0 Copyright© 1979by AcademicPre~s,Inc. All rightsof reproductionin any formreserved. Printedin GreatBritain
458
CLIN
ET AL.
b C \
\ I | | I | |
V F~G. 1. (b) is obtained by adding (a) and (c). The new line shape is in fact obtained by addition of two dispersion Lorentzian functions of two different variables (zr/AT+A¢o) and (zr/AT-A¢o). This can be represented (Fig. 1) as two Lorentzian dispersion functions of the variable Aoj, shifted with respect to the origin by an amount of (+~r/AT), and 180 degrees out of phase. This result is quite general and using the theorem on Fourier transform and convolution products (5), the Fourier transform of an FID multiplied by a sinuso'/dal function can be written: FT(FID x sin
at) = FT(FID) ®
FT(sin at)
I-3]
where ® means "convolution product": 4-00
f(x)~) g(x)= f f(xt)g(x-xt) dxl
[4]
--oo
The Fourier transform of the function (sin at) represents the difference between the two delta functions 8(A~o + a ) and 8 ( A ~ 0 - a ) (5). Consequently the Fourier transform of the FID, filtered by a sinusoidal window of period 2rr/a will be the result of subtraction between the transform of initial FID, shifted in the frequency domain by an amount of a/2rr and the same, shifted by an amount of -a/27r. Note that the convolution of any function by a delta function leads to a change in the value
RESOLUTION ENHANCEMENT IN FOURIER TRANSFQRM
459
of the origin of the abscissa: +OO
f f(x)6(x-Xo) dx = f(xo).
[5]
--o0
The number of data points acquired for an FID is 2N. There is a relationship between the spectra width (SW), the acquisition time (AT) and this total number of samples taken (2N): 2 N = ~ × SW × A T
[6]
i.e., N = SW x AT. As, after Fourier transformation, the real part of the spectrum occupies only N data points, the frequency difference between two successive addresses is
kv = SW/N
= 1/AT
(=&o/2rr).
[7]
This value corresponds to the shift between the two Lorentzian curves (formula [2]) in the special case studied by De Marco and W/ithrich, where the period of the sinusoidal window is twice the acquisition time. APPLICATION The preceding remarks lead to a new method for obtaining the results of (4). From a practical point of view, one carries out the following procedure: (1) Calculate the FT of the original FID, and then perform the classical phase correction. It is the dispersion part of the signal that is of interest, because the sinusoidal function introduces a 90 ° phase shift. (2) Duplicate the signal in another block of memory. Then perform a left shift of this block by one address. (3) Subtract from this signal the original dispersion signal. (For users of Nicolet computers with any of a variety of programs, the procedure is to set DC-- - 1, define D F and D T blocks and issue the commands MV, LS and AT). DISCUSSION The method we have described, easy and fast to perform, also permits one to understand the non-universality of the "sine-bell" routine of De Marco and Wfithrich. For a signal with well-defined physical characteristics (i.e., resolution), the result of the "sine-bell" routine will depend upon the size of the block reserved for the acquisition, i.e., upon the acquisition time, for a given spectral width (formula [6]). As an example, we show in Fig. 2a a Lorentzian line obtained after Fourier transformation of a given FID. In Fig. 2b we show the result of the "sine-bell' routine in this case: it's clear that the loss of signal to noise is too severe. (In fact, the "sine-bell" routine has been simulated using the routine described in the preceeding paragraph). On Fig. 2c we show the result of the "sine-bell" routine applied to an
b
c
d
e
FIG. 2. (a) Conventional spectrum. (b) Resolution enhancement obtained by difference of two dispersed lorentzian curves shifted by one address, the acquisition having been performed on 16 K points (same result as "sine-bell" routine). (c) Resolution enhancement obtained by difference of two dispersed lorentzian curves shifted by one address, the acquisition having been performed on 4 K points (same result as "sine-bell" routine). (d) Resolution enhancement obtained by difference of two dispersed lorentzian curves shifted by four addresses, the acquisition having been performed on 16 K points. (e) Same results as (d) or (c) after addition of spectrum (a) multiplied by a factor 0.5.
a
RESOLUTION ENHANCEMENT IN FOURIER TRANSFORM
461
42,
fHOD 8
b
t
FIG. 3. Part of the spectrum of a polypeptide. (a) Conventional spectrum. (b) Resolution enhanced spectrum, following a classical sequence (two shifts, and add the real part, multiplied by a factor 0.3).
462
CLIN E T AL.
FID corresponding to the same sample, with the same characteristics for the acquisition, but for an acquisition time that is one fourth as long. On Fig. 2d we show the result of a more general ~ipplication of our routine to the same FID as the one used for the case of Fig. 2b. To obtain a result comparable with the one of Fig. 2c, we had only to perform four successive shifts before subtracting a Lorentzian curve from the other. It was obvious that, the only difference between the two cases being the two acquisition times ATa = 4ATe), the same result could be obtained by performing in one case a shift of four addresses, and in the other, a shift of one address. The technique presented here is in fact more versatile than one might infer from the paper of De Marco and Wiithrich. The influence of the resolution enhancement on the decrease of the signal to noise ratio and on the distortion of the lineshape has already been discussed (4) by De Marco and Wfithrich. It appears from Fig. 3 of their manuscript that the "sine-bell" routine fails for A T / T * < 5, a condition that corresponds to numerous cases of well resolved spectra. In fact, with the procedure presented here, one can adjust the shift between the lorentzian curves in order to compensate for lower values of A T / T ~ . We wish to point out that if the loss of signal to noise is too drastic, one can always add some suitable proportion of the absorption signal to the results obtained with the simple routine. Figure 2e can be compared to the original lineshape of Fig. 2a obtained from the same FID. We chose not to develop in this paper the detailed mathematical analysis of the influence of the adjustable parameter in our treatment, that is the number of one-address shifts. We believe in fact that this influence is quite easy to visualise. Nor do we develop in this paper a detailed discussion of the different precautions one has to take when resolution enhancement is used in NMR. One has to keep in mind that, for any routine leading to an improvement in resolution, there will be (i) a loss of signal to noise ratio or (ii) a lineshape distortion, or (iii) both of them. This has to be pointed out because, if some lines of a spectrum have not the same shape (same T~), they will undergo different corrections. In fact, this last point should be kept in mind by anyone using a resolution enhancement routine, as well a simple exponential multiplication as a somewhat more sophisticated routine. Last, it should be noted that the method can be used, without any modification, for quadrature detected data, since it only requires the knowledge of the imaginary (or 90 ° out of phase) part of the spectrum. The authors have written a program, overlay to the B~8-51218 from Bruker, for BNC-12 computers, that permits one to visualise the result of the correction without using supplementary memory; this allows one to fit the data before using additional memory or destroying the original data. Copies of the program are available upon request. REFERENCES I. I . D . CAMPBELL,C. M. DOBSON, R. J. P. WILLIAMS,AND A. V. XAVIER, jr. Magn. Reson. 11, 172 (1973). 2. W. B. MONIZ, C. F. PORANSKI, JR, AND S. A. SOJKA, J. Magn. Reson. 13, 110 (1974).
RESOLUTION ENHANCEMENT IN F O U R I E R TRANSFORM
463
3. N. N. SEMENDYAEV, Stud. Biophys. 47, 151 (1974). 4. A. DE MARCO AND K. WOTHRICH, J. Magn. Reson. 24, 201 (1976). 5. D. C. CHAMPENEY, "Fourier Transforms and Their Physical Applications," p. 29, Academic Press, New York, 1973.