Correlation NNW spectroscopy

JOURNAL

OP MAGNETIC

RESONANCE

13,243-248

(1974)

Correlation NMR Spectroscopy* In 1965 Ernst and Anderson (I) showed that, under suitable conditions of excitation, a nuclear spin system can be treated as a linear system, thereby showing a way to obtain an order of magnitude in enhancement of sensitivity in high-resolution pulsed NMR experiments. However, the problems associated with the high peak radio frequency power required for adequate excitation bandwidth and the special filtering required to define the receiver bandwidth led to the exploration of excitations other than pulses. One of those was the stochastic excitation proposed by Ernst (2) and Kaiser (3). Some of the properties desired in an ideal excitation are low peak power, readily variable bandwidth, and uniform power spectral density in the specified bandwidth. We thus require a function which has an autocorrelation function of the form sin (X)/X. A fast linear sweep meets all of these requirements almost idea1ly.l We have previously reported the use of a fast linear sweep for signal to noise enhancement (4) where the spectrum was obtained by cross-correlation of a continuous wave spectrum recorded in fast passage with a single reference line recorded under the same conditions. As we learned later, a similar technique was used by Peterson (5) for filtering NMR spectra. Here we wish to discuss the use of a calculated reference line, clarify the physical interpretation of the experiment, and present one experimental example of the use of the technique. If we express the time function which describes a circularly polarized radio frequency field which has a frequency varying at a constant rate (a), we obtain Eq. [l]: Ui (t) = A exp (j&‘/2). The Fourier transform of tit(t) is given by Eq. 121, co f --CO

Ui(t) exp (-jwt)dt

= A(27C/Q)“’ exp [j(71/4 - W2/2a)]

PI

and implies a constant energy spectral density equal to A22x/a. If we limit the sweep so that ui runs from w, to w,, then the spectrum of ui(t) is given by Eq. [3]: F.T.(u, (t)} = A(7~/2a)l’~ exp [j(71/4 - w2/2a)]. [erf (2,) - erf (.a,)],

PI

where erf(z) is the error function with a complex argument and z, = (2~7)~~(w, - w) exp(-jn/4). Investigation of Eq. [3] will show that the power spectrum is approximately constant for w between w, and w, and approximately zero outside that interval. The transition regions near w, and w, are of the order of I// tp - ta / in width. For the purposes * Presented in tems, New York, April 1973. 1 Another way as was suggested

part at the 5th International Conference on Magnetic Resonance in Biological SysNY, December 1972, and at the 14th Experimental NMR Conference, Boulder, CO, to obtain such a function is to synthesize it from the prescribed power spectral density by B. L. TOMLINSON AND H. D. W. HILL, J. Chem. Phys. 59,1775 (1973).

Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

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of analysis, it will be easier to follow the experiment for a single sweep from w = -CD to w = +co and then investigate the influence of a finite sweep width. The fast linear sweep excitation can be used in two ways. First, it can replace the pulse of a standard Fourier transform pulse spectrometer or the stochastic excitation of a stochastic resonance Fourier transform spectrometer. In order to accumulate repeated spectra, we need a phase lock between the swept frequency and the constant reference frequency supplied to the synchronous detector. The spectrum [H(W)] is recovered from the Fourier transform of the response [U,(w)] through the relationship given by Eq. [4], H(w) = uO Cw>/ ui Cw>, E41 where Ui (w) can be readily calculated from Eq. [2]. Second, it can replace the slow swee in a conventional continuous wave spectrometer which in addition incorporates a crosscorrelator. This configuration has many new and interesting properties which follow

q(t) Ui(WI

u,(t) u,(4 FIG.

u&t) U&l

1. A block diagram for a practical correlation

U,(t) U&4

spectrometer.

from the analysis of signal flow in the spectrometer represented by Fig. I. The time functions at the indicated points in the spectrometer are related through the expressions given in Table 1. The corresponding relationships in the spectral representation are also given there, but in a condensed notation in which * represents the complex iconjugate and * represents correlation. From the expressions for uO(t), we see that such TABLE SIGNALS

PRESENT IN A PRACTICAL TIME

k i

s d 0

1 CORRELATION

AND FREQUENCY

u!i(t)

exp (j&/2) m ui (x)h(t s --m

(2n/~)‘/~exp (jn/4 -jwz/2a) - x)dx

vi (WI. H(w) ui (WI * u,(w)

ut (X)/L: (x + t)dx (2n/~)~l”

c.H(w)

IN

G(w)

& @I. u: (t> co J --m

SPECTROMETER

DOMAINS

=

exp (jn/4))H(at)

u, (-w) =

. u,* (-w)

=

(27~/a)~‘~ exp (j7c/4)h(-w/a) (Znc/u)/z(-t)

=

FIG.

A

-W

2. Signals at various stages of analysis in a practical correlation spectrometer using single phase sensitive detector. (a) Thirty accumulated scans of the NMR spectrum of o-chloronitrobenzene, 50 mg/ml in CD&, TMS. Sweep width = 290 Hz, sweep time = 12 set, time constant = 10 msec [Reti,(i (b) Fourier coethcients of (a). The first 500 coefficients are plotted. The left half are the cosine terms, the right half are the sine terms. [(1/2)(/,(w)]. (c) Terms derived from Eq. ]2] displayed as in (b). [exp(-j(w*/2u))]. (d) The product of (b) and the complex conjugate of (c) displayed as in (b). @*(w/a)]. (e) The inverse transformation of (d), a real function plotted at 40% points. [ReH(w)]. (f) A slow passage spectrum of the same sample as (a) with sweep time -5000 sec.

f

c

I

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a device can yield H(W) directly. That function is related to Green’s function h(t) for the system by a Fourier transformation. In a practical spectrometer, one might decide to accumulate am as a sampled function and average it over repeated scans of ui (t) in order to improve the signal to noise ratio. In such a case, am must be coherent from sweep to sweep and this is guaranteed by the synchronous detector. Once one has the function am in sampled mtmerical

1040

960

800

600

720

640

560

480Hz from

Hz0

3. Part of proton spectrum of a 10 mA4 lysine-Vasopressin solution in H20. (a) Single scan CW, 900 Hz in 250 sec. (b) 83 scans at 3 set in correlation mode, sensitivity enhanced by applying calculated reference line of 0.75 Hz line width. FIG.

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247

form, the correlation is readily accomplished using Fourier transformations, as indicated in Table 1, and Ui (-w) as given by Eq. [2]. We obtain U,(w) as the intermediate product. It has the same form and meaning as h(t) has in a pulsed NMR experiment, and therefore at this point, we may use any of the resolution enhancement functions or filter functions which are routinely used in pulsed Fourier transform spectroscopy. The simplest treatment could be an exponential filtering. Then we would recognize the complex conjugate of the Fourier transform of a reference line at zero frequency with a linewidth of 2w, in the partial product of U,(-w) and the filter function of the form exp (-wZ w/a). The response of each line in the spectrum is transformed by the synchronous detector to a Jacobsohn-Wangsness decay (6), so each line suffers the same distortion by the low-pass filter. If the filter is used only to eliminate signals which violate the Nyquist criterion, the distortion is negligible in most instances. An incorrect settin of the phase at the synchronous detector by 4d may be compensated by multiplication of the intermeditae product by exp(-j4d). The transition to a discrete Fourier transformation poses no problems if the Nyquist criterion is imposed on am by a suitable cutoff frequency on the low-pass filter, and the signal from the last line decays to a small value before the sweep ends. In the derivation of the equations in Table 1, we have assumed a complex signal, implying a spectrometer which contains two phase detectors, It is easy to prove that a single detector will supply the same information with a loss in signal to noise ratio of 3 db. Proton spectra were recorded at 250 MHz on the time sharing spectrometer at the NIH Facility for Biomedical Research in Pittsburgh, Pennsylvania (7). The spectrometer was operated in the linear frequency sweep mode with an internal homonuclear lock on TMS. The sweep was under program control using a Xerox Data Systems (XDS) Sigma 5 computer with a 15-bit digital to analog converter. The data were collected at the output of a single Princeton Applied Research Model 121 synchronous detector using the 12 db per octave low-pass filter in that device, and an XDS Model MD 51 analog to digital converter. A total of 4096 sample points were collected on each sweep and accumulated in memory. The data were analyzed2 using a fast Fourier transform program by Singleton (8). The resulting spectrum was displayed on a Cal Comp Model. 504 drum plotter. The slow passage spectrum was recorded by the continuous wave technique for comparison purposes. The results are shown at various stages of the analysis in Fig. 2. We feel that this new technique, which has been used now for over one year in our laboratory, offers several advantages over pulse techniques. The most important of them is the readily created rectangular power spectrum with practically no power outside the sweep width. This makes the accumulation of spectra in the presence of strong solvent lines without overdriving the dynamic range of the spectrometer rather easy (Fig. 3). If the sweep time chosen is of the order of the relaxation time T,, then the overall optimum sensitivity of the Correlation NMR Spectroscopy is similar to the sensitivity of Pulsed FT Spectroscopy. The same is true for the Fast Sweep FT Spectroscopy based on Eq. 4. We hope to explore these and other questions in a full paper. ACKNOWLEDGMENTS We would like to thank the Division of Research Resources of the National Institutes of Health for ’ A Fortran subroutine which transforms Ud(w) to UO(w>is available from RFS on request.

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continued support through Grant No, RR00292, and one of us (RFS) wouid like to thank the Sarah Mellon Scaife Foundation for a generous unrestricted grant.

1. R. R. ERNST

AND W. A. ANDERSON,

REFERENCES Rev. Sci. In&r. 37, 93 (1966).

2. R. R. ERNST, J. Mugn. Resonance 3, 10 (1970). 3. R. KAISER, J. Magn. Resonance 3,28 (1970). 4. J. DADOK AND R. F. SPRECHER,Paper presented at the 13th Experimental NMR Conference, -Asilomar, CA, April 1972. 5. 6. A. PETERSSON, Thesis, California Institute of Technology, 1970. 6. B. A. JACOBSOHN AND R. K. WANGSMTSS, Phys. Rev. 73, 942 (1948). 7. J. DADOK, R. F. SPRECHER, A. A. BOTHNER-BY, AND T. LINK, Paper presented at the 11th Experimental NMR Conference. Pittsburgh, PA, April 1970. 8. R. C. SINGLETON, IEEE Trans. Audio and Electroacoust. AU-17, 93 (1969). JOSEF RICHARD

Carnegie-Mellon University Pittsburgh, Pennsylvania 15213 Received November 9, 1973

DADOK

F. SPRKXER