Determination of Coupling Constants by Deconvolution of Multiplets in NMR 1 Damien Jeannerat* and Geoffrey Bodenhausen† ,2 *De ´partement de chimie, Universite ´ de Gene `ve, 30 Quai Ernest Ansermet, CH-1211 Gene `ve 4, Switzerland; and †De ´partement de chimie, Ecole Normale Supe ´rieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France Received March 22, 1999; revised June 10, 1999 The structures of multiplets in one- and two-dimensional NMR spectra can be simplified by recursive deconvolution in the fre- quency domain. Deconvolution procedures are described for in- phase and antiphase doublets of delta functions. Recursive sim- plification is illustrated by applications to double-quantum-filtered correlation spectra (DQF-COSY) and selective correlation spectra (soft-COSY).Coupling constants can be measured reliably even if signals of opposite signs lead to partial cancellation. © 1999 Academic Press Key Words: deconvolution of multiplets; correlation spectros- copy (COSY); cross-peak multiplets. The structure of multiplets in one- and two-dimensional NMR spectra contains a wealth of information about scalar and dipolar couplings. The coupling constants can be determined by deconvolution in the frequency or in the time domain (1–3), by maximum entropy techniques (4–6), or by taking advantage of properties of trigonometric functions (7). In this paper, we shall focus on frequency-domain analysis (8), pursuing the work by Huber and Bodenhausen (9). Some of the difficulties encountered in determining the inverse of convolution have been overcome and the criteria for the success of simplification have been improved. The convolution product of two functions f and g may be defined as follows: h  x  =  -  f  u  g  x - u  du . [1] If the functions are discrete, the integral may be replaced by a sum. Thus, convolution of two arrays a = { a 0 , a 1 ,..., a n } and b = { b 0 , b 1 ,..., b m } gives an array c = { c 0 , c 1 ,..., c n+m } containing n + m + 1 elements, where c i =  k=0 i a k b i -k . [2] We shall refer to the operation which allows one to derive the array a from the knowledge of b and c as deconvolution. As shown by Bracewell (10), this can be expressed as a k = b 0 -1  c k -  j =0 k-1 a j b k-j  . [3] This equation can be verified by substitution into Eq. [2]. We shall only consider multiplets that are composed of in-phase and antiphase doublets. An in-phase doublet of delta functions can be represented by an array b, b i =  +1, i = 0 +1, i = m 0, elsewhere [4] and an antiphase doublet by another array b, b i =  +1, i = 0 -1, i = m 0, elsewhere. [5] For such doublets, Eq. [3] can be simplified since there is only one nonzero element in the summation, so that we obtain for in-phase doublets, a k = c k - a k-m , [6] and for antiphase doublets, a k = c k + a k-m . [7] Thus each point of the deconvoluted spectrum can be com- puted by taking the sum or difference of only two numbers. In the case of in-phase doublets we obtain 1 This work was carried out in part at the Center for Interdisciplinary Magnetic Resonance, National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive, Tallahassee, FL 32310. 2 To whom correspondence should be addressed. Geoffrey.Bodenhausen@ ens.fr. Journal of Magnetic Resonance 141, 133–140 (1999) Article ID jmre.1999.1845, available online at http://www.idealibrary.com on 133 1090-7807/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.