Hadamard NMR

Hadamard NMR is a method for reducing the acquisition time of two-dimensional experiments by only acquiring the rows in the 2D that contain signals. By using a Hadamard matrix, multiple selective one-dimensional spectra can by acquired simultaneously and deconvoluted post-acquisition.

Most of the data points in a typical 2D NMR spectrum are noise. Only a few of the points in the processed data matrix contain the signals of interest. If one could collect only the signals then data collection would be much faster. This can be done by collecting a series of 1D selective spectra, but doing this for every resonance is tedious, and more importantly, one only acquires signals for one resonance at a time, leading to a loss of signal-to-noise per unit time. If the 1D selective spectra could be acquired simultaneously then signals for all resonances would be collected throughout the entire process, making it much more efficient. Simultaneous selective excitation of multiple sites can be achieved using complex shaped pulses that modern spectrometers are capable of generating. The problem, however, is how to separate the signals originating from each of the selective excitation sites. This is where a Hadamard matrix is used.

A Hadmard matrix is a square matrix containing only two different values, typically +1 and -1. One can use a Hadamard matrix to assign the phase (e.g. +180,-180) of the selective pulses. The figure below shows an 8x8 Hadamard matrix with blue upward arrows representing +180^o pulses and red downward arrows representing -180^o pulses. This is the Hadamard matrix one would use if there were no more than eight resonances involved.

Each row in the Hadamard matrix represents the selective pulses for one of the excitation sites. The Hadamard selective pulse is a combination of all the selective pulses in one column. When performing a Hadamard experiment a series of spectra is collected using different phases of the selective pulses to enable separation of the signals from each site. In this example we would use eight different Hadamard pulses generated from the eight different columns. The top row indicates that for the first selective excitation site the sign of the selective pulse alternates with each Hadamard pulse. The fifth row shows the sign switching after every second Hadamard pulse. This phase encoding of the selective excitation pulse allows each site to be distinguished from the others by post-acquisition addition and subtraction of the spectra. For example, if we consider the first row, taking every pair across the matrix and subtracting the second spectrum from the first one finds the signals increase on every step. Doing the same for any other row in the matrix finds the signals cancel upon reaching the end of the row. Different addition and subtraction patterns allow the signal from each site to be extracted from the others.

Some restrictions imposed by the use of the Hadmard matrix are immediately obvious. Firstly, the size of the matrix dictates the number of scans. For example, if one can obtain sufficient signal in a single scan a larger number of scans (eight in this example) is still required to record all the combinations of the phase of the selective pulses and enable signal deconvolution. Secondly, the order of the Hadamard matrix must be larger than the number of resonances, and must be a power of two, e.g. 4x4, 8x8,16x16.... For these reasons as the number of resonances increases the benefits of Hadamard NMR start to decrease, but for lower concentration samples with a small number of resonances Hadamard NMR can provide a dramatic reduction in acquisition time.

A small molecule (MW 371.22) with ten ¹H resonances was used to acquire the TOCSY spectra below. The same number of scans was used for both spectra and they are plotted at the same contour levels. The standard TOCSY, at top with blue contours, took 97.5 minutes to acquire while the Hadamard TOCSY, below with red contours, took just 3.25 minutes.

In addition to the dramatic time savings, the Hadamard TOCSY is much cleaner and has better resolution. The large water resonance is missing since it was not excited, and t₁ noise does not exist because there is no t₁ period. Resolution in the horizontal directly-detected dimension is improved since a larger number of points than normal (16384 vs 4096 in this case) can be used to digitize the spectrum, while resolution in the indirect dimension is limited only by the processing capacity of the computer.

The Facility has Hadamard versions of COSY, TOCSY, NOESY, ROESY, HSQC and INADEQUATE experiments available. Generation of the Hadamard pulses and processing of the spectra can all be done within TopSpin using macros supplied by Bruker and is only slightly more complicated than standard processing.