th  - 1988 - 51st ENC Conference

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-- [ DISCRETE ANALYSIS OF STOCHASTIC NMR USING WIENER SERIES:
S.T.S. Wong*, R.D. Newmark & M.S. Roos, Donner Laboratory, Lawrence Berkeley Laboratory, University
of California, Berkeley, California 94720.
Stochastic NMR is an efficient alternative to conventional NMR techniques for spectroscopy and imaging
that can reduce the peak RF power requirement by several orders of magnitude. The stochastic experiment
is analysed by a Wiener series expansion of the non-linear NMR system with a discrete Gaussian white noise
process for the input. This diifers from Kaiser's analysis for continuous excitation (JMR 48, 293, 1982). A
method for correcting the distortions in spectra (images) reconstructed by simple 1D cross-correlation due to
the NMR system non-linearity will be presented.
The experiment consists of a series of RF pulses with Rip angles being a sample of a discrete Gaussian
white noise process. One data point is sampled after every RF pulse. Spectral (image) information is obtained
by Fourier transforming the 1D cross-correlation of the sampled data points with the white noise sequence.
By modeling the system with a set of difference equations, analytic expressions for the signal power and
the reconstructed spectra (projections) were obtained. These expressions allow us to choose the Rip angle
variance which maximizes signal-to-noise ratio. Unfortunately, the Rip angle variance that gives the maximum
signal-to-noise ratio is large enough to cause the magnetization response to saturate, giving rise to distortions
in the reconstructed spectra (projections).
When the non-linear magnetization response is expanded into a Volterra series, the desired non-distorted
spectrum corresponds to the linear kernel, hl, of the series. However, there is no easy way to obtain hl
theoretically or experimentally. The non-linear response cam be expanded into a series of Wiener functionals
which can be obtained by multi-dimensional cross-correlations of the sampled data with the Gaussian white
noise excitation. The 1D cross-correlation gives the first order Wiener kernel, kl, which approaches hl as
the Rip angle variance approaches zero. As the system becomes more non-linear, hl becomes dispersed into
Wiener functionals of order higher than one: hl = kl + ki(3) + k1(s) + ...... , where k](3), k1(s) etc., are the linear
components of the higher order Wiener functionals. Expressions for kl, ki(3) and k1(s) have been derived.
The sum/=i + ki(3) reduces the distortions significantly for the Rip angle variance which gives the maximum
signal-to-noise ratio.
11 1 J STOCHASTIC NMR IMAGING WITH OSCILLATING GRADIENTS:
S.T.S. Wong, M.S. Roos*, R.D. Newmark, Donner Laboratory, Lawrence Berkeley Laboratory, University of
California, Berkeley, California 94720.
The analysis of NMR imaging and volume selective spectroscopy with oscillating gradients (A. Macovski,
J. Mag. Res. Med., vol. 2, p. 29-39, 1985) has been extended to include stochastic excitation. Advantages of
the method described relative to conventional techniques are a large reduction in the peak RF power required
relative to deterministic pulse excitation and the elimination of switching transients using oscillating gradients.
The stochastic experiment consists of a sequence of RF pulses where the Rip angles are a sample of a
discrete Gaussian white noise process. One data point is sampled after every RF pulse in the presence of a B0
gradient which varies sinusoidally throughout the experiment. The magnetization response to RF excitations
is assumed to be linear, which is valid in the limit of small Rip angles. The sampled signal is then cross-
correlated with the product of the Gaussian white noise sequence and a phase demodulation kernel derived
from the time-varying gradient waveform in order to reconstruct an image.
The expected value of the reconstructed image has the form of a localization function convolved with the
spin distribution. This function is
e -°/T' 3o (2~Gsin(WmO'/2) z~,
\ Wm /
where 3o is the zero-order Bessel function, G and co. are, respectively, the amplitude and frequency of the
siausoidal gradient. The time lag of the cross-correlation, o, is a free parameter that may be used to manipulate
T2 contrast. Integrating over o also improves localization, resulting in Jo(TGz/a~,n). The sidelobes of the
localization function can be reduced by including harmonics of the gradient frequency in the kernel, allowing
synthesis of a localization function from a series of Bessel functions of increasing order.
The variance of a given pixel of the image obtained by cross-correlation does not approach zero in the
limit of infLnite observation time. It becomes part of the overall image noise in additon to observation noise.
Both types of noise can be reduced by averaging images reconstructed using separate samples of the excitation
process.
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