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Selective excitation with the DANTE sequence. The baseline syndrome
JOIJRNALOFMAGNETICRESONANCE8~,206-211
(1989)
Selective Excitation with the DANTE The Baseline Syndrome XI-LI W u , PINGXU, Department
Sequence.
JANFRIEDRICH,ANDRAYFREEMAN
of Physical Chemistry, Cambridge University, Cambridge, England Received August 19, 1988
The simplest type of selective radiofrequency pulse is the single long weak pulse (I, 2) which we shall refer to as a soft pulse. If it has a rectangular envelope, the frequency-domain excitation spectrum contains a set of side-lobe responsesflanking the m a in response, approximating a sine function. For practical reasons it is often convenient to replace this soft pulse with a repetitive sequence of hard pulses separated by free precession delays, the so-called DANTE sequence(3, 4). If all the hard pulses are identical, the responseapproximates that of the rectangular soft pulse, except that there are sideband responsesat intervals of the pulse repetition rate 1/(At). Alternatively the hard pulses can be a m p litude m o d u lated (either in intensity or in pulse width) to tailor the effective pulse envelope according to some suitable shape (4). Gaussian (5)) hyperbolic secant (6)) sine function ( 7)) and half-Gaussian (8) shaping functions have been used for various purposes. The soft pulse and the corresponding DANTE sequencehave very similar properties provided that they both have the same total duration and overall flip angle. In a sense,DANTE is merely a digitized version of the soft pulse. W h e n the corresponding magnetization trajectories are calculated from the Bloch equations, starting on the 3-Z axis, they terminate very close to the same point. The soft pulse trajectory is a smooth curve and the DANTE trajectory follows a zigzag path (representing the effects of alternate pulsing and free precession) which crosses and recrosses the smooth curve. W h e n the DANTE sequencecontains a large number of hard pulses (fine digitization) then the results obtained with the two kinds of selective pulse are virtually indistinguishable. However, for a DANTE sequencewith a lim ited number of pulses(coarse digitization) there are two important discrepancies.In the frequency d o m a in, they take the form of a positive displacement of the baseline and an oscillatory contribution. Both effects were evident in the first simulations of DANTE excitation spectra (3). For selective excitation experiments, baseline distortions of this kind are particularly unfortunate, leading to weak excitation of the NMR spectrum across the entire frequency range. The purpose of this communication is to point out that these baseline artifacts may be corrected by halving the intensity of the first and last pulses of the DANTE sequence. W e take as an example a rectangular soft pulse (Fig. 1a) compared with a IO-pulse DANTE sequence(Fig. 1b) of the same total duration ( T = 9At) and the 0022-2364189 $3.00 Copyrieht 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
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FIG. I. (a) Soft pulse; a single weak pulse of duration T = 9At. (b) A IO-pulse DANTE sequence with intervals of At. (c) DANTE sequence with first and last pulses halved in amplitude. (d) DANTE sequence of 9 pulses with initial and final intervals of At/2 s.
same overall flip angle (90”). When the first and last pulses are halved in amplitude (all pulses being resealed to maintain the same total flip angle) then the DANTE sequence and the soft pulse produce virtually the same result. When we consider the magnetization trajectories, this makes sense geometrically. The trajectory for the uncorrected DANTE sequence diverges from the smooth curve and terminates with an appreciably larger + Y component (Fig. 2a). For the corrected DANTE sequence the first and last segments of the zigzag have half the amplitude (Fig. 2b), ensuring that the DANTE trajectory straddles the soft pulse trajectory in a symmetrical fashion. We may think of the unmodified DANTE sequence as being the superposition of the “corrected” DANTE sequence together with half-amplitude pulses at times -T and 0 s. In the Fourier spectrum, the former creates an oscillatory component at the frequency T-' while the latter creates a positive baseline offset. The oscillatory component merges with the sine function wiggles, extending them across the entire excitation spectrum (3). Halving the fast pulse of a DANTE sequence cancels the baseline offset (Fig. 3b). A full-amplitude final pulse would cause a slight overshoot of the magnetization trajectory, always in the direction of an increased Y component of magnetization. Provided the overall hip angle is small compared with 90”, this positive baseline displacement is the same for all resonance offsets. The correction is most easily visualized for an offset m idway between the centerband and the sideband response, where 180”of free precession occurs between each pulse of the DANTE sequence, converting +M, into -Mr. The final half-amplitude pulse simply carries this magnetization vector back to the +Z axis.
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FIG. 2. Magnetization trajectories for the soft pulse (smooth curve) and the equivalent DANTE sequence (zigzag). (a) Unmodified 1O-pulse DANTE; the two trajectories terminate some distance apart. (b) DANTE with first and last pulses halved in amplitude. (c) Nine-pulse DANTE with initial and final Af/2 intervals. Note that in(b) and (c)the two trajectories terminate at essentially the same point.
Halving thefirst pulse reduces the oscillations in the baseline (Fig. 3c) so that they tend to disappear in the region midway between centerband and sideband responses. Consider an arbitrary offset in this central region. Just prior to the last pulse there is a residual Y component of magnetization proportional to the amplitude of the initial pulse. When this first pulse is set at half-amplitude, the final (half-amplitude) pulse largely cancels the residual Y component. Thus, halving both the first and the last pulse cancels both kinds of baseline artifact (Fig. 3d). Since both effects combine to generate undesirable excitation of off-resonance spins, DANTE pulse sequences should be programmed with the first and last pulses at half their normal amplitudes. The effect is strongest for coarsely digitized sequences,but is still appreciable when there are 50 pulses in the sequence (3). Alternatively the DANTE sequence may be started and ended with a delay At / 2. Figure 2c compares the trajectory for a soft pulse with that for a DANTE sequence consisting of nine equal pulses, beginning and ending with free precession intervals of At/2 s. This is the sequence sketched out in Fig. Id; the pulse amplitudes have
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I
b 0
d
FIG. 3. Frequency-domain excitation patterns calculated by solving the Bloch equations for a 1O-pulse DANTE sequence. (a) Unmodified DANTE. (b) Last pulse halved in amplitude, showing suppression of baseline offset. (c) First pulse halved in amplitude, showing suppression of oscillatory component. (d) First and last pulses halved in amplitude, showing both corrections.
been scaled to maintain a constant total flip angle. Note that in Fig. 2c, the soft pulse and DANTE trajectories terminate at essentially the same point on the unit sphere. One consequence of the At/2 final delay is that all odd-numbered sideband responses are inverted in phase. For many experiments where the magnetization is initially along the +Z axis, the first period At / 2 of free precession has no effect, but for experiments where the DANTE sequence is preceded by some other excitation pulse, this interval is important. We first realized the extent of the baseline offset effect while using a DANTE sequence modulated according to the first half of a Gaussian curve (8). This is a particularly clear example of baseline offset because the frequency-domain excitation pattern is approximately Gaussian in shape and has no side-lobe responses. Since the initial time-domain ordinate is vanishingly small, there is a negligible oscillatory component on the baseline. However, the effect of the final pulse is very large indeed. Figure 4a shows the excitation pattern calculated for a nine-pulse DANTE sequence tailored according to a half-Gaussian envelope; there is an appreciable baseline offset.
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FIG. 4. Centerband and two sideband responses in the frequency-domain excitation pattern of a ninepulse DANTE sequence with pulse flip angles modulated according to the first half of a Gaussian curve. (a) Unmodified DANTE sequence, showing a positive baseline offset. (b) DANTE sequence with final pulse halved in amplitude, showing no baseline offset.
Figure 4b has been calculated for the same sequence with the last pulse reduced in amplitude by 50%, giving an excitation spectrum which has no baseline offset. In some early experiments with shaped selective pulses (8)) computer hardware limitations restricted the fineness of digitization of the DANTE sequence, so it was particularly important to correct the baseline offset. Consequently, in later experiments (9, 10) with half-Gaussian-shaped DANTE sequences, we always halved the intensity of the final pulse. This phenomenon is of course quite general, being associated with the representation of a smooth function by discrete samples, and it is all the more serious the coarser the digitization. Each sampling point should represent the area of the appropriate trapezium; the first and the last points represent trapezia that are approximately half the area of the others. A related effect occurs in the discrete Fourier transform of a free induction decay, leading to a baseline offset of the resulting NMR spectrum unless the value of the initial data point is halved ( I I ). Another way of looking at this baseline offset phenomenon is in terms of discontinuities in the time-domain function, for example, the sharp voltage step at the begin-
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ning of a free induction decay. For the purposes of Fourier transformation, the timedomain function is assumed to be periodic, so the first and the last data points merge and must be replaced by their mean value. If this is not done, there is an error equivalent to a false positive signal for the first ordinate of the free induction decay, corresponding to a baseline offset in the frequency-domain spectrum. In practice, bandlim iting filters often achieve a similar effect by smoothing the discontinuity, and it is not then necessary to halve the first data point, but in two-dimensional spectroscopy there can be no such filtration of interferograms in the evolution period, so baseline offsets do occur, appearing as ridges in the appropriate frequency dimension ( 2 1) . We may conclude that whenever selective pulses are to be implemented by a sequence of discrete hard pulses separated by periods of free precession (a DANTE sequence), any discontinuities at the beginning and end of the pulse envelope should be smoothed by halving the amplitudes of the initial and final pulses. ACKNOWLEDGMENT ‘The authors are pleased to acknowledge illuminating discussions with Dr. James Keeler. REFERENCES I. 2. 3. 4. 5.
S. ALEXANDER, Rev. Sci. Instrum. 32, 1066 (1961). R. FREEMAN AND S. WITTEKOEK, J. Magn. Reson. 1,238 (1969). G. BODENHAUSEN, R. FREEMAN, AND G. A. MORRIS, J. Magn. Reson. 23,17 l(l976). G. A. MORRIS AND R. FREEMAN, J. Magn. Reson. 29,433 (1978). C. J. BAUER, R. FREEMAN, T. FRENKIEL, J. KEELER, AND A. J. SHAKA, J. Magn. Reson. 58,442
(1984). 6. J. BAUM, R. TYCKO, AND A. PINES, J. Chem. Phys. 79,4643 (1983). 7. L. E. CROOKS, J. HOENNINGER, M. ARAKAWA, L. KAUFMAN, R. MCREE, J. WATTS, AND J. R. SINGER, SPIE Recent Future Dev. Med. Imaging II 206,120 (1979). 8. J. FRIEDRICH, S. DAVIES, AND R. FREEMAN, J. Magn. Reson. 75,390 (1987). 9. S. DAVIES, J. FRIEDRICH, AND R. FREEMAN, J. Magn. Reson. 75,540 (1987). 10. S. DAVIES, J. FRIEDRICH, AND R. FREEMAN, J. Magn. Reson. 76,555 (1988). II. G. OTIING, H. WIDMER, G. WAGNER, ANDK. WUTHRICH, J. Magn. Reson. 66,187 (1986).
(1989)
Selective Excitation with the DANTE The Baseline Syndrome XI-LI W u , PINGXU, Department
Sequence.
JANFRIEDRICH,ANDRAYFREEMAN
of Physical Chemistry, Cambridge University, Cambridge, England Received August 19, 1988
The simplest type of selective radiofrequency pulse is the single long weak pulse (I, 2) which we shall refer to as a soft pulse. If it has a rectangular envelope, the frequency-domain excitation spectrum contains a set of side-lobe responsesflanking the m a in response, approximating a sine function. For practical reasons it is often convenient to replace this soft pulse with a repetitive sequence of hard pulses separated by free precession delays, the so-called DANTE sequence(3, 4). If all the hard pulses are identical, the responseapproximates that of the rectangular soft pulse, except that there are sideband responsesat intervals of the pulse repetition rate 1/(At). Alternatively the hard pulses can be a m p litude m o d u lated (either in intensity or in pulse width) to tailor the effective pulse envelope according to some suitable shape (4). Gaussian (5)) hyperbolic secant (6)) sine function ( 7)) and half-Gaussian (8) shaping functions have been used for various purposes. The soft pulse and the corresponding DANTE sequencehave very similar properties provided that they both have the same total duration and overall flip angle. In a sense,DANTE is merely a digitized version of the soft pulse. W h e n the corresponding magnetization trajectories are calculated from the Bloch equations, starting on the 3-Z axis, they terminate very close to the same point. The soft pulse trajectory is a smooth curve and the DANTE trajectory follows a zigzag path (representing the effects of alternate pulsing and free precession) which crosses and recrosses the smooth curve. W h e n the DANTE sequencecontains a large number of hard pulses (fine digitization) then the results obtained with the two kinds of selective pulse are virtually indistinguishable. However, for a DANTE sequencewith a lim ited number of pulses(coarse digitization) there are two important discrepancies.In the frequency d o m a in, they take the form of a positive displacement of the baseline and an oscillatory contribution. Both effects were evident in the first simulations of DANTE excitation spectra (3). For selective excitation experiments, baseline distortions of this kind are particularly unfortunate, leading to weak excitation of the NMR spectrum across the entire frequency range. The purpose of this communication is to point out that these baseline artifacts may be corrected by halving the intensity of the first and last pulses of the DANTE sequence. W e take as an example a rectangular soft pulse (Fig. 1a) compared with a IO-pulse DANTE sequence(Fig. 1b) of the same total duration ( T = 9At) and the 0022-2364189 $3.00 Copyrieht 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
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FIG. I. (a) Soft pulse; a single weak pulse of duration T = 9At. (b) A IO-pulse DANTE sequence with intervals of At. (c) DANTE sequence with first and last pulses halved in amplitude. (d) DANTE sequence of 9 pulses with initial and final intervals of At/2 s.
same overall flip angle (90”). When the first and last pulses are halved in amplitude (all pulses being resealed to maintain the same total flip angle) then the DANTE sequence and the soft pulse produce virtually the same result. When we consider the magnetization trajectories, this makes sense geometrically. The trajectory for the uncorrected DANTE sequence diverges from the smooth curve and terminates with an appreciably larger + Y component (Fig. 2a). For the corrected DANTE sequence the first and last segments of the zigzag have half the amplitude (Fig. 2b), ensuring that the DANTE trajectory straddles the soft pulse trajectory in a symmetrical fashion. We may think of the unmodified DANTE sequence as being the superposition of the “corrected” DANTE sequence together with half-amplitude pulses at times -T and 0 s. In the Fourier spectrum, the former creates an oscillatory component at the frequency T-' while the latter creates a positive baseline offset. The oscillatory component merges with the sine function wiggles, extending them across the entire excitation spectrum (3). Halving the fast pulse of a DANTE sequence cancels the baseline offset (Fig. 3b). A full-amplitude final pulse would cause a slight overshoot of the magnetization trajectory, always in the direction of an increased Y component of magnetization. Provided the overall hip angle is small compared with 90”, this positive baseline displacement is the same for all resonance offsets. The correction is most easily visualized for an offset m idway between the centerband and the sideband response, where 180”of free precession occurs between each pulse of the DANTE sequence, converting +M, into -Mr. The final half-amplitude pulse simply carries this magnetization vector back to the +Z axis.
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FIG. 2. Magnetization trajectories for the soft pulse (smooth curve) and the equivalent DANTE sequence (zigzag). (a) Unmodified 1O-pulse DANTE; the two trajectories terminate some distance apart. (b) DANTE with first and last pulses halved in amplitude. (c) Nine-pulse DANTE with initial and final Af/2 intervals. Note that in(b) and (c)the two trajectories terminate at essentially the same point.
Halving thefirst pulse reduces the oscillations in the baseline (Fig. 3c) so that they tend to disappear in the region midway between centerband and sideband responses. Consider an arbitrary offset in this central region. Just prior to the last pulse there is a residual Y component of magnetization proportional to the amplitude of the initial pulse. When this first pulse is set at half-amplitude, the final (half-amplitude) pulse largely cancels the residual Y component. Thus, halving both the first and the last pulse cancels both kinds of baseline artifact (Fig. 3d). Since both effects combine to generate undesirable excitation of off-resonance spins, DANTE pulse sequences should be programmed with the first and last pulses at half their normal amplitudes. The effect is strongest for coarsely digitized sequences,but is still appreciable when there are 50 pulses in the sequence (3). Alternatively the DANTE sequence may be started and ended with a delay At / 2. Figure 2c compares the trajectory for a soft pulse with that for a DANTE sequence consisting of nine equal pulses, beginning and ending with free precession intervals of At/2 s. This is the sequence sketched out in Fig. Id; the pulse amplitudes have
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I
b 0
d
FIG. 3. Frequency-domain excitation patterns calculated by solving the Bloch equations for a 1O-pulse DANTE sequence. (a) Unmodified DANTE. (b) Last pulse halved in amplitude, showing suppression of baseline offset. (c) First pulse halved in amplitude, showing suppression of oscillatory component. (d) First and last pulses halved in amplitude, showing both corrections.
been scaled to maintain a constant total flip angle. Note that in Fig. 2c, the soft pulse and DANTE trajectories terminate at essentially the same point on the unit sphere. One consequence of the At/2 final delay is that all odd-numbered sideband responses are inverted in phase. For many experiments where the magnetization is initially along the +Z axis, the first period At / 2 of free precession has no effect, but for experiments where the DANTE sequence is preceded by some other excitation pulse, this interval is important. We first realized the extent of the baseline offset effect while using a DANTE sequence modulated according to the first half of a Gaussian curve (8). This is a particularly clear example of baseline offset because the frequency-domain excitation pattern is approximately Gaussian in shape and has no side-lobe responses. Since the initial time-domain ordinate is vanishingly small, there is a negligible oscillatory component on the baseline. However, the effect of the final pulse is very large indeed. Figure 4a shows the excitation pattern calculated for a nine-pulse DANTE sequence tailored according to a half-Gaussian envelope; there is an appreciable baseline offset.
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FIG. 4. Centerband and two sideband responses in the frequency-domain excitation pattern of a ninepulse DANTE sequence with pulse flip angles modulated according to the first half of a Gaussian curve. (a) Unmodified DANTE sequence, showing a positive baseline offset. (b) DANTE sequence with final pulse halved in amplitude, showing no baseline offset.
Figure 4b has been calculated for the same sequence with the last pulse reduced in amplitude by 50%, giving an excitation spectrum which has no baseline offset. In some early experiments with shaped selective pulses (8)) computer hardware limitations restricted the fineness of digitization of the DANTE sequence, so it was particularly important to correct the baseline offset. Consequently, in later experiments (9, 10) with half-Gaussian-shaped DANTE sequences, we always halved the intensity of the final pulse. This phenomenon is of course quite general, being associated with the representation of a smooth function by discrete samples, and it is all the more serious the coarser the digitization. Each sampling point should represent the area of the appropriate trapezium; the first and the last points represent trapezia that are approximately half the area of the others. A related effect occurs in the discrete Fourier transform of a free induction decay, leading to a baseline offset of the resulting NMR spectrum unless the value of the initial data point is halved ( I I ). Another way of looking at this baseline offset phenomenon is in terms of discontinuities in the time-domain function, for example, the sharp voltage step at the begin-
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ning of a free induction decay. For the purposes of Fourier transformation, the timedomain function is assumed to be periodic, so the first and the last data points merge and must be replaced by their mean value. If this is not done, there is an error equivalent to a false positive signal for the first ordinate of the free induction decay, corresponding to a baseline offset in the frequency-domain spectrum. In practice, bandlim iting filters often achieve a similar effect by smoothing the discontinuity, and it is not then necessary to halve the first data point, but in two-dimensional spectroscopy there can be no such filtration of interferograms in the evolution period, so baseline offsets do occur, appearing as ridges in the appropriate frequency dimension ( 2 1) . We may conclude that whenever selective pulses are to be implemented by a sequence of discrete hard pulses separated by periods of free precession (a DANTE sequence), any discontinuities at the beginning and end of the pulse envelope should be smoothed by halving the amplitudes of the initial and final pulses. ACKNOWLEDGMENT ‘The authors are pleased to acknowledge illuminating discussions with Dr. James Keeler. REFERENCES I. 2. 3. 4. 5.
S. ALEXANDER, Rev. Sci. Instrum. 32, 1066 (1961). R. FREEMAN AND S. WITTEKOEK, J. Magn. Reson. 1,238 (1969). G. BODENHAUSEN, R. FREEMAN, AND G. A. MORRIS, J. Magn. Reson. 23,17 l(l976). G. A. MORRIS AND R. FREEMAN, J. Magn. Reson. 29,433 (1978). C. J. BAUER, R. FREEMAN, T. FRENKIEL, J. KEELER, AND A. J. SHAKA, J. Magn. Reson. 58,442
(1984). 6. J. BAUM, R. TYCKO, AND A. PINES, J. Chem. Phys. 79,4643 (1983). 7. L. E. CROOKS, J. HOENNINGER, M. ARAKAWA, L. KAUFMAN, R. MCREE, J. WATTS, AND J. R. SINGER, SPIE Recent Future Dev. Med. Imaging II 206,120 (1979). 8. J. FRIEDRICH, S. DAVIES, AND R. FREEMAN, J. Magn. Reson. 75,390 (1987). 9. S. DAVIES, J. FRIEDRICH, AND R. FREEMAN, J. Magn. Reson. 75,540 (1987). 10. S. DAVIES, J. FRIEDRICH, AND R. FREEMAN, J. Magn. Reson. 76,555 (1988). II. G. OTIING, H. WIDMER, G. WAGNER, ANDK. WUTHRICH, J. Magn. Reson. 66,187 (1986).