text: a method in accordance with a presently preferred exemplary embodiment of the invention will be described below with reference to fig1 - 5 . it will be appreciated by those of ordinary skill in the art that the description given herein with respect to those figures is for exemplary purposes only and is not intended in any way to limit the scope of the invention . all questions regarding the scope of the invention may be resolved by referring to the appended claims . in u . s . pat . no . 5 , 153 , 515 , the present inventor described how to generate a b 1 profile which will give the desired excitation profile for nmr imaging and the like . in particular , a technique was described which allows an ideal pulse to be approximated as closely as possible . tailored hard pulse sequences or soft pulse sequences were designed which excite a narrow bandwidth of resonant frequencies for clarity of imaging for better clinical diagnosis . the techniques described in that patent describe how to generate hard pulse sequences or soft pulses which achieve frequency responses as close as possible to the ideal or optimal pulse . in accordance with the method described in u . s . pat . no . 5 , 153 , 515 , the contents of which were incorporated by reference above , one starts with a physically realizable z magnetization which may be achieved by a hard pulse sequence and then determines a hard pulse sequence which will actually yield that z magnetization . in particular , an n hard pulse sequence to be applied around the x axis within a total duration of t has a magnetization response m z which is written as a fourier cosine series in which . linevert split . m z ( ω ). linevert split . is always less than or equal to one . then , if a desired m z ( ω ) is given , the equation for calculating m z ( ω ) reduces to a phase retrieval or deconvolution problem in which m z ( ω ) may be converted to a complex polynomial in s = exp ( iωt /( n - 1 )) and then solved for the roots of the polynomial 1 - m z 2 ( ω ), or equivalently , for the two polynomials 1 - m z ( ω ) and 1 + m z ( ω ). one then groups the roots of the polynomials by means of symmetry considerations and chooses half of them . there are also techniques such as cepstral deconvolution which allow for the solution of m xy ( ω ) as an ( n - 1 ) degree polynomial in s . this is a representation of m xy ( ω ) as an ( n - 1 ) th degree complex fourier series . as will be noted in more detail below , by representing these magnetization vectors as spinors and selecting different roots for the polynomials , constraints such as peak power and total energy over the pulse may be placed on the input pulses used to generate the desired frequency dependent excitation . the next step in the process is to find a pulse sequence which will yield the fourier series which will provide the desired magnetization . generally , m z is a complex fourier series . as noted in u . s . pat . no . 5 , 153 , 515 , the resulting problem is similar to a problem in digital filter design , namely , the design of finite impulse response filters . using finite impulse response filter theory , a desired hard pulse sequence may be created by specifying the desired z magnetization , the number of pulses , and the duration of the pulse sequence . one then views m z as a fourier series in ωt /( n - 1 ) and uses a finite impulse response filter design technique to obtain a fourier series which specifies m z as having certain desired values over certain specified ranges . one then normalizes the fourier series obtained for m z so that it is always less than or equal to one in absolute value and then one uses the techniques described in u . s . pat . no . 5 , 153 , 515 to synthesize the consistent m xy and a pulse sequence to actually generate that magnetization , b 1 . the present invention is an extension of the technique described in u . s . pat . no . 5 , 153 , 515 in that now one may specify further constraints on the input pulses such as the maximum total energy over the pulse duration and / or that the peak power should be reduced . to explain how this may be accomplished in accordance with the present invention , the magnetization system will be described using the spinor formulation of the bloch equation without relaxation . in this version of the formulation , a classical description of nuclear magnetic resonance is used in which the effect of being off resonance or of a radio frequency pulse is to induce a rotation . a counterclockwise rotation of θ radians around an axis having directional cosines ( c x , c y , c z ) can be written as the following complex unitary matrix : ## equ5 ## this matrix can be represented by its first column ( α , β ). thus , free precession at a frequency ω hz around the z axis for time t is represented by the spinor ( e - i πωt , 0 ). the effect of a radio frequency pulse on a spin at frequency ω can be described by a spinor [ α ( ω ), β ( ω )]. when applied to a system at equilibrium ( m z = 1 ), then the resultant magnetization can be written as : thus , the frequency specification consists of specifying either m z ( ω ), m xy ( ω ), α ( ω ), β ( ω ), or β ( ω )/ α ( ω ). the direct inversion method described in u . s . pat . no . 5 , 153 , 515 proceeds by specifying one of these parameters , generating α ( ω ) or β ( ω ) by using a consistency criterion and a phase retrieval algorithm , and then generating a rf pulse . as noted in u . s . pat . no . 5 , 153 , 515 , each hard pulse can be considered to be a rotation around an axis in the xy plane , lasting for an infinitesimal time . the sequence consists of ( n + 1 ) hard pulses , separated by time t / n , when the system precesses freely around the z axis . the total duration of the sequence is t . it was also shown that the spinor corresponding to such a pulse sequence can be written as : ## equ6 ## using such notation , the spinor corresponding to a rotation by angle θ around an axis in the xy plane with phase θ is [ cos ( θ / 2 ), e i θ sin ( θ / 2 )]. over any bounded frequency range , the frequency response of any shaped pulse can be approximated to arbitrary accuracy by an n hard pulse sequence , for large n . furthermore , the flip angles and phases of the hard pulses may be derived from the shaped pulse amplitude . in other words , if the shaped pulse is described by ω 1 ( t ), where ω 1 ( t ) is in radians / sec , then the jth hard pulse has amplitude ## equ7 ## and phase ## equ8 ## minimization of total pulse energy the present inventor has discovered that the total energy of the shaped pulse is encoded in one of the fourier coefficients of α ( ω ). if the total energy e of the shaped pulse lasting from time 0 to time t is defined as : ## equ9 ## where . linevert split . ω 1 ( t ). linevert split . is the amplitude ( in hz ) of the radio frequency pulse , then using the notation of equation 4 , it can be shown that : ## equ10 ## the proof of this is relatively simple . in the hard pulse sequence : ## equ11 ## for large n , a hard pulse sequence derived from a shaped pulse sequence has a small θ j which can be written as ( o ( 1 / n )) using landau &# 39 ; s o notation . as a result , ## equ12 ## taking the log of both sides of equation 10 , we get ## equ13 ## since the last term can be neglected , substituting for θ , we get : ## equ14 ## however , using a riemann sum approximation for the integral defining the energy e , we get : ## equ15 ## therefore : ## equ16 ## which is the same as equation 9 . the relationship of equation 15 has a number of immediate consequences . first , it establishes that the total pulse energy is proportional to the fourier coefficient , a , and that everything else relates to the shape of the pulse &# 39 ; s profile . second , for a hard pulse sequence derived from a shaped pulse , it establishes that 1 - a n is o ( 1 / n ). this can serve as a rough check to see if n is large enough , which also implies that the proportionality is valid for large n . third , if α ( ω ) is specified , the total energy has also been specified . therefore , all the rf pulses which can be generated with the same α ( ω ) but different β ( ω ) will have the same energy . in accordance with the invention , if only m z ( ω ) is specified , then one has only specified . linevert split . α ( ω ). linevert split . 2 . generally , α ( ω ) is generated using a phase retrieval algorithm . given a desired frequency response , it is known that one should choose the minimum phase solution for α ( ω ) corresponding to all roots outside the unit circle in order to obtain the pulse with the least energy . for example , pauly et al . teach in the afore - mentioned article entitled &# 34 ; parameter relations for the shinnar - le roux selective excitation pulse design algorithm &# 34 ;, ieee trans . med . imaging , vol . 10 , no . 1 , march 1991 , that the minimum phase solution minimizes the power requirements . indeed , pauly et al . derive a similar equation for the total energy . however , pauly et al . do not illustrate that the frequency response could be modified by choosing a different α ( ω ). in other words , given α ( ω ) with the same . linevert split . α ( ω ) 2 , pauly et al . demonstrated that the minimum phase α ( ω ) was the minimum energy . however , the comparison of two different α ( ω ) was not made by pauly et al ., and pauly et al . did not suggest that such a consideration should be introduced into the pulse design protocol . if m z ( ω ) is symmetric ( i . e ., m z ( ω )= m z (- ω )), then the minimum phase solution has all real coefficients . therefore , a pulse with constant rf phase can be synthesized which has minimum energy . in other words , for symmetric . linevert split . α ( ω ). linevert split . 2 , one cannot reduce the energy requirements by using variable phase pulses . however , for this relationship to be valid , it is necessary that n be sufficiently large that approximations for the cosine of the flip angle are valid . to illustrate this , a series of pulses have been synthesized with different frequency responses . the energy of these pulses vary by a factor of almost 40 . fig1 shows the profile of the typical pulse , and fig2 shows the frequency response caused by the pulse of fig1 when applied to an mr imaging apparatus . in fig2 the z magnetization is normalized to be 1 at equilibrium . table 1 below lists the frequency characteristics resulting from the application of these pulses . in table 1 , shaped pulses lasting for 5 msec were synthesized to desired frequency requirements using n ( column 1 ) hard pulses . from table 1 , it can be appreciated that as n becomes larger and the individual flip angles become smaller , the approximation becomes even more accurate . table 1__________________________________________________________________________ energynumber excita - energy ( fromof tion width (± ( from equationpulses angle hz ) pulse ) 15 ) % error__________________________________________________________________________40 90 500 96 . 1462 96 . 3247 . 18640 90 1000 165 . 537 166 . 572 . 62540 90 2000 301 . 328 307 . 025 1 . 89180 90 500 96 . 3913 96 . 4867 . 09980 90 1000 166 . 401 166 . 725 . 19580 90 2000 305 . 921 307 . 148 . 40180 180 500 988 . 438 999 . 140 1 . 08380 180 1000 1822 . 55 1866 . 4 2 . 40680 180 2000 3377 . 18 3636 . 91 7 . 691120 180 2000 3515 . 5 3622 . 96 3 . 057__________________________________________________________________________ in summary , the present inventor has shown that if one converts a shaped pulse into an n + 1 hard pulse sequence , with n sufficiently large so that each flip angle of the hard pulse sequence is small , then a n , or the coefficient e - i πωt in the fourier expansion of α ( ω ) is proportional to the power ( equation 15 ). therefore , in the design of a hard pulse sequence in accordance with the present invention , one specifies α ( ω ) so that it is close to the desired frequency response , subject to the constraint that log ( a n ) or 1 - a n is less than ( π 2 t * e )/ 2n , where e is the maximum total energy to be allowed for the pulse . given such an α ( ω ), an appropriate β ( ω ) can be synthesized , and a hard pulse sequence constructed which can be used as the basis for a shaped ( soft ) pulse . however , in accordance with the invention , the shaped pulse will have the designated maximum total energy and , as will be described more fully below , a reduced peak power . thus , the relationship between the fourier transform of the frequency response of the pulse and the total energy of the pulse is encoded in the lowest fourier coefficients of the frequency response . in particular , the total energy of the shaped pulse is encoded in one of the fourier coefficients of α ( ω ). this observation allows the pulse synthesis algorithm described in u . s . pat . no . 5 , 153 , 515 to use the total energy of the pulse as a design parameter , further elucidating the complex relationship between an rf pulse and its frequency response . in other words , the problem of synthesizing reduced energy pulses is reduced to the problem of synthesizing a fourier series whose frequency profile is close to a desired profile , subject to a constraint on one fourier coefficient . using the technique of the invention , reductions in energy of inversion pulses of the order of 10 - 20 % have been achieved with minimal effects on the inversion profile . further reductions are achievable with proportionately more effect on the inversion profile . for many rapid imaging schemes , this reduction in sar per pulse can be highly significant . the above approach illustrates how to reduce the total energy required by the pulse . however , because of transmitter power limitations , it is often desirable to reduce the maximum power required by the rf pulse as well . the following description illustrates how this may be done . as noted above , in order to synthesize the rf pulse with the desired frequency response , one has to specify the frequency response as a finite fourier series . typically , one specifies either m z ( ω ), or equivalently . linevert split . α ( ω ). linevert split . 2 or . linevert split . β ( ω ). linevert split . 2 . one then constructs α ( ω ) and β ( ω ) using a phase retrieval algorithm and then constructs the desired pulse . because algorithms , using the cepstral transform , exist for rapidly obtaining the minimum and maximum phase solutions , such techniques may be readily used for calculating β ( ω ) as well as α ( ω ). this approach has the additional advantage that if one specified a symmetric m z ( ω ) m z (- ω )= m z ( ω ), one gets a constant phase pulse . in accordance with the method of the invention , on the other hand , the basis for reducing the peak power of the pulse is the idea that using variable phase pulses may require less peak power than constant phase pulses because each frequency can be excited by a radio frequency pulse close to it . the pulse synthesis algorithm of u . s . pat . no . 5 , 153 , 515 is thus modified . one starts , as in u . s . pat . no . 5 , 153 , 515 , by constructing a filter for m z ( ω ), or equivalently , for . linevert split . α ( ω ). linevert split . 2 . the minimum phase solution for α ( ω ) is then determined so that the total energy of the pulse is unchanged . where the present method differs is in the choice of β ( ω ). in accordance with the method of the invention , if an α ( ω ) is specified which automatically specifies the total energy , then a β ( ω ) is synthesized which is compatible . if β ( ω ) is chosen such that it is not symmetric in ω , the pulse synthesized will be a variable phase pulse . since the maximum power requirements of a variable phase pulse are generally less than that of a constant phase pulse ( because it can effect a frequency sweep ), the peak power may be minimized by choosing a β ( ω ) which best approximates a frequency sweep . this is accomplished by appropriately selecting the roots for β ( ω ). as noted in the u . s . pat . no . 5 , 153 , 515 , a phase retrieval algorithm may be used to solve for the roots of . linevert split . β ( ω ). linevert split . 2 . these roots generally come in pairs ( z , 1 / z *) as described in that patent , each pair is ordered so that . linevert split . z . linevert split .≧ 1 . for each pair , one chooses one element of the pair , z j , and : ## equ17 ## where c is a suitable normalizing constant . the result is a solution for β (•). if m z ( ω ) is symmetric ( i . e ., m z ( ω )= m z (- ω )), then if z is a root , so is z *. to obtain a constant phase pulse , whenever a root is chosen , its conjugate is also chosen . this will lead to a β ( ω ) with real coefficients in the expansion and a constant phase pulse . the cepstral deconvolution algorithm yields such constant phase pulses . however , in order to obtain reduced peak power pulses in accordance with the invention , a different root selection scheme for β ( ω ) is chosen . for each ordered pair , z is selected as the solution for β ( ω ) when the imaginary component of z is greater than or equal to 0 ; otherwise 1 / z * is chosen . on the other hand , z may be selected as a solution for β ( ω ) when the imaginary component of z is less than or equal to 0 ; otherwise 1 / z * is selected . such selections of the roots of β ( ω ) lead to pulses with variable phase . because the same α ( ω ) is used , roughly the same total energy is used . also , the larger the number of pulses used , the less change there is in the total energy . those skilled in the art will appreciate that unlike the minimum phase solution , the roots are not always outside the unit circle . linevert split . z . linevert split .& gt ; 1 . those skilled in the art will also appreciate that minor modifications to the process may be made where a few roots are chosen by different criteria , so long as substantially similar results are yielded . the modified algorithm described herein has been applied to the synthesis of reduced power inversion and excitation pulses . as shown in fig3 the b 1 amplitude ( in hz ) for a constant phase pulse is reduced by the use of a reduced power , variable phase pulse in accordance with the invention . for example , in fig3 the peak of b 1 is reduced from 2301 hz for a constant phase inversion pulse to 1374 hz for a reduced power , variable phase pulse . fig4 illustrates the phase of the variable phase pulse of fig3 where peak b 1 is reduced by 40 % and the total energy by 64 %. fig5 illustrates a comparison of the inversion profiles of the resulting z magnetizations . as illustrated , the two curves are superimposable since they are substantially identical . thus , it is possible to reduce the peak power demands of a pulse by properly selecting the roots for β ( ω ), without any effect on the excitation or inversion profile , time of the pulse , or the total energy of the pulse sequence . indeed , numerous constraints on the desired frequency response for the fourier transform in such a response may be introduced in accordance with the invention in order to , for example , minimize the total energy of the pulse or to provide reduced peak powers . peak power reductions on the order of 60 - 70 % have been obtained using the techniques of the present invention . as noted in u . s . pat . no . 5 , 153 , 515 , the frequency response of every soft pulse can be uniformly approximated on every bounded frequency range by hard pulse sequences . in other words , given a soft pulse lasting for time t , for a given n , the jth pulse of an n hard sequence lasting for time t can be constructed . as described therein , one can reverse the procedure by constructing the soft pulse from a given hard pulse sequence . that is , given an n hard pulse sequence lasting for time t , one can construct a soft pulse lasting for t *( n /( n - 1 )). the soft pulse would have n segments , each lasting for a time t /( n - 1 ). during each segment , one applies a constant b 1 field which has the effect of rotating the 0 frequency spins the same amount as the hard pulses did . if the flip angles of the hard pulses are relatively small , the frequency responses of the hard pulse sequence and the soft pulse , over a limited frequency range , are similar . as described in u . s . pat . no . 5 , 153 , 515 , the soft pulse with a similar frequency response to a hard pulse sequence may be formed from a hard pulse sequence by choosing a large number of pulses n for the hard pulse sequence and then choosing a hard pulse sequence which has relatively small flip angles . in order to get a hard pulse sequence with relatively small flip angles , one chooses the roots of 1 - m z 2 which are used to construct m xy in accordance with the techniques described in u . s . pat . no . 5 , 153 , 515 . in accordance with the invention , a reduced power soft pulse is generated by defining the spinor components of a pulse , α ( ω ) and β ( ω ), and assuming that the pulse lasts for time t . then , for a soft pulse , it can be shown that the fourier transform of the e . sup . πi ωt α ( ω )= g ( s ) is 0 for s ( the fourier transform variable ) as s approaches 0 . in other words , the limit as s approaches 0 for g ( s ) is equal to e * π 2 , where e is the total energy of the soft pulse . also , a ( s ) is zero for s & lt ; 0 or s & gt ; t . thus : ## equ18 ## on the other hand , for an infinite duration pulse , the pulse can be defined to be the limit of the pulse lasting from - t to t , as t approaches infinity . the power of the pulse is the limit of the power of the finite duration pulses . therefore , given the frequency response of the pulse , the power is encoded in the value of the fourier transform of α ( ω ) at a particular point . certain embodiments of the invention have been described in detail above which will provide methods of pulse generation for optimally exciting specified frequency bands with certain constraints on the input pulse . while preferred embodiments have been shown and described , modifications are within the spirit and scope of the invention and will be recognized by those with ordinary skill in the art . for example , the invention need not be implemented using spinors but may be represented using conventional nomenclature . accordingly , all such modifications are intended to be included within the scope of the invention as defined by the following claims .