e., centered at sufficiently high |B1+|), the process can start with a conventional single-band linear-phase
finite impulse response filter designed using a weighted-least squares method. That filter is then duplicated, and the duplicates are frequency modulated selleck chemicals llc to opposite center frequencies and subtracted from each other. This is equivalent to modulation of the single-band filter by a sine function at the center frequency. For very close passbands (i.e., passbands close to |B1+|=0) however, ripples from one band can distort the other. In these cases, an odd, dual-band ββ filter can be designed directly using weighted-least-squares. The distortions could also be mitigated using a phase-correction method . Once the ββ filter is designed, assuming small excitation angles the inverse SLR transform reduces to a simple scaling of the filter coefficients to obtain the ΔωRF(t)ΔωRF(t) waveform. The SLR algorithm conventionally designs an RF pulse that accompanies a constant gradient waveform. In |B1+|-selective selleck compound pulse design, A(t)A(t) replaces the gradient waveform. In the small-excitation angle regime, the αα profile at the end of a pulse with duration T is : equation(6) α(|B1+|)=e-ıγ2|B1+|∫0TA(t)dt,and the ββ profile is: equation(7) β(|B1+|)=ı2eıγ2|B1+|∫0TA(s)ds∫0TΔωRF(t)e-ıγ|B1+|∫tTA(s)dsdt.
equation(8) =ı2eı2|B1+|k(0)∫0TΔωRF(t)e-ı|B1+|k(t)dt,where k(t)≜γ∫tTA(s)ds is the pulse’s |B1+|-frequency trajectory. From Eq. (6), it is evident that if A(t)A(t) is constant and comprises no pre- or rewinder lobes before or after the ΔωRF(t)ΔωRF(t) waveform to achieve zero total area, then αI≠0, which is unacceptable. Zero total area could be achieved by adding a negative rewinder lobe to A(t)A(t) with the same area as the main lobe, but according to Eq. (8) this would create a nonzero βIβI since ΔωRF(t)ΔωRF(t) would deposit energy at negative frequencies only, as depicted in the middle column of Fig. 3. A real and odd ββ profile can only be produced if ΔωRF(t)ΔωRF(t) deposits energy anti-symmetrically
as a function of frequency, and therefore cannot be produced with this trajectory. Placing the rewinder lobe at the beginning of the pulse would also lead to nonzero βIβI. The desired symmetric k(t)k(t) can be restored Thiamet G by splitting the rewinder lobe, so that half is played at the beginning and half at the end, as shown in the right column of Fig. 3. With this configuration, α=1α=1 and βI=0βI=0 as required. This A(t)A(t) waveform configuration is analogous to a balanced gradient waveform configuration for conventional slice-selective excitation, which is commonly used for refocusing pulses in spin echo sequences and for excitation pulses in balanced steady-state free precession sequences . Fig. 4a shows that as a |B1+|-selective pulse is scaled to excite a large tip-angle, αIαI grows and degrades the excited profile by creating a large unwanted MyMy component (Eq. (4)), particularly in the stopband.