Table 1.

Properties of gapped HSn and square pulses

Parameter	Gapped HSn pulses (dc-duty cycle)	Square pulse
RF driving function, fn(t)	sech(β(2t/Tp-1)n)	Constant
Relative RF integral, I(n)	$\frac{2d_{\mathrm{c}}}{T_{\mathrm{p}}}{\int }_0^{Tp}{f}_{\mathrm{n}}\left(\tau \right)d\tau \approx d_{\mathrm{c}}{\left(\frac{\pi }{2\beta }\right)}^{1∕n}$	$\underset{n\to \infty }{1}$
Relative RF power, P(n)	$\frac{2d_{\mathrm{c}}}{T_{\mathrm{p}}}{\int }_0^{T\mathrm{p}}{f_{\mathrm{n}}}^2\left(\tau \right)d\tau \approx d_{\mathrm{c}}{\left(\frac{1}{\beta }\right)}^{1∕n}$	$\underset{n\to \infty }{1}$
Total pulse length, Tp, s	$\frac{R}{b_{\mathrm{w}}}$	$\frac{1}{3b_{\mathrm{w}}}$
Amplitude modulation function, ω1(t)	ω1maxfn(t)[comb(bwt)⊕rect(bwdct)]	${\omega }_{1\max }rect(t∕T_{\mathrm{p}})$
Frequency modulation function, ωRF(t) ,rad/s	${\omega }_{\mathrm{c}}+2A\left(\frac{{\int }_0^t{f_{\mathrm{n}}}^2\left(\tau \right)d\tau }{{\int }_0^{T_{\mathrm{p}}}{f_{\mathrm{n}}}^2\left(\tau \right)d\tau }-\frac{1}{2}\right)$	Constant
Flip angle, θ, rad	$\approx {\omega }_{1\max }{d}_{\mathrm{c}}{\beta }^{-1∕2\mathrm{n}}\frac{\sqrt{R}}{b_{\mathrm{w}}}$	$\approx \frac{{\omega }_{1\max }}{3b_{\mathrm{w}}}$
Peak RF amplitude, ω1max, rad/s	$\approx \frac{{\beta }^{1∕2\mathrm{n}}}{d_{\mathrm{c}}\sqrt{R}}\theta b_{\mathrm{w}}$	≈ 3θbw
Relative RF energy, J, rad2/s	$\approx \frac{1}{d_{\mathrm{c}}}{\theta }^2{b}_{\mathrm{w}}$	≈ 3θ2bw
Relative RF energy (Ernst angle), J, rad2/s	$\approx \frac{2T_{\mathrm{R}}{b}_{\mathrm{w}}}{T_1{d}_{\mathrm{c}}}$	$\approx \frac{6T_{\mathrm{R}}{b}_{\mathrm{w}}}{T_1}$
Relative SAR (Ernst angle), SAR, rad2/s2	$\approx \frac{2b_{\mathrm{w}}}{T_1{d}_{\mathrm{c}}}$	$\approx \frac{6b_{\mathrm{w}}}{T_1}$
Relative amplitudes of sidebands, Am	≈ sinc(mdc)	-