Minimum and Maximum Phase RF Pulses
The SLR designs so far have been linear phase:
- They refocus (almost) perfectly
- They can be used as spin echo pulses
If you don’t care about phase, there are much more selective pulses:
- Saturation pulses
- Inversion pulses
- Slab select pulses
These can be almost twice as selective as linear phase pulses.
You can design a minimum phase pulse by first designing a mimimum phase $\beta$, scaling it to $\sin(\phi/2)$, where $\phi$ is your flip angle.
The function dzmp(n,tb,d1,d2) designs a minimum phase waveform with n samples, a time-bandwdith product of 8, and a passband ripple of d1, and a stopband ripple of d2. These are equalripple designs, which can end up with spikes (“Conolly wings”) at the ends. The output of dzmp is scaled to 1, so this corresponds to an inversion pulse.
%use octave
try
cd rf_tools_octave
catch
try
cd ../rf_tools_octave
catch
try
cd ../rf_tools_octave
catch
cd ../../rf_tools_octave
end
end
end
pkg load signal
%use octave
bm = dzmp(256,8,0.001,0.001);
rfm = b2rf(bm);
t = [1:256]/32;
re_rfscale_rfm = real(rfscale(rfm,8));
(View plot code)
The linear phase inversion with the same time bandwidth product from the pevious demos is
%use octave
bl = msinc(256,2);
rfl = b2rf(bl);
re_rfscale_rfl = real(rfscale(rfl,8));
Comparing the two rf pulses
(View plot code)
We can compare the profiles as
%use octave
x = [-64:64]/4;
mzm = ab2inv(abr(rfm,x));
mzl = ab2inv(abr(rfl,x));
(View plot code)
Click and drag to zoom in between the transition bands (time = 0 to 5 ms)
This can be useful for other pulses, such as slab select pulses
Something to try:
- Design a minimum phase excitation pulse (scale $\beta$ to $\sqrt{2}/2$)
- Compare the complex excitation profile of the min phase pulse, and a max phase pulse (min phase reversed in time).