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If I understand correctly, to select a particular slice (of an object, during an MRI scan) that is orthogonal to the direction of the main magnetic field applied, we apply a magnetic field that varies linearly in strength in the direction of the main magnetic field (conventionally the z direction). Subsequently, in either the x or y directions within that slice we apply another linearly increasing magnetic field. This means that within that slice, protons (of the same molecules) precess at frequencies linearly varying in a chosen direction. This means that we can approximate the slice as a bunch of rectangles, each of which induces different frequency signals in the RF coil (although in fact no two points along the direction of the gradient have the same frequency). Now, instead of applying a third linearly varying B field in the next dimension (x or y, depending on the previously chosen one), magnetic pulses are applied in this dimension for varying periods of time so as to shuffle the phases of protons precessing at the same frequency due to the previous 2 magnetic fields. I understand that by fourier transforming the sum of all these guys (all the different signals recorded at different phase differences, for each frequency band, by means of tipping the magnetisation into the x-y plane) we can isolate particular voxels' contribution to the signal. But why, instead of doing this complicated phase pulsing, don't we just apply the third magnetic field in the left-over dimension and do business as usual?

To my mind this method sounds a lot simpler, and the only clarification I can find for phase encoding is that I am actually mistaken in my understanding of the entire process. Someone please explain!

I'll include a paragraph of what I'm talking about (I can't copy paste the text and I am definitely not transcribing it): enter image description here enter image description here

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