Magnetic Resonance Imaging

Fourier Transform

A mathematical technique capable of converting a time domain signal to a frequency domain signal and vice versa.

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The + and - Frequency Problem

M, starting at +x, CW rotating about the Z axis,

M_x(t)

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FT

 

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M_y(t)

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FT

 

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Solution: input M_x and M_y as RE and IM

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Signal Detection

Linear Detection

Detecting just M_x or M_y, the computer discards half of the frequency domain data.

 

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Quadrature Detection

Detection of both M_x and M_y.

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The Fourier Transform

An FT is defined by the integral

Think of f(w) as the overlap of f(t) with a wave of frequency w .

This is easy to picture by looking at RE part of f(w) only.

Consider f( t ) = cos( 4t ) + cos( 9t ).

To understand the FT, examine the product of f(t) with cos(w t) for 1 < w < 10,
then the summation of the values of this product between 1 and 10 seconds.

w=1
w=2
w=3
w=4
w=5
w=6
w=7
w=8
w=9
w=10
f(w)

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inverse Fourier transform (IFT)

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Phase Correction

The FT makes use of RE and IM inputs:
    M_x = RE input
    M_y = IM input

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Consider:   f(t) = e^-at e^-i2pnt

M_X (Real)

M_Y (Imaginary)

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If the inputs are switched... cosine part input as IM, sine part input as RE

M_X (Real)

M_Y (Imaginary)

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To obtain an absorption spectrum...

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FID with 40^o phase shift in the RE and IM

M_X (Real)

M_Y (Imaginary)

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The coordinate transformation matrix can be used with f = - 45^o.

The phase corrected FIDs:

M_X (Real)

M_Y (Imaginary)

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FT phase corrected FIDs gives an absorption spectrum for RE output of the FT.

M_X (Real)

M_Y (Imaginary)

This correction can be done in the frequency domain as well as in the time domain.

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Constant (b) and Linear (m) phase corrections.

f = m n + b

Constant phase corrections arise from the inability of the spectrometer to detect the exact M_xand M_y.
Linear phase corrections, m, arise from the inability of the spectrometer to detect transverse magnetization starting immediately after the RF pulse.

In MRI, M_x or M_y signals rarely displayed. Instead a magnitude signal is used. magnitude = square root (M_x² and M_y²)

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Copyright © 2000 J.P. Hornak.
All Rights Reserved.