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fourier transforms: theory and usage, mostly usage The ft page gives the basic functionality needed for a variety of data sets. A homodyne recon for fractional echoes is implemented following Doug Noll's paper in IEEE medical imaging 10(2): 154-163, 1994. The following topics are covered : chopping before the ft, chopping after the ft, inverse vs forward ft, "echo center at"( homodyne/partial nex), scaling, zip As an example consider the following 1d plot of a MR signal: Chop before ft: MR slang for applying a p phase roll to each data point. The Fourier Shift Theorem says a phase roll in one space will give a shift in the transformed space. This is needed since we acquire data in the range of [-nyquist .. nyquist] in k space. While the Fourier Transform assumes sampling in the range of [0 .. 2*nyquist]. As can be seen, chopping the data before the ft transform shifts the result 1/2 fov: no chop before ft chop after ft Chopping after the ft: Another consequence of sampling k space in the range of [-nyquist .. nyquist] is seen in the phase of the transformed signal. To obtain a smooth phase signal chopping must be applied after the ft as well: no chop after ft chop after ft inverse ft: This simply sets the sign of the exponent M(x) = S S(t)e-j2pi x/n This can be seen to flip the result. The same result occurs from changing the sign of the readout gradient during acquisition. forward ft inverse ft echo center at: for scans which have been performed with a fractional echo or half nex acquisitions, the center of k space is not the center of the acquisition matrix. A homodyne algorithm is used to approximate the missing data. In This example the left side of k space has 16 samples with the right side having 64. To correctly ft this data check this option and enter 16 to designate the center. The transformed image will be contained in the i channel of the result. The q channel will contain the left over phase error. Displaying the magnitude will defeat the whole purpose of this algorithm Scale the ft to unity: When an ft is performed the additional scale factor of (1/n)1/2 is applied. This allows for a unitary transform when doing forward, inverse ft pairs: zip 0,2,4 : Transformed data can be interpolated by padding the data with zeros prior to the Fourier transform. If the resolution of a data set is a power of 2 then the standard fft algorithm is used. If the resolution is not a power of 2 then a mixed radix ft is used. The following table shows the resulting resolution:
| resolution is a power of 2 | resolution is not a power of 2
zip 0 | unchanged | unchanged
zip 2 | 2*resolution | next power of 2
zip 4 | 4*resolution | 2 times the zip 2 resolution