# How You Perform the Fourier Transform in Practice

There are two ways to ask iNMR to perform the FT. Both ways pass through the command ‘Process/Fourier Transform’ (or its keyboard equivalent, Cmd-R). In the standard way you are confronted with a dialog which shows the parameters of the operation and let you change them. In the fast way you skip the dialog and have no control on the parameters. To selected this fast way, press the ‘Option’ key. The first time you perform the FT on a given document, iNMR try to guess the parameters for you. This is presently done only when the original file contains all the needed information, in other words with recent file formats. If you repeat the processing on the same document, then iNMR just keeps in memory the parameters previously used.

You may be appalled by the number of options shown by the dialog. They are all important, to various degrees. Here we discuss most of them, one by one, while weighting options are dealt with into another chapter. If your spectrometer (or its software) is recent, there are good chances you can process spectra using exclusively the parameters that iNMR chooses for you. It doesn't hurt, though, to read what follows. In other cases, where you are requested to enter the parameters, two approaches are possible, also simultaneously. The first approach is to exploit any information you have about the origin of the spectrum, like instrument brand or name of the pulse sequence. In a second approach you first try to perform the FT in its most basic variant, then, observing where you were wrong, and having learned in advance the effect that each option can have, you can tell which option needs to be added to the basic FT.

- The Undo command does not work after the FT. If you want to change parameters and retry this operation, use ‘File/Repeat’ instead. This is certainly faster than undoing, in 1D spectroscopy at least. In 2D spectroscopy, if you are not sure about which options to set, do not work with the whole matrix, but on an extract instead, that is a single column or subplane. What happens is quite easy to understand in the homonuclear case, where you already know where the tallest peak (the diagonal peak) will fall. If it happens to fall at the expected position and there is no counter-diagonal, then your parameters are correct. If you work with an extract, the Repeat command will bring you back to the last extract. If, instead, you work with the whole matrix, the Repeat command brings you back to the raw, time domain, data.
- You may be confused the first times you are using iNMR, because it Transposes a matrix before performing the FT along an indirect dimension. Try remembering that the just transformed dimension is drawn along the x axis.

# Real

Modern spectrometers acquire spectra in a way that avoids the use of this option.
In direct dimensions, it was once required by old Bruker instruments (many of which are still running today).
Otherwise, it is used, for indirect dimensions, when the TPPI protocol is implemented.
Beware: the so-called “States-TPPI” protocol is actually incompatible with Real FT.
We are just speaking of pure TPPI.

If you reverse the correct setting of the real option, the resulting spectrum will contain
the double of the peaks.
Today it is possible for you to do NMR without ever using Real FT and TPPI altogether; you lose
nothing but troubles.

# Swap Sides / Mirror Image

These two options are better understood together. Suppose you expect to read the word “Molfetta”,
but instead you read “ettaMolf”. This is swapping sides. In another case you may read “attefloM”:
this is the mirror image. Finally, you can apply both operations and get “floMatte”.
Needless to say, the two operations are reflexive and commute. For an expert eye, it is quite easy to recognize
their wrong setting, yet sometimes other options are wrong (like Real, Shuffling or even Weight).
In that cases the spectrum becomes chaotic.

It is useful to know that, in direct dimensions, both options must be selected for Varian spectra, only
Swap sides for Bruker spectra. The Ruben-States protocol also require swapping sides, while States-TPPI
does not.

Generally speaking, all NMR spectra should *swap sides*, because any signal that falls at the transmitter
frequency will appear, after FT, to have frequency zero. The spectrometer does not know about TMS. It just
understands the carrier frequency. Because the transmitter is placed in the middle of the spectrum,
we move the instrumental zero in the middle of the screen, that is, we swap sides.
With today's digital instruments
many other things actually happen, yet the result is the same. An acquisition scheme may already include
the step corresponding to the swapping, that's why there are exceptions.

# Shuffling

Phase Sensitive, multidimensional spectra always require shuffling. All 1D spectra and the simplest 2D do not. In case you don't know which kind of shuffling is required, choose “phase sensitive”, because it is the most common one. “echo-antiecho” was first introduced in the gradient-enhanced HSQC experiment. In case you know that your spectrum is not phase sensitive, then the correct choice is: “do not shuffle”.

# Size

You cannot change the spectral width, but you can (artificially) increase or reduce the sampling
frequency of the spectrum. The method of increasing the number of points is quite trivial: you add
some zeroes at the end of the FID.
The FT will mix uniformly the zeroes with the rest.
In case you wonder why all NMR software limit your choice to the powers of zero, that's the reason:
in 1965 Cooley and Tukey found out an algorithm to speed up FT considerably, and it only worked
when the size of the problem was a power of 2. Their discovery was instrumental in making FT a popular technique.
Today we still use their idea.

When you arrive at deciding the size of your spectrum
it's better if you are an ignorant!

Adding zeroes at the end of the FID is called **Zero-Filling**. In the standard case, when you don't implement it,
there is a certain waste of information. Remember that half of the acquired spectrum
is stored into the imaginary component. Those points are true experimental points, independent from the real
part, so they can contribute to increase the signal to noise ratio.
After phase correction, however, the imaginary part has no use, so that possible gain in S / N is not achieved.
Zero-filling avoids that information loss. The net result, if you double the size of the FID through zero-filling, is that
you have a (theoretical) advantage, because the information contained into the imaginary part
falls back into the real part. If you zero-fill more than once, you have no further advantage,
but it doesn't hurt, though. Obviously, computation times and RAM requirements considerably grow.

For zero-filling to work smoothly, the acquisition time must be quite longer than the relaxation time. In simpler words, the tail of the FID must already have a value near to zero. Otherwise we have the side effects of wiggles around peaks, in the transformed spectrum. When the FID is quite short, as usually happens in 2D spectroscopy, we observe wiggles even in the absence of zero-filling. This is not a reason to renounce to zero-filling. You save the day by applying an apodization function. The FT dialog offers a choice between 4 such functions.

# Use Linear Prediction

Linear Prediction, abbreviated as **LP**, was, along with other acronyms like MEM and MLM,
a fade of the 1980s, in NMR at least. All of them were proposed like alternatives to the FT.
They always suffered from slowness and/or numerical instability and/or the need of some a priori information, like the
number of expected signals or the noise volume. Today they also suffer from being out of fashion!

In some limited context, nonetheless, LP is a better alternative to zero-filling. In the essence,
instead of prolonging the FID with meaningless zeroes, you can prolong it with a matching, FID-like tail. LP is a must in
the gradient-enhanced HSQC, but can also be applied, at your discretion, along the indirect dimension
of any phase-sensitive multi-dimensional spectrum. In the best case you increase both resolution and sensitivity.
In the worst case you create some wiggles along the diagonal (States spectra) or the counter-diagonal
(TPPI spectra). One sensible option is to try with and without LP, then choose for the best.

When you use LP instead of zero-filling, it is necessary to apply a strong apodization function, like
a squared cosine bell (as a minimum). In the stated cases, this is the standard choice also
if you opt for zero-filling.

# HyperComplex

Phase sensitive multidimensional spectra require as many phase corrections as the number of their axes. Taking for example a 2D spectrum, a possible workflow is: FT along f2; phase correction along f2; FT along f1. At this point you can phase correct along f1, but no more along f2. To phase along f2, you would need the imaginary part along that axis. Unfortunately, during the last FT, you chose a shuffling, which, by default, deletes the imaginary part. Quite often, if not always, phase correction is better performed first along f1, then along f2. What you have to do is to specify the HyperComplex option along with a shuffling protocol. At the cost of doubling processing time and RAM requirement, you keep the imaginary part of the spectrum. If you don't plan to phase correct, however, there is no need to select this option.

# Subtract D. C. (drift current)

This option is almost outdated and applies exclusively on 1D spectra acquired on old analog spectrometers. It eliminates a small glitch that often appears at the center of the spectral width. What this option does in practice is to calculate the average value of the FID tail and subtract this value from the whole FID. The result is a FID that decays exactly to zero, as expected from theory. Do not apply, however, this option neither to digital spectra nor to 2D-3D spectra!

# Magnitude

This option is actually extraneous to FT. You find it as a command in the Process menu. Having it inside the dialog is simply a convenience for you. If you check the box, when FT is completed the spectrum is put in magnitude representation. Choose this option only for the final FT, in the case of non phase-sensitive multi-dimensional experiments.

# Apply These Weights

Weighting Functions are described in their own page.