# Overview¶

There are lots of times we would like to do mathematical operations in the frequency domain, then convert back to space or time and continue with other operations, or the other way around, or do calculations in both domains. Both gmtmath and grdmath currently operate only in one domain, but with some changes could become much more powerful modules. Tools like grdfft allows only for a tiny selection of preset operators and are mostly useful for filtering, etc. spectrum1d allows for no calculations on the spectrum. Some of the motivation for this proposal comes from issue #495.

# Common issues for both tools¶

## Operators¶

We need to add two operators: FFT (forwards FFT) and IFFT (inverse FFT). The modules will know their state so FFT cannot be called after another FFT, etc. However, there are times when we wish to set things up directly in the frequency domain, do some math, then call IFFT and save the result. We need a way to let the two modules know which domain we start in. We will pre-scan the command-line and if the first FFT operator is IFFT then we know we are staring in the frequency domain [time ro space].

## Options¶

We need to make the FFT-setup specific option -N as used in grdfft and gravfft avaialble in grdmath and gmtmath, and perhaps also in spectrum1d. This begs the question if we should introduce a new common GMT option for FFT setup. Because -N is already used in some of these programs I suggest -z as in

-z[f|q|s|<nx>[/<ny>]][+a|d|l][+e|m|n][+t<width>][+w<suffix>][+z[p]]

Obviously, ny is not needed for the 1-D case and could also be skipped for square 2-D setups.

## Variables¶

While T (in gmtmath) and X,Y (grdfft) have been used for spatial coordinates, these will return frequencies or wavenumbers in that mode. We could instead insist that users use F or K (frequency or wavenumber – same thing), and KX, KY, KR (wavenumbers) instead as these names are still available.

## Complex mathematics¶

When in complex mode (i.e., after calling FFT) we may need to divert some operators like MUL to shadow operators like ZMUL which will do a proper complex mutiplication. Same for DIV and many others. Could start by just supporting the basics and add more later.

# Specific issues for gmtmath¶

GMT data tables are normally real-valued only, with one user-selected column designated to hold time (or distance). When a GMT data table is read by gmtmath and FFT is called, we will double the number of columns and use the new columns to store the imaginary components, and let the duplicate time-column hold the frequency vector. When IFFT is called the output is placed in the real columns and the no-longer-used imaginary columns are removed. There are some special cases: (a) While in the frequency domain we assume any input data tables are already complex with all extra columns present. (b) If we call = (the output operator) while still in frequency mode then we output all the real and imaginary columns as they are. Because the 1-D FFT function expects a single multiplexed array we may need to duplicate the real and imaginary data columns into a temp vector for the purpose of doing the FFT, then copy the result back into the data table. Some of this may be hidden in the calling sequences.

# Specific issues for grdmath¶

GMT grids are real-valued only; there is no complex GMT grid, and most of the grid formats could not handle such data (netCDF is one exception, of course). However, when reading programmatically we can tell GMT that a grid is either the real or imaginary part of a complex grid and store both in a single enlarged GMT_GRID structure. Thus, complex grids in GMT are handled externally using two separate grids but internally (in grdmath and others dealing with FFTs) as a single enhanced grid. This has numerous advantages such as any GMT tool can also operate or make plots of the imaginary part, and grdmath can convert the complex grid to polar form, etc. Internally, the format feeds naturally into and out of the FFT machinery (we have code in place for the mux and demux). While in the frequency domain we assume any input grids are given in pairs, separated by a comma, i.e., Re-grid.nc,Im-grid.nc. Any of the two names in the pair could be a hyphen (–), meaning the component is zero [if only one grid is given we assume it is a real component with zero imaginary component. (b) If we call = (the output operator) while still in frequency mode then we output the real and imaginary component grids as they are, separately.

# Examples¶

## 2-D¶

Read in a complex grid via its two component grids, then take inverse transform and write the result

grdfft real.nc,imag.nc IFFT = model.nc

Read in a regular real grid, then take forward transform, multiply by radial wavenumber and write the complex result
grdfft mydataset.nc FFT KR MUL = real.nc,imag.nc

## 1-D¶

Read in a data table with time and three observable series, remove the means, take the FFT, multiply all by the frequency, take inverse transform and save.

gmtmath mydata.txt DUP MEAN SUB FFT F MUL IFFT = newstuff.txt

# Notes¶

1. Programs like Matlab also have an FFTSHIFT operator (reshuffle array or grid layout to place zero frequency in the middle; it is a toggle operator). However, in GMT we will ensure the zero frequency is always in the middle so that frequencies have a natural range from negative Nyquist to +positive Nyquist. This eliminates the need for an explicit FFTSHIFT.
2. The common –N option for FFT specifications allows for the saving of a raw spectrum as real and imaginary grids. This is a bit different from what happens when using = but related and the code will take advantage of that.
3. I decided against adding a SPECTRUM operator to mimic what spectrum1d and grdfft –E does as it would (a) kind of eliminate spectrum1d from the mix and (b) also be messy since spectrum1d has many options about windowing, output values, etc.). Likewise, grdfft can do cross-spectra so I think we leave these capabilities where they currently are.