Window functions

Window functions (also called apodization functions) are applied to time-domain NMR data before Fourier transformation to modify the lineshape and signal-to-noise characteristics of the resulting spectrum. NMRTools automatically detects and stores the window function used during processing, whether by TopSpin or NMRPipe for loaded spectra, or by apodize for raw data processed in NMRTools (see Processing a 1D spectrum). When a window is passed to apodize without tmax, for example Cos2Window(), its length is set from the acquisition time of each dimension.

Overview

Window functions are represented by subtypes of the abstract type WindowFunction. Each window function type stores:

Parameters specific to that window function
The acquisition time tmax (calculated after linear prediction but before zero filling)

Window functions are stored as axis metadata and can be accessed through the :window key:

using NMRTools

# Load example spectrum
spec = exampledata("2D_HN")

# Get the window function for a specific dimension
window = spec[F2Dim, :window]

CosWindow(0.04783359999999993)

Available window functions

t is the time from t = 0 and tmax the acquisition time. Omit tmax when passing a window to apodize and it is set from each dimension's acquisition time.

Window	w(t)	Notes
`NullWindow(tmax)`	$1$	no apodization
`UnknownWindow(tmax)`	–	the window could not be determined from the processing parameters
`ExponentialWindow(lb, tmax)`	$\exp(-\pi \, \mathrm{lb} \, t)$	line broadening `lb` (Hz): improves S/N at the cost of resolution; negative `lb` narrows lines
`CosWindow(tmax)`	$\cos\left(\frac{\pi t}{2 t_\mathrm{max}}\right)$	cosine (sine bell shifted by 90°); common in indirect dimensions
`Cos2Window(tmax)`	$\cos^2\left(\frac{\pi t}{2 t_\mathrm{max}}\right)$	squared cosine; falls to zero more smoothly than `CosWindow`. Alias `Cos²Window`
`GeneralSineWindow(offset, endpoint, power, tmax)`	$\sin^{\mathrm{power}}\left(\pi\,\mathrm{offset} + \frac{(\mathrm{endpoint} - \mathrm{offset})\,\pi t}{t_\mathrm{max}}\right)$	general sine bell; `offset` and `endpoint` (0 to 1) set the phase at the start and end. `offset=0.5, endpoint=1` gives `CosWindow` (`power=1`) or `Cos2Window` (`power=2`)
`LorentzToGaussWindow(expHz, gaussHz, tmax)`	$\exp(\pi \, \mathrm{expHz} \, t) \exp\left(-\frac{(\pi \, \mathrm{gaussHz} \, t)^2}{2}\right)$	Lorentz-to-Gauss transformation: removes `expHz` (Hz) of Lorentzian broadening and adds `gaussHz` (Hz) of Gaussian broadening; resolution enhancement
`GeneralGaussWindow(expHz, gaussHz, center, tmax)`	$\exp(\pi \, \mathrm{expHz} \, t) \exp\left(-\frac{(\pi \, \mathrm{gaussHz} \, (t - t_\mathrm{center}))^2}{2}\right)$	as `LorentzToGaussWindow`, with the Gaussian maximum at `center` (0 to 1) of the acquisition time; `center=0` gives `LorentzToGaussWindow`

ExponentialWindow(5.0, 0.1), Cos2Window(0.1), GeneralSineWindow(0.5, 1.0, 2.0, 0.1)

(ExponentialWindow(5.0, 0.1), Cos2Window(0.1), GeneralSineWindow(0.5, 1.0, 2.0, 0.1))

Working with window functions

Extracting window parameters

Window function objects store their parameters as fields:

# Load spectrum
spec = exampledata("2D_HN")

# Get window for F2 dimension
win = spec[F2Dim, :window]

# For ExponentialWindow
if win isa ExponentialWindow
    println("Line broadening: ", win.lb, " Hz")
    println("Acquisition time: ", win.tmax, " s")
end

# For sine windows
if win isa CosWindow
    println("Acquisition time: ", win.tmax, " s")
end

Computing the apodization function

The apod function returns the time-domain apodization vector:

# Load a spectrum
spec = exampledata("2D_HN")

# Get apodization function for F2 dimension
apod_function = apod(spec, F2Dim)

println("Apodization vector length: ", length(apod_function))
println("First few values: ", apod_function[1:5])

Apodization vector length: 512
First few values: [1.0, 0.9996988186962042, 0.9987954562051724, 0.9972904566786902, 0.9951847266721969]

You can also compute it directly from a dimension:

dim = dims(spec, F2Dim)
apod_function = apod(dim)

By default, the apodization vector is zero-filled to match tdzf. To get only the acquired points:

apod_function = apod(spec, F2Dim, false)  # zerofill=false

Simulating lineshapes

The lineshape function generates a simulated spectrum including the effect of the window function:

# Get the ¹H axis
axis = dims(spec, F1Dim)

# Simulate a peak at 8.5 ppm with R2 = 10 s-1
chemical_shift = 8.5  # ppm
R2 = 10.0            # s-1
simulated = lineshape(axis, chemical_shift, R2)

println("Simulated lineshape length: ", length(simulated))

Simulated lineshape length: 1216

This is useful for:

Fitting experimental peaks
Understanding window function effects
Quality control and validation

You can also request complex-valued lineshapes:

# Get complex lineshape
simulated_complex = lineshape(axis, chemical_shift, R2, ComplexLineshape())