Basis functions


Name

Type

Expression

Derivative

Usage
 

Sinusoid

Complex

e
^{iωt}

ite
^{iωt}

Always
 
Damping function

Real

{e}^{a{t}^{\eta}}

{t}^{\eta}{e}^{a{t}^{\eta}}

Used except along indirect dimensions of constant time experiments
 
Parameters

Parameter

Symbol

Variable

Basis function

Initial value

Constrained

Frequency

Ω

Yes

Sinusoid

From peak position

No

Decay Rate

Α

Yes

Damping function

From “prototype” signal

Yes

Decay Power

Η

No

Damping function

Assigned based on profiling of data sets. Fixed per analysis on single data set

No (fixed)

 The time domain basis functions along each model dimension are the complex product of a sinusoid basis function with a damping function. The corresponding frequency domain functions are obtained from Fourier transformation using a digital operator derived from the acquisition and processing parameters. Multidimensional basis functions are derived from the product of the orthogonal component basis functions along each dimension. For gradientbased (nonlinear) optimization of the parameters, the derivative basis functions are used. The exponent η appearing in the decay rate term is a value that modulates the signal between a Lorentzian (η = 1) and Gaussian (η = 2) decay profile. This value is adjusted to fit a similar class of peak shapes and is left constant throughout the optimization of any given data set. The corresponding derivative basis functions are used to calculate the derivative of the basis function with respect to the angular frequency (sinusoid) and decay rate (damping function).