dcmri.nexpconv#

dcmri.nexpconv(n, T, t)[source]#

Convolve n identical normalised exponentials analytically

Parameters:

  • n (float) – number of exponentials. Since an analytical formula is used this can also be non-integer.

  • T (float) – the characteristic time of the exponential.

  • t (array_like, optional) – the time points where the values of f are defined, in the same units as T.

Returns:

The result of the convolution as a 1D array.

Return type:

numpy.ndarray

Raises:

ValueError – if n<1 and if T<0

See also

conv, expconv, biexpconv, stepconv

Notes

nexpconv returns the exact analytical result of the following n convolutions:

\[g(t) = \frac{e^{-t/T}}{T} \otimes \ldots \otimes \frac{e^{-t/T}}{T}\]

The result is a gamma variate function with unit area:

\[g(t) = \frac{1}{\Gamma(n)}\left(\frac{t}{T}\right)^{n-1} \frac{e^{-t/T}}{T}\]

Note the gamma-variate function becomes unstable due to numerical overflow at large n and/or short T. This function handles that situation by moving to a numerical solution, but this slows down the computations significantly. Callers should if possible avoid this situation by placing suitable bounds on the parameters.

Example

Import package and create a vector of uniformly sampled time points t with spacing 5.0s:

>>> import dcmri as dc
>>> t = 5.0*np.arange(4)

Calculate the convolution of 4 normalised exponentials with time constants 5s:

>>> g = dc.nexpconv(4, 5, t)
array([0.        , 0.01226265, 0.03608941, 0.04480836])

Examples using dcmri.nexpconv#

A comparison of convolution functions

A comparison of convolution functions