# Laplace Transform Real (G Dataflow)

Version:

Computes the real Laplace transform of a sequence.

## x

The array describing the evenly sampled time signal.

The first element of this array belongs to t = 0, the last to t = end.

## end

The instant in time of the last sample.

The entire sample interval is between 0 and end.

## error in

Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

Default: No error

## Laplace{x}

The result of the Laplace transform as an array.

## error out

Error information. The node produces this output according to standard error behavior.

## The Continuous Version of the Laplace Transform

The real Laplace transform of a real signal x(s) is defined by the following equation:

$\mathrm{Laplace}\left\{X\right\}\left(s\right)={\int }_{0}^{\infty }x\left(t\right)\mathrm{exp}\left(-st\right)dt$

for real $s\ge 0$.

## The Discrete Version of the Laplace Transform

The discrete version of the Laplace transform of a discretely and evenly-sampled signal is a generation of the above continuous version.

The definition of the Laplace transform is not of much use if the time signal increases rapidly with the time. The discrete version of the Laplace transform cannot fully detect the convergence behavior of the original definition.

The discrete version of the Laplace transform is computationally expensive. An efficient strategy for the discrete Laplace transform is based on the fast fractional Fourier transform (FFFT). The definition of the FFFT is as follows:

$\mathrm{FFFT}\left\{X\right\}\left(t\right)={\int }_{0}^{\infty }x\left(s\right)\mathrm{exp}\left(-iast\right)ds$

with an arbitrarily chosen complex alpha.

The following image shows the real Laplace transform of the function $f\left(t\right)=\mathrm{sin}\left(t\right)$ in the interval (0, 6).

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported