# Inverse Fast Hilbert Transform (G Dataflow)

Last Modified: January 9, 2017

Computes the inverse fast Hilbert transform of a sequence using Fourier identities.

## Hilbert{x}

The input signal.

## error in

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

Default: No error

## x

The inverse Hilbert Transform of the input signal.

## error out

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

## Algorithm for Calculating the Inverse Fast Hilbert Transform

The following equation defines the inverse Hilbert transform of a function h(t):

$x\left(t\right)={H}^{-1}\left\{h\left(t\right)\right\}=-\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{h\left(\tau \right)}{t-\tau }d\tau$

Using the definition of the Hilbert transform

$h\left(t\right)=H\left\{x\left(t\right)\right\}=\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{x\left(\tau \right)}{t-\tau }d\tau$

you can obtain the inverse Hilbert transform by negating the forward Hilbert transform

$x\left(t\right)={H}^{-1}\left\{h\left(t\right)\right\}=-H\left\{h\left(t\right)\right\}$

## Algorithm for Discrete Implementation of the Inverse Hilbert Transform

This node performs the discrete implementation of the inverse Hilbert transform with the aid of the Hilbert transform by taking the following steps:

1. Hilbert transform the input sequence X.
Y = H{X}

2. Negate Y to obtain the inverse Hilbert transform.
H-1{X} = -Y

The Hilbert transform works best with AC coupled, band-limited signals.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported