# Difference between revisions of "Contributions:HilbertFilter"

## Synopsis

This filter computes the envelope or the phase of a signal using Hilbert transform. The discrete input signal x(n) is first transformed to its analytic representation (i.e., analytic signal), which is composed of real and imaginary parts.

$x_{a}(n) = \operatorname{Re}(x(n)) + \operatorname{Im}(x(n))$

The real part is the same input signal, and the imaginary part is the Hilbert transform of the input signal. The Hilbert transform is implemented as the convolution of the input signal with the filter h(n).

$h(n) = \begin{cases} \ \ {2 \over \pi n}, & \mbox{for } n \mbox{ odd}\\ \ \ 0, & \mbox{for } n \mbox{ even}\\ \end{cases}$

To get an ideal Hilbert transform, n must be infinitely long $(-\infty < n < \infty)$. However, for real time implementations, h(n) must be truncated and delayed to guarantee a causal filter. Thus, the FIR filter is defined as

$h(n) = \begin{cases} \ \ {2 \over \pi (n-\frac{N-1}{2})}, & \mbox{for } n \mbox{ odd}\\ \ \ 0, & \mbox{for } n \mbox{ even},\\ \end{cases}$

for $0\leq n \leq N-1$, where N must be an odd number representing the length of the filter. The resulting Hilbert transform is delayed by $\delta = \frac{N-1}{2}$ implying that the real part of the analytic signal must be delayed by the same amount too in order to estimate the envelope and phase of the input signal.

## Location

Not yet released: will be released at http://www.bci2000.org/svn/trunk/src/contrib/SignalProcessing/HilbertSignalProcessing/HilbertFilter.cpp

## Versioning

### Authors

Cristhian Potes, Jeremy Hill

### Source Code Revisions

• Initial development:
• Tested under:
• Known to compile under:
• Broken since:

## Parameters

### OutputSignal

This parameter may be one of

0 - Copy input signal
no processing,
1 - Magnitude
Hilbert envelope amplitude,
2 - Phase
Hilbert phase,
3 - Real part
original input signal, but with a delay to match its timing to the imaginary part.
4 - Imaginary part
original signal filtered with an FIR-Hilbert transformer.

### Delay

The delay of the Hilbert transform can be calculated as $\delta = \frac{N-1}{2}$, where N is an odd number representing the length of the filter.

None.