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 a real and an imaginary part.

$x_a(n)=\mathrm{Re}(x(n))+\mathrm{Im}(x(n))$

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

$h(n)=\{\begin{array}{ll}\frac{2}{\pi n},& \text{for }n\text{ odd}\\ 0,& \text{for }n\text{ even}\\ \end{array}$

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

$h(n)=\{\begin{array}{ll}\frac{2}{\pi (n-\frac{N-1}{2})},& \text{for }n\text{ odd}\\ 0,& \text{for }n\text{ even},\\ \end{array}$

for $0\le n\le N-1$, where N must be an odd number representing the length of the filter. The resulting Hilbert transform is delayed by $\delta =\frac{N-1}{2}$

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.

States

None.

See also

User Reference:Filters, Contributions:SignalProcessing, Contributions:HilbertSignalProcessing