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In Mathematics and in Signal Processing , the Hilbert transform, here denoted \mathcal{H}, of a real-valued function, s(t)\,, is obtained by Convolving signal s(t) with 1/(\pi t) to obtain \widehat s(t). Therefore, the Hilbert transform \widehat s(t) can be interpreted as the output of a Linear Time Invariant system
with input s(t), and a system impulse response given as 1/(\pi t). It is a useful
mathematical tool to describe the Complex Envelope of a real-valued carrier modulated signal in communication
theory (see below for more on applications). The precise definition is as follows:
  • s)(t) = rac{1}{\pi}\int_{-\infty}^{\infty} rac{s( au)}{t- au}\, d au.\,


where

:h(t) = rac{1}{\pi t}\,

and considering the integral as a Cauchy Principal Value (which avoids the singularity at au = t\,).

It follows that the Hilbert transform has a frequency response given by the Fourier Transform :

And since:

:\mathcal{F}\{\widehat s\}(\omega) = H(\omega )\cdot \mathcal{F}\{s\}(\omega),

the Hilbert transform has the effect of shifting the Negative Frequency components of s(t)\, by +90° and the positive frequencies components by −90°.

We also note that H^2(\omega ) = -1\,. So multiplying the above equation by -H(\omega )\, gives

:\mathcal{F}\{s\}(\omega) = -H(\omega )\cdot \mathcal{F}\{\widehat s\}(\omega)

from which the inverse Hilbert transform is apparent''':'''

  • \widehat s)(t) = -\mathcal{H}\{\widehat s\}(t).\,



HILBERT TRANSFORM EXAMPLES


Notice: Some authors, e.g., Bracewell, use our -\mathcal{H} as their definition of the forward transform. A consequence is that the right column of this table would be negated.

  <math>\delta(t)</math><br>''' "http://wwwinformationdelightinfo/encyclopedia/entry/Delta_function" class="copylinks">Delta Function ''' <math> {1 \over \pi t}</math>