mesa@pdn.paradyne.com (Osvaldo Mesa) (05/24/91)
Does anyone know of a good practical reference to discrete Hilbert transform design and implementation (I have seen chapter 7 of Oppenheim and Schafer)? My first search through the library did not show anything that looked practical to me. Using chapter 7 of Op. and Sch. I could design a Hilbert transform using the same methodology but are there more alternatives? Thanks, -- Osvaldo A. Mesa AT&T Paradyne uunet!pdn!mesa Mail Stop LG-132 mesa@pdn.paradyne.com P. O. Box 2826 (813)530-8648 Largo, FL USA 34649-2826
jbuck@forney.berkeley.edu (Joe Buck) (05/26/91)
In article <1991May24.132426.15237@pdn.paradyne.com>, mesa@pdn.paradyne.com (Osvaldo Mesa) writes: |> Does anyone know of a good practical reference to discrete Hilbert transform |> design and implementation (I have seen chapter 7 of Oppenheim and Schafer)? The ideal discrete Hilbert transform of x(n) is, using eqn notation x hat (n) = { 2 over pi } sum from { i=- inf } to inf h(i)x(n-i) where h(i) is 1/i for odd n, and 0 for even n. Note that this is a two-sided filter, so in most cases you can't implement it exactly. One thing you can do is simply truncate the sequence at some distance from zero and delay it. For example, you can implement it as a 63-tap FIR filter (with half the coefficients zero) by setting h(i) to zero for i outside [-31,+31]. This gives acceptable results and corresponds to a delayed version of the Hilbert transform. So, for instance, if you're using the Hilbert transform to reconstruct an analytic signal from its real part, use the above filter to get the imaginary part and delay the original signal by 31 samples to get the real part. In some cases this is more work than you need. In many modem applications you need a response corresponding to the Hilbert transform only in the narrow band corresponding to the signal; you don't need that abrupt transition at zero frequency because you have no signal there. This allows you to use a method such as the Parks-McClellan method to design a better filter with lower order, since the response can be anything in frequencies where there is no signal. -- Joe Buck jbuck@galileo.berkeley.edu {uunet,ucbvax}!galileo.berkeley.edu!jbuck