🚀 go-pugleaf

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