Hilbert IIR filter implementation

Al:

[snip]
"Al Clark" <dsp@danvillesignal.com> wrote in message > Peter,
>
> I think I'm still missing something. Doesn't rotating the low pass
> prototype just create a bandpass filter? How do you structure the filter
> to get the hilbert pair? Maybe I need pictures or coffee. Can you create
> a simple example?
> -- 
> Al Clark
> Danville Signal Processing, Inc.
[snip]

Of course it's a "complex" bandpass filter.

Isn't that *exactly* what a Hilbert Transformer in "parallel" with a
constant delay
does?  i.e. the "pair" forms a *complex* bandpass filter which rejects
negative
frequencies.

The "rotated" low pass filter is really two real filters sharing a common
input sequence
with separate output sequences.  Each of the output sequences is a real
sequence and
the two are Hilbert Transforms of each other i.e. in the frequency domain
they
differ by 90 degrees in their phase angles.

You don't have to take the complex signal viewpoint if you are more
comfortable with
real sequences, but a complex sequence is nothing more than a pair of real
sequences.

If you wish to take the view of complex signals and complex frequencies
[includes the
concept of  positive and negative frequencies] it will help with the
understanding of
the 90 degree phase splitter, in this *complex* view then the two output
sequences
would be simply "labeled" the "real" sequence" and the other the "imaginary
sequence".

i.e.  One way of looking at a Hilbert Transformer placed in parallel with a
pure constant
delay network equal to the average latency of the Hilbert transformer is to
view the parallel
combination of the Hilbert Transformer and the pure constant delay pair as a
complex filter.
[In this case a filter that has one real input sequence and two output
sequences, call one
output sequence "real" and the other output sequence "imaginary" if you
will.]

The complex filter formed by a constant delay network in parallel with the
Hilbert
Transfromer then can be viewed as a *complex* bandpass filter passing a band
of
[complex] positive frequencies from [approximately] w = 0 up to its'
specified upper
cutoff  frequency whilst suppressing all the negative frequencies which fall
in its' lower
stopband.  i.e. a complex band  pass filter.

The pair of cross-coupled real filters resulting from the rotation of a real
low pass
filter about z = 0 in the z-plane produces a pair of real filters whose
output sequences
are Hilbert Transforms of each other.

The advantages of this approach to producing a pair of sequences with a 90
degree
phase difference, when compared to an FIR Hilbert Transformer in
parallel with a constant delay is that the pair of IIR filters is *much*
more efficient at
producing the 90 degree phase shifted output sequences.

By more effiecient I mean:

The IIR method I proposed when designed to meet the same phase shift error
specification will have:

a)  Considerably less over all latency than the Hilbert Transformer
technique.
b)  Considerably fewer multiply-adds.
c) Considerably less memory requirements.

Sometimes the improvement in these implementation factors is almost an order
of magnitude!

Nice when you are short of power, memory and latenency, say in a mobile or
span powered
application.

On the other hand the IIR implementation produces a non-linear phase angle
when compared
to the linear phase of the Hilbert Transformer constant delay pair.  Often
that simply doesn't
matter.  But if it does, it can be corrected by using all-pass delay
equalizers in the two output
paths, but that is a subject for another posting!  :-)

--
Peter
Consultant
Indialantic By-the-Sea, FL.