# Hilbert transform question

Started by April 17, 2005
```Hi Guys,

I have a question about Hilbert transformer applications.
First, we can build Hilbert transformers using
a tapped-delay line structure (like a tapped-delay
line FIR filter.)

An ideal Hilbert transformers (HT) would have a freq
magnitude response that's flat over the HT's entire
operating frequency range.  However, practical HTs
have a magnitude null at zero Hz and a freq mag response
that slowly rises in magnitude as we go up in
frequency.  Like this:

|       ****************************
|      *
|     *
|    *
|   *
|  *
|-*---------------------------------------
0          Freq -->

Adding more taps (coefficients) to the HT's implementation
allows us to increase the slope of the HT's mag response
just above zero Hz.  You guys know all of this.

OK, here's my question: When would we want to increase
the the slope of the HT's mag response?
I know that increasing the slope widens the usable
bandwidth of the HT, but I can't think of any applications
where that's desirable.  The only HTs I've designed
operated on signals centered at Fs/4, so the HT's passband
didn't have to be wide.  To rephrase my question,
for what applications (for what real-world signals)
would we want an HT that operates very near zero Hz?

I ask that question because a professor sent me
a manuscript describing a "HT Implementation Trick"
for building an HT whose usable passband extends down
close to zero Hz.  I haven't modeled his scheme yet to
see if it actually works, but I'm just wondering if
his idea is useful or not.

[-Rick-]

```
```Rick Lyons wrote:
> Hi Guys,
>
>   I have a question about Hilbert transformer applications.
> First, we can build Hilbert transformers using
> a tapped-delay line structure (like a tapped-delay
> line FIR filter.)
>
> An ideal Hilbert transformers (HT) would have a freq
> magnitude response that's flat over the HT's entire
> operating frequency range.  However, practical HTs
> have a magnitude null at zero Hz and a freq mag response
> that slowly rises in magnitude as we go up in
> frequency.  Like this:
>
>
>  |       ****************************
>  |      *
>  |     *
>  |    *
>  |   *
>  |  *
>  |-*---------------------------------------
>    0          Freq -->
>
> Adding more taps (coefficients) to the HT's implementation
> allows us to increase the slope of the HT's mag response
> just above zero Hz.  You guys know all of this.
>
> OK, here's my question: When would we want to increase
> the the slope of the HT's mag response?
> I know that increasing the slope widens the usable
> bandwidth of the HT, but I can't think of any applications
> where that's desirable.  The only HTs I've designed
> operated on signals centered at Fs/4, so the HT's passband
> didn't have to be wide.  To rephrase my question,
> for what applications (for what real-world signals)
> would we want an HT that operates very near zero Hz?
>
> I ask that question because a professor sent me
> a manuscript describing a "HT Implementation Trick"
> for building an HT whose usable passband extends down
> close to zero Hz.  I haven't modeled his scheme yet to
> see if it actually works, but I'm just wondering if
> his idea is useful or not.
>
> Thanks for your opinions guys.
>
> [-Rick-]
>
Phasing method SSB (single side band) receivers and transmitters require
a hilbert transform to be done on the audio channel, and it needs to
extend down to the lowest audio frequency that is to be usable.  So
there's value in getting that corner frequency lower.

Off the top of my head the limitations I can see to a hilbert transform
are on the low-frequency end you need a longer impulse response which
translates directly to a longer FIR filter (but which could perhaps be
stretched with a clever combination of FIR and IIR filters) and on the
high-frequency end you have larger and larger gains in your taps, to the
detriment of general system precision.  These limits come purely from
the shape of the transform, so they'd apply regardless of how the thing
were implemented.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```
```Rick Lyons wrote:
> Hi Guys,
>
>   I have a question about Hilbert transformer applications.
> First, we can build Hilbert transformers using
> a tapped-delay line structure (like a tapped-delay
> line FIR filter.)
>
> An ideal Hilbert transformers (HT) would have a freq
> magnitude response that's flat over the HT's entire
> operating frequency range.  However, practical HTs
> have a magnitude null at zero Hz and a freq mag response
> that slowly rises in magnitude as we go up in
> frequency.  Like this:
>
>
>  |       ****************************
>  |      *
>  |     *
>  |    *
>  |   *
>  |  *
>  |-*---------------------------------------
>    0          Freq -->
>
> Adding more taps (coefficients) to the HT's implementation
> allows us to increase the slope of the HT's mag response
> just above zero Hz.  You guys know all of this.
>
> OK, here's my question: When would we want to increase
> the the slope of the HT's mag response?
> I know that increasing the slope widens the usable
> bandwidth of the HT, but I can't think of any applications
> where that's desirable.  The only HTs I've designed
> operated on signals centered at Fs/4, so the HT's passband
> didn't have to be wide.  To rephrase my question,
> for what applications (for what real-world signals)
> would we want an HT that operates very near zero Hz?
>
> I ask that question because a professor sent me
> a manuscript describing a "HT Implementation Trick"
> for building an HT whose usable passband extends down
> close to zero Hz.  I haven't modeled his scheme yet to
> see if it actually works, but I'm just wondering if
> his idea is useful or not.
>
> Thanks for your opinions guys.
>
> [-Rick-]

Generating single sideband is best done with a pair of audio signals in
quadrature. With analog, I've always seen it done -- and done it -- with
a pair of all-pass filters whose outputs are in quadrature but bear no
specified relation to the input phase.* The same can be done digitally,
but a Hilbert transformer makes linear phase possible. For certain
instrumentation schemes that can be imperative; in any event, it's a
nice feature. Somewhere, I have a design of an HT with about 160 taps
and a usable band from 50 Hz to 3950 Hz with an 8 KHz sample rate.

Jerry
_________________________
* Three R-C sections in each filter span a decade with about 1/4 degree
match.
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```in article 116590lhcfqvuf1@corp.supernews.com, Tim Wescott at
tim@wescottnospamdesign.com wrote on 04/17/2005 13:59:

> Phasing method SSB (single side band) receivers and transmitters require
> a hilbert transform to be done on the audio channel, and it needs to
> extend down to the lowest audio frequency that is to be usable.

in article qI-dnXyXY8t4NP_fRVn-1Q@rcn.net, Jerry Avins at jya@ieee.org wrote
on 04/17/2005 14:24:

> Generating single sideband is best done with a pair of audio signals in
> quadrature. With analog, I've always seen it done -- and done it -- with
> a pair of all-pass filters whose outputs are in quadrature but bear no
> specified relation to the input phase.*

you know, it's kinda funny guys.  i'm not as old as Jerry but at one time in
my life, dating back to '68, i was a ham radio operator (my call sign was
WB0CCA for "W B 0 Casselton's Commie Adolescent") and was doing SSB (with a
Heathkit "Hot Water" HW-100, the cheapest 5-band SSB a kid could afford).
although there were a lot of tubes, and 2 or 3 transistors in the VFO, there
were no all-pass filters nor direct Hilbert transformers in that box.  the
SSB was generated the old fashioned way with AM (at some IF frequency that i
forget) and an nasty sharp crystal-lattice filter that was 2.1 kHz wide.
when translated down to the audio baseband, i recall it was 350 Hz to 2450
Hz.  that's where all of our voices fit into.  not hi-fi but it was enough.

anyway, i didn't understand the Hilbert transform or even quadrature
modulation, but i did understand AM and had some concept of what filtering

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```Rick Lyons wrote:
...
> OK, here's my question: When would we want to increase
> the the slope of the HT's mag response?
> I know that increasing the slope widens the usable
> bandwidth of the HT, but I can't think of any applications
> where that's desirable.  The only HTs I've designed
> operated on signals centered at Fs/4, so the HT's passband
> didn't have to be wide.  To rephrase my question,
> for what applications (for what real-world signals)
> would we want an HT that operates very near zero Hz?
>
> I ask that question because a professor sent me
> a manuscript describing a "HT Implementation Trick"
> for building an HT whose usable passband extends down
> close to zero Hz.  I haven't modeled his scheme yet to
> see if it actually works, but I'm just wondering if
> his idea is useful or not.
>

An important use for the HT is in encoding/decoding horizontal Ambisonic
B-Format signals to/from the stereo-compatible "UHJ" format. UHJ requires that
signals are phase-shifted by 90deg over the audio range. There are numerous
pages on the web detailing B-Format and UHJ; this is quite a nice one:

http://www.music.gla.ac.uk/~george/audio/spatial_intro/spatial_intro.html

Most of the CDs produced by the Nimbus label were recorded in B-Format and UHJ
encoded.

Richard Dobson

```
```On Sun, 17 Apr 2005 15:55:23 -0400, robert bristow-johnson
<rbj@audioimagination.com> wrote:

>Heathkit "Hot Water" HW-100, the cheapest 5-band SSB a kid could afford).
>although there were a lot of tubes, and 2 or 3 transistors in the VFO, there
>were no all-pass filters nor direct Hilbert transformers in that box.  the
>SSB was generated the old fashioned way with AM (at some IF frequency that i
>forget) and an nasty sharp crystal-lattice filter that was 2.1 kHz wide.

Wouldn't those have had to be DSB modulators, and to have just
stripped the unwanted sideband? Wouldn't the carrier from an
AM modulator have been too large as a filter residual?
'Course, times have changed!

Chris Hornbeck
6x9=42 April 29
```
```in article nrc661pc5c9hln4onnv156qnod5qfvtvv1@4ax.com, Chris Hornbeck at
chrishornbeckremovethis@att.net wrote on 04/18/2005 00:16:

> On Sun, 17 Apr 2005 15:55:23 -0400, robert bristow-johnson
> <rbj@audioimagination.com> wrote:
>
>> Heathkit "Hot Water" HW-100, the cheapest 5-band SSB a kid could afford).
>> although there were a lot of tubes, and 2 or 3 transistors in the VFO, there
>> were no all-pass filters nor direct Hilbert transformers in that box.  the
>> SSB was generated the old fashioned way with AM (at some IF frequency that i
>> forget) and a nasty sharp crystal-lattice filter that was 2.1 kHz wide.
>
> Wouldn't those have had to be DSB modulators, and to have just
> stripped the unwanted sideband? Wouldn't the carrier from an
> AM modulator have been too large as a filter residual?

it could have been that.  with 1960s technology, you could have had the IF
AM modulator create something like

(1 + a*x(t)) * cos(w0*t)         (w0 = intermediate frequency)

with circuitry to coherently subtract the cos(w0*t).  then what you got left
would be DSB.  but if the carrier was just 350 Hz away from the passband,
the other sideband was only 700 Hz away.  it would still need a good and
sharp filter.  i remember that the crystal lattice filter was a component
mounted off of the circuit boards and 2 cables connected to it.  also, when
switching from USB to LSB, the IF was offset by, i s'pose, 2.8 kHz since the
filter was fixed.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```On Mon, 18 Apr 2005 00:37:54 -0400, robert bristow-johnson
<rbj@audioimagination.com> wrote:

>  it would still need a good and
>sharp filter.  i remember that the crystal lattice filter was a component
>mounted off of the circuit boards and 2 cables connected to it.

And in those days the filter was PFM ( pure ... magic ) although some
intrepid souls made their own, largely by (literally) cut-and-try,
by filing on quartz crystals and inserting them into some hoped-for
band pass filter.

In that era the Collins LC (? is my memory correct?) filters at
455 kHz were considered to be "The Shit" because they had some
really deep slopes.

Probably nobody worried too much about residual carrier anyway;
a small beat somewhere wasn't the end of the world.

Thanks,

Chris Hornbeck
6x9=42 April 29
```
```Rick Lyons wrote:
> Hi Guys,
>
>   I have a question about Hilbert transformer applications.
> First, we can build Hilbert transformers using
> a tapped-delay line structure (like a tapped-delay
> line FIR filter.)
>
> An ideal Hilbert transformers (HT) would have a freq
> magnitude response that's flat over the HT's entire
> operating frequency range.  However, practical HTs
> have a magnitude null at zero Hz and a freq mag response
> that slowly rises in magnitude as we go up in
> frequency.  Like this:
>
>
>  |       ****************************
>  |      *
>  |     *
>  |    *
>  |   *
>  |  *
>  |-*---------------------------------------
>    0          Freq -->
>
> Adding more taps (coefficients) to the HT's implementation
> allows us to increase the slope of the HT's mag response
> just above zero Hz.  You guys know all of this.
>
> OK, here's my question: When would we want to increase
> the the slope of the HT's mag response?
> I know that increasing the slope widens the usable
> bandwidth of the HT, but I can't think of any applications
> where that's desirable.  The only HTs I've designed
> operated on signals centered at Fs/4, so the HT's passband
> didn't have to be wide.  To rephrase my question,
> for what applications (for what real-world signals)
> would we want an HT that operates very near zero Hz?
>
> I ask that question because a professor sent me
> a manuscript describing a "HT Implementation Trick"
> for building an HT whose usable passband extends down
> close to zero Hz.  I haven't modeled his scheme yet to
> see if it actually works, but I'm just wondering if
> his idea is useful or not.
>
> Thanks for your opinions guys.

I would be most interested in such a HT. When dealing
with large volumes of broad-band data, one pushes the
limits of signal sampling, to save system/storage costs.
So even if the signal is centered at Fs/4, the bandwidth
may in some applications be as large as 0.05*Fs - 0.4*Fs.

To what extent I have used HTs with seismic data, I have
implemented it as an IFFT(FFT(X)[0:Fs/2])) transform,
which only works in off-line applications. Having a HT
that potentially could work in an on-line application
would be very interesting for "broadish-band" applications.

Rune

```
```I've worked on a broadband wireless system that generates a real
baseband signal, but uses a Hilbert transform to filter the negative
frequencies before digital modulation to the passband. In this case we
needed a good Hilbert filter, with passband coming right down to close
to 0Hz. In the end, this SSB scheme was implemented with a 41 tap FIR
Hilbert filter, using a truncated ideal Hilbert filter, and a Kaiser
smoothing window.

\$0.02

```