need a book: Hilbert transform

[John Larkin]

Hi,

Can anybody recommend a book that has good stuff about implementing
the discrete Hilbert transform, preferably in an FPGA? I need
practical stuff, like accuracy over frequency ranges for given tap
count, windowing, truncation effects, etc.

I need to take a stream of digitized (16 bit) analog samples and
produce an I-Q data stream pair over roughly a 20:1 frequency range,
to maybe 1 degree accuracy. Max signal frequency will be low, 1 KHz
maybe.

Any other resource tips would be appreciated. We could pay for a bit
of consulting maybe, too.

Thanks,

John

 

[Joerg]

Hello John,

Skolnik's "Radar Handbook" has some I/Q error budgeting in chapter 3 but
I don't think it'll suffice here. I have never seen a book that goes
this far into practical matters on Hilbert (except for the analog
version), maybe too small a market for publishers. If there is anything
out there it'll be most likely for Doppler applications.

Artech House may have some good stuff and you could check them. Most
books about transforms come with a CD full of helpful routines but often
they will be plain C code:

http://www.artechhouse.com/default.asp?Frame=Book.asp&Book=C5P10&Country=&Continent=ME&State=

Possibly you could start at one of the math simulator vendors, like here:

http://www.mathworks.com/access/helpdesk/help/toolbox/signal/filterd9.html

Simulators are almost a must when doing this kind of stuff. My turf is
med imaging and there usually every company invents the wheel again,
simulates until smoke comes out of the PC and then it all becomes a
trade secret. They rarely publish.

What do you want to do? Does it have to be poured into the FPGA?

Regards, Joerg

http://www.analogconsultants.com

 

[John Larkin]

On Wed, 01 Jun 2005 00:44:18 GMT, Joerg
<notthisjoergsch@removethispacbell.net> wrote:

Hello John,

Skolnik's "Radar Handbook" has some I/Q error budgeting in chapter 3 but
I don't think it'll suffice here. I have never seen a book that goes
this far into practical matters on Hilbert (except for the analog
version), maybe too small a market for publishers. If there is anything
out there it'll be most likely for Doppler applications.

Artech House may have some good stuff and you could check them. Most
books about transforms come with a CD full of helpful routines but often
they will be plain C code:

http://www.artechhouse.com/default.asp?Frame=Book.asp&Book=C5P10&Country=&Continent=ME&State=

Possibly you could start at one of the math simulator vendors, like here:

http://www.mathworks.com/access/helpdesk/help/toolbox/signal/filterd9.html

Simulators are almost a must when doing this kind of stuff. My turf is
med imaging and there usually every company invents the wheel again,
simulates until smoke comes out of the PC and then it all becomes a
trade secret. They rarely publish.

What do you want to do? Does it have to be poured into the FPGA?

Suppose I have an AC power system, and I can digitize a pair of
voltage and current waveforms. I want to report everything: trms
volts/amps, true power, reactive power, phase angle. The line
frequency could vary from maybe 20 to 80 Hz for a stationary
generator, or 200-800 for an aircraft system (including startup and
weird situations.) I'll digitize to 16 bits, at maybe 20K
samples/second or something. I'm considering doing all the signal
processing in an FPGA, crunching maybe 8 voltage+current pairs.

For the rms volts/amps, we could just square the samples, filter, and
allow my pokey uP to occasionally pick up that and square root.

True power is just the product of the e*i samples, lowpass filtered.
Easy.

What's tricky is the reactive power/phase angle thing. The ideal thing
would be to delay the voltage samples 90 degrees and then multiply by
the current samples, then filter to get the signed reactive power. The
trick is to delay the voltage sample data stream 90 degrees. A
discrete (fir) Hilbert would give me the phase-shifted voltage signal
(actually 135 deg, not 90, so I'd have to delay the current samples,
too, but that's OK.) I just need to quantify how good a given
implementation might be.

The other way to do it would be to use a fifo clocked at a multiple of
the waveform frequency, delaying the voltage or current samples by 90
degrees. That would take a digital PLL to track the voltage waveform
frequency and generate a 128x or something tracking clock. Maybe a
dds/nco clock gen with some fancy digital phase detector? That's
complex, too, but has the advantage of acquiring the waveform
frequency essentially for free. I could have a range bit the user sets
for the 60 vs 400 Hz situations, so I'd only need about a 4:1 tracking
range on the clock.

Either way, it sounds like I'm in for some simulation. PowerBasic!


John

 

[Terry Given]

John Larkin wrote:

On Wed, 01 Jun 2005 00:44:18 GMT, Joerg
notthisjoergsch@removethispacbell.net> wrote:


Hello John,

Skolnik's "Radar Handbook" has some I/Q error budgeting in chapter 3 but
I don't think it'll suffice here. I have never seen a book that goes
this far into practical matters on Hilbert (except for the analog
version), maybe too small a market for publishers. If there is anything
out there it'll be most likely for Doppler applications.

Artech House may have some good stuff and you could check them. Most
books about transforms come with a CD full of helpful routines but often
they will be plain C code:

http://www.artechhouse.com/default.asp?Frame=Book.asp&Book=C5P10&Country=&Continent=ME&State=

Possibly you could start at one of the math simulator vendors, like here:

http://www.mathworks.com/access/helpdesk/help/toolbox/signal/filterd9.html

Simulators are almost a must when doing this kind of stuff. My turf is
med imaging and there usually every company invents the wheel again,
simulates until smoke comes out of the PC and then it all becomes a
trade secret. They rarely publish.

What do you want to do? Does it have to be poured into the FPGA?



Suppose I have an AC power system, and I can digitize a pair of
voltage and current waveforms. I want to report everything: trms
volts/amps, true power, reactive power, phase angle. The line
frequency could vary from maybe 20 to 80 Hz for a stationary
generator, or 200-800 for an aircraft system (including startup and
weird situations.) I'll digitize to 16 bits, at maybe 20K
samples/second or something. I'm considering doing all the signal
processing in an FPGA, crunching maybe 8 voltage+current pairs.

For the rms volts/amps, we could just square the samples, filter, and
allow my pokey uP to occasionally pick up that and square root.

True power is just the product of the e*i samples, lowpass filtered.
Easy.

What's tricky is the reactive power/phase angle thing. The ideal thing
would be to delay the voltage samples 90 degrees and then multiply by
the current samples, then filter to get the signed reactive power. The
trick is to delay the voltage sample data stream 90 degrees. A
discrete (fir) Hilbert would give me the phase-shifted voltage signal
(actually 135 deg, not 90, so I'd have to delay the current samples,
too, but that's OK.) I just need to quantify how good a given
implementation might be.

The other way to do it would be to use a fifo clocked at a multiple of
the waveform frequency, delaying the voltage or current samples by 90
degrees. That would take a digital PLL to track the voltage waveform
frequency and generate a 128x or something tracking clock. Maybe a
dds/nco clock gen with some fancy digital phase detector? That's
complex, too, but has the advantage of acquiring the waveform
frequency essentially for free. I could have a range bit the user sets
for the 60 vs 400 Hz situations, so I'd only need about a 4:1 tracking
range on the clock.

Either way, it sounds like I'm in for some simulation. PowerBasic!


John

why not a synchronous demodulator (or lock-in amp

if you have 3 phases, its trivial. measure all 3 phases, and do a 3-2
phase transform to create an equivalent rotating vector a + jb. for a
single-phase system, build a 90 degree phase shifter so you have
Vpeak*sin(theta) + j*Vpeak*cos(theta).

then multiply by exp(-jtheta) (decompose into real & imaginary calcs)

You will then get 2 outputs, call them Vd and Vq. If you have a pure
sinusoidal system, and theta = integral(w_line.dt) is the correct phase,
one of these will be zero. Use this as the feedback to a PI controller,
whose reference is zero. PI output has ideal line frequency added to it
(if you want, not necessary) and is then integrated to produce theta.

once it syncs up, you have your real & imaginary V,I components in the
stationary reference frame, IOW they are DC quantities. easy to figure
out reactive power etc.

Cheers
Terry

 

[Mike Monett]

John Larkin wrote:

[...]

Suppose I have an AC power system, and I can digitize a pair of
voltage and current waveforms. I want to report everything: trms
volts/amps, true power, reactive power, phase angle. The line
frequency could vary from maybe 20 to 80 Hz for a stationary
generator, or 200-800 for an aircraft system (including startup and
weird situations.) I'll digitize to 16 bits, at maybe 20K
samples/second or something. I'm considering doing all the signal
processing in an FPGA, crunching maybe 8 voltage+current pairs.

For the rms volts/amps, we could just square the samples, filter, and
allow my pokey uP to occasionally pick up that and square root.

True power is just the product of the e*i samples, lowpass filtered.
Easy.

Isn't that Volt-Amperes? You need the phase angle to separate the
components into true power and reactive power.

What's tricky is the reactive power/phase angle thing.

Can't you just detect the zero crossings in the voltage and current
waveforms and get the phase angle between them?

Then solve the relationship:

cos(phi) = True Power / Total Power

Any waveform distortion could mess things up a bit trying to find the
zero crossings. A simple boxcar integrator is very fast in software and
might help improve the accuracy.

Is this what you are looking for, or did I miss something fundamental in
your post?

[...]

John

Mike Monett

 

[Fred Bloggs]

John Larkin wrote:

Suppose I have an AC power system, and I can digitize a pair of
voltage and current waveforms. I want to report everything: trms
volts/amps, true power, reactive power, phase angle. The line
frequency could vary from maybe 20 to 80 Hz for a stationary
generator, or 200-800 for an aircraft system (including startup and
weird situations.) I'll digitize to 16 bits, at maybe 20K
samples/second or something. I'm considering doing all the signal
processing in an FPGA, crunching maybe 8 voltage+current pairs.

For the rms volts/amps, we could just square the samples, filter, and
allow my pokey uP to occasionally pick up that and square root.

True power is just the product of the e*i samples, lowpass filtered.
Easy.

What's tricky is the reactive power/phase angle thing. The ideal thing
would be to delay the voltage samples 90 degrees and then multiply by
the current samples, then filter to get the signed reactive power. The
trick is to delay the voltage sample data stream 90 degrees. A
discrete (fir) Hilbert would give me the phase-shifted voltage signal
(actually 135 deg, not 90, so I'd have to delay the current samples,
too, but that's OK.) I just need to quantify how good a given
implementation might be.

The other way to do it would be to use a fifo clocked at a multiple of
the waveform frequency, delaying the voltage or current samples by 90
degrees. That would take a digital PLL to track the voltage waveform
frequency and generate a 128x or something tracking clock. Maybe a
dds/nco clock gen with some fancy digital phase detector? That's
complex, too, but has the advantage of acquiring the waveform
frequency essentially for free. I could have a range bit the user sets
for the 60 vs 400 Hz situations, so I'd only need about a 4:1 tracking
range on the clock.

Either way, it sounds like I'm in for some simulation. PowerBasic!

Pathetically trivial when you use Parseval's Identity and various
elementary results relating multiplication in the time domain to
convolution in the frequency domain and take note that the frequency
domain representation is on an orthogonal basis...

 

[Terry Given]

John Larkin wrote:

On Wed, 01 Jun 2005 16:21:36 +1200, Terry Given <my_name@ieee.org
wrote:


John Larkin wrote:

On Wed, 01 Jun 2005 00:44:18 GMT, Joerg
notthisjoergsch@removethispacbell.net> wrote:



Hello John,

Skolnik's "Radar Handbook" has some I/Q error budgeting in chapter 3 but
I don't think it'll suffice here. I have never seen a book that goes
this far into practical matters on Hilbert (except for the analog
version), maybe too small a market for publishers. If there is anything
out there it'll be most likely for Doppler applications.

Artech House may have some good stuff and you could check them. Most
books about transforms come with a CD full of helpful routines but often
they will be plain C code:

http://www.artechhouse.com/default.asp?Frame=Book.asp&Book=C5P10&Country=&Continent=ME&State=

Possibly you could start at one of the math simulator vendors, like here:

http://www.mathworks.com/access/helpdesk/help/toolbox/signal/filterd9.html

Simulators are almost a must when doing this kind of stuff. My turf is
med imaging and there usually every company invents the wheel again,
simulates until smoke comes out of the PC and then it all becomes a
trade secret. They rarely publish.

What do you want to do? Does it have to be poured into the FPGA?



Suppose I have an AC power system, and I can digitize a pair of
voltage and current waveforms. I want to report everything: trms
volts/amps, true power, reactive power, phase angle. The line
frequency could vary from maybe 20 to 80 Hz for a stationary
generator, or 200-800 for an aircraft system (including startup and
weird situations.) I'll digitize to 16 bits, at maybe 20K
samples/second or something. I'm considering doing all the signal
processing in an FPGA, crunching maybe 8 voltage+current pairs.

For the rms volts/amps, we could just square the samples, filter, and
allow my pokey uP to occasionally pick up that and square root.

True power is just the product of the e*i samples, lowpass filtered.
Easy.

What's tricky is the reactive power/phase angle thing. The ideal thing
would be to delay the voltage samples 90 degrees and then multiply by
the current samples, then filter to get the signed reactive power. The
trick is to delay the voltage sample data stream 90 degrees. A
discrete (fir) Hilbert would give me the phase-shifted voltage signal
(actually 135 deg, not 90, so I'd have to delay the current samples,
too, but that's OK.) I just need to quantify how good a given
implementation might be.

The other way to do it would be to use a fifo clocked at a multiple of
the waveform frequency, delaying the voltage or current samples by 90
degrees. That would take a digital PLL to track the voltage waveform
frequency and generate a 128x or something tracking clock. Maybe a
dds/nco clock gen with some fancy digital phase detector? That's
complex, too, but has the advantage of acquiring the waveform
frequency essentially for free. I could have a range bit the user sets
for the 60 vs 400 Hz situations, so I'd only need about a 4:1 tracking
range on the clock.

Either way, it sounds like I'm in for some simulation. PowerBasic!


John

why not a synchronous demodulator (or lock-in amp

if you have 3 phases, its trivial. measure all 3 phases, and do a 3-2
phase transform to create an equivalent rotating vector a + jb. for a
single-phase system, build a 90 degree phase shifter so you have
Vpeak*sin(theta) + j*Vpeak*cos(theta).

then multiply by exp(-jtheta) (decompose into real & imaginary calcs)

You will then get 2 outputs, call them Vd and Vq. If you have a pure
sinusoidal system, and theta = integral(w_line.dt) is the correct phase,
one of these will be zero. Use this as the feedback to a PI controller,
whose reference is zero. PI output has ideal line frequency added to it
(if you want, not necessary) and is then integrated to produce theta.

once it syncs up, you have your real & imaginary V,I components in the
stationary reference frame, IOW they are DC quantities. easy to figure
out reactive power etc.

Cheers
Terry


The only tricky part here is the "once it syncs up" bit.

John

because Vq == 0 (ish) (and Vd == Vpeak) is the very definition of
locked, a sync detector is pretty easy. |Va| diode-ORed with |Vb| gives
a nice big DC signal whenever Vac exists, ensuring you can quickly and
easily tell when an AC supply voltage exists of a suitable magnitude,
and whether or not lock-in has occurred is then a simple Vq thresholding
exercise (given suitable PI gains).

Cheers
Terry

 

[Mike Monett]

John Larkin wrote:


[...]

True power is just the product of the e*i samples, lowpass
filtered. Easy.

Isn't that Volt-Amperes? You need the phase angle to separate the
components into true power and reactive power.

VA would be the product of the RMS-averaged voltage and current
values, which throws away phase information. The average of the
instantaneous e/i sample pairs is true power, just as if you'd
used an analog multiplier to calculate power.

Good explanation - thanks.

What's tricky is the reactive power/phase angle thing.

Can't you just detect the zero crossings in the voltage and
current waveforms and get the phase angle between them?

No, because I only have the digitized samples, and because the
current waveform has a huge dynamic range and might be very ugly.
The voltage waveform in a power system is usually pretty close to
a clean sine.

John

Yes, the current waveform could be difficult, especially with
rectifier loads feeding capacitors. The peak current can be
enormous. I couldn't find any references that discuss this, but you
can see the peaks of the AC waveform are flattened because of it.

I did come across an article that discusses different methods of
measuring reactive power. One method is to shift the voltage
waveform 90 degrees in software. Another method used in the Analog
Devices E77xx series is the phase shift of a simple rc filter. This
would certainly cause errors with short current spikes such as DC
power supplies. Here's the url:

http://www.metering.com/archive/021/52_1.htm

Looks like you are going to have fun

Mike Monett

 

[new.online.de]

Suppose I have an AC power system, and I can digitize a pair of
voltage and current waveforms. I want to report everything: trms
volts/amps, true power, reactive power, phase angle. The line
frequency could vary from maybe 20 to 80 Hz for a stationary
generator, or 200-800 for an aircraft system (including startup and
weird situations.) I'll digitize to 16 bits, at maybe 20K
samples/second or something. I'm considering doing all the signal
processing in an FPGA, crunching maybe 8 voltage+current pairs.

For the rms volts/amps, we could just square the samples, filter, and
allow my pokey uP to occasionally pick up that and square root.

True power is just the product of the e*i samples, lowpass filtered.
Easy.

What's tricky is the reactive power/phase angle thing.

reactive power is defined:

reactive power = squareroot (apparent power ^2 - true power ^2)
cos phi = true power / apparent power
reactive power = sin phi * apparent power

This was done in this way because if voltage and current are not purely
sinusoidal, the reactive power
contains power products of harmonics. They cannot be calculated using the
hilbert transform of the fundamental only.
Reactive power is the square root of the square sums of fundamental rp,
reactive products of equal harmonics, and products of unequal harmonics.
Reactive products of equal harmonics are products of equal harmonics with
integral zero ( i.e. harmonic reactive power).
The time integrals of nonequal harmonics products are always zero. This is
called distortion power, but its is a part of the reactive power.

True power is the square root of the square sums of fundamental real power
and real products of equal harmonics. This is the same as your

True power is just the product of the e*i samples, lowpass filtered

Most important is to use a sampling rate which is high enough to saisfy
Nyquist's theorem. A sampling frequency 100* line frequency may be a
starting point, but *1000 or above may be necessary.

hope this can help to simplify your problem! regards ...gerhard

 

[Joerg]

Hello John,

Actually, no. Nyquist is irrelevant: we're not trying to reconstruct
the waveform, but merely gather statistics on it. One can make a very
nice, accurate power meter that samples at a fraction of the line
frequency.

As long as the goal isn't to meter cheap printers. Their cheap-as-can-be
"power supplies" sound like a moped with fouled spark plugs and their
current plot looks like 4th of July fireworks ;-)

Regards, Joerg

http://www.analogconsultants.com

 

[Terry Given]

Joerg wrote:

Hello John,

Actually, no. Nyquist is irrelevant: we're not trying to reconstruct
the waveform, but merely gather statistics on it. One can make a very
nice, accurate power meter that samples at a fraction of the line
frequency.


As long as the goal isn't to meter cheap printers. Their cheap-as-can-be
"power supplies" sound like a moped with fouled spark plugs and their
current plot looks like 4th of July fireworks ;-)

Regards, Joerg

ROTFLMAO! dont beat about the bush Joerg, tell us what you really think.

Cheers
Terry

 

[John Larkin]

On Thu, 02 Jun 2005 21:11:27 GMT, Joerg
<notthisjoergsch@removethispacbell.net> wrote:

Hello John,

Actually, no. Nyquist is irrelevant: we're not trying to reconstruct
the waveform, but merely gather statistics on it. One can make a very
nice, accurate power meter that samples at a fraction of the line
frequency.

As long as the goal isn't to meter cheap printers. Their cheap-as-can-be
"power supplies" sound like a moped with fouled spark plugs and their
current plot looks like 4th of July fireworks ;-)

But longterm, you'll still get the right number of KWHs!


John

 

[Joerg]

Hello John,

As long as the goal isn't to meter cheap printers. Their cheap-as-can-be
"power supplies" sound like a moped with fouled spark plugs and their
current plot looks like 4th of July fireworks ;-)

But longterm, you'll still get the right number of KWHs!

Yes, although it could be very long term if they are constantly printing
something.

Regards, Joerg

http://www.analogconsultants.com

 

[Joerg]

Hello Terry,

ROTFLMAO! dont beat about the bush Joerg, tell us what you really think.

No hidden agendas here. Seriously, I have a new printer where the power
supply sounds like a Madagascan hissing cockroach, the scope won't sync
on anything when checking the current intake and the topper was that it
was able to swamp the X10 signal in that area. These signals were a
whopping 3V. I wonder how it ever passed the smog check. It needed two
Dollar-sized toroids just to make X10 work again.

Regards, Joerg

http://www.analogconsultants.com

 

[Mike Monett]

John Larkin wrote:

[...]

In most ac power systems, the voltage waveform is reasonably
sinusoidal, so I can get away with defining reactive power as the
averaged product of current * (90 degree shifted voltage) waveforms.

(A good Hilbert shifts all frequencies 90 degrees!)

You don't even need it. Just lock a DDS or pll to 360 times the line
frequency to drive the ADC's. You can get a 90 degree shift by looking 90
clocks ahead (or behind) in memory. You can multiply the actual ADC
values and not have to apologize for approximations. This also gives the
phase relations needed to handle 3-phase power.

But I do need the phase angle. Arc-cos is ambiguous.

When you average the reactive power, capacitive and inductive loads give
the opposite average values. So you automatically get the right answer.

Averaging also helps eliminate noise in the current waveform and improves
accuracy. You can change the average in software to suit conditions.

[...]

Most important is to use a sampling rate which is high enough to saisfy
Nyquist's theorem. A sampling frequency 100* line frequency may be a
starting point, but *1000 or above may be necessary.
Actually, no. Nyquist is irrelevant: we're not trying to reconstruct
the waveform, but merely gather statistics on it. One can make a very
nice, accurate power meter that samples at a fraction of the line
frequency.

Yes, but you have to average for a very long time. You can provide much
faster response to changing conditions by using a faster clock. This may
be significant when debugging 400Hz a/c power systems during engine
start, for example. Silicon and software are cheap, and will be even
cheaper in the future. You can add a lot of benefit for the user with a
very small increase in cost.

A clock at 360X the line frequency should give plenty of resolution. It
is not unreasonably high for a 16-bit ADC at 400Hz.

The nice thing about using sine and cosine is the narrow current pulses
from a rectifier in a power supply only affect the real power average.

I was pleasantly surprised to find the pulses occur at the zero crossings
for the reactive part, so they average to zero and have minimal affect on
the result. I have some SPICE plots that show this and could post them if
you like.

John

Mike Monett

 

[Mac]

On Tue, 31 May 2005 17:20:07 -0700, John Larkin wrote:

Hi,

Can anybody recommend a book that has good stuff about implementing
the discrete Hilbert transform, preferably in an FPGA? I need
practical stuff, like accuracy over frequency ranges for given tap
count, windowing, truncation effects, etc.

I need to take a stream of digitized (16 bit) analog samples and
produce an I-Q data stream pair over roughly a 20:1 frequency range,
to maybe 1 degree accuracy. Max signal frequency will be low, 1 KHz
maybe.

Any other resource tips would be appreciated. We could pay for a bit
of consulting maybe, too.

Thanks,

John

It seems like it would be more cost effective and easier to do this using
a DSP.

The Hilbert transform of a real-valued signal is as easy as two FFT's,
basically, and DSP's are good at FFT's.

In any event, Xilinx has tons of notes about doing FFT's and convolution
and correlation and so on on their website.

--Mac