need a book: Hilbert transform

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

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

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

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

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

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...

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

On Wed, 01 Jun 2005 00:44:02 -0400, Mike Monett <no@spam.com> wrote:

>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.
> 

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.


>> 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

On Wed, 01 Jun 2005 13:36:24 GMT, Fred Bloggs <nospam@nospam.com>
wrote:


>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...

Thanks, Fred, you're always so helpful.

John

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