On 7 Jan 2006 05:21:21 -0800, "Rune Allnor" <allnor@tele.ntnu.no>
wrote:

>
>Rick Lyons wrote:
>> Hi Rune,
>>
>>   I haven't followed all the details of this thread,
>> but if Paul's modulated signal is low frequency
>> relative to the signal sample rate (Fs), then perhaps
>> the "absolute value followed by low pass filtering"
>> scheme will work OK for him.
>
>I remember you mentioned that somebody had submitted
>a Hilbert transformer for very low reltive frequencies to
>your column in Signal Processing Magazine.
>
>Did that article ever get published?
>
>Rune

Hi Rune,

   no it didn't.  
Rune, I'll send ya' a private E-mail with 
more info on this subject.

Regards,
[-Rick-]

Atmapuri wrote:
> Hi!
> 
> 
>>You're still welcome, but I think you ought to get a book, or at least 
>>look up the exact meanings of some of the terms you use.
> 
> 
> There is nothing like a good book recommendation.
> (or several of them). I have pretty much all the standard
> DSP books, but many times I discover that people that wrote
> them assume that the reader knows everything about the
> analog logic and systems that the DSP builds upon.
> (because DSP is just another level to the analog EE systems).
> 
> Thanks!
> Atmapuri 

The ARRL handbook is an excellent compendium that outlines the essences 
of major branches of basic communications approaches. It is updated 
every year. I buy a new one every so often. 
http://www.arrl.org/catalog/?item=NO-HB2006

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Hi!

> You're still welcome, but I think you ought to get a book, or at least 
> look up the exact meanings of some of the terms you use.

There is nothing like a good book recommendation.
(or several of them). I have pretty much all the standard
DSP books, but many times I discover that people that wrote
them assume that the reader knows everything about the
analog logic and systems that the DSP builds upon.
(because DSP is just another level to the analog EE systems).

Thanks!
Atmapuri 

Atmapuri wrote:
> Hi!
> 
> 
>>What is the ratio of the signal bandwidth to the carrier frequency?
> 
> 
> No carrier.  Entire spectrum (from 0 to FS/2) was raising and falling
> with time. I was interested in the envelope.

If there is no carrier, what is being modulated? It seems to me (if I 
understand you right) that your signal is passing through a channel with 
varying gain, like the output of a radio would be if the volume control 
were constantly being varied. If that's the case, then every frequency 
component of the unvaried-amplitude signal has sidebands at the changing 
volume frequency(s).

>>still be an envelope, but not the same envelope. What is often referred to 
>>as single sideband is in fact single sideband, suppressed carrier -- 
>>SSSC. Then there is no envelope at all with a single modulating frequency. 
>>We are left with M*A*[cos((wc + wm)*t)].
>>
>>The modulation is there, but not the envelope. That's one reason the 
>>distinction is not merely semantic.
> 
> 
> Humm. I think I had SSSC, but there was also the envelope. This
> is because you think of modulation in terms of creating it, and not
> recording it as a real world signal.

Modulation is a process that impresses a signal on a carrier. AM, SSSB, 
  and PM and its cousin FM are all examples. Both AM and SSSB have 
envelopes when the modulating signal is not a single frequency, but only 
AM's envelope is an image of the modulating signal. FM and PM have no 
envelopes. You don't store envelopes, only modulating and demodulated 
signals, or the entire modulated signal.


>                                     If you take a hammer and
> beat with it on a plate, how do you determine the frequency of
> the beating by using an accelerometer? When the hammer hits
> you get a very wide band signal and the frequency of the beating
> is a very low frequency.

The repetition rate is the beating frequency. For many purposes, the 
spectrum produced by the plate is irrelevant.

>>>>>What is polarity?  (not an electrical engineer).
>>>>
>>>>The envelope has both negative and positive parts; an AM envelope is 
>>>>symmetrical about the time axis -- there are two curves. The demodulated 
>>>>signal is either the all-positive or all-negative part.
>>>
>>>
>>>Humm... But abs(analytical signal) gives only one part.
>>
>>The magnitude of the analytic signal is sqrt(re^2 + im^2). What would abs 
>>mean here?
> 
> 
> Absolute value = sqrt(re^2 + im^2)   for complex numbers.

That is also the magnitude of a complex number. One reason we bother to 
make an analytic signal with a Hilbert transforms is to be able to find 
its magnitude. An analytic signal is conveniently represented by a 
complex number.

> Thanks!

You're still welcome, but I think you ought to get a book, or at least 
look up the exact meanings of some of the terms you use. At least one of 
us is hampered by misconceptions.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Hi!

> What is the ratio of the signal bandwidth to the carrier frequency?

No carrier.  Entire spectrum (from 0 to FS/2) was raising and falling
with time. I was interested in the envelope.

> still be an envelope, but not the same envelope. What is often referred to 
> as single sideband is in fact single sideband, suppressed carrier -- 
> SSSC. Then there is no envelope at all with a single modulating frequency. 
> We are left with M*A*[cos((wc + wm)*t)].
>
> The modulation is there, but not the envelope. That's one reason the 
> distinction is not merely semantic.

Humm. I think I had SSSC, but there was also the envelope. This
is because you think of modulation in terms of creating it, and not
recording it as a real world signal. If you take a hammer and
beat with it on a plate, how do you determine the frequency of
the beating by using an accelerometer? When the hammer hits
you get a very wide band signal and the frequency of the beating
is a very low frequency.

>>>>What is polarity?  (not an electrical engineer).
>>>
>>>The envelope has both negative and positive parts; an AM envelope is 
>>>symmetrical about the time axis -- there are two curves. The demodulated 
>>>signal is either the all-positive or all-negative part.
>>
>>
>> Humm... But abs(analytical signal) gives only one part.
>
> The magnitude of the analytic signal is sqrt(re^2 + im^2). What would abs 
> mean here?

Absolute value = sqrt(re^2 + im^2)   for complex numbers.

Thanks!
Atmapuri 

Atmapuri wrote:
> Hi!
> 
> 
>>The signal consists of frequencies that are all close to the carrier. 
>>Those terms in the computation that involve the product of two of those 
>>frequencies yield frequencies approximately double the carrier's, and 
>>possibly DC. Those are easily filtered out. What remains is what you want.
> 
> 
> Ah.. Very narrow band signal. That is not what I had.
> Now I understand :)

What is the ratio of the signal bandwidth to the carrier frequency?

> So one method for envelope detection would be:
> 1.) bandpass filter
> 2.) abs(analytical signal).

You need magnitude(analytical signal), so abs() is inherent.

> And the bandpass filter would probably be a pair
> of two filters multiplied one with cosine and the other
> with sine of FS/4 so that the resulting two time series
> would be exactly 90 degrees a part. All done without
> a hilbert transform and probably fairly wide bandpass
> filter.
> 
> But that still does not explain why using abs(real signal)
> would not be good enough, if the envelope has a low
> enough frequency. Although the bandpass design really
> does seem to give a very clean result without the need for
> very sharp lowpass filters (optionally implemented with a pair of
> decimation/interpolation.)

Regardless of the frequency of the envelope, what assurance have you 
that enough of your samples will fall near a peak of the carrier?

>>The envelope of a single-sideband signal is not its modulation. (That's 
>>why an ordinary AM (envelope) detector won't demodulate it.
 >
> I dont see why. If the envelope is there, it is there... (looking only
> at values bigger than zero).

An AM signal (carrier amplitude A, carrier frequency wc, modulating 
frequency wm, (M/100)% modulation) is represented by

      f(t) = A*[1 + M*cos(wm*t)]*cos(wc*t).

Trig identities show another form,

      f(t) = A*cos(wc*t) + M*A*[cos((wc + wm)*t) + cos((wc + wm)*t)],

from which the spectrum is immediately evident. Now, if one of the 
sidebands -- say the lower, cos((wc + wm)*t) -- is removed, there will 
still be an envelope, but not the same envelope. What is often referred 
to as single sideband is in fact single sideband, suppressed carrier -- 
SSSC. Then there is no envelope at all with a single modulating 
frequency. We are left with M*A*[cos((wc + wm)*t)].

The modulation is there, but not the envelope. That's one reason the 
distinction is not merely semantic.

>>>What is polarity?  (not an electrical engineer).
>>
>>The envelope has both negative and positive parts; an AM envelope is 
>>symmetrical about the time axis -- there are two curves. The demodulated 
>>signal is either the all-positive or all-negative part.
> 
> 
> Humm... But abs(analytical signal) gives only one part.

The magnitude of the analytic signal is sqrt(re^2 + im^2). What would 
abs mean here?

> So that implies that you would really have to follow the peaks
> to see the difference in the envelope between all-positive and
> all-negative part. Where in real world do you need to have
> info like that? Is upper and lower envelope related to upper
> and lower sideband?

Using both parts by means of abs() doubles the number of samples 
available, making it more likely that enough of them will fall near a 
peak of the carrier. When the sampling frequency is high enough, that's 
an unnecessary embellishment, but abs() is probably cheaper than 
selecting out the positive samples.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Hi!

> The signal consists of frequencies that are all close to the carrier. 
> Those terms in the computation that involve the product of two of those 
> frequencies yield frequencies approximately double the carrier's, and 
> possibly DC. Those are easily filtered out. What remains is what you want.

Ah.. Very narrow band signal. That is not what I had.
Now I understand :)

So one method for envelope detection would be:
1.) bandpass filter
2.) abs(analytical signal).

And the bandpass filter would probably be a pair
of two filters multiplied one with cosine and the other
with sine of FS/4 so that the resulting two time series
would be exactly 90 degrees a part. All done without
a hilbert transform and probably fairly wide bandpass
filter.

But that still does not explain why using abs(real signal)
would not be good enough, if the envelope has a low
enough frequency. Although the bandpass design really
does seem to give a very clean result without the need for
very sharp lowpass filters (optionally implemented with a pair of
decimation/interpolation.)

> The envelope of a single-sideband signal is not its modulation. (That's 
> why an ordinary AM (envelope) detector won't demodulate it.

I dont see why. If the envelope is there, it is there... (looking only
at values bigger than zero).

>> What is polarity?  (not an electrical engineer).
>
> The envelope has both negative and positive parts; an AM envelope is 
> symmetrical about the time axis -- there are two curves. The demodulated 
> signal is either the all-positive or all-negative part.

Humm... But abs(analytical signal) gives only one part.
So that implies that you would really have to follow the peaks
to see the difference in the envelope between all-positive and
all-negative part. Where in real world do you need to have
info like that? Is upper and lower envelope related to upper
and lower sideband?

Thanks!
Atmapuri

Atmapuri wrote:
> Hi!
> 
> 
>>f(t) never exceeds 1, and the envelope is +/-A*[1 + f(t)]. The
>>mathematical definition of "envelope" applies here.
> 
> 
> Ok. I never tried to overlay the results of demodulation
> with or without the hilbert transform with the actual
> time series. I always assumed that it is a match, but
> a little averaged.
> 
> 
>>>At a given moment in time you have:
>>>
>>>a1*sin(w1*t) + a2*sin(w2*t) + .... +an*sin(wn*t)
>>>
>>>Depending on their phase relations you would get non-constant
>>>value of the envelope for a signal with two or more frequencies
>>>after applying absolute value to the analytical signal.
>>
>>At any given instant, there is one sample and one Hilbert transformed
>>sample. How can there be a non-constant sum of squares?
> 
> 
> Where is my error:
> 
> Original signal:Re=  a1*sin(w1*t) + a2*sin(w2*t)
> Hilbert transform: Im = a1*cos(w1*t) + a2*cos(w2*t)
> ---------------------------------------------------
> result = re^2 + im^2 =
>          = a1^2 + a2^2   +
> 2*a1*a2*((sin(w1*t)*sin(w2*t)+cos(w2*t)*cos(w1*t))
> 
> The first part a1^2 + a2^2 is a constant. But the second part:
> 
> 2*a1*a2*(sin(w1*t)*sin(w2*t)+cos(w2*t)*cos(w1*t))
> 
> Is not.

The signal consists of frequencies that are all close to the carrier. 
Those terms in the computation that involve the product of two of those 
frequencies yield frequencies approximately double the carrier's, and 
possibly DC. Those are easily filtered out. What remains is what you want.

>>>(envelope = the time series obtained after applying absolute
>>>value to the analyitical signal.)
>>
>>The envelope is that curve which is tangent to all cycles of the carrier.
>>The points of tangency will not be exactly at the peaks unless the
>>envelope is at a maximum or minimum. The time series is a sampling of
>>(usually the positive part) of that curve.
> 
> 
> Ok. Its new to me that actual accurate values are required
> and not just a result that is scaled.

I don't see where scaling enters in.. That's just gain.

>>You aren't filtering the signal; you propose filtering the envelope of the
>>signal. That's not the same at all.
> 
> 
>>You're getting muddled, at least with language. The carrier alone is
>>modulated, usually by a signal with bandwidth greater than zero. The
>>bandwidth of the modulating signal is half the bandwidth of the modulated
>>carrier. I repeat my question: what is the meaning of "bandwith of the
>>envelope"?
 >
> It must be that for me the envelope and the amplitude demodulation
> dont carry much difference in the meaning, but it does for you.

Well, if "envelope" and "modulation" mean the same to you, we may lose a 
little precision in discussing things, but I'm happy to do it your way.

> One reason is because it is possible to "map" or convert the result
> of amplitude demodulation which uses recitification to the
> result obtained when the using the analytical signal.

The envelope of a single-sideband signal is not its modulation. (That's 
why an ordinary AM (envelope) detector won't demodulate it.

> The relation is approx: y[i] = x[i]^1.06
> y[i] = amplitude demodulated signal using abs on analytical signal
> x[i]  = amplitude demodulated signal using abs on real signal.
> 
> So, except for lowpass filtering, there is very little difference
> between envelope detection and amplitude demodulation.
> 
> And the second mapping:
> If the envelope of the signal is a very low frequency
> (lower than the lowpass cutoff), then there is no difference at all
> between amplitude demodulated signal obtained by using
> abs on a real signal and the actual "envelope" expressed as:
> 
> A* [1 + f(t)]
> 
> 
>>>When you apply abs to the real signal and then lowpass filtter with
>>>decimation you get the envelope. That envelope has a bandwidth.
>>>It does not contain a single frequency if you take a look at it
>>>in the frequency domain.
>>
>>Decimation isn't necessary. What you get is the demodulated signal, with
>>some slight but inevitable distortions. It has one polarity, whereas the
>>envelope has two.
> 
> 
> What is polarity?  (not an electrical engineer).

The envelope has both negative and positive parts; an AM envelope is 
symmetrical about the time axis -- there are two curves. The demodulated 
signal is either the all-positive or all-negative part. (The carrier 
changes rapidly from positive to negative. Since it has no DC component, 
it is symmetrical about the time axis. The envelope is those two 
mirror-image curves of lower frequency which are tangent to the carrier 
near but rarely at its maximum excursions.)

> Thanks!

Again, you're welcome.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Hi!

> f(t) never exceeds 1, and the envelope is +/-A*[1 + f(t)]. The
> mathematical definition of "envelope" applies here.

Ok. I never tried to overlay the results of demodulation
with or without the hilbert transform with the actual
time series. I always assumed that it is a match, but
a little averaged.

>> At a given moment in time you have:
>>
>> a1*sin(w1*t) + a2*sin(w2*t) + .... +an*sin(wn*t)
>>
>> Depending on their phase relations you would get non-constant
>> value of the envelope for a signal with two or more frequencies
>> after applying absolute value to the analytical signal.
>
> At any given instant, there is one sample and one Hilbert transformed
> sample. How can there be a non-constant sum of squares?

Where is my error:

Original signal:Re=  a1*sin(w1*t) + a2*sin(w2*t)
Hilbert transform: Im = a1*cos(w1*t) + a2*cos(w2*t)
---------------------------------------------------
result = re^2 + im^2 =
         = a1^2 + a2^2   +
2*a1*a2*((sin(w1*t)*sin(w2*t)+cos(w2*t)*cos(w1*t))

The first part a1^2 + a2^2 is a constant. But the second part:

2*a1*a2*(sin(w1*t)*sin(w2*t)+cos(w2*t)*cos(w1*t))

Is not.

>> (envelope = the time series obtained after applying absolute
>> value to the analyitical signal.)
>
> The envelope is that curve which is tangent to all cycles of the carrier.
> The points of tangency will not be exactly at the peaks unless the
> envelope is at a maximum or minimum. The time series is a sampling of
> (usually the positive part) of that curve.

Ok. Its new to me that actual accurate values are required
and not just a result that is scaled.

> You aren't filtering the signal; you propose filtering the envelope of the
> signal. That's not the same at all.

> You're getting muddled, at least with language. The carrier alone is
> modulated, usually by a signal with bandwidth greater than zero. The
> bandwidth of the modulating signal is half the bandwidth of the modulated
> carrier. I repeat my question: what is the meaning of "bandwith of the
> envelope"?

It must be that for me the envelope and the amplitude demodulation
dont carry much difference in the meaning, but it does for you.

One reason is because it is possible to "map" or convert the result
of amplitude demodulation which uses recitification to the
result obtained when the using the analytical signal.

The relation is approx: y[i] = x[i]^1.06
y[i] = amplitude demodulated signal using abs on analytical signal
x[i]  = amplitude demodulated signal using abs on real signal.

So, except for lowpass filtering, there is very little difference
between envelope detection and amplitude demodulation.

And the second mapping:
If the envelope of the signal is a very low frequency
(lower than the lowpass cutoff), then there is no difference at all
between amplitude demodulated signal obtained by using
abs on a real signal and the actual "envelope" expressed as:

A* [1 + f(t)]

>> When you apply abs to the real signal and then lowpass filtter with
>> decimation you get the envelope. That envelope has a bandwidth.
>> It does not contain a single frequency if you take a look at it
>> in the frequency domain.
>
> Decimation isn't necessary. What you get is the demodulated signal, with
> some slight but inevitable distortions. It has one polarity, whereas the
> envelope has two.

What is polarity?  (not an electrical engineer).

Thanks!
Atmapuri

Rick Lyons wrote:
> Hi Rune,
>
>   I haven't followed all the details of this thread,
> but if Paul's modulated signal is low frequency
> relative to the signal sample rate (Fs), then perhaps
> the "absolute value followed by low pass filtering"
> scheme will work OK for him.

I remember you mentioned that somebody had submitted
a Hilbert transformer for very low reltive frequencies to
your column in Signal Processing Magazine.

Did that article ever get published?

Rune