FIR Hilbert transformers

Atmapuri wrote:
> Hi!
> 
> 
>>One cannot read amplitude from a single sample of A*sin(theta), but
> 
> 
> Of course not, but that is why you apply "abs". It is true that this
> will give you a series of strange spikes and not a a nice line
> as it would with analytical signal, but the differences are
> not in the frequency range which is not used anyway.

Absolute value simply gets you twice as many points on the positive half 
of the envelope, making it more likely that one of them will be near the 
peak. That's a good move in practice, but beside the point of this 
discussion.

>>simultaneous values of A*sin(theta) and A*cos(theta) allow the computation 
>>in any of several ways. The most straightforward way is 
>>sqrt{[A*sin(theta)]^2 + [A*cos(theta)]^2}.
> 
> 
> This is the abs(complex number). It gives a range of harmonics
> (distortion) in the resulting time series as it does when applied to
> real time series also.

It gives the instantaneous magnitude of the envelope In A*sin(wt), is 
the 'A' ("A" by convention positive). Instantaneous magnitudes don't 
have harmonics. In a modulated wave, the number won't be exact, but the 
error will usually be less that the quantization error of a 16-bit ADC.

> My point was that looking at frequency response there is no difference 
> between applying absolute value on the real time series or on the
> analytical signal, if the expected bandwidth of the envelope is less
> than about 1/4 of the bandwidth of the original signal.

With two samples per cycle of the carrier -- the minimum Nyquist allows 
-- it matters little how narrow the modulation bandwidth. The chance 
that one of the samples will be even within 90% (26 degrees) of the peak 
is too small for decent performance.

> Paul probably does not need Hilbert. Only abs and a good
> lowpass filter (decimation).

With his very high sample rate, he doesn't even need abs(), although 
it's cheap enough to throw in.

> The funny thing is, that there is a function that maps envelope obtained
> with applying abs to real signal and applying abs to analytical signal.
> 
> Its not linear but still a fairly simple power function.
> 
> 
>>An envelope is the slowly varying magnitude of the carrier, close enough 
>>to a single frequency.
> 
> 
> Humm... In my case the envelope has relatively narrow band in compare
> to really wide bandwidth of the modulated signal.

What is the "bandwidth" of an envelope? It the term means anything, it 
must mean the bandwidth of the modulated signal. Please explain.

> Thanks!

Sure.

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

Hi!

> Absolute value simply gets you twice as many points on the positive half 
> of the envelope, making it more likely that one of them will be near the 
> peak. That's a good move in practice, but beside the point of this 
> discussion.

The envelope is interpoloated between those points with the lowpass filter.
So if you have two peaks, the resulting envelope can be more accurate.

Of you course you could also clip everyting below zero and it would still
work, but probably the resulting envelope would not be as accurate.

>> This is the abs(complex number). It gives a range of harmonics
>> (distortion) in the resulting time series as it does when applied to
>> real time series also.
>
> It gives the instantaneous magnitude of the envelope In A*sin(wt), is the 
> 'A' ("A" by convention positive). Instantaneous magnitudes don't have 
> harmonics. In a modulated wave, the number won't be exact, but the error 
> will usually be less that the quantization error of a 16-bit ADC.

Instantaneous magnitudes for a single frequency not. But the signal does
not have a single frequency and you are not applying the abs in frequency
domain but in time domain.

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.

(envelope = the time series obtained after applying absolute
value to the analyitical signal.)

The signal can be a sum of frequencies. Not neccessarily
modulated. And you suddenly get extra frequencies in the
resulting envelope signal.

If the signal would be modulated, the resulting envelope
of the signal would be a mix of two effects:

1.) actual modulation
2.) variations due to presence of multiple frequencies
in the modulated signal.

>> My point was that looking at frequency response there is no difference 
>> between applying absolute value on the real time series or on the
>> analytical signal, if the expected bandwidth of the envelope is less
>> than about 1/4 of the bandwidth of the original signal.
>
> With two samples per cycle of the carrier -- the minimum Nyquist allows --  
> it matters little how narrow the modulation bandwidth. The chance that one 
> of the samples will be even within 90% (26 degrees) of the peak is too 
> small for decent performance.

That is because you are thinking that the peak must be
found "by hand" and that the envelope must be actually
constructed by going from peak to peak and drawing a line.

That is not needed.  Low frequency cutoff lowpass filter can also
be used to  "fill-in" the space between peaks. And the filter cares litle
if there is an actual maximum hit on the peak or not.

> What is the "bandwidth" of an envelope? It the term means anything, it 
> must mean the bandwidth of the modulated signal. Please explain.

The signal has a bandwidth. Entire bandwidth is modulated. The
frequency of modulation is not a single frequency, but again
a signal with a given bandwidth.

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.

Thanks!
Atmapuri 

Atmapuri wrote:
> Hi!
> 
> 
>>Absolute value simply gets you twice as many points on the positive half 
>>of the envelope, making it more likely that one of them will be near the 
>>peak. That's a good move in practice, but beside the point of this 
>>discussion.
> 
> 
> The envelope is interpoloated between those points with the lowpass filter.
> So if you have two peaks, the resulting envelope can be more accurate.

Taking the absolute value is a non-linear operation. You don't want to 
do any filtering after that with a cut-off much higher than the 
modulating frequency. Now I see where all the distortion you mentioned 
comes from.

> Of you course you could also clip everyting below zero and it would still
> work, but probably the resulting envelope would not be as accurate.

The OP didn't ask how to do AM demodulation. He asked how to find the 
envelope.

>>>This is the abs(complex number). It gives a range of harmonics
>>>(distortion) in the resulting time series as it does when applied to
>>>real time series also.
>>
>>It gives the instantaneous magnitude of the envelope In A*sin(wt), is the 
>>'A' ("A" by convention positive). Instantaneous magnitudes don't have 
>>harmonics. In a modulated wave, the number won't be exact, but the error 
>>will usually be less that the quantization error of a 16-bit ADC.
> 
> 
> Instantaneous magnitudes for a single frequency not. But the signal does
> not have a single frequency and you are not applying the abs in frequency
> domain but in time domain.

For practical purposes, the single frequency is the carrier frequency. 
An envelope can be expressed either of two ways. For present purposes, 
the simplest instantaneous representation is [1 + f(t)]*A*cos(wt) where 
f(t) never exceeds 1, and the envelope is +/-A*[1 + f(t)]. The 
mathematical definition of "envelope" applies here.

> 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?

> (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.

> The signal can be a sum of frequencies. Not neccessarily
> modulated. And you suddenly get extra frequencies in the
> resulting envelope signal.

Certainly not.

> If the signal would be modulated, the resulting envelope
> of the signal would be a mix of two effects:
> 
> 1.) actual modulation
> 2.) variations due to presence of multiple frequencies
> in the modulated signal.

Do the math. You won't find any.

>>>My point was that looking at frequency response there is no difference 
>>>between applying absolute value on the real time series or on the
>>>analytical signal, if the expected bandwidth of the envelope is less
>>>than about 1/4 of the bandwidth of the original signal.
>>
>>With two samples per cycle of the carrier -- the minimum Nyquist allows --  
>>it matters little how narrow the modulation bandwidth. The chance that one 
>>of the samples will be even within 90% (26 degrees) of the peak is too 
>>small for decent performance.
> 
> 
> That is because you are thinking that the peak must be
> found "by hand" and that the envelope must be actually
> constructed by going from peak to peak and drawing a line.
> 
> That is not needed.  Low frequency cutoff lowpass filter can also
> be used to  "fill-in" the space between peaks. And the filter cares litle
> if there is an actual maximum hit on the peak or not.

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

>>What is the "bandwidth" of an envelope? It the term means anything, it 
>>must mean the bandwidth of the modulated signal. Please explain.
> 
> 
> The signal has a bandwidth. Entire bandwidth is modulated. The
> frequency of modulation is not a single frequency, but again
> a signal with a given bandwidth.

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"?

> 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.

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

On 4 Jan 2006 08:26:31 -0800, "Rune Allnor" <allnor@tele.ntnu.no>
wrote:

   (snipped)

>If the requirements to the
>result are so strict you can't decimate, you will need to find
>the elaborate implementation, maybe Rick can help with that.
>If you settle for FIRPM you may have to accept some larger errors
>in the processed data.
>
>> Please correct me if I am wrong.. 
>> Paul
>
>Rune

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.

However, if the modulated signal contains spectral 
components over a wide range of the 0-to-Fs/2 band 
then quadrature processing using the Hilbert transform 
will yield a much more accurate result for envelope
detection.

I wonder what is the spectral nature of Paul's signal
(relative to Fs),
and if Paul's processing is "real-time", or is it 
"off-line" processing of a single block of time domain 
samples.  That info would be helpful in order for us 
to provide suggestions to Paul.

Rune, your:

    xi = 2*imag(ifft(x(1:floor(N/2)),N));

does give the HT of x... as far as I know.
However, I think the command

    xa = ifft(x(1:floor(N/2)),N)

will provide an analytic (complex) signal sequence, xa, 
whose magnitude is equal to the envelope that Paul 
is seeking.

[-Rick-]

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!

> 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

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!

> 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!
> 
> 
>>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!

> 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