robert bristow-johnson wrote:

  ...

> in a sense, that's a better comparison since the integrator and H.T.
> are both 90 degree lag.

Is an inverter 180-degree lag, or 180-degree lead? Suppose I invert the
output of an HT; that sure looks like a lead if uninverted is a lag.

An inverted integrator makes negative output with DC in.  An inverted
differentiator makes negative spikes with positive steps in. Those
behaviors establish a reference. I know no comparable test for an
inverted HT, so it can lead or lag as one pleases.

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

I. R. Khan wrote:
> Thanks RBJ for explaining. I have another related question, and hope you
> or some one else can explain.
>
> We can transform an even length FIR differentiator to a Hilbert
> transformer by just making few changes to the signs of the impulse
> response coefficients. If we just assign negative sign to the first half
> of the coefficients and positive sign to the second half, we get a HT.
> Why does it happen? What actually happens to the frequency (magnitude)
> response of differentiator by making these changes to the impulse
> response coefficients?

i dunno.  is the sign change happening in some kind of pattern?  is it
every other coefficient that gets multiplied by -1?  perhaps, in the
frequency domain there is a sorta H1(f) = f getting added to H(f) = 1-f
which results in a sorta constant magnitude.  but none of this is
something i have thought of before.

> I can understand the transformation of differentiator to halfband
> lowpass filter, that we take derivative of the differentiator's
> frequency response (multiply impulse response coefficients with
> indices), squeeze to half (insert zeros in impulse response) and scale
> (set middle coefficient to 1/2 etc). How can we explain transformation
> of differentiator to HT?

boy, there are a lot of concepts here that i have never thunked about.
halfband design of the H.T. is something i have thought about (it needs
to be odd lengthed FIR and nearly half of the taps have zero
coefficients making it a little more efficient).  but this
transformation of differentiator to H.T. is not something i had thought
about.  but there is some "ring of truth" to what you are alluding to
(differentiating the linear gain will get you constant gain).  it does
not sound like crap.

> Ishtiaq.

now we know what the "I" stands for.

Tim Wescott wrote:
> robert bristow-johnson wrote:
>
> -- snip --
>
> > if the magnitude response is approximately linear with
> > frequency, consider calling it an approximation to a differentiator.
...
> >
> Don't forget our friend the integrator!

in a sense, that's a better comparison since the integrator and H.T.
are both 90 degree lag.

r b-j

I. R. Khan wrote:

> Jerry Avins wrote:
> 
>> throws the communication out of order.
>> Top post is confusing because it
>>
>> I. R. Khan wrote:
>>
>>> But odd symmetric coefficients do not necessarily make a HT. A
>>> differentiator also has odd symmetric coefficients.
>>>
>>> Clay wrote:
>>>
>>>> By simple inspection you can see if the phase repsonse matches that of
>>>> a Hilbert transformer. The coefs need to have odd symmetry. This is
>>>> necessary in order to have a 90 degree phase shift.
>>
>>
>> I assume you want the HT to occupy the widest bandwidth between DC and
>> Fs/2. Number the coefficients c[n] according to their position from the
>> center, negative for one side, positive. For an odd number of
>> coefficients, c[0] is zero, and so are all other even-numbered ones. The
>> odd numberer coefficients, multiplied by their indices, will be a
>> constant.
> 
> 
> Are you talking about windows based design?

Yes. Before windowing.

>                                This does not seem true for
> other designs. For example, for the following HT
> {-1/16, 0, -9/16, 0, 9/16, 0, 1/16}

Did you plot the magnitude response? What is the useful bandwidth?
Compare it to [-6 0 -10 0 -30 0 +30 0 +10 0 +6]

> I found a very nice chapter on HTs in Rick's book, and got the
> information that I needed. Thanks for your reply.

That book is good for a lot of things; you're welcome.

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

Thanks RBJ for explaining. I have another related question, and hope you 
or some one else can explain.

We can transform an even length FIR differentiator to a Hilbert 
transformer by just making few changes to the signs of the impulse 
response coefficients. If we just assign negative sign to the first half 
of the coefficients and positive sign to the second half, we get a HT. 
Why does it happen? What actually happens to the frequency (magnitude) 
response of differentiator by making these changes to the impulse 
response coefficients?

I can understand the transformation of differentiator to halfband 
lowpass filter, that we take derivative of the differentiator's 
frequency response (multiply impulse response coefficients with 
indices), squeeze to half (insert zeros in impulse response) and scale 
(set middle coefficient to 1/2 etc). How can we explain transformation 
of differentiator to HT?

Ishtiaq.

robert bristow-johnson wrote:

-- snip --

> if the magnitude response
> is not sufficiently constant, consider calling it a "crappy Hilbert
> transformer".

Chuckle

> if the magnitude response is approximately linear with
> frequency, consider calling it an approximation to a differentiator.
> 
> r b-j
> 
Don't forget our friend the integrator!

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

I. R. Khan wrote:
> But odd symmetric coefficients do not necessarily make a HT.

most certainly true.  all the odd symmetric thing does is give you a 90
degree phase shift (ignoring the delay needed for causality).

> A differentiator also has odd symmetric coefficients.

yes, that 90 degree phase shift (and odd symmetry) is what an
differentiator and H.T. have in common.  what they *don't* have in
common is the magnitude of the frequency response.  the differentiator
will (within a certain margin of error) have a magnitude that is
increasing linearly with frequency while the H.T. has (withing a
certain margin or error) a constant magnitude w.r.t. frequency.

there is no perfect H.T. that is realizable, but you *can* have a
perfect 90 degree phase shift (plus a constant delay).  if the
magnitude response between DC and Nyquist is constant enough for your
tolerance, call it a "Hilbert Transformer".  if the magnitude response
is not sufficiently constant, consider calling it a "crappy Hilbert
transformer".  if the magnitude response is approximately linear with
frequency, consider calling it an approximation to a differentiator.

r b-j

Jerry Avins wrote:
> throws the communication out of order.
> Top post is confusing because it
> 
> I. R. Khan wrote:
> 
>> But odd symmetric coefficients do not necessarily make a HT. A
>> differentiator also has odd symmetric coefficients.
>>
>> Clay wrote:
>>
>>> By simple inspection you can see if the phase repsonse matches that of
>>> a Hilbert transformer. The coefs need to have odd symmetry. This is
>>> necessary in order to have a 90 degree phase shift.
> 
> I assume you want the HT to occupy the widest bandwidth between DC and
> Fs/2. Number the coefficients c[n] according to their position from the
> center, negative for one side, positive. For an odd number of
> coefficients, c[0] is zero, and so are all other even-numbered ones. The
> odd numberer coefficients, multiplied by their indices, will be a
> constant.

Are you talking about windows based design? This does not seem true for 
other designs. For example, for the following HT
{-1/16, 0, -9/16, 0, 9/16, 0, 1/16}
I found a very nice chapter on HTs in Rick's book, and got the 
information that I needed. Thanks for your reply.

I leave the cases of an even number of taps and the proper
> constant for unity gain as an exercise.
> 
> Jerry

throws the communication out of order.
Top post is confusing because it

I. R. Khan wrote:

> But odd symmetric coefficients do not necessarily make a HT. A
> differentiator also has odd symmetric coefficients.
> 
> Clay wrote:
> 
>> By simple inspection you can see if the phase repsonse matches that of
>> a Hilbert transformer. The coefs need to have odd symmetry. This is
>> necessary in order to have a 90 degree phase shift.

I assume you want the HT to occupy the widest bandwidth between DC and
Fs/2. Number the coefficients c[n] according to their position from the
center, negative for one side, positive. For an odd number of
coefficients, c[0] is zero, and so are all other even-numbered ones. The
odd numberer coefficients, multiplied by their indices, will be a
constant. I leave the cases of an even number of taps and the proper
constant for unity gain as an exercise.

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

But odd symmetric coefficients do not necessarily make a HT. A 
differentiator also has odd symmetric coefficients.

Clay wrote:
> By simple inspection you can see if the phase repsonse matches that of
> a Hilbert transformer. The coefs need to have odd symmetry. This is
> necessary in order to have a 90 degree phase shift.
> 
> Clay
>