Complex FIR coefficients

On 30 Mar, 18:24, Tim Wescott <t...@seemywebsite.now> wrote:
> Randy Yates wrote:
> > HardySpicer <gyansor...@gmail.com> writes:
>
> >> So I suppose it is complex because the imaginary part also has
> >> frequency-selective properties as well as real.
>
> > Not exactly. It is complex because F(w) != F*(-w), i.e., the frequency
> > response isn't Hermitian symmetric. Note I use "*" here to denote
> > conjugation.
>
> That's not a _physical_ interpretation, because no physical system has a
> frequency response that isn't Hermitian symmetric.

Wrong.

1) 2D signals from antenna arrays are physical.
2) After DFT along the time axis the physical signal exist
   in (w,x) domain
3) Each vector along the x direction consits of complex-valued
   samples
4) Spatial narrow-band filters, e.g. for DoA or velocity filtering,
   needs to be complex-valued

Physical as anything. Complex as it comes. In any sense of the word.

Rune

Tim:

[snip]
> That's not a _physical_ interpretation, because no physical system has a 
> frequency response that isn't Hermitian symmetric.
>
> -- 
> Tim Wescott
[snip]

That is a common misconception but untrue.

'Physical' systems with non Hermitian symmetry are not only possible, indeed 
they have practical uses.

Complex physical systems are rarely addressed in common textbooks and so 
remain somewhat obscure.  Just because they are uncommon does not mean they 
don't exist!

So called complex systems have been synthesized, designed, prototyped and 
even manufactured.

Addmittedly they are uncommon, especially in natural form, but man can make 
them easily.  This is more difficult to do (exactly) in analogue form than 
in digital form, but even complex analogue systems have been built.

There have  been numerous [OK... several] professional technical papers 
written about such systems over the years, beginning way back 40-50 years 
ago.

Search for subjects such as "complex analogue filters", etc... will turn up 
some references.  I have worked on and wrtten about complex analogue filters 
myself.

-- Pete
    Indialantic By-the-Sea, FL 

On 31-03-2010 o 06:53:53 Rune Allnor <allnor@tele.ntnu.no> wrote:

> Wrong.
>
> 1) 2D signals from antenna arrays are physical.
> 2) After DFT along the time axis the physical signal exist
>    in (w,x) domain
> 3) Each vector along the x direction consits of complex-valued
>    samples
> 4) Spatial narrow-band filters, e.g. for DoA or velocity filtering,
>    needs to be complex-valued
>
> Physical as anything. Complex as it comes. In any sense of the word.
>
> Rune


I can just put 'j' or 'i' befor any physical value
and than interpret it at will.
I can fit j to anything I want.

But when we have description of something
than partial imaginary result numbers
can have no physical interpretation.

Can you interpret imaginary impulse response?
I can travel back in time on a paper.

-- 
Mikolaj

on 07-04-2010 o 13:53:36 glen herrmannsfeldt <gah@ugcs.caltech.edu> wrote:

(...)
> I believe that in some cases complex physical quantities
> that are in exponents have a physical interpretation.
> As examples, the dielectric constant and its square root,
> the index of refraction.  Other than in exponents,
> the use of complex numbers for physical quantities,
> such as describing phase shifts, seems more of a
> convenience, and not something with a physical
> interpretation.
(...)

It seems that sometimes, luckily
when you use complex (compressed, packed, combined, compact)
way of describing few dependent physical things
their imaginary part (additional dimension used for compression)
can be human understandable
and could have interpretation.

But you can always decompress complex matrix
to it's scalar version equations.


-- 
Mikolaj

Mikolaj <sterowanie_komputerowe@poczta.onet.pl> wrote:
(snip)
 
> I can just put 'j' or 'i' befor any physical value
> and than interpret it at will.
> I can fit j to anything I want.
 
> But when we have description of something
> than partial imaginary result numbers
> can have no physical interpretation.

I believe that in some cases complex physical quantities
that are in exponents have a physical interpretation.
As examples, the dielectric constant and its square root,
the index of refraction.  Other than in exponents,
the use of complex numbers for physical quantities,
such as describing phase shifts, seems more of a
convenience, and not something with a physical
interpretation.
 
> Can you interpret imaginary impulse response?
> I can travel back in time on a paper.

-- glen
 

on 07-04-2010 o 13:03:04 Mikolaj <sterowanie_komputerowe@poczta.onet.pl>  
wrote:


> It seems that sometimes, luckily
> when you use complex (compressed, packed, combined, compact)
> way of describing few dependent physical things

the imaginary part of that complex representation (additional dimension  
used for compression)

> can be human understandable
> and could have interpretation.
>
> But you can always decompress complex matrix
> to it's scalar version equations.


-- 
Mikolaj

glen wrote:
>Mikolaj <sterowanie_komputerowe@poczta.onet.pl> wrote:
>(snip)
> 
>> I can just put 'j' or 'i' befor any physical value
>> and than interpret it at will.
>> I can fit j to anything I want.
> 
>> But when we have description of something
>> than partial imaginary result numbers
>> can have no physical interpretation.
>
>I believe that in some cases complex physical quantities
>that are in exponents have a physical interpretation.
>As examples, the dielectric constant and its square root,
>the index of refraction.  Other than in exponents,
>the use of complex numbers for physical quantities,
>such as describing phase shifts, seems more of a
>convenience, and not something with a physical
>interpretation.

Complex eigenvalues often have physical interpretation.  A non-hermitian
hamiltonian is sometimes used when particles leave the group of states
being considered (for example, atoms that become ionized, and assume that
you no longer care about those as part of your ensemble, so you don't
consider those states).  It is a bit of a hack, but the point is that the
complex eigenvalues in that case then have the interpretation of loss over
time (where, for other situations, "loss" might be of either sign).

Michael Plante <michael.plante@n_o_s_p_a_m.gmail.com> wrote:
(snip, I wrote)

>>I believe that in some cases complex physical quantities
>>that are in exponents have a physical interpretation.
(snip)
 
> Complex eigenvalues often have physical interpretation.  A non-hermitian
> hamiltonian is sometimes used when particles leave the group of states
> being considered (for example, atoms that become ionized, and assume that
> you no longer care about those as part of your ensemble, so you don't
> consider those states).  It is a bit of a hack, but the point is that the
> complex eigenvalues in that case then have the interpretation of loss over
> time (where, for other situations, "loss" might be of either sign).

Are these solutions of differential equations such that the
complex value is in an exp()?  If so, then the imaginary terms
(multiplied by i) are the decay (or absorption) term.

-- glen 

>Michael Plante <michael.plante@n_o_s_p_a_m.gmail.com> wrote:
>(snip, I wrote)
>
>>>I believe that in some cases complex physical quantities
>>>that are in exponents have a physical interpretation.
>(snip)
> 
>> Complex eigenvalues often have physical interpretation.  A
non-hermitian
>> hamiltonian is sometimes used when particles leave the group of states
>> being considered (for example, atoms that become ionized, and assume
that
>> you no longer care about those as part of your ensemble, so you don't
>> consider those states).  It is a bit of a hack, but the point is that
the
>> complex eigenvalues in that case then have the interpretation of loss
over
>> time (where, for other situations, "loss" might be of either sign).
>
>Are these solutions of differential equations such that the
>complex value is in an exp()?  If so, then the imaginary terms
>(multiplied by i) are the decay (or absorption) term.

It's similar to your example.  One could find a basis where the time
evolution operator U=exp(-i.H.t/hb) is diagonal.  So it could be seen that
way.

Michael Plante wrote:
>glen wrote:
>>Michael Plante <michael.plante@n_o_s_p_a_m.gmail.com> wrote:
>>(snip, I wrote)
>>
>>>>I believe that in some cases complex physical quantities
>>>>that are in exponents have a physical interpretation.
>>(snip)
>> 
>>> Complex eigenvalues often have physical interpretation.  A
>non-hermitian
>>> hamiltonian is sometimes used when particles leave the group of states
>>> being considered (for example, atoms that become ionized, and assume
>that
>>> you no longer care about those as part of your ensemble, so you don't
>>> consider those states).  It is a bit of a hack, but the point is that
>the
>>> complex eigenvalues in that case then have the interpretation of loss
>over
>>> time (where, for other situations, "loss" might be of either sign).
>>
>>Are these solutions of differential equations such that the
>>complex value is in an exp()?  If so, then the imaginary terms
>>(multiplied by i) are the decay (or absorption) term.
>
>It's similar to your example.  One could find a basis where the time
>evolution operator U=exp(-i.H.t/hb) is diagonal.  So it could be seen
that
>way.


While what I wrote was inspired by your post, my point was not so much
about the ability to reduce it to that form, which depends on being able to
find a clean solution, since this sort of perturbation is probably less
necessary in simple cases.  Rather, the value is in being able to directly
interpret perturbations to the Hamiltonian without trying to find a
solution.

An interesting application is when a suitable "gain" mechanism is present
(not explicitly included) to balance this loss, but the the gain introduces
unpolarized atoms, whereas the "loss" removes from consideration whatever's
available.  Then the interpretation of this perturbation is depolarization
of the ensemble over time.

Michael