Why cplex filters?

"Randy Yates" <yates@ieee.org> wrote in message
news:vfmy836v.fsf@ieee.org...
> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
>
> > "Randy Yates" <yates@ieee.org> wrote in message
> > news:ptd74i85.fsf@ieee.org...
> >> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
> >>
> >> > "ZedToe" <acoustictech_zhangtao@yahoo.com.sg> wrote in message
> >> > news:7c4bf533.0401231957.22326456@posting.google.com...
> >> >> Dear all,
> >> >>
> >> >> Why need we design complex filters? What are the advantages they
have
> >> >> against the real-coefficient fitlers? Except the unsymmetric
spectrum,
> >> >> there indeed nothing special. Also it can be derived from a real
> >> >> lowpass filter with an appropriated phase shift. What do you think?
> >> >>
> >> >> Thanks.
> >> >>
> >> >> Zedtoe
> >> >
> >> > I think you can use a complex FIR filter to correct for both
amplitude
> > as
> >> > well as phase (you only get amplitude changes using a real FIR
> >> > filter...linear phase of course).
> >>
> >> Wrong. You can design real, non-linear-phase FIR filters. The amount of
> > control
> >
> > But can you use them on complex signals and achieve a desired phase and
> > amplitude correction? (I'm not challenging, but asking).
>
> I'm pretty sure you can, but I can't really think of any canned filter
> design packages that let you do this. You could try specifying a
> vector of N complex numbers representing magnitude/phase at
> frequencies from -Fs/2 to +Fs/2 and inverse transforming the result,
> but there is certain to be some constraints on the length N versus the
> amount of magnitude and phase control.

I always thought that, for complex signals, you need a complex filter to do
amplitude and phase correction. Your statement says otherwise and I'm
struggling to grasp it. I certainly do understand that real FIR filters of
doing phase correction on real signals.
Perhaps the part I have difficulty with is that the phase in a complex
signal is determined by the relationship between the I,Q components and
unless the phase of each component is changed by a different amount (complex
filter) you can only achieve a shift in the overall phase of the complex
signal but not change the shape of the phase response.

Cheers
Bhaskar



> --
> %  Randy Yates                  % "And all that I can do
> %% Fuquay-Varina, NC            %  is say I'm sorry,
> %%% 919-577-9882                %  that's the way it goes..."
> %%%% <yates@ieee.org>           % Getting To The Point', *Balance of
Power*, ELO
> http://home.earthlink.net/~yatescr

"Bhaskar Thiagarajan" <bhaskart@deja.com> writes:

> "Randy Yates" <yates@ieee.org> wrote in message
> news:vfmy836v.fsf@ieee.org...
>> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
>>
>> > "Randy Yates" <yates@ieee.org> wrote in message
>> > news:ptd74i85.fsf@ieee.org...
>> >> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
>> >>
>> >> > "ZedToe" <acoustictech_zhangtao@yahoo.com.sg> wrote in message
>> >> > news:7c4bf533.0401231957.22326456@posting.google.com...
>> >> >> Dear all,
>> >> >>
>> >> >> Why need we design complex filters? What are the advantages they
> have
>> >> >> against the real-coefficient fitlers? Except the unsymmetric
> spectrum,
>> >> >> there indeed nothing special. Also it can be derived from a real
>> >> >> lowpass filter with an appropriated phase shift. What do you think?
>> >> >>
>> >> >> Thanks.
>> >> >>
>> >> >> Zedtoe
>> >> >
>> >> > I think you can use a complex FIR filter to correct for both
> amplitude
>> > as
>> >> > well as phase (you only get amplitude changes using a real FIR
>> >> > filter...linear phase of course).
>> >>
>> >> Wrong. You can design real, non-linear-phase FIR filters. The amount of
>> > control
>> >
>> > But can you use them on complex signals and achieve a desired phase and
>> > amplitude correction? (I'm not challenging, but asking).
>>
>> I'm pretty sure you can, but I can't really think of any canned filter
>> design packages that let you do this. You could try specifying a
>> vector of N complex numbers representing magnitude/phase at
>> frequencies from -Fs/2 to +Fs/2 and inverse transforming the result,
>> but there is certain to be some constraints on the length N versus the
>> amount of magnitude and phase control.
>
> I always thought that, for complex signals, you need a complex filter to do
> amplitude and phase correction. Your statement says otherwise and I'm
> struggling to grasp it. I certainly do understand that real FIR filters of
> doing phase correction on real signals.

I may have been a bit imprecise - a real filter can correct a
*portion* of a complex signal's phase response. Essentially, a real
filter will have a Hermitian-symmetric frequency response, so you can
only control one-half of the bandwidth (i.e., Fs/2 of the bandwidth),
the other half being necessarily a reflection of the first half. In
other words, you can control the phase from 0 to Fs/2, but then the
phase from 0 to -Fs/2 will be odd symmetric with that of 0 to +Fs/2
(i.e., phi(-f) = -phi(f), 0 <= f < Fs/2).

> Perhaps the part I have difficulty with is that the phase in a complex
> signal is determined by the relationship between the I,Q components and
> unless the phase of each component is changed by a different amount (complex
> filter) you can only achieve a shift in the overall phase of the complex
> signal but not change the shape of the phase response.

Can you please rephrase, Bhaskar? I'm not understanding you here.
-- 
%  Randy Yates                  % "Though you ride on the wheels of tomorrow,
%% Fuquay-Varina, NC            %  you still wander the fields of your
%%% 919-577-9882                %  sorrow."
%%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO
http://home.earthlink.net/~yatescr

"Randy Yates" <yates@ieee.org> wrote in message
news:ad49xbb5.fsf@ieee.org...
> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
>
> > "Randy Yates" <yates@ieee.org> wrote in message
> > news:vfmy836v.fsf@ieee.org...
> >> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
> >>
> >> > "Randy Yates" <yates@ieee.org> wrote in message
> >> > news:ptd74i85.fsf@ieee.org...
> >> >> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
> >> >>
> >> >> > "ZedToe" <acoustictech_zhangtao@yahoo.com.sg> wrote in message
> >> >> > news:7c4bf533.0401231957.22326456@posting.google.com...
> >> >> >> Dear all,
> >> >> >>
> >> >> >> Why need we design complex filters? What are the advantages they
> > have
> >> >> >> against the real-coefficient fitlers? Except the unsymmetric
> > spectrum,
> >> >> >> there indeed nothing special. Also it can be derived from a real
> >> >> >> lowpass filter with an appropriated phase shift. What do you
think?
> >> >> >>
> >> >> >> Thanks.
> >> >> >>
> >> >> >> Zedtoe
> >> >> >
> >> >> > I think you can use a complex FIR filter to correct for both
> > amplitude
> >> > as
> >> >> > well as phase (you only get amplitude changes using a real FIR
> >> >> > filter...linear phase of course).
> >> >>
> >> >> Wrong. You can design real, non-linear-phase FIR filters. The amount
of
> >> > control
> >> >
> >> > But can you use them on complex signals and achieve a desired phase
and
> >> > amplitude correction? (I'm not challenging, but asking).
> >>
> >> I'm pretty sure you can, but I can't really think of any canned filter
> >> design packages that let you do this. You could try specifying a
> >> vector of N complex numbers representing magnitude/phase at
> >> frequencies from -Fs/2 to +Fs/2 and inverse transforming the result,
> >> but there is certain to be some constraints on the length N versus the
> >> amount of magnitude and phase control.
> >
> > I always thought that, for complex signals, you need a complex filter to
do
> > amplitude and phase correction. Your statement says otherwise and I'm
> > struggling to grasp it. I certainly do understand that real FIR filters
of
> > doing phase correction on real signals.
>
> I may have been a bit imprecise - a real filter can correct a
> *portion* of a complex signal's phase response. Essentially, a real
> filter will have a Hermitian-symmetric frequency response, so you can
> only control one-half of the bandwidth (i.e., Fs/2 of the bandwidth),
> the other half being necessarily a reflection of the first half. In
> other words, you can control the phase from 0 to Fs/2, but then the
> phase from 0 to -Fs/2 will be odd symmetric with that of 0 to +Fs/2
> (i.e., phi(-f) = -phi(f), 0 <= f < Fs/2).
>
> > Perhaps the part I have difficulty with is that the phase in a complex
> > signal is determined by the relationship between the I,Q components and
> > unless the phase of each component is changed by a different amount
(complex
> > filter) you can only achieve a shift in the overall phase of the complex
> > signal but not change the shape of the phase response.
>
> Can you please rephrase, Bhaskar? I'm not understanding you here.

Never mind Randy...I can't even make out what I was trying to say here - I
guess I can't really think in phase domain.


> --
> %  Randy Yates                  % "Though you ride on the wheels of
tomorrow,
> %% Fuquay-Varina, NC            %  you still wander the fields of your
> %%% 919-577-9882                %  sorrow."
> %%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO
> http://home.earthlink.net/~yatescr

"Bhaskar Thiagarajan" <bhaskart@deja.com> writes:

> "Randy Yates" <yates@ieee.org> wrote in message
> news:ad49xbb5.fsf@ieee.org...
>> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
>>
>> > "Randy Yates" <yates@ieee.org> wrote in message
>> > news:vfmy836v.fsf@ieee.org...
>> >> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
>> >>
>> >> > "Randy Yates" <yates@ieee.org> wrote in message
>> >> > news:ptd74i85.fsf@ieee.org...
>> >> >> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
>> >> >>
>> >> >> > "ZedToe" <acoustictech_zhangtao@yahoo.com.sg> wrote in message
>> >> >> > news:7c4bf533.0401231957.22326456@posting.google.com...
>> >> >> >> Dear all,
>> >> >> >>
>> >> >> >> Why need we design complex filters? What are the advantages they
>> > have
>> >> >> >> against the real-coefficient fitlers? Except the unsymmetric
>> > spectrum,
>> >> >> >> there indeed nothing special. Also it can be derived from a real
>> >> >> >> lowpass filter with an appropriated phase shift. What do you
> think?
>> >> >> >>
>> >> >> >> Thanks.
>> >> >> >>
>> >> >> >> Zedtoe
>> >> >> >
>> >> >> > I think you can use a complex FIR filter to correct for both
>> > amplitude
>> >> > as
>> >> >> > well as phase (you only get amplitude changes using a real FIR
>> >> >> > filter...linear phase of course).
>> >> >>
>> >> >> Wrong. You can design real, non-linear-phase FIR filters. The amount
> of
>> >> > control
>> >> >
>> >> > But can you use them on complex signals and achieve a desired phase
> and
>> >> > amplitude correction? (I'm not challenging, but asking).
>> >>
>> >> I'm pretty sure you can, but I can't really think of any canned filter
>> >> design packages that let you do this. You could try specifying a
>> >> vector of N complex numbers representing magnitude/phase at
>> >> frequencies from -Fs/2 to +Fs/2 and inverse transforming the result,
>> >> but there is certain to be some constraints on the length N versus the
>> >> amount of magnitude and phase control.
>> >
>> > I always thought that, for complex signals, you need a complex filter to
> do
>> > amplitude and phase correction. Your statement says otherwise and I'm
>> > struggling to grasp it. I certainly do understand that real FIR filters
> of
>> > doing phase correction on real signals.
>>
>> I may have been a bit imprecise - a real filter can correct a
>> *portion* of a complex signal's phase response. Essentially, a real
>> filter will have a Hermitian-symmetric frequency response, so you can
>> only control one-half of the bandwidth (i.e., Fs/2 of the bandwidth),
>> the other half being necessarily a reflection of the first half. In
>> other words, you can control the phase from 0 to Fs/2, but then the
>> phase from 0 to -Fs/2 will be odd symmetric with that of 0 to +Fs/2
>> (i.e., phi(-f) = -phi(f), 0 <= f < Fs/2).
>>
>> > Perhaps the part I have difficulty with is that the phase in a complex
>> > signal is determined by the relationship between the I,Q components and
>> > unless the phase of each component is changed by a different amount
> (complex
>> > filter) you can only achieve a shift in the overall phase of the complex
>> > signal but not change the shape of the phase response.
>>
>> Can you please rephrase, Bhaskar? I'm not understanding you here.
>
> Never mind Randy...I can't even make out what I was trying to say here - I
> guess I can't really think in phase domain.

I'll take that as a rephrasal! :)

Let me give you my understanding of the term "phase domain" or "phase
response." If we have a filter with impulse resonse h(t) or h(n*Ts),
then we can find its frequency response by taking the Fourier
transform of its impulse response

  H(w) = F{h(n)}. 

The frequency response H(w) is, in general, a complex function of the
real variable w, therefore we can express it as

  H(w) = A(w) * e^{j*phi(w)},

where A(w) is a non-negative, real function and phi(w) is a real
function. This is simply the polar form of a complex number, but it is
a function of frequency w.  Then A(w) is said to be the "magnitude
response" and phi(w) the "phase response" of the filter.

Any sine wave sin(w*t + theta) with phase theta that passes through
the filter will have a phase (theta + phi(w)) when it reaches the
output, i.e., the output sine wave will be A(w) * sin(w*t + theta +
phi(w)).

Since 2*pi radians corresponds to one cycle at the frequency w, we can
translate phase to time.  Since any input signal can be decomposed
into a sum of sinusoids we can then use the filter's phase response
phi(w) to see the amount of time it will delay each sinusoid.

For example, a linear-phase filter has a constant *time delay* at all
frequencies. If we denote that time delay as Td, then we can
convert time delay into phase according to the following simple
formula

  phi(w) = 2 * pi * (Td / T)
         = ((2 * pi) / T) * Td
         = (2 * pi * f) * Td
         = w * Td,

where w = 2*pi*f and T = 1/f. Since Td is a constant, phi(w) is a
linear function of w - hence the phrase "linear phase."

Does that help? 
-- 
%  Randy Yates                  % "Rollin' and riding and slippin' and
%% Fuquay-Varina, NC            %  sliding, it's magic."
%%% 919-577-9882                %  
%%%% <yates@ieee.org>           % 'Living' Thing', *A New World Record*, ELO
http://home.earthlink.net/~yatescr

Randy Yates wrote:

   ...

> Since 2*pi radians corresponds to one cycle at the frequency w, we can
> translate phase to time.  Since any input signal can be decomposed
> into a sum of sinusoids we can then use the filter's phase response
> phi(w) to see the amount of time it will delay each sinusoid.

Caveat: From that, the time is known with an ambiguity of n*2*pi/w, 
where n is an arbitrary integer. The ambiguity can often be resolved by 
knowing a boundary condition and the continuity of the function.

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

"Randy Yates" <yates@ieee.org> wrote in message
news:65ew8z0u.fsf@ieee.org...
> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
>
> > "Randy Yates" <yates@ieee.org> wrote in message
> > news:ad49xbb5.fsf@ieee.org...
> >> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
> >>
> >> > "Randy Yates" <yates@ieee.org> wrote in message
> >> > news:vfmy836v.fsf@ieee.org...
> >> >> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
> >> >>
> >> >> > "Randy Yates" <yates@ieee.org> wrote in message
> >> >> > news:ptd74i85.fsf@ieee.org...
> >> >> >> "Bhaskar Thiagarajan" <bhaskart@deja.com> writes:
> >> >> >>
> >> >> >> > "ZedToe" <acoustictech_zhangtao@yahoo.com.sg> wrote in message
> >> >> >> > news:7c4bf533.0401231957.22326456@posting.google.com...
> >> >> >> >> Dear all,
> >> >> >> >>
> >> >> >> >> Why need we design complex filters? What are the advantages
they
> >> > have
> >> >> >> >> against the real-coefficient fitlers? Except the unsymmetric
> >> > spectrum,
> >> >> >> >> there indeed nothing special. Also it can be derived from a
real
> >> >> >> >> lowpass filter with an appropriated phase shift. What do you
> > think?
> >> >> >> >>
> >> >> >> >> Thanks.
> >> >> >> >>
> >> >> >> >> Zedtoe
> >> >> >> >
> >> >> >> > I think you can use a complex FIR filter to correct for both
> >> > amplitude
> >> >> > as
> >> >> >> > well as phase (you only get amplitude changes using a real FIR
> >> >> >> > filter...linear phase of course).
> >> >> >>
> >> >> >> Wrong. You can design real, non-linear-phase FIR filters. The
amount
> > of
> >> >> > control
> >> >> >
> >> >> > But can you use them on complex signals and achieve a desired
phase
> > and
> >> >> > amplitude correction? (I'm not challenging, but asking).
> >> >>
> >> >> I'm pretty sure you can, but I can't really think of any canned
filter
> >> >> design packages that let you do this. You could try specifying a
> >> >> vector of N complex numbers representing magnitude/phase at
> >> >> frequencies from -Fs/2 to +Fs/2 and inverse transforming the result,
> >> >> but there is certain to be some constraints on the length N versus
the
> >> >> amount of magnitude and phase control.
> >> >
> >> > I always thought that, for complex signals, you need a complex filter
to
> > do
> >> > amplitude and phase correction. Your statement says otherwise and I'm
> >> > struggling to grasp it. I certainly do understand that real FIR
filters
> > of
> >> > doing phase correction on real signals.
> >>
> >> I may have been a bit imprecise - a real filter can correct a
> >> *portion* of a complex signal's phase response. Essentially, a real
> >> filter will have a Hermitian-symmetric frequency response, so you can
> >> only control one-half of the bandwidth (i.e., Fs/2 of the bandwidth),
> >> the other half being necessarily a reflection of the first half. In
> >> other words, you can control the phase from 0 to Fs/2, but then the
> >> phase from 0 to -Fs/2 will be odd symmetric with that of 0 to +Fs/2
> >> (i.e., phi(-f) = -phi(f), 0 <= f < Fs/2).
> >>
> >> > Perhaps the part I have difficulty with is that the phase in a
complex
> >> > signal is determined by the relationship between the I,Q components
and
> >> > unless the phase of each component is changed by a different amount
> > (complex
> >> > filter) you can only achieve a shift in the overall phase of the
complex
> >> > signal but not change the shape of the phase response.
> >>
> >> Can you please rephrase, Bhaskar? I'm not understanding you here.
> >
> > Never mind Randy...I can't even make out what I was trying to say here -
I
> > guess I can't really think in phase domain.
>
> I'll take that as a rephrasal! :)
>
> Let me give you my understanding of the term "phase domain" or "phase
> response." If we have a filter with impulse resonse h(t) or h(n*Ts),
> then we can find its frequency response by taking the Fourier
> transform of its impulse response
>
>   H(w) = F{h(n)}.
>
> The frequency response H(w) is, in general, a complex function of the
> real variable w, therefore we can express it as
>
>   H(w) = A(w) * e^{j*phi(w)},
>
> where A(w) is a non-negative, real function and phi(w) is a real
> function. This is simply the polar form of a complex number, but it is
> a function of frequency w.  Then A(w) is said to be the "magnitude
> response" and phi(w) the "phase response" of the filter.
>
> Any sine wave sin(w*t + theta) with phase theta that passes through
> the filter will have a phase (theta + phi(w)) when it reaches the
> output, i.e., the output sine wave will be A(w) * sin(w*t + theta +
> phi(w)).
>
> Since 2*pi radians corresponds to one cycle at the frequency w, we can
> translate phase to time.  Since any input signal can be decomposed
> into a sum of sinusoids we can then use the filter's phase response
> phi(w) to see the amount of time it will delay each sinusoid.
>
> For example, a linear-phase filter has a constant *time delay* at all
> frequencies. If we denote that time delay as Td, then we can
> convert time delay into phase according to the following simple
> formula
>
>   phi(w) = 2 * pi * (Td / T)
>          = ((2 * pi) / T) * Td
>          = (2 * pi * f) * Td
>          = w * Td,
>
> where w = 2*pi*f and T = 1/f. Since Td is a constant, phi(w) is a
> linear function of w - hence the phrase "linear phase."
>
> Does that help?

Absolutely. I appreciate your taking the time and I guess this helps me get
back to what I was trying to say/ask.
The phase response of a complex signal phi(w) is essentially
tan_inv(Q(w)/I(w)) right?
Now, when a real filter is applied (say with linear phase resp) to this
signal (essentially, the same filter is applied separately to the I and Q
components) the Q(w) and I(w) each go through a linear shift in phase as a
function of w. So the final phase response of the signal tan_inv(Q(w)/I(w))
doesn't really change because the numerator and denominator went through the
same change. So even if the real filter has non-linear phase response, it
wouldn't help in 'shaping' the phase response of the complex signal.
A complex filter on the other hand is basically 2 different real filter with
2 different phase responses applied to the I,Q components. Controlling how
each of these 2 filters modify the phase response of the I, Q individually,
you can now control the overall output phase response of the complex signal.
This is what I was trying to say initially, but I suspect I got something
wrong in my statements regarding how the real filter affects the complex
signal's phase response.

Cheers
Bhaskar

> --
> %  Randy Yates                  % "Rollin' and riding and slippin' and
> %% Fuquay-Varina, NC            %  sliding, it's magic."
> %%% 919-577-9882                %
> %%%% <yates@ieee.org>           % 'Living' Thing', *A New World Record*,
ELO
> http://home.earthlink.net/~yatescr

On Mon, 26 Jan 2004 02:29:07 GMT, Randy Yates <yates@ieee.org> wrote:

  (snipped)
>
>Zedtoe, to answer your question, I think you must ask another
>question: When is it necessary or convenient to process complex
>signals? (once you have a complex signal, then obviously you need
>complex filters to filter them with)
>
>In my experience, the answer to this has been "when we are processing
>a signal with modulation." The reason is that such a signal is almost
>always modulated on a carrier frequency Fc with bandwidth B, i.e., the
>signal information is from Fc - B/2 to Fc + B/2 (Hz). In that case
>it makes sense to utilize the whole digital bandwidth from -Fs/2 to +Fs/2
>and sample at Fs = B, quadrature downconverting the comm signal so that
>the carrier center frequency Fc is translated to DC. If you translated 
>this down to a real signal, you'd be wasting have the bandwidth. 
>

Yep,
  there are *all sorts* of applications where 
it's useful to monitor (measure) the 
instantaneous time-domain phase of a sinusoid, 
instantaneous frequency of a sinusoid, or 
the relative phase of spectral components. 

  * digital communications systems (modulation & 
        demodulation)
  * radar systems,
  * time difference of arrival processing in radio 
        direction finding schemes,
  * coherent pulse measurement systems,
  * antenna beamforming applications,

[-Rick-]