How do All-Pass Filters Work?

All a highpass, lowpass, bandpass, or band stop filters is, is a 
discrete numerical solutions to differential equation.  Is this also 
true for all-pass filters?  Is there some continuous time equation that 
functions as an all-pass filter?  What I am getting at is: How does an 
all pass filter change the phase of a signal?

On Mon, 21 May 2007 11:29:20 -0400, Chris Barrett wrote:

> All a highpass, lowpass, bandpass, or band stop filters is, is a 
> discrete numerical solutions to differential equation. 

Mostly, yes.  To be precise, in continuous time such a filter can be _modeled_ by a
differential equation, and in discrete time a filter can be _implemented_
as a _difference_ equation.

> Is this also 
> true for all-pass filters? 

Yes.

> Is there some continuous time equation that
> functions as an all-pass filter? 

Yes, and there are physical realizations, also.  One can do this with
op-amps, or even with capacitors and coils -- although you'll need at
least some of those coils to be transformers, if you want an entirely
passive network.

> What I am getting at is: How does an
> all pass filter change the phase of a signal?

An all pass filter changes the phase (but not the amplitude) of a signal
by pairing a stable pole (be it resonant or not) with an unstable zero. 
In a continuous-time system this boils down to a Laplace transfer function
with pairs that go like:

T(s) = product of ((s - a_k) / (s + a_k))

where every a_k is stable (i.e. it has a positive real part) and, if it's
complex, a conjugate to keep the description real.

In a discrete-time IIR filter this boils down to a z-domain transfer
function with pairs that go like

T(z) = (gain adjust) * product of ((z - 1/a_k) / (z - a_k))

Again, every a_k must be stable (in this case it must follow |a_k| < 1),
and any complex a_k must have a conjugate.

For an FIR filter your job is conceptually easier: just gin up the phase
vs. frequency response you want, accept the fact that you're going to need
a really, really long FIR, and go to it.

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

Chris Barrett wrote:
> All a highpass, lowpass, bandpass, or band stop filters is, is a 
> discrete numerical solutions to differential equation.  Is this also 
> true for all-pass filters?  Is there some continuous time equation that 
> functions as an all-pass filter?  What I am getting at is: How does an 
> all pass filter change the phase of a signal?

Do you understand how an analog all-pass works? If not (and you can hack 
simple circuit theory) you might find it instructive.

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

Chris Barrett wrote:
> All a highpass, lowpass, bandpass, or band stop filters is, is a
> discrete numerical solutions to differential equation.  
> Is this also true for all-pass filters?  Is there some
> continuous time equation that functions as an all-pass filter?  

Sure. Just take the (discrete) impulse response of some allpass
examples that you can find here,

http://www-ccrma.stanford.edu/~jos/waveguide/Example_Allpass_Filters.html

then use sinc-interpolation to get the impulse response of a
continuous-time allpass system.

> What I am getting at is: How does an
> all pass filter change the phase of a signal?

Here is a simple allpass:

http://www-ccrma.stanford.edu/~jos/waveguide/Allpass_Two_Combs.html

The trick is to cancel every pole inside the unit circle with a
corresponding zero reflected about the unit circle. The recursive part
of the allpass is minimum phase, the transversal part is maximum
phase.

Regards,
Andor

Tim Wescott wrote:
> On Mon, 21 May 2007 11:29:20 -0400, Chris Barrett wrote:

   ...

>> Is there some continuous time equation that
>> functions as an all-pass filter? 
> 
> Yes, and there are physical realizations, also.  One can do this with
> op-amps, or even with capacitors and coils -- although you'll need at
> least some of those coils to be transformers, if you want an entirely
> passive network.

For a simple example, consider a transformer with a center-tapped 
secondary. Ground the secondary and connect one end to the (infinite 
impedance) load with a resistor, the other end with a capacitor. The 
output amplitude is constant for constant-amplitude input to the 
primary. High and low frequencies have their relative phase shifted 180 
degrees. The phase is quadrature at w = 1/RC.

   ...

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

I wrote:
> Chris Barrett wrote:
> > All a highpass, lowpass, bandpass, or band stop filters is, is a
> > discrete numerical solutions to differential equation.  
> > Is this also true for all-pass filters?  Is there some
> > continuous time equation that functions as an all-pass filter?  
>
> Sure. Just take the (discrete) impulse response of some allpass
> examples that you can find here,
>
> http://www-ccrma.stanford.edu/~jos/waveguide/Example_Allpass_Filters....
>
> then use sinc-interpolation to get the impulse response of a
> continuous-time allpass system.

Oops. That's not an allpass system ...

Jerry Avins wrote:
> Tim Wescott wrote:
>> On Mon, 21 May 2007 11:29:20 -0400, Chris Barrett wrote:
> 
>   ...
> 
>>> Is there some continuous time equation that
>>> functions as an all-pass filter? 
>>
>> Yes, and there are physical realizations, also.  One can do this with
>> op-amps, or even with capacitors and coils -- although you'll need at
>> least some of those coils to be transformers, if you want an entirely
>> passive network.
> 
> For a simple example, consider a transformer with a center-tapped 
> secondary. Ground the secondary and connect one end to the (infinite 
> impedance) load with a resistor, the other end with a capacitor. The 
> output amplitude is constant for constant-amplitude input to the 
> primary. High and low frequencies have their relative phase shifted 180 
> degrees. The phase is quadrature at w = 1/RC.

I got an email requesting that I provide a  mathematical justification 
of my example above. I won't, because there's a more elegant way.

Plot the output across the open-circuit load on the complex voltage 
plane in the usual orientation: positive real to the right, positive 
imaginary up. Connect the capacitor to the secondary end marked positive 
and the resistor to the other. (The center tap is at the origin, of 
course.) The phasor from the origin to the load represents the output 
voltage in amplitude and phase. At very high frequencies, the capacitor 
is a short, and the output is on the real axis at a distance represented 
by the signal's magnitude, say A. At very low frequencies, the output 
phasor again lies on the real axis, this time pointing left.

The currents through the resistor and capacitor are of course the same, 
so the phasors representing their voltages must be perpendicular. The 
resistor phasor starts from [-A, 0], the capacitor phasor starts from 
{+A, 0], and they meet at the output node at right angles. Clearly, the 
locus of possible meeting points is a semicircle. The output phasor runs 
from the origin to a point on that semicircle. It's length is clearly 
constant.

I could have proved the same with a little algebra and less typing, but 
the ideas expressed wouldn't have been as compelling or memorable.

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

Jerry Avins wrote:
> Chris Barrett wrote:
> 
>> All a highpass, lowpass, bandpass, or band stop filters is, is a 
>> discrete numerical solutions to differential equation.  Is this also 
>> true for all-pass filters?  Is there some continuous time equation 
>> that functions as an all-pass filter?  What I am getting at is: How 
>> does an all pass filter change the phase of a signal?
> 
> 
> Do you understand how an analog all-pass works? If not (and you can hack 
> simple circuit theory) you might find it instructive.
> 
> Jerry

No.  I was hoping I could figure this out mathematically.  It seemed to 
worked for basic equalization filters.

Chris Barrett wrote:
> Jerry Avins wrote:
>> Chris Barrett wrote:
>>
>>> All a highpass, lowpass, bandpass, or band stop filters is, is a 
>>> discrete numerical solutions to differential equation.  Is this also 
>>> true for all-pass filters?  Is there some continuous time equation 
>>> that functions as an all-pass filter?  What I am getting at is: How 
>>> does an all pass filter change the phase of a signal?
>>
>>
>> Do you understand how an analog all-pass works? If not (and you can 
>> hack simple circuit theory) you might find it instructive.
>>
>> Jerry
> 
> No.  I was hoping I could figure this out mathematically.  It seemed to 
> worked for basic equalization filters.

Well, circuit theory is math applied to Kirchoff's laws. Of course, EEs 
learn some shortcuts so they don't have to go back to first principles 
every time. Make two voltage dividers using three equal resistors and 
one capacitor. Calculate the open-circuit voltage as a function of 
frequency that can be measured from tap to tap. You will find that you 
have written the equation of an analog all-pass filter very similar to 
the one I described. (You deserve for the world to know that it was not 
you who asked for its mathematical analysis.)

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

Jerry Avins wrote:
> Chris Barrett wrote:
> 
>> Jerry Avins wrote:
>>
>>> Chris Barrett wrote:
>>>
>>>> All a highpass, lowpass, bandpass, or band stop filters is, is a 
>>>> discrete numerical solutions to differential equation.  Is this also 
>>>> true for all-pass filters?  Is there some continuous time equation 
>>>> that functions as an all-pass filter?  What I am getting at is: How 
>>>> does an all pass filter change the phase of a signal?
>>>
>>>
>>>
>>> Do you understand how an analog all-pass works? If not (and you can 
>>> hack simple circuit theory) you might find it instructive.
>>>
>>> Jerry
>>
>>
>> No.  I was hoping I could figure this out mathematically.  It seemed 
>> to worked for basic equalization filters.
> 
> 
> Well, circuit theory is math applied to Kirchoff's laws. Of course, EEs 
> learn some shortcuts so they don't have to go back to first principles 
> every time. Make two voltage dividers using three equal resistors and 
> one capacitor. Calculate the open-circuit voltage as a function of 
> frequency that can be measured from tap to tap. You will find that you 
> have written the equation of an analog all-pass filter very similar to 
> the one I described. (You deserve for the world to know that it was not 
> you who asked for its mathematical analysis.)
> 
> Jerry

Is this in any filter design web pages or books?  What I would like to 
see is some example formulas in the s-domain along with corresponding 
phase plots.