Talk:All-pass filter

This page doesn't include the possibility of $$z_0$$ being complex, which, in audio terms at least, are the most useful all-pass filters. To accommodate complex $$z_0$$, the z-transform implementation equation would need to be altered to change $$z_0^{-1}$$ to $$(z_0^*)^{-1}$$ in the second term of the denominator.

E hu man (talk) 22:03, 19 February 2008 (UTC)
 * Sometime in the past, someone fixed this problem by putting the transfer function in terms of the reciprocal of z. Then someone "fixed" the conjugate problem "again," which made the page nonsensical. I've gone ahead and fixed things once again. I think the page as it stands should satisfy everyone. --TedPavlic | (talk) 14:48, 5 September 2008 (UTC)

'Too technical' tag
I see that a 'Too technical' tag has been put on this article. I should have thought that this topic - like many articles on engineering matters is inherently of interest only to readers who have enough background knowledge in the subject area to follow the exposition. I suggest that this in no way diminishes its validity as a subject to be included in Wikipedia, or its value to readers who do have the appropriate background. DaveApter (talk) 15:32, 7 March 2018 (UTC)
 * Well yes and no. An advanced technical subject will always be difficult for a reader with no knowledge of the subject.  Against that, Wikipedia articles should be written with the general reader in mind.  The lead at least, should be understandable to the general reader.  Someone who has heard the phrase and comes to this article should go away satisfied they know what it is.  This article could make a better job of explaining why phase is an issue in electronics with example applications.  Too often, technical writers here tend to dive straight in to the equations with little explanation in prose.  In all probability, practising engineers will go to their textbooks for the design equations, not Wikipedia.  Sometimes, the equations are not really necessary at all and do little more than show off the knowledge of the writer rather than add to the readers understanding.  Encyclopaedias are not textbooks. SpinningSpark 16:59, 7 March 2018 (UTC)


 * Thank you. Yes I agree that the lead could be a lot clearer and more accessible. I will have a go at re-writing it shortly. Also the opening paragraph of the main section would benefit from some introductory explanation before diving into the equations. DaveApter (talk) 18:29, 7 March 2018 (UTC)
 * Put an applications section before any of the circuit implementations is what I suggest. Stereo music lines and analogue television are the two I am most familiar with, but I am sure there are many others. See  and . SpinningSpark 18:56, 7 March 2018 (UTC)
 * I added the tag because despite some of my college education in signal processing and general background in music, I still didn't really internalize what this article is trying to say. If you click on the link in the technical tag, the first sentence says: "The content in articles in Wikipedia should be written as far as possible for the widest possible general audience."  Ideally I would be able to understand the main idea within a matter of seconds (perhaps with a picture or an audio demo).  If I were more interested/more educated in this area I could then study the article in more depth, including more technical sections of the article.  I agree with the guidance provided by Spinningspark. Intcreator (talk) 23:19, 7 March 2018 (UTC)


 * I've added some material to the lead, and will come back and expand that over the next few days. DaveApter (talk) 15:20, 9 March 2018 (UTC)


 * I've moved the discussion of applications from the lead into a newly created section of the body. I think the introduction should not be too intimidating now, and anyone not understanding the terms can follow the wikilinks for an explanation. I there any objection to the removal of the tag now? DaveApter (talk) 17:11, 24 January 2019 (UTC)
 * No objections, but the lede is probably too short. A pure delay is an easy to understand example of an all pass filter. Constant314 (talk) 20:44, 24 January 2019 (UTC)
 * You do know that a pure delay can't be achieved with a rational function, right? SpinningSpark 00:44, 25 January 2019 (UTC)
 * Well. I cannot say that I was cognizantly aware of that fact, but it seems right. But, is that an objection?  The article does include digital filters; the Z ( or Z-1 ) in the Z-transform is a pure delay.   An ideal transmission line that is terminated with its characteristic impedance is a pure delay.  And yes, I shamelessly invoke idealness and point out all-pass filters are only all pass in their ideal forms, too. Constant314 (talk) 02:55, 25 January 2019 (UTC)
 * One has to ask whether an ideal transmission line can be classed as a filter at all. What exactly is it supposed that it is filtering?  An actual filter built out of transmission lines (that is a maximally flat time-delay distributed element filter) won't be a pure delay either.  That is what is usually meant in RS by transmission line (or microwave) all-pass network. SpinningSpark 15:31, 25 January 2019 (UTC)
 * Definitely a filter. Its output shows frequency dependent phase shift.  Its Lapalce transform is e-sd where d is the delay. Constant314 (talk) 23:22, 25 January 2019 (UTC)
 * In view of the above, I'll remove the tag, and will think about adding a bit more detail to the lead over the next few days. DaveApter (talk) 14:09, 3 February 2019 (UTC)

And the citations tag
Agreed, the article is short of citations; I will look for some references and add them.DaveApter (talk) 14:14, 3 February 2019 (UTC)

Inversion
You have now twice tried to insert something about inversion at high (or low) frequency. You can only make such statements about a specific circuit, not as a general statement in the lead. For instance, the passive lattice circuit can show an inversion at either high or low frequency, just swap the capacitors and inductors to achieve this. The inversion can even be arranged to happen at both high and low frequencies – even with a passive circuit. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 14:52, 7 February 2019 (UTC)
 * Thanks - it was my (obviously all to casual) attempt to address the objection by the editor who reverted my previous amendment. I will think about formulating the point in a sufficiently generalised form. DaveApter (talk) 18:34, 7 February 2019 (UTC)
 * Dave, you are still getting it wrong. You are trying to add a general statement about all all-pass filters in lede that is simply not true of all all-pass filters.  You could add that statement to the caption of figure 1. Constant314 (talk) 19:07, 7 February 2019 (UTC)
 * OK so please propose an articulation here that you would find acceptable. DaveApter (talk) 18:03, 8 February 2019 (UTC)
 * Regarding Fig. 1, it is obvious to me, as an experienced op-amp circuit designer, that the DC gain is -1 and the infinite AC gain is +1. I would have no problem with a statement to that effect added to the caption of fig 1.
 * But, before we get to that, the phase of :$$H(s) = \frac{ s - \frac{1}{RC} }{ s + \frac{1}{RC} }, \,$$ is given as $$ \quad \angle H(i\omega)  =  \arctan( -\omega RC ) - \arctan( \omega RC ). \,$$ which evaluates to zero when $$ \omega=0$$.  I think the correct expression is $$ \quad \angle H(i\omega)  =  \arctan( \frac 1 {-\omega RC} ) - \arctan(  \frac 1 {\omega RC} ) = -2 \arctan( \frac 1 {\omega RC} ) $$ which evaluates to -180 in the limit as at $$ \omega$$ approaches DC.  Can anybody verify? Constant314 (talk) 18:36, 8 February 2019 (UTC)
 * I'm not sure how you derived that. I think the original expression ($$-2 \tan^{-1} \omega \tau$$) is correct.  The tangent of $$\angle H(\omega)$$ is imaginary part over real part right?  So $$\omega \tau$$ has to be on top.  I get your point about $$H(0) = -1$$, but that is due to the inversion of the op-amp and an inversion is not the same as a phase change.  Now I know that inverting a sinusoid gets the same result as a 180° phase shift, but that they are different things is obvious if you view the output on an oscilloscope of an asymmetrical waveform. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 23:13, 8 February 2019 (UTC)
 * When you determine the angle of a complex number, it is Arctan of the imaginary part divided by the real part. Example: $$ \quad \angle {(1 + j0)}  =  \arctan( \frac 0 1 ) =0.$$
 * The real part of the numerator of H(s) would be $$ \frac {-1} \tau$$ and the imaginary part would be $$ j \omega $$ so you would taking the arctan of $$ \frac {-1} { \tau \omega}$$ Constant314 (talk) 23:40, 8 February 2019 (UTC)
 * "...the imaginary part divided by the real part...", "...the imaginary part would be $$ j \omega $$..." Read what you wrote. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 00:40, 9 February 2019 (UTC)

I made a mistake and retract my suggested correction. However, there is still something wrong. The definition I gave about the angle of a complex number is only correct when the real part is positive. The real part of this numerator is negative. I copied the following out of the discussion for atan2. If the number is x + jy then the angle In terms of the standard $arctan$ function, whose range is $(−π⁄2, π⁄2)$, it can be expressed as follows: $$\operatorname{atan2}(y,x) = \begin{cases} \arctan(\frac y x) &\text{if } x > 0, \\ \arctan(\frac y x) + \pi &\text{if } x < 0 \text{ and } y \ge 0, \\ \arctan(\frac y x) - \pi &\text{if } x < 0 \text{ and } y < 0, \\ +\frac{\pi}{2} &\text{if } x = 0 \text{ and } y > 0, \\ -\frac{\pi}{2} &\text{if } x = 0 \text{ and } y < 0, \\ \text{undefined} &\text{if } x = 0 \text{ and } y = 0. \end{cases}$$

So, the angle of the numerator is $$  \arctan( \frac {\omega} {-\frac 1 {RC}}) + \pi = \arctan(  {- \omega RC})  + \pi$$

So, the expression in the article should be $$ \quad \angle H(i\omega) =  \arctan( -\omega RC ) - \arctan( \omega RC ) +\pi = \pi - 2 \arctan( \omega RC ). \,$$

The phase at { $$ 0, \frac 1 {RC}, \infin $$ } ,  is { $$   \pi, \frac \pi 2 , 0 $$ }. That confirms my expectation. Constant314 (talk) 02:26, 9 February 2019 (UTC)

By the way, the circuit in the reference you added has R and C swapped relative to figure 1 of this article. I have references for the circuit in your reference that agree with the formula for phase shift with your reference. We're good on that.(talk) 02:36, 9 February 2019 (UTC)
 * I'm good with the $$+ \pi$$ bit if you want to put that in. And well spotted. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 02:57, 9 February 2019 (UTC)

rather than simply deleting every attempt I make to provide a succinct accessible summary, would it not be more constructive to edit it so that it meets your satisfaction. Alternatively, perhaps you could suggest a form of words that achieves this? DaveApter (talk) 17:18, 11 February 2019 (UTC)
 * There is nothing unconstructive about removing information that is incorrect. We are primarily here to serve our readers, not to avoid hurting each other's feelings.  Leaving wrong information in the article is very bad on that account.  Your contributions are not based on reliable sources.  This is the entire problem.  If you had provided a source, I would at least have checked out what the source said and whether or not it could be considered reliable.  If the source was reliable and you had just misinterpreted it then I might edit what you wrote to comply with the source.  However, if you insist on adding your own original research then you can expect it be perfunctorily reverted if we think you are wrong.  You wouldn't have had multiple reverts if you had made the alternative suggestions here first after the initial revert. Here's what our policy says on this "Any material lacking a reliable source directly supporting it may be removed and should not be restored without an inline citation to a reliable source." Please pay attention to that in future.
 * And to answer you question directly, I don't believe that it is possible to summarise in the way that you want. There are numerous variations, some of which I mentioned above. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 18:28, 11 February 2019 (UTC)

Filter in fig. 1 produces phase lead
The filter in fig 1 produces a phase lead at $&omega;$ = 1/RC of +90$&deg;$. The output should lead the input. I included a SPICE simulation of the filter with the input being a 100 Hz sine burst. The quadrature frequency is 100Hz.



You see that the output (yellow) settles to a quarter cycle ahead of the input (cyan).

Note, E1 is equivalent to a very very good opamp with a gain of 109 and infinite bandwidth.

So, I change "delayed quarter wavelength" to "advanced quarter period". Maybe cycle would be better than period.Constant314 (talk) 05:02, 9 February 2019 (UTC)

Padé approximation
The Padé approximation uses the other all-pass filter, the one with the low pass filter. Maybe we should replace fig 1 with the other filter and start over.

Anyway, swapping "Interpretation as a Padé approximation to a pure delay" section with "Implementation using low-pass filter"section. Constant314 (talk) 05:11, 9 February 2019 (UTC)

Phase, phase shift, phase lag, phase lead and delay
I’m in a quandary about the use of the terms phase shift and delay when the filter produces phase lead.

Consider for a moment, an ordinary one pole low pass filter. The phase of the transfer function is 0° at DC and become increasingly negative, reaching -45° at the 3 dB point and approaches -90°  at frequencies well above the 3 dB point. The output lags the input. We say the output is delayed from the input. We would commonly say that this filter produces a phase shift of 90° at high frequencies. So, in common usage, if the transfer function has a negative phase, we say that the phase shift is positive and the delay is positive.

Now consider a high pass filter (zero at DC and a single real pole). The phase of the transfer function is +90° at DC and become decreasingly positive, reaching +45° at the 3 dB point and approaches 0°  at frequencies well above the 3 dB point. The output leads the input. Is the phase shift negative? Is the delay negative? A high pass filter, below the 3dB point is a sort of differentiator or predictor, so a negative delay is not unreasonable, but it certainly could be confusing of the general reader.

So, what do we say when the output leads the input? When the phase of the transfer function is positive, do we say that the phase shift is negative? Do we say that the delay is negative? I'm leaning toward avoiding the use of the terms phase shift and delay in the context of a filter that produces phase lead.

Your comments would be appreciated. Constant314 (talk) 21:30, 9 February 2019 (UTC)


 * Phase delay has no bearing on causality. It is group delay that must be positive for causality reasons.  To some extent, it is meaningless to talk about a delay of a single sinusoid.  Any practical method of determining the delay of a single sinusoid will involve limiting it to a finite burst, at which point you no longer have a pure single sinusoid since it now has an envelope. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 22:04, 9 February 2019 (UTC)


 * Then at the 90° point we could use apparent delay or apparent advance? Constant314 (talk) 22:11, 9 February 2019 (UTC)
 * Not sure that that terminology has any provenance in sources. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 22:40, 9 February 2019 (UTC)
 * Well, if delay is wrong and apparent delay has no support, shall we just remove any discussion of delay? But do we really need support for apparent delay since both apparent and delay are used in their ordinary sense? Constant314 (talk) 22:45, 9 February 2019 (UTC)

Implementation using the low pass filter
I have added a sub-section for the implementation using the low pass filter, since has at least one reliable ref that discuss it and I have at least one and maybe more that also discuss that version. Also, the low pass version produces phase lag and the terms delay and phase shift have their ordinary meanings.

I have put the Pade sub-section under the low-pass implementation sub-section because it uses the low-pass implementation.

My intent is to flesh out the low-pass section to the point that it is equivalent to the section on the high-pass filter implementation and then swap them so that the low pass implementation is discussed first. Then, after swapping, I will work to eliminate the terms phase shift and delay in the high-pass version so that it isn't necesary to describe them as having negative values.

Note, this is my fourth new talk topic in a row. If you missed them you might want to review them also. Constant314 (talk) 21:37, 9 February 2019 (UTC)

General usage sub-section needs reference
Although it seems plausible, it is not obvious and needs a reference. It also is misleading as written. Merely putting an all-pass filter in cascade with an unstable system, won’t stabilize the unstable system. However, an all-pass used in this way may make it easier to stabilize the unstable system with feedback. Constant314 (talk) 23:01, 12 February 2019 (UTC)


 * Well the first thing I would say about that is that it is an application, not an implementation, so at the very least the section needs to be moved. Pole-zero cancellation using all-pass filters is a theoretically possible way of stabilising a system, but it is difficult to achieve in practice.  Only slight mistuning or drift results in imperfect cancellation and can reintroduce instability.  This is discussed in Signals and Systems using MATLAB, pages 374-375. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 10:32, 13 February 2019 (UTC)


 * Instead of moving it, perhaps it should be removed. Constant314 (talk) 18:30, 13 February 2019 (UTC)
 * I wouldn't object. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 19:42, 13 February 2019 (UTC)
 * I'll wait a little while longer to see if a citation turns up.Constant314 (talk) 21:07, 13 February 2019 (UTC)
 * If you just want a citation, the book I linked will do for that. <b style="background:#FAFAD2;color:#C08000">Spinning</b><b style="color:#4840A0">Spark</b> 21:33, 13 February 2019 (UTC)