Announcement

Collapse
No announcement yet.

Results of the Minimum Phase / Off-axis test from yesterday (jay, dlr)

Collapse
X
 
  • Filter
  • Time
  • Show
Clear All
new posts

  • Results of the Minimum Phase / Off-axis test from yesterday (jay, dlr)

    Yesterday, in that way too long of a thread, there was discussion about how a transient perfect crossover can remain minimum phase in its summation off axis or even with some offset delay added.

    For back ground, it mostly began with this exchange between Jay and me (Please feel free to scroll past the quotes if you wish they are just reassembled here for reference):

    One more minor thing. I think a transient perfect system is minimum phase, not only precisely on axis but also in a narrow, nearby off-axis region beyond which a phase shift due to acoustic offeset finally brings about a non-M-P summed response.

    -jAy

    Regarding a transient perfect system's off-axis window: Sure, there is a narrow window in which the summed phase errors are small enough to be inconsequential, so it may still be considered minimum phase / transient perfect. However, in reality errors begin to present themselves at some level with any movement off-axis. Now, the question becomes estabilshing a tolerance for the phase error that defines the window you are referring to. It is the same as asking how much do we turn down the volume before we say it's now quieter, or quite enough. Or how much can we turn down the lights before it is now darker. From a technical persepective though transient perfect isn't "perfect" as soon as the relative path lengths change.

    Jeff B.

    Then there was this series:

    There is an important difference between MP and TP. TP implies MP, MP does not imply TP. If you have a system which is TP on the design axis and move off axis slightly there likely hood is that the system will remain MP but will now have non-flat amplitude, thus will not be TP. This still usually give better transient behavior that a non-TP system.
    __________________
    john k....

    This is exaclty what I thought initially, but I was a bit confused after seeing Jeff and Dave's posts. Now I'm going back to my original position, not only affected by your post but also based on my simulation. A 1st order T-P system deviates from being T-P, but still remains M-P (not nearly M-P, as we might think), even when a small amount (the limit of which depends on Fc) of relative acoustic offset occurs between drivers (e.g., woofer's AC a bit behind tweeter's by changing the mic axis upward). I verified this via simulation.

    -jAy

    I Did not intend to confuse. John is correct about the difference between transient perfect and minimum phase as systems go. One is a subset of the other and to be transient perfect requires stricter conditions to be met than simply summing to minimum phase. However, the technical differences do not change any of the things Dave and I were saying as we used the terms in our discussion regarding driver phase response and even system phase behavior.
    Jeff B.

    The confusion arose because you and Dave's posts implied that a 1st order transient perfect system is strictly minimum-phase only when the drivers' relative acoustic offset is zero, without necessarily distinguishing between transient-perfect and minimum-phase when it's really necessary. You'll see it if you read yours and Dave's post again. The difference between T-P and M-P was the central point of my original post on this issue, if you read it carefully again:

    One more minor thing. I think a transient perfect system is minimum phase, not only precisely on axis but also in a narrow, nearby off-axis region beyond which a phase shift due to acoustic offeset finally brings about a non-M-P summed response.

    -jAy
    Things finally wrapped up with this:

    First, you are correct you did make the distinction between the two and I didn't pick up on it - mainly because, as I stated above, it changes nothing with respect to my reply and what I was trying to communicate. Yes, there is difference between the two which I stated above, but that difference doesn't really enter into what I was trying to say. We apparently won't agree - that's OK. (Although I am sure we actually do. )

    Jeff B.

    Your graphs posted above clarify what you were trying to say, and I agree. It remains though that errors do begin as soon as you go off-axis or introduce offset as you did. At first they may be too small to notice, so yes, there is a window that works just as you stated, but I said this too above. The phase doesn't suddenly switch from minimum phase to "phase all out of whack" there is a continuum that it goes through as you increment this angle. Of course, I am sure you realize that.
    Jeff B.

    I know what you mean, Jeff. But I don't think this notion applies here. Though I didn't post more simulations I did, this idealized BW1 T-P crossover doesn't exhibit any degree of non-M-P behavior until the acoustic offset becomes quite large and the summed response quite wavy. I used 1 kHz Fc to see it better.
    -jAy

    I understand what you are saying, and I am sure it stays fairly minimum phase looking for quite a while, enough that comparing line may not reveal it, but comparing numerical values may show the difference. But just to be clear, even though it looks correct, the phase change is a continuous thing not something that waits until the delay reaches a certain point before the phase begins to shift. So, although it may stay very close to minimum phase for quite a wide arc or with some offset misadjustment there is still some degree of error, even if it is small. Adding offset and saying that it is still minimum phase would be the same as saying that adding a small amount of delay doesn't add any excess delay to the phase at all, and we know that technically that isn't right, right?

    Jeff B.

    It was simply that if a system's summed response is minimum phase on a design axis (and this alone represents a fairly special case of crossovers), that moving off that axis technically changes the relative path lengths between the two drivers and the mic from what we had on the design axis. This change in path lengths adds delay to one of the drivers with respect to the other. It will change the summed response, yes - but, the added delay will also cause the phase to roll more in one driver pulling the summed phase off from minimum phase as defined by the summed response compared to what the HB Transform says the minimum phase would be. It may be a very small difference, but technically there has to be some difference none-the-less.

    We can not add delay to an individual driver's output without impacting the existing minimum phase behavior of the summed system response. Don't you agree?

    Jeff B.


    I am almost ready to prove out my point mathematically, and I am confident in the results and would certainly stand corrected if I proved to be wrong. However, I don't have the time to do that at the moment because we have a family outing in a little while. But it does seem to me that we are straining at gnats here, don't you think?

    I am really not just trying to just defend myself either. I am OK with being wrong. That's why I am willing to do the math and see.
    Jeff B.
    OK, so most of you skipped all of that above. Here's the results of the mathmatical test.

    I used the same example given by Jay. I took a textbook First Order Butterworth with zero offset that results in flat summed response and a flat minimum phase response as well - in a phrase - it's Transient Pefect.

    Next I offset the woofer back by 25mm. This created ripples in the response and makes the summation no longer Transient Perfect, but Jay's point was that it was still minimum phase.

    Then I extracted the minimum phase from the summed response and compared it to the system phase response to see if they were still identical or not.

    I was saying that although it appears to be minimum phase there are still phase errors present due to the delay added to the woofer which pulls the phase reponse off of what would be perfect minimum phase. I agreed though that there is a window where the system can still be described as MP but we would need to define tolerances to the phase deviation in order to say whether we still considered it minimum phase or not. In a post above I said it was like gradually turning down the lights to make it darker, but we needed to define when dark was dark enough.

    Here is a plot of the test I referred to above. It include the summed acoustic response with the 25mm offset, the phase, and the extracted minimum phase.



    It looks very close. Here it is zoomed in on the phase.



    And here is the deviation of the phase from minimum phase.



    This shows exactly what I was trying to describe. However, it may show what Jay was trying to say too, because I think there is some subjective opinion about what stills fall into the realm of minimum phase.

    Is this minimum phase behavior? Not perfectly - which is exactly what I was saying. However, I concede that for practical purposes the maximum deviation is only 9.5 degrees from MP and much less than this most of the time, so I would be willing to place this within that window that is very near minimum phase behavior that I mentioned above. So, in that I would be happy to agree with Jay from a practical standpoint. I was just saying that there is some delay added, so there has to be some deviation too. I will also add that it is really not as much deviation as I thought there might be for a 25mm offset. Keep in mind that is equivalent to only a 12.5mm offset at a 2kHz crossover, since this is more in the range we typically crossover.

    Jeff B.
    Click here for Jeff Bagby's Loudspeaker Design Software

  • #2
    Re: Results of the Minimum Phase / Off-axis test from yesterday (jay, dlr)

    Jeff,

    Thanks for the follow-up. As I said yesterday, this is a simple mathematical problem. What you did above is not a mathematical proof but a simulation-based example. In fact I almost proved that a theoretical BW1+ transient perfect crossover is exactly (not nearly) minimum-phase even with a time delay (or a physical offset) added to LP or HP response. A reason why I didn't posted it here yet is that, due to my lack of math skills in functional anlaysis, I couldn't prove that the inverse transfer function in the time domain of BW1 (with a time delay added to LP component) integrates to a finite number---I already showed the regular transfer function does. These two are conditions for stability which must be met for being a minimum-phase system. Another condition, causality, is already met. (EDIT: the conditions I'm talking about here are ones described in the time domain, not in the frequency domain. I used time-domain ones for the proof since they were easier for me to handle in the current situation where a fixed time delay is added).

    There are two reasons why you obtained a simulation result showing some deviation from minimum phase. One is that you didn't use a response summed at least up to 50 kHz for accurate HBT computation. Using a response only to 20 kHz doesn't give an accurate result. Simulation results I showed last time used responses to 50 kHz. Another reason is that the HBT algorithm in your PCD is the same one as "Quick Phase" in the FRC. This down-sized formula doesn't give a precise result. Here are what I got from the same simulation condition that you used. That is, BW1 crossover with 1 kHz Fc with a 25 mm offset added to LP.




    As you can see there's no discrepancy between the summed phase through crossover (magenta) and the HBT computed phase (cyan). Compare this to your first graph in your post.

    Below is the same thing but I used the "Quick" HBT algorithm to show that this algorithm gives a different result.



    A small difference from your result occurs because I used a response up to 50 kHz while you used one to 20 kHz.

    After all this analytical and simulational investigation, I'm now pretty sure that theoretical BW1 T-P crossovers remain perfectly minimum-phase (as defined mathematically) even with a physical offset added to a driver up to a point where the added phase creates a perfect null which occurs when LP and HP responses' amplitudes are equal and they are 180 degrees out of phase. Beyond this point, they are no longer minimum-phase.
    -jAy
    Last edited by jkim; 01-03-2009, 05:48 PM.

    Comment


    • #3
      Re: Results of the Minimum Phase / Off-axis test from yesterday (jay, dlr)

      Jeff,

      Not to be a stickler but there has to be something wrong with you MP response. First, since the response is flat at 10 Hz the phase must go to zero at low frequency. Second, you show a turn up of the phase around 2 hundred Hz. A MP response for the amplitude should must turn down when the response starts to turn down.

      I use SPL trace to copy your amplitude response from one figure and then imported to SoundEasy and extracted the MP. As you can see the SE MP (wide dark gray) is almost Identical to what you label "phase". I could have dicked with it a little more to get a perfect match.

      John k.... Music and Design NaO dsp Dipole Loudspeakers.

      Comment


      • #4
        Re: Results of the Minimum Phase / Off-axis test from yesterday (jay, dlr)

        Originally posted by johnk... View Post
        Jeff,

        Not to be a stickler but there has to be something wrong with you MP response. First, since the response is flat at 10 Hz the phase must go to zero at low frequency. Second, you show a turn up of the phase around 2 hundred Hz. A MP response for the amplitude should must turn down when the response starts to turn down.
        Right, John. I gave two reasons why his simulation came out like that.

        Comment


        • #5
          Re: Results of the Minimum Phase / Off-axis test from yesterday (jay, dlr)

          Originally posted by jkim View Post
          Right, John. I gave two reasons why his simulation came out like that.
          You can actually tell it's MP by inspection of the phase response. We know that the 1st order crossover is flat at DC and as frequency goes to infinity. Therefore the net phase differences from DC to infinity must be zero.

          By the way. I believe I posted earlier that an MP system must be stable and casual. It must also have a stable, causal inverse. I think I left the last part out. I did in a discussion over at DIY Audio.
          John k.... Music and Design NaO dsp Dipole Loudspeakers.

          Comment


          • #6
            Re: Results of the Minimum Phase / Off-axis test from yesterday (jay, dlr)

            Originally posted by johnk... View Post
            You can actually tell it's MP by inspection of the phase response. We know that the 1st order crossover is flat at DC and as frequency goes to infinity. Therefore the net phase differences from DC to infinity must be zero.
            Right, eyeball inspection should be sufficient in case clear non-M-P phase behavior is present like the above.

            By the way. I believe I posted earlier that an MP system must be stable and casual. It must also have a stable, causal inverse. I think I left the last part out. I did in a discussion over at DIY Audio.
            No worry, I got it from other text(s).

            Comment


            • #7
              Re: You're Right.....

              Originally posted by jkim View Post
              Jeff,

              Thanks for the follow-up. As I said yesterday, this is a simple mathematical problem. What you did above is not a mathematical proof but a simulation-based example. In fact I almost proved that a theoretical BW1+ transient perfect crossover is exactly (not nearly) minimum-phase even with a time delay (or a physical offset) added to LP or HP response. A reason why I didn't posted it here yet is that, due to my lack of math skills in functional anlaysis, I couldn't prove that the inverse transfer function in the time domain of BW1 (with a time delay added to LP component) integrates to a finite number---I already showed the regular transfer function does. These two are conditions for stability which must be met for being a minimum-phase system. Another condition, causality, is already met. (EDIT: the conditions I'm talking about here are ones described in the time domain, not in the frequency domain. I used time-domain ones for the proof since they were easier for me to handle in the current situation where a fixed time delay is added).

              There are two reasons why you obtained a simulation result showing some deviation from minimum phase. One is that you didn't use a response summed at least up to 50 kHz for accurate HBT computation. Using a response only to 20 kHz doesn't give an accurate result. Simulation results I showed last time used responses to 50 kHz. Another reason is that the HBT algorithm in your PCD is the same one as "Quick Phase" in the FRC. This down-sized formula doesn't give a precise result. Here are what I got from the same simulation condition that you used. That is, BW1 crossover with 1 kHz Fc with a 25 mm offset added to LP.




              As you can see there's no discrepancy between the summed phase through crossover (magenta) and the HBT computed phase (cyan). Compare this to your first graph in your post.

              Below is the same thing but I used the "Quick" HBT algorithm to show that this algorithm gives a different result.



              A small difference from your result occurs because I used a response up to 50 kHz while you used one to 20 kHz.

              After all this analytical and simulational investigation, I'm now pretty sure that theoretical BW1 T-P crossovers remain perfectly minimum-phase (as defined mathematically) even with a physical offset added to a driver up to a point where the added phase creates a perfect null which occurs when LP and HP responses' amplitudes are equal and they are 180 degrees out of phase. Beyond this point, they are no longer minimum-phase.
              -jAy

              I stand corrected. You are right, and I see that now. See, I'm OK with that

              You are correct - I only went to 20khz (and only 10 Hz on the low end). It did not occur to me that I needed to go higher (and lower) since I was using textbook filters in the example, but you're right, it makes a difference.

              You are also correct that I used "Quick Phase". PCD does not have a Hilbert-Bode function at all, in this case I used the FRC, but I did used Quick Phase and did not extend the response since it was flat at the extremes. I assumed it would be accurate with the textbook responses. After reviewing what you and John posted in reply I see where my errors were made and your original position was a correct one.

              I momentarily questioned the funky blip in the extracted phase beginning at 200Hz, but since it only showed itself in the zoomed in version I went on. I should have dug into it a little more. Being in a hurry did not benefit me here. :o

              It is a bit counterintuitive that you can add this much delay in the form of an offset to a first order filter and it remains minimum phase - but apparently it does, and all of the negative affect of the offset is in the summation of the amplitude response. The phase doesn't go non-minimum until the offset become "large" is relation to the crossover point. Just as you were saying :rolleyes: It would seem that impulse response would still not line up in time though.

              I really never thought of this feature of a minimum phase summation crossover before in this way. Thanks Jay, dlr, and John for enlightening me and graciously pointing out where my errors were. It's fun to learn new things.

              Jeff B.
              Click here for Jeff Bagby's Loudspeaker Design Software

              Comment


              • #8
                Re: You're Right.....

                Our thought process can't always be accurate. I, too, make this kind of mistakes all the time, just like I did in the previous thread about the non-effect of diffraction and non-uniform radiation upon a driver being minimum-phase.

                Yeah, this property of BW1+ crossover is interesting. I didn't think of it before, either. An implication is that T-P (M-P in practice) crossovers can still remain M-P off axis, which means that system FR distortion is reflected precisely in system phase distortion. Of course, in practice an off-axis limit for maintaining this behavior should be less than that of an idealized system due to drivers' natural rolloff and beaming.

                This makes me rethink of T-P crossover systems. Of course, the audibility of phase distortion of usual LR2 and LR4 crossovers hasn't been well validated. This is a difficult problem because it's not just physical but psycho-acoustical. Scientific data are not sufficient. BTW, I found this article on the internet. I wonder if you've seen it. Not much new info, though.

                http://www.ocf.berkeley.edu/~ashon/a.../phaseaud2.htm

                This also talks about research results of the following thesis:

                http://www.music.miami.edu/programs/.../chapter_5.htm


                Anyway, in the future, I'll try a simple BW1 2-way using drivers like Jordan JX92s and RS28 crossed around 5 kHz. But a tweeter with a smaller flange will be better if it has good low-end performance.

                -jAy

                Comment


                • #9
                  Re: You're Right.....

                  Originally posted by jkim View Post
                  Our thought process can't always be accurate. I, too, make this kind of mistakes all the time, just like I did in the previous thread about the non-effect of diffraction and non-uniform radiation upon a driver being minimum-phase. Yeah, this property of BW1+ crossover is interesting. I didn't think of it before, either. The implication is that T-P (M-P in practice) crossovers can still remain M-P off axis, which means that system FR distortion is reflected precisely in system phase distortion. Of course, in practice an off-axis limit for maintaining this behavior should be less than that of an idealized system due to drivers' natural rolloff and beaming.

                  This makes me rethink of T-P crossover systems. Of course, the audibility of phase distortion of usual LR2 and LR4 crossovers hasn't been well validated. This is a difficult problem because it's not physical but psycho-acoustical. Scientific data are not sufficient. BTW, I found this article on the internet. I wonder if you've seen it. Not much new info, though.

                  http://www.ocf.berkeley.edu/~ashon/a.../phaseaud2.htm

                  This also talks about research results of the following thesis:

                  http://www.music.miami.edu/programs/.../chapter_5.htm


                  Anyway, in the future, I'll try a simple BW1 2-way using drivers like Jordan JX92s and RS28 crossed around 5 kHz. But a tweeter with a smaller flange will be better if it has good low-end performance.

                  -jAy
                  I have read several articles on the audibility of phase, and have even done some experimentation myself. Testing absolute phase then reversing it was certainly more audible than I anticipated. Other than that, I can't say that I hear anything wrong with a good LR4 design. However, my favorite sounding systems, they ones that seem the most coherent and seamless, have been LR2, and we have speculated that it is in part due to the lesser phase rotation.

                  I have tried building systems that were first order time aligned - but had a very difficult time getting everything right - mainly due to the fact that it is difficult to find drivers with an acceptably wide bandwidth to keep from having excess phase roll in their own response. I have worked with the Jordan in two systems. It is an interesting driver, but I have liked several other better. I gave mine to a friend who is still enjoying them.

                  A much easier method to achieve a transient perfect system is with a three-way where the midrange is used as what we call a "filler-driver" over lapping a two-way with an LR2 crossover. The filler-driver uses first order slopes on each side. The mid's output is padded down to flatten the overall response. It works very well for achieving your goal. It is best to use a very wide bandwidth mid or full-range driver in this application too. The Jordan would make an excellent filler and the RS28 would work well in the tweeter application too. I believe John used to have an article on the filler-driver set-up on his website. If you want to stick to a two-way you might want to try one of John's higher order transient perfect crossover designs. They work better with real drivers, and they give you higher roll-off rates too.

                  Jeff
                  Click here for Jeff Bagby's Loudspeaker Design Software

                  Comment


                  • #10
                    Re: Results of the Minimum Phase / Off-axis test from yesterday (jay, dlr)

                    Good timing that you guys bring up the audibility of phase distortion. Take the test yourself with single-driver headphones that don't add their own phase distortion.

                    http://www.htguide.com/forum/showthread.php4?t=32280
                    Dennis

                    Comment

                    Working...
                    X