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  • Cauer / Elliptic Filter based crossover

    Recently I started playing around with elliptic filters (also called "Cauer" filters) for crossovers. I started a thread about it over at DIY Audio, but I thought it would be of interest here as well. I will be implementing these using some active filter boards that I recently designed and posted about, also at DIY Audio.

    For those not familiar with them, Elliptic filters have a very sharp transition band and one or more zeros (notches) in the stop band. Attenuation does not continue to increase in the stop band, but rises and falls below some minimum amount of attenuation. There is some ripple in the passband.

    Like Chebyshev filters, Elliptic filters are not necessarily useful "as is" for crossovers. When you create the analogous HP filter from the LP one and sum up their signals, there is often a sharp peak of several dB at the corner frequency. To get around this, I "detune" the high Q stage of the filter so that the HP and LP stages overlap smoothly and their sum is more or less flat.

    Below is an example that I made up, using some FRD files I downloaded from Roman Bednarek's web site. The crossover is 5th order, with 1 dB of passband ripple. The first plot is zoomed out to show the notches in the HP and LP filters. Three lines indicate the response of the tweeter + HP filter, the woofer + LP filter, and the combined response. The HP filter looks like it continues to roll off as frequency decreases, but this is not the case. The stopband attenuation flattens out for Elliptic filters, in contrast to all pole filters that one is used to seeing. The additional "roll off" is caused by the tweeter's response falling at low frequencies, contributing an extra 12dB/octave attenuation there. For the woofer, the stop band contains the woofers breakup peaks, which cause the stop band attenuation to degrade by about 6dB around 4k-9k Hz, but this area is still at least 66 dB below the tweeter's level.



    This plot shows a nice flat response, about +/- 1dB 200Hz-10k Hz (on-axis). I'm not sure what the off axis response looks like, but because of the high slope the interference between drivers should be minimal.


    I know that there were a few threads from around 2006 or so on what were then called "elliptic" filters, but which were really some filter function (e.g. Butterworth or Linkwitz-Riley) plus a couple of zeros (notches) thrown in for fun, to increase attenuation in the stop band. Jason Cuadra came up with one of these implementations. Not really an elliptic filter, but it's in the same spirit I guess. I have not seen much discussion on this topic in more recent times. I'd like to hear about people's experiences with this kind of filter for loudspeaker crossovers. If that's you please post a follow up. Thanks.

    -Charlie
    Charlie's Audio Pages: http://audio.claub.net

  • #2
    Re: Cauer / Elliptic Filter based crossover

    (old thread, but good topic)

    Charlie, did the detuning/relaxation cause any bigger problems in the sum on another axis other than the design axis?
    "Our Nation’s interests are best served by fostering a peaceful global system comprised
    of interdependent networks of trade, finance, information, law, people and governance."
    - from the October 2007 U.S. Naval capstone doctrine
    A Cooperative Strategy for 21st Century Seapower
    (a lofty notion since removed in the March 2015 revision)

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    • #3
      Re: Cauer / Elliptic Filter based crossover

      Originally posted by JRT View Post
      (old thread, but good topic)

      Charlie, did the detuning/relaxation cause any bigger problems in the sum on another axis other than the design axis?
      It shouldn't since the highpass and lowpass sections appear to be in phase at the crossover. Therefore, there should be no peaks on any axis and a narrow dip at the crossover in the power response. As for how the lobing is tilted, I don't know enough to surmise that.

      This has to be the longest wait ever for a first reply to a thread. I wonder how it's possible that Charlie posted this in 2011 and no one replied at all.
      Click here for Jeff Bagby's Loudspeaker Design Software

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      • #4
        Re: Cauer / Elliptic Filter based crossover

        Links to Thiele's patent in case anybody wants a good source of info on the topic. The second link is easier to read. The first link is quick and has pics that can be easilly uploaded and posted.

        Patent: US 6854005 B2
        Inventor: Albert Neville Thiele
        Original Assignee: Techstream Pty Ltd.

        http://www.google.com/patents/US6854005

        http://worldwide.espacenet.com...
        "Our Nation’s interests are best served by fostering a peaceful global system comprised
        of interdependent networks of trade, finance, information, law, people and governance."
        - from the October 2007 U.S. Naval capstone doctrine
        A Cooperative Strategy for 21st Century Seapower
        (a lofty notion since removed in the March 2015 revision)

        Comment


        • #5
          Re: Cauer / Elliptic Filter based crossover

          Originally posted by Jeff B. View Post
          It shouldn't since the highpass and lowpass sections appear to be in phase at the crossover. Therefore, there should be no peaks on any axis and a narrow dip at the crossover in the power response. As for how the lobing is tilted, I don't know enough to surmise that.

          This has to be the longest wait ever for a first reply to a thread. I wonder how it's possible that Charlie posted this in 2011 and no one replied at all.
          I found the thread searching for ellipticals. Figured nobody would mind the old thread since all of the participants are active here.
          "Our Nation’s interests are best served by fostering a peaceful global system comprised
          of interdependent networks of trade, finance, information, law, people and governance."
          - from the October 2007 U.S. Naval capstone doctrine
          A Cooperative Strategy for 21st Century Seapower
          (a lofty notion since removed in the March 2015 revision)

          Comment


          • #6
            Re: Cauer / Elliptic Filter based crossover

            Click image for larger version

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            k is the ratio of lower notch frequency fNL in the high-pass response to the crossover or transition frequency fX

            k = fNL /fX = fX /fNH
            "Our Nation’s interests are best served by fostering a peaceful global system comprised
            of interdependent networks of trade, finance, information, law, people and governance."
            - from the October 2007 U.S. Naval capstone doctrine
            A Cooperative Strategy for 21st Century Seapower
            (a lofty notion since removed in the March 2015 revision)

            Comment


            • #7
              Re: Cauer / Elliptic Filter based crossover

              Interesting Charlie, to bad nobody commented 3 years ago :o

              Originally posted by charlielaub View Post
              The HP filter looks like it continues to roll off as frequency decreases, but this is not the case. The stopband attenuation flattens out for Elliptic filters, in contrast to all pole filters that one is used to seeing. The additional "roll off" is caused by the tweeter's response falling at low frequencies, contributing an extra 12dB/octave attenuation there. For the woofer, the stop band contains the woofers breakup peaks, which cause the stop band attenuation to degrade by about 6dB around 4k-9k Hz, but this area is still at least 66 dB below the tweeter's level.
              So, there's no high pass or low pass? Just eq basically? So then at say 100hz in your example, there's zero attenuation on the tweeter? So then at high volumes the tweeter will certainly experience over excursion?

              Seems to me there should be at least a 12db/oct high pass on the tweeter at about 500hz.


              Also, what's the advantage here? What are you achieving that a steep high or low pass filter can't achieve?

              Thanks
              https://www.youtube.com/channel/UCm2...oSKdB448TTVEnQ

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              • #8
                Re: Cauer / Elliptic Filter based crossover

                Originally posted by ryanbouma View Post
                Interesting Charlie, to bad nobody commented 3 years ago :o



                So, there's no high pass or low pass? Just eq basically? So then at say 100hz in your example, there's zero attenuation on the tweeter? So then at high volumes the tweeter will certainly experience over excursion?

                Seems to me there should be at least a 12db/oct high pass on the tweeter at about 500hz.


                Also, what's the advantage here? What are you achieving that a steep high or low pass filter can't achieve?

                Thanks
                No, that's not correct. That tweeter is not down 90 dB at 200 Hz on its own. It's that the slope changes at low frequencies to a much lower slope, but there's already a lot of attenuation by this point, more than you will get with most other crossovers, so the tweeter is well protected.
                Click here for Jeff Bagby's Loudspeaker Design Software

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                • #9
                  Re: Cauer / Elliptic Filter based crossover

                  To split hairs, this is NOT an elliptical filter based crossover:


                  What you are seeing is a set of filters (high pass and low pass) that result when the inventor (Thiele) added one or more notches to an existing filter type (Butterworth, LR, etc.) of modest order in an effort to make it steeper near the corner frequency. It's all there in the patent.

                  A true elliptical filter is formed only by highpass notch filters, lowpass notch filters, and for odd orders there is an additional first order stage. There are exactly zero highpass or lowpass notch filters in the NTM crossover, thus it's not an elliptic filter. Please understand that a notch filter is not the same as a lowpass notch filter or highpass notch filter. Sure the NTM crossover is nice and all, but not elliptic! Also what is significantly different about elliptic filters compared to the type of filters that we normally use in audio is that the "stopband" either levels off at some attenuation or decreases at 6dB per octave if odd order. This is because the highpass or lowpass notch filters are like "shelf" filters in that they transition between two "levels" between the frequency where there is a possible peak (where the Q of the filter comes into play) and the filter where there is a notch (the frequency of the zero of the filter). The more that these two frequencies are separated, the greater the difference in the "shelf" levels that the filter can create for a given order.

                  The elliptic filter gets its name from the functions that are used to describe the amplitude response of the filter. Here is some good info on the subject:
                  http://www.ece.rutgers.edu/~orfanidi/ece521/notes.pdf
                  Good luck trying to digest this stuff... It's rather dense material. But let's talk for a bit not about the design and mathematics behind the filter, but about the properties of elliptic filters:

                  One of the interesting properties of the elliptic filter type is that it provides the steepest transition (slope) for a given filter order. For instance, in the paper I linked to above, the same attenuation that a 5th order elliptic filter could provide required a 35th (!) order Butterworth filter. Also, unlike all pole filters, the filter group delay is spread out over the various pole frequencies taken on by the HP/LP notch sections, and this tends to result in a lower peak group delay compared to Butterworth or even LR type filters of the same order. For example, the NTM filter in the figure above has a huge group delay peak at a single frequency, and this can be audible and not in a good way. Elliptic filters should be better in this regard, because the peak(s) in group delay are lower and spread out, and a lower order filter is required to provide similar attenuation. On the other hand, elliptic filters do have passband ripple, that is to say that their response is not flat in the frequency range where the filter is not designed to attenuate. The amplitude of this ripple is a design variable, and this makes the filter more complicated to design. You often see tables of prototype elliptic filters having a 3dB passband ripple amplitude, but this is practically unusable for loudspeakers. So you need to be able to design your own filter with lower passband ripple in such a way that you can actually implement them. This is not easy given the math involved.

                  Back in 2010 I was able to find a very useful DOS program (unfortunately it's no longer available) that lets me design an elliptic filter such that it is specified as a series of HP/LP notch filters plus a first order stage. I can choose the passband ripple amplitude that I desire, as well as the steepness and filter order. After I played around with this design program for awhile I realized that elliptic high pass and low pass filters are NOT complementary, that is if you sum them you get a large peak or dip around the crossover point (for normal or reversed phase) or some other ugly set of dips and peaks (at other phase angles). This type of filter was never really intended to be used in "pairs". The amplitude of these peaks and dips is directly related to the passband ripple amplitude so keeping this low helps the situation somewhat but that alone is not enough to make it useful for audio work. What I realized was that I could "detune" the highest Q section so that the response at the "corner" was 6dB down. I then found that an elliptic HP and LP filter pair would add to very nearly flat amplitude when in phase with each other, with an error of only a fraction of a dB (like the NTM crossover). This depends somewhat on all the parameters that went into the design of the filter and the amount of detuning applied. Using this approach, I have been able to come up with several filter prototypes with low ripple and low addition error of order 4-7 having various stopband attenuation levels (60-80dB) and other parameters that are very useful for loudspeakers. I can implement these as "biquad" sections either using an analog active or digital active (DSP) crossover hardware. I currently do this with the miniDSP hardware, which is convenient for this kind of implementation using the advanced biquad programming.

                  I have built a couple of loudspeakers using these detuned elliptical filters and I think that they work well and provide very steep cutoff. As I showed in the plots in the first posts in this rediscovered thread, the driver's own rolloff usually comes into play after a couple of octaves, providing additional rolloff to the otherwise flat stopband. Since the rolloff is steep the off-axis summation issue tends to be minimal since the frequency band where there is any real "overlap" is quite narrow. But you do have to pay attention to the relative phase of the drivers just like with other filter types so that they add in a desirable way.

                  When I first posted this thread I thought that people would be interested and excited about elliptic filters. I even thought I might try and write a white paper about it. But since there has not been a single post until now I kind of put that on the back burner. If there is enough interest I will do a write up and describe some of the filters that I have been using.

                  -Charlie
                  Charlie's Audio Pages: http://audio.claub.net

                  Comment


                  • #10
                    Re: Cauer / Elliptic Filter based crossover

                    Originally posted by charlielaub View Post
                    When I first posted this thread I thought that people would be interested and excited about elliptic filters. I even thought I might try and write a white paper about it. But since there has not been a single post until now I kind of put that on the back burner. If there is enough interest I will do a write up and describe some of the filters that I have been using.

                    I would find it interesting, and would encourage your effort in that.

                    The subject matter is certainly worthwhile.
                    "Our Nation’s interests are best served by fostering a peaceful global system comprised
                    of interdependent networks of trade, finance, information, law, people and governance."
                    - from the October 2007 U.S. Naval capstone doctrine
                    A Cooperative Strategy for 21st Century Seapower
                    (a lofty notion since removed in the March 2015 revision)

                    Comment


                    • #11
                      Re: Cauer / Elliptic Filter based crossover

                      Originally posted by charlielaub View Post
                      The elliptic filter gets its name from the functions that are used to describe the amplitude response of the filter. Here is some good info on the subject:
                      http://www.ece.rutgers.edu/~orfanidi/ece521/notes.pdf
                      Good luck trying to digest this stuff... It's rather dense material. But let's talk for a bit not about the design and mathematics behind the filter, but about the properties of elliptic filters:
                      That's some heavy math going on in that paper!

                      Just a newb question, is this style of filter only meant for an electronic crossover?
                      See my projects on Instagram and Facebook

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                      • #12
                        Re: Cauer / Elliptic Filter based crossover

                        Passive versions of the Cauer elliptic work just fine derekj, if that's what you were asking.

                        Comment


                        • #13
                          Re: Cauer / Elliptic Filter based crossover

                          When I started using CALSOD and its optimizer in the late 90's, I was surprised at times at just how well the crossover would integrate and at other times look very close, yet the summed response would be worse. It took a while for me to realize that CALSOD was optimizing for the summed target well into the area out of view below the range displayed. What I found was that with most of the components allowed to float during the optimization, the result was not the target I expected, such as an L-R for the tweeter. Quite often it created a large trap from certain components that was effectively a form of what Thiele described, but it was on the highpass leg only. It wasn't targeted to it by any means, the highpass below the trap would converge back to the original target, but the trap would bring in the summed response. It was the optimizer going through iterations when the target was for the summed response, not an individual driver. I never built any of these, after learning of this I later locked components in a way to maintain the desired target for the tweeter as much as possible. Even a woofer would occasionally show a tendency to having a trap as well, but it usually looked more like fine tuning the lowpass.

                          dlr
                          WinPCD - Windows .NET Passive Crossover Designer

                          Dave's Speaker Pages

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                          • #14
                            Re: Cauer / Elliptic Filter based crossover

                            Originally posted by JRT View Post
                            I would find it interesting, and would encourage your effort in that.

                            The subject matter is certainly worthwhile.
                            +1, indeed... We'd all benefit from that, not just closet 8" two way developers like myself...

                            Comment


                            • #15
                              Update to Old thread:
                              In the September 2016 issue of audioxpress, Perry Marshall details how he improved Bass response by implementing Elliptic Transfer Functions utilizing miniDSP high-pass shelf function.
                              "Not a Speaker Designer - Not even on the Internet"
                              “Pride is your greatest enemy, humility is your greatest friend.”
                              "If the freedom of speech is taken away, then dumb and silent we may be led, like sheep to the slaughter."

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