[BC] Correct URL for the Symmetra-Peak

[Robert Orban]

At 11:58 AM 8/16/2006, you wrote:
>From: Robert Meuser <Robertm at broadcast.net>
>Subject: Re: [BC] Correct URL for the Symmetra-Peak
>To: "Broadcasters' Mailing List" <broadcast at radiolists.net>
>Message-ID: <44E33D11.4000500 at broadcast.net>
>Content-Type: text/plain; charset=ISO-8859-1; format=flowed
>
>
>An opamp version would be easy if gyrators were used. I bet with the
>adjustability of the gyrator you could tweak more customized
>performance.  Next questions, how would this compare to the less complex
>Orban implementation in the 8100?  Are digital implementations better?
>Would the gyrators make for a better "sound" than all those inductors.


For a given filter to be allpass, it is necessary and sufficient for each 
s-plane pole at a + j b to have a mirrored s-plane zero at -a + j b, where 
j is the imaginary operator. The poles and zeros can be real or complex. 
All of this is straightforward to realize with opamps. The topologies are 
simple compared to the passive LC realizations and the design equations are 
simpler. Also, unlike passive LC lattices. efficient opamp designs are 
minimum-reactance, meaning that there is only reactive element (capacitor, 
this case) per pole.

That being said, I modeled Kahn's original circuit in a network analysis 
program, and it turns out that the network, even with ideal components, is 
not strictly allpass. It has a slight equi-ripple magnitude unflatness of 
about 0.0015 dB total peak-to-peak. The pattern of its poles and zeros are 
more complicated than a classic allpass would be. (Note that I removed 8 
"parasitic" poles and zeros that had real parts than were essentially zero 
and which were basically artifacts of the pole/zero calculation.)

We note that four of the zeros are non-minimum phase (i.e., have positive 
real parts), as we would expect to achieve the phase rotation along with 
essentially flat magnitude. This defines the allpass function as producing 
720 degrees of total phase rotation, which could be realized with a 
four-pole active RC realization. However, in the Kahn circuit the 
non-minimum phase zeros do not have corresponding poles having the same 
imaginary part and a negative real part. Instead, there are four 
minimum-phase zeros that interact with the poles to achieve the "almost 
allpass" magnitude response.

Because the Kahn circuit is not really allpass, realizing it in active-RC 
form would be more expensive than realizing a true active-RC allpass.

  NETWORK ZEROS

      REAL PART        IMAG.PART           ZERO FREQ.         ZERO Q
        (HZ)             (HZ)                (HZ)

    3.47349983D+02  -4.54363292D-02      3.47349986D+02  -5.00000004D-01
    3.47349983D+02   4.54363292D-02      3.47349986D+02  -5.00000004D-01
    3.47259135D+02  -4.54124603D-02      3.47259138D+02  -5.00000004D-01
    3.47259135D+02   4.54124603D-02      3.47259138D+02  -5.00000004D-01
   -3.47339484D+02  -3.49233614D-02      3.47339486D+02   5.00000003D-01
   -3.47339484D+02   3.49233614D-02      3.47339486D+02   5.00000003D-01
   -3.47269634D+02  -3.49262854D-02      3.47269636D+02   5.00000003D-01
   -3.47269634D+02   3.49262854D-02      3.47269636D+02   5.00000003D-01


  NETWORK POLES

      REAL PART        IMAG.PART           POLE FREQ.         POLE Q
        (HZ)             (HZ)                (HZ)

   -6.57610569D+02   0.00000000D+00      6.57610569D+02   5.00000000D-01
   -4.77014135D+02  -2.30264272D+02      5.29683037D+02   5.55206857D-01
   -4.77014135D+02   2.30264272D+02      5.29683037D+02   5.55206857D-01
   -2.86838858D+02  -1.95816052D+02      3.47304559D+02   6.05400122D-01
   -2.86838858D+02   1.95816052D+02      3.47304559D+02   6.05400122D-01
   -1.83422321D+02   0.00000000D+00      1.83422321D+02   5.00000000D-01
   -2.05078479D+02  -9.89954869D+01      2.27721955D+02   5.55206857D-01
   -2.05078479D+02   9.89954869D+01      2.27721955D+02   5.55206857D-01

Bob Orban