Ask the Doctors!
Drs. David P. Berners and Jonathan S. Abel Answer Your Signal Processing Questions.
...I love the sound of the "Pultec EQ" plug-in but I've never met anybody who's even seen a real one. What's the big deal about the real one, and why is the plug 'so real'???"
-"daveyj" via email
Doctors David P. Berners
& Jonathan S. Abel
One of the distinctive features of the Pultec EQP-1A is that it is a passive EQ, using only resistors, capacitors, and inductors to accomplish equalization. Since no amplifiers or followers are used within the EQ circuit, the different sections of the EQ interact with each other more than they might if active components were to be used. A good example of this is the way the low shelving works: if you dial in a 3dB boost while also applying a 3dB cut, the two filters do not cancel. Rather, they interact in a new, interesting way.
"...every passive component has parasitic properties which do not show up on a schematic."
In terms of modeling, in order to get the behavior right, the whole EQ circuit had to be modeled at the same time; if we modeled each stage of the EQ separately, the interaction wouldn't have come out right. We started by obtaining the closed-form transfer function for the EQ using a schematic. This gave us a good starting point for the behavior of the EQ, and the general way in which the circuits interact.
What remained was to find appropriate component values for the resistors, inductors, and capacitors within the circuit. We could, of course, have simply read the values from the schematic and used those in our model. However, we instead chose to use measurements from actual units to provide our component values.
There were several arguments for using measurements: First, tolerances in the component values lead to differences between units, and we wanted to model a particularly good-sounding unit. Second, every passive component has parasitic properties which do not show up on a schematic. For example, capacitors have varying amounts of series resistance depending upon how they are made. In order to incorporate these properties into our model, we had to augment our schematic and resort to measurement. Third, there is some dispute as to what component values were used in original units, because components were potted in order to prevent discovery of their values. Finally, the tapers of the potentiometers must be recorded, and this can only be done by experiment.
After selecting a "golden" unit, we developed a set of control settings explicitly designed to expose, collectively, the values of the components within the EQ circuit based on the transfer function we had discovered earlier. This involved creating settings which allowed us to observe the interaction between components, as well as the characteristics of the pots. After running computer-based optimizations, we were able to recover values for all of the components including parasitics to within an absolute impedance factor. This gave us a precise model of one particular Pultec unit.
After obtaining estimates of the nonlinearities and equalization imparted by transformers and make-up amplification, we were ready to carry our design into the discretized domain. Upsampling allowed a transformation to discrete-time with minimal warping of the filter characteristics. For upsampling, we chose a linear-phase filter with close to 100dB of rejection in the stopband. Using linear-phase filtering for the upsampling process allowed us to match the phase of the Pultec response throughout the audio band; the large amount of stopband rejection was necessary in order to prevent aliased signal components from degrading the system response. The tight constraints placed on phase response and anti-aliasing properties led to a tradeoff between system latency and high-frequency magnitude response. Thus, in order to maintain acceptable overall system latency, there is a 2db rolloff at 18kHz when running the plugin at 44.1kHz, and a .01db rolloff at 18kHz when running the plug-in at 48kHz. For an analog system, a comparable magnitude rolloff could be problematic, because there would be an associated phase warping which would extend down to the 2kHz range. However, since the anti-aliasing filter is linear-phase, this problem does not occur in the plugin.
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