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Volume 1, Number 2, May 2003
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Ask the Doctors!
Drs. David P. Berners and Jonathan S. Abel Answer Your Signal Processing Questions.

I have heard that the Cambridge EQ sounds like an analog equalizer. What's under the hood to make this happen?

Doctors David P. Berners
& Jonathan S. Abel
There are two parts to designing a digital EQ: First, you've got to design a digital filter which has the desired frequency response. That's really important---it's an EQ, after all. Then, you have to develop a filter structure which lets you hear the frequency response you worked so hard to get; numerics are key, as is controllability. When we designed Cambridge, we wanted an analog-sounding EQ, so we took steps in both the design and the implementation of our filters to try to match as closely as possible the response of a high-quality analog EQ.

"In fact, in many cases the difference between two different filter structures in terms of distortion can be equivalent to a difference in bit-depth of twelve bits or more."

Many digital EQs (such as the UA EX-1) are designed using analog prototype filters, and then carried over to the digital domain by means of the bilinear transform. The bilinear transform has many desirable properties: it preserves filter order in many cases, and also preserves filter stability. As long as the sampling rate exceeds the frequencies of interest by a comfortable margin, the bilinear transform provides a very accurate way to design a digital filter.

However, the bilinear transform introduces frequency warping into the filter response---different frequencies are displaced by different amounts as they are mapped into the digital domain. It is possible to match any one frequency in the analog and digital domains; every other point in the spectrum will be displaced. At low frequencies, the amount of displacement is small, but at frequencies approaching one-half the sampling rate, the amount of displacement increases more and more.

In actuality, the entire response of the analog filter is mapped between DC and one-half the sampling rate, so that if the analog filter rolls off at very high frequencies, the rolloff will be "transposed" below one-half the sampling rate for the digital filter. For the case of parametric and shelf filters, although no high-frequency rolloff is introduced, the response of the filter will change drastically if the filter has nonconstant features at high frequencies.

One way to minimize this problem is to use upsampling so that the EQ operates at a very high sampling rate. This way, the amount of frequency warping is reduced because all of the signal content is comfortably below the sampling frequency used within the EQ. The drawbacks to upsampling are processing latency, increased DSP power, and artifacts introduced by the up/downsampling process.

For Cambridge, we chose to operate the EQ at the original sampling rate, but to use a method other than the blinear transform to take our analog prototype filters into the digital domain. For our shelf filters we designed a new method to "digitize" the filters which allows a faithful rendering of the analog filter response, even at high frequencies. This part of the design is decoupled from the choice of analog prototype; using this method, we can take any analog EQ shelf or parametric circuit we wish to model, and directly produce a digital filter. What remains at this point is the implementation of the filter.

One fact that is not widely known is that it is possible to implement many different signal processing structures in the digital domain which have the same transfer function, or EQ. Choice of filter structure can have a dramatic effect on levels of distortion. In fact, in many cases the difference between two different filter structures in terms of distortion can be equivalent to a difference in bit-depth of twelve bits or more.

Because we wanted to be very careful about preserving signal integrity in the Cambridge, we chose a filter structure that requires a bit more DSP than some other structures (to test this, compare DSP usage to the EX-1 using the same number of bands), but results in very low distortion. This leads to a filter which is more true to its analog counterpart.

Why does Cambridge have so many cut filters?

We decided to have such a large palette of filter types because each of these filters has a different sound, and has historically proven to be valuable. The "slope" filters are equivalent to using separate, cascaded first-order filters for cutting lows or highs. The Bessel filters have been a popular choice in the past, because, while they aren't incredibly sharp in terms of rejection, they have gentle phase properties. The Butterworth filters are what is known as a "maximally flat" design, and provide a good tradeoff between solid rejection and nice phase properties. Finally, the Elliptic filter provides a rolloff which is as sharp as possible for a given filter order (expense). The best way to choose between these filters in a particular situation is to try them out and see how they sound--

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