Audio Engineering Society UK

Title: ‘How I Does Filters: An uneducated person’s way to design highly-regarded digital equalisers and filters’
Location: Royal Academy of Engineering, London
Description: Christmas Lecture by Peter Eastty of Oxford Digital
Start Time: 18:30 for 19:00
Date: Tuesday 8th December 2009

Abstract

“Much has been written in many learned papers about the design of audio filters and equalisers – this is not another one of those. The presenter is a bear of little brain and has over the years had to reduce the subject of digital filtering into bite-sized lumps containing a number of simple recipes that have got him through most of his professional life. Complete practical implementations of bell (or presence) filters, high pass and low pass multi-order filters and shelving filters will be given. The infrequently seen higher order shelving filters will also be used to generate minimum phase IIR filters of arbitrary shape. The tutorial is designed for the complete novice, is light on mathematics and heavy on explanation and visualization – even so, the provided code works and can be put to practical use.”

The audio recording of this lecture is available to download here (MP3, 15MB).

Meeting Report

The PowerPoint visuals for this lecture are available from the website, and it’s highly recommended that you view them while reading this report or listening to the recording, because many of the key concepts are graphical and make no sense without the pictures. Download the visuals here (4MB PDF).

Peter opened his lecture with the assertion that it would not be mathematical, an interesting proposition for a topic notorious for the complexity and abstraction of its mathematics. He also pointed out that he has never designed an analogue filter in his life: instead of approaching digital filter design by designing analogue filters and translating them to the digital domain, he has always considered it from an exclusively discrete-time, sampled perspective. The first design aims are to create a bell-shaped presence filter, and a shelf filter: essentially the same as EQ controls on a mixing console (see page 3 of the visuals). Both of these filter types are defined by three independent parameters, as illustrated: gain, frequency, and Q (for the bell filter) or overshoot for the shelf. To achieve this, there are just three building blocks available (page 4): multiplication, addition, and a delay of one of more samples. So, how do we go about arranging these building blocks to make filters?

Peter took us on a rapid yet simple-to-follow tour of the effect of combining simple combinations of multiplication, addition and delay, with the most intuitive explanation I’ve ever heard of visualising filter responses in the z-plane. It’s difficult to paraphrase to convey the meaning without simply writing down the whole lecture, but listening to the recording whilst viewing the visuals will convey the message. He started by considering the behaviour of the simplest possible combination of a delay and adder – effectively a one-tap FIR, derived the z-plane representation and frequency response entirely graphically and intuitively, then proceeded to extend by adding a multiplier (for coefficient values other than 1), resulting in the insight that the coefficient moves a zero along the x-axis of the z-plane (page 19).

Next, it was shown that responses could be combined by cascading filters together (page 20 and following), but that the elements of cascaded filters can also be combined together into one structure with a single adder (accumulator) with identical behaviour, with simple mathematical relationships between the multiplier values (coefficients) in the cascaded filters and the combine structure. Based on this, a relationship was derived between the coefficients and the positions of zeroes on the x-axis – a little maths involving a square root, but still pretty straightforward (page 23). Of course, square roots often tend to give rise to roots of negative numbers (page 26 – looks remarkably like the quadratic formula) – so what do you do then? Well, in a move highly reminiscent of complex numbers, each zero moves off the x-axis in relation to a couple of new equations, to create a symmetrical pair (pages 27-29), and all the findings so far are summarised on page 30.

So far, the lecture had focussed on things that reduce gain (zeroes) – what about things that increase it, pulling that 3D surface upwards (page 31) instead of downwards? Just like with op-amps, positive gain is created using feedback loops, and the feedback loop contains very similar topologies to the filters already discussed (pages 32-34). Second-order filters with only negative gain response (all-zero) can be combined with second order filters with only positive gain response (all-pole) (pages 35-36), and the resulting structure is often known as a biquad, beloved of digital mixing console designers for (among many other things) digital versions of traditional parametric equalisers and filters. It is shown to have two symmetrical pole/zero pairs, and when the frequency response is plotted against log frequency, it can give rise to the familiar bell-curve EQ frequency response (page 41) if the pole and zero are associated with the same frequency. The distance between the pole and the zero was shown to be related to the Q or bandwidth of the filter (pages 43-50), and the geometry for the curves of constant frequency is calculated.

Curves of constant gain were also shown (pages 51-55), and it was then shown (pages 56-58) that the curves of constant frequency and curves of constant gain are orthogonal at all points – important for independent control of them. All these equations were pulled together, and with the addition of a gain correction term (page 60), resulted in the definitive equations for biquads (page 61). It was demonstrated with code snippets that these equations are directly implemented in Oxford Digital’s products. The effect of a biquad with same gain for the pole and zero, but at different frequencies, was illustrated (pages 65-66): it was shown that the perfectly-damped response is achieved when the gain circle has its origin on the unit circle (page 67).

Higher-order filters can be created by adding extra pole/zero pairs on the same constant frequency curves, but on carefully-chosen constant-gain curves (pages 76-81). It was then demonstrated how to make non-integer-order filters, by using the fact that a coincident pole/zero pair cancel each other, so by introducing such a pair (no effect on the filter) and then slowly moving them apart, the pole/zero configurations for integer orders can be interpolated between. This is truly novel, and although the graphics illustrating the configurations are not in the visuals linked to above, they can be seen in Peter’s convention paper presented at the 125th AES Convention, entitled “Accurate IIR Equalization to an Arbitrary Frequency Response, with Low Delay and Low Noise Real-Time Adjustment“.

This ability to have non-integer order IIR filters permits the construction of arbitrary filter responses, but without the usual penalties of FIR filters (namely, long processing delays and poor phase performance). Peter demonstrated his real-time filter software, running on a laptop with a frequency response curve that is manipulated by attaching handles and moving them arbitrarily as desired. Naturally, some extreme frequency responses result in filter orders in the hundreds, and CPU power is limited, so the filter order can be limited and the response gracefully falls away from the handles if an impractical response is requested. The filter changes response quickly and completely smoothly (to the audio) in real time, even with rapid changes to extreme filter responses with orders greater than 100 – how this is achieved, Peter declined to elaborate further! Controlling the coefficients of IIR filters such that smooth changes in gain/frequency/bandwidth are achieved without artefacts or (worse) instability is regarded as a challenging task for simple conventional filter designs, so achieving this for Peter’s much more sophisticated arbitrary-response EQ with extremely high orders is impressive.

Peter concluded his fascinating lecture with the observation – made possible by his EQ – that if one creates a dramatic comb-like filter response (in this case, alternating 12dB gain boost at roughly octave intervals), then shifts the frequencies of all the gain/cut points together in logarithmic frequency (i.e. group all the handles together and drag them left/right at once), the resulting effect sounds like playback pitch is being increased or decrease, despite the audio remaining at constant pitch and playback speed. Peter makes the entirely plausible suggestion that the rapid scaling of a complex frequency-domain structure in log frequency creates a psychoacoustic illusion of pitch shift, because it sounds like the frequency scaling of harmonic structures characteristic of pitch shift.

Many thanks to Peter Eastty for a fascinating and entertaining Christmas lecture, which delivered fascinating insights for both seasoned digital audio engineers and those new to the field, and revealed genuinely groundbreaking technology.

Meeting report by Michael Page