Filter Science with Cytomic’s Andrew Simper

What Andrew Simper doesn’t know about filters isn’t worth knowing. As the coding expert behind Cytomic, he’s spent years analysing analogue circuits and systems with the aim to bring those nonlinearities to the digital domain. His lauded DSP filter algorithms can be found in Ableton Live 9’s bundled samplers, EQ Eight and Auto Filter; as well as in Cytomic’s The Drop. We sat down with Andrew for an in-depth chat about the science and theory behind filters. To kick off, we asked him how a filter works, exactly…

“All filters work by storing energy and adding to this store in some way from the input signal. Common stores of energy in analogue electronics are capacitors and inductors. Inside computers these stores are numbers in the RAM of the computer. The speed the store changes is determined by a gain factor, which is commonly called the cutoff frequency (Fig.1).

“That all sounds a bit abstract, so here’s an example. A basic low-pass filter is a device that smooths out sudden changes in a signal. It does this by storing energy and updating this store over time in a very particular way. To work out how much to add, you take the difference between the input and how much energy is currently stored, and add a bit of this difference to the store. In a high-pass filter, the store is updated in the same way, but instead of outputting the store itself like in a low-pass, you instead output the difference between the input and the store. The cutoff of the filter is determined by how much you add of the difference to the store. I find it much easier to express this as some basic pseudo code: store += cutoff * (input - store) low = store high = input - store “As you can see from the diagram, the low-pass filter removes the sudden changes but keeps everything else from the input, and the high-pass filter keeps the sudden changes but removes everything else. This is called a ‘one-pole’ filter because there is ‘one’ store of energy being updated at the cutoff gain amount.

“This basic example shows a linear filter. You may quite rightly say, “How is it a linear filter? I can clearly see curves that are definitely not straight lines!”. The linear part refers to harmonic content being added that wasn’t there before – linear filters don’t add any harmonics, they only make what is already there louder or quieter. This is a little difficult to tell just from looking at the plots, but one easy way to test how linear the filter is to stick in a sine wave and see if you only get a sine wave as the output, then you can be sure we haven’t introduced any nonlinearities (Fig.2).

“One thing you may have noticed is that the low-pass filter is both a little quieter than the input and is also delayed a little. The engineering terms for these are “attenuation” and “phase” of the filter – the phase bit just means how much delay happens to a single frequency from the input. This basic filter delays every input frequency by a slightly different amount, so this is called a non-linear phase filter.

“If two one-pole filters are placed one after another, then the input signal attenuation of frequencies is steeper: this is called a ‘two-pole’ filter. Likewise, if four one-pole filters are cascaded, this is called a ‘four-pole’ filter. With multiple poles, you can also add another gain factor to generate resonance, which is a way of boosting the cutoff frequency. The only way to generate resonance is to delay the cutoff frequency enough that it lines up with itself again so they are back in phase, and then gain it up so they match in amplitude in a feedback loop.”

Analogue Filter Circuits

We’ve all heard of filters based on famous synthesiser circuitry. So why aren’t all filters created equal? What is it about certain types that give them their charm and character? Why does a Moog filter differ from an MS-20 filter? We asked Andrew Simper of Cytomic to explain.

“There are so many different filter structures in analogue and digital, and each one works differently and produces a different sound. There are definitely good-sounding and bad-sounding filters in both analogue and digital. The MS-20 Sallen Key and the Moog Transistor Ladder are definitely worth talking about.

“A few of the main differences include the number of poles, the amount of resonance possible, how feedback is used for resonance, how the different responses like high and low-pass are formed, the types of gain elements used and their associated nonlinearities, the gain staging of the input to change how much the filter can be driven, and how strong the resonance is and how it interacts with the input signal to create ‘growl’.

“Nonlinearities are bunched up into three main types: the filter core, the buffers, and the resonance. In the case of the MS-20 and the Moog there are multiple different circuits used for each, so we really have to get down to specific models of specific synths otherwise things are totally different!

“Let’s take an MS-20 mk1 low-pass filter for starters. It’s a two-pole filter (-12 dB / octave), and it uses single biased transistors at the filter’s core to vary the cutoff. These are asymmetric in nature, which introduces a lot of even and odd harmonics to the signal when driven. The two low-pass stages are ‘unbuffered’: in other words, the two stages interact with each other. The signal is then buffered by a JFET kept in its linear region so that the cutoff gain isn’t affected by the next stage. The signal is then put through a

distortion stage most likely copied from the Boss OD-1 or Ibanez Tube Screamer TS808: a diode clipper in the negative feedback loop of on op-amp. This creates a parallel distortion, and helps limit the resonance, especially when in self oscillation, but this overdrive distortion is in the main signal path, so you hear it directly at the output.

“Now, the Moog Prodigy filter. It’s a four-pole low-pass filter, so will sound less bright above the cutoff. It uses the famous transistor ladder structure, which has a fully differential core, so a positive and negative version of the signal appear up each side of the ladder, and certain parts of the noise and nonlinearities are cancelled. These differential transistors buffer each of the one-pole stages from each other, and introduce symmetric distortion, which has strong odd harmonics. There are five stages of this differential distortion, then the signal is buffered at the output. This buffer differs greatly between models, but in the Moog Prodigy this is done with another two additional stages of differential transistor pairs, which introduces more distortion, but at a lower amplitude than the filter core distortion. This output is sent to the amp section and also to the input of the resonance gain op-amp and fed back into the input of the filter. Because the input stage clips more than the output stage, most of the resonance limiting distortion is done at the input.

“In summary, the MS-20 is a two-pole low-pass, so sounds brighter above the cutoff, has an asymmetric filter core, and has the main resonance limiting done by a stomp box overdrive type structure right at the output, so you hear this drive directly, which makes things even brighter. The Moog is a steeper four-pole low-pass, so sounds much duller above the cutoff, the core saturation is symmetric, and the resonance limiting is done at the input, which then gets low-pass filtered four times before you hear it, which makes things sound even deeper. They are both classics, and both sound brilliant, but they are almost polar opposites in terms of tone and operation!

“Both of these filters share something in common: the low-pass output is being saturated to limit the resonance. This means that the low frequencies in the input signal will fight with the resonance, especially when the cutoff is also low. Not all filters do this, for instance if the band-pass output is used to limit resonance, then any frequencies in the input lower than the cutoff won’t fight the resonance as strongly. This leads to a completely different sounding filter, but only when resonance is used.”

Cutoff instability

“Another important thing to note about non-linear filters is that generally the cutoff frequency changes based on how large the input signal is. This is a form of self FM, and it is clearly audible if you drive into a non-linear filter with self oscillating resonance. Linear EQ type filters don’t do this at all: cutoff will remain exactly where you put it, which is great for some things, but sounds pretty boring. Normally the cutoff gets modulated downwards with a loud input signal. A picture

(Fig.3) makes things much clearer, in the diagram you can see the spacing of the self oscillating wobbles is further apart when the input signal is higher.

“When the cutoff of a non-linear filter is swept, the resonance tends to latch onto each of the harmonics of the input as it passes nearby. It’s like a frequency snapping mechanism, and this is the key to the cool growling sounds you get with non-linear filters.

“There are also various ways of forming different responses in filters. Some, like a State Variable Filter, output the low-pass, band-pass, and high-pass signal all at once, so you can easily mix them together and generate any different shapes. Some filters allow you to put the input signal into different points of the filter to generate different responses, so you could actually filter multiple different signals at the same time, low-passing one signal, and high-passing another, and band-passing the last and the summed result appears at the output of the filter.

“Another way to generate different responses is by using four one-pole low-pass filters in a row, and then adding up different amounts of the input signal and each of the individual low-pass outputs. In this way you can actually generate band-pass, notch, and high-pass responses all from multiple low-pass outputs. This method was made famous by the Oberheim X-pander.”