Making sounds with analogue electronics - Part 2: Subtractive synthesis

A filter is an amplifier whose gain changes with frequency. It is usually the convention to have filters whose maximum gain is one, and so it is more correct to say that for a filter, the attenuation changes with frequency. A VCF is one where one or more parameters can be altered using a CV. Filters are powerful modifiers of timbre, because they can change the relative proportions of harmonics in a sound.

Filters come in many different forms. One classification method is based on the shape of the attenuation curve. If a sine wave test signal is passed through a filter, then the output represents the attenuation of the filter at that frequency; this is called the frequency response of the filter. An alternative method injects a noise signal into the filter and then monitors the output spectrum, but the sine wave method is easier to carry out. The major types of frequency response curve are

• low-pass
• band-pass
• high-pass
• notch.

Low-pass
A low-pass filter has more attenuation as the frequency increases. The point at which the attenuation is 3 dB is called the cut-off frequency, since this is the frequency at which the attenuation first becomes apparent. It is also the point at which half of the power in the audio signal has been lost and so it is sometimes called the half-power point. Below the cut-off frequency, a low-pass filter has no effect on the audio signal and it is said to have a fl at response (the attenuation does not change with frequency). Above the cut-off frequency, the attenuation increases at a rate which is called a slope. The slope of the attenuation varies with the design of the filter.

Simple filters with one resistor and capacitor (RC) will have slopes of 6 dB/octave, which means that for each doubling of frequency, the attenuation increases by 6 dB. Each pair of RC elements is called a pole and the slope increases as the number of poles increases. A two-pole filter will have an attenuation of 12 dB/octave, whilst a four-pole filter will have 24 dB/octave. Audibly, a four-pole filter has a more 'synthetic' tone and makes much larger changes to the timbre of the sound as the cut-off frequency is changed. A two-pole filter is usually associated with a more 'natural' sound and more subtle changes to the timbre (Figure 3.3.10).

FIGURE 3.3.10 Filter responses are normally shown on a log frequency scale since a dB/octave cut-off slope then appears as a straight line. But harmonics are based on linear frequency scales and on these graphs the filter appears as a curve. Low-pass filtering a sawtooth waveform with the cut-off frequency set to four different values:

(i) At 100 Hz, the filter cut-off frequency is the same as the fundamental frequency of the sawtooth waveform. The second harmonic is 30 dB below the fundamental and so the ear will hear an impure sine wave at 100 Hz.
(ii) At 300 Hz, the first three harmonics are in the pass-band of the filter and the output will sound considerably brighter.
(iii) At 500 Hz, the first five harmonics are in the filter pass-band, and so the output will sound like a slightly dull sawtooth waveform.
(iv) At 1 kHz, the first ten harmonics are all in the pass-band of the filter and the output will sound like a sawtooth waveform.

Low-pass VCFs usually have the cut-off frequency as the main controlled parameter. A sweep of cut-off frequency from high to low frequencies makes any audio signal progressively 'darker', with the lower frequencies emphasized and less high frequencies present. A filter sweeping from high frequency to low frequency of cut-off is often referred to as changing from 'open' to 'closed'. When the cut-off frequency is set to maximum, and the filter is 'open', then all frequencies can pass through the filter.

As the cut-off frequency of a low-pass filter is raised from zero, the first frequency that is heard is usually the fundamental. As the frequency rises, each of the successive harmonics (if any) of the sound will be heard. The audible effect of this is an initial sine wave (the fundamental), followed by a gradual increase in the 'brightness' of the sound as any additional frequencies are allowed through the filter. If the cut-off frequency of a low-pass filter is set to allow just the fundamental to pass through the filter, then the resulting sine wave will be identical for any input signal waveform. It is only when the cut-off frequency is increased and additional harmonics are heard, the differences between the different waveforms will become apparent. For example, a sawtooth will have a second harmonic, whilst a square wave will not.

High-pass
A high-pass filter has the opposite filtering action to a low-pass filter: it attenuates all frequencies that are below the cut-off frequency. As with the low-pass VCF, the primary parameter that is voltage controlled is the cut-off frequency. High-pass filters remove harmonics from a signal waveform, but as the frequency is raised from zero, it is the fundamental which is removed first. As additional harmonics are removed, the timbre becomes 'thinner' and brighter, with less low-frequency content and more high-frequency content, and the perceived pitch of the sound may change because the fundamental is missing.

Some subtractive synthesizers have a high-pass (not voltage-controlled) filter connected either before or after the low-pass VCF in the signal path. This allows limited additional control over the low frequencies that are passed by the low-pass filter. It is usually used to remove or change the level of the fundamental, which is useful for imitating the timbre of instruments where the fundamental is not the largest frequency component.

Band-pass
A band-pass filter only allows a set range of frequencies to pass through it unchanged – all other frequencies are attenuated. The range of frequencies that are passed is called the bandwidth, or more usually, the pass-band, of the filter. Band-pass VCFs usually have control over the cut-off frequency and the bandwidth.

Band-pass (and notch) filters are the equivalent of the resonances that happen in the real world. A wine-glass can be stimulated to oscillate at its resonant frequency by running a wet finger around the rim.

A band-pass filter can be thought of as a combination of a high-pass and a low-pass filters, connected in series, one after the other in the signal path. By using the same CV to the cut-off frequency inputs of two VCFs (one high-pass and the other low-pass), the cut-off frequencies will 'track' each other and the effective bandwidth of the band-pass filter will stay constant as the cut-off frequencies are changed. The width of the band-pass filter's pass-band can be controlled by adding an extra CV offset to one of the filters. If the cut-off frequency of the low-pass filter is set below that of the high-pass filter, then the pass-band does not exist, and no frequencies will pass through the filter (Figure 3.3.11).

FIGURE 3.3.11 A band-pass filter only passes frequencies in a specific range. This is normally the two points at which the filter attenuates by 3 dB. It can be thought of as a low-pass and a high-pass filter connected in series (one after the other). In the example shown, the lower cut-off frequency is about 0.6f (for the high-pass filter), whilst the upper cut-off frequency is about 1.6f (for the low-pass filter). The bandwidth of the filter is the difference between these two cut-off frequencies. Small differences are referred to as 'narrow', whilst large differences are known as 'wide'.

Band-pass filters are often described in terms of the shape of their pass-band response. Narrow pass-bands are referred to as 'narrow' or 'sharp', and they produce marked changes in the frequency content of an audio signal. Wider passbands have less effect on the timbre, since they merely emphasize a range of frequencies. The middle frequency of the pass-band is called the center frequency.

Very narrow band-pass filters can be used to examine a waveform and determine its frequency content. By sweeping through the frequency range, each harmonic frequency will be heard as a sine wave when the center frequency of the band-pass filter is the same as the frequency of the harmonic (Figure 3.3.12).

FIGURE 3.3.12 If a narrow band-pass filter is used to process a sound that has a rich harmonic content, then the harmonics which are in the pass-band of the filter will be emphasized, whilst the remainder will be attenuated. This produces a characteristic resonant sound. If the band-pass filter is moved up and down the frequency axis, then a characteristic 'wah-wah' sound will be heard – this is sometimes used on electric guitar sounds.