ON SYNTHESIZERS: Filter Modulation and Phase Shift
By Bob Moog
The voltage-controlled filter is the basic tone-coloring sound modifier in most synthesizers. A lowpass-resonant filter, the most popular type of voltage-controlled filter, allows frequencies up to its cutoff frequency to pass, and attenuates those frequencies above cutoff. Fig. 1 shows how the gain of a typical synthesizer lowpass-resonant filter varies with the frequency of the signal passing through it. The vertical axis represents gain, the ratio of the amplitude of a frequency component at the filter's output to its amplitude at the filter's input. 0dB is a gain of unity (output amplitude is the same as the input amplitude); a loss of 12dB is a gain reduction of a factor of 4.
This type of gain-versus-frequency characteristic is called 24dB per octave lowpass. At low frequencies, the gain is 0dB. The gain drops to -12dB at cutoff and continues to drop at a rate approaching 24dB per octave at frequencies above cutoff. The cutoff frequency is a parameter of the filter, which can be voltage-controlled over the entire audio frequency range.
The gain-versus-frequency response is an obvious characteristic of a lowpass filter. When harmonically rich material is passed through the filter and the cutoff frequency is slowly varied, we hear the tone color change clearly. However, another property of a voltage-controlled filter which may not be as obvious at first, but which assumes great importance in filter modulation, is the phase shift. Phase is the position of a wave in time. All filters introduce phase shift. Fig. 2 shows how the phase shift goes from 0° (output wave not delayed with respect to input) at low frequencies, through 180° at the cutoff frequency, and finally to 360° (output wave delayed a full cycle) at frequencies well above cutoff.
Filter phase shift becomes important to our ears when the filter cutoff is modulated and the amount of phase shift therefore changes rapidly. A continuously changing phase shift is the same thing as a frequency shift. Say for instance that a 1kHz sine wave passes through the filter, and that a modulating signal moves the filter's cutoff frequency from 10kHz down to 100Hz in a tenth of a second. During this tenth of a second, the phase shift goes from nearly zero to nearly one whole cycle. One complete cycle is effectively taken from the filter output signal. In other words, during the one-tenth second modulation period, the average frequency of the output is shifted downward by one cycle in one-tenth second, or ten cycles per second. If in the next tenth second the modulating signal goes back up and the filter's phase shift drops to its original low value, the average frequency of the output is then shifted upward by 10Hz! Thus, if the modulating signal is a 5Hz triangular wave (one-tenth second up, then one-tenth second down), the filter will impart a true vibrato (periodic frequency variation) of ±1% width (±10Hz in 1kHz) to a 1kHz sine wave. Of course, this vibrato does not sound the same as simple frequency modulation of a oscillator.
Fig. 3 shows a typical patch for filter modulation. The basic cutoff frequency of the filter is determined by the cutoff control, which feeds a steady voltage to the filter control input. Added to that are the modulating oscillator signal that varies the filter cutoff around this basic value, and connections to additional control voltage sources such as a contour generator, a second modulating oscillator, a sample-and-hold circuit, or an external foot pedal. Over the filter block is a connection from the filter's output, through an inverter and attenuator, to the audio input. This is called a feedback loop, and provides a way of getting some of the output signal back into the input. If the fed back signal is in phase with the audio input (positive feedback) then these two signals add and the gain of the filter increases. If the fed back signal is out of phase with respect to the input signal (negative feedback), then the gain of the filter goes down. The inverter in Fig. 3 reverses the phase of the filter's output so that, at frequencies well below cutoff, the feedback is negative. At the cutoff frequency, however, the filter itself delays the signal one half cycle, thereby inverting it with respect to the input. When this delayed signal is flipped over again by the inverter, the fed back signal is in phase with the input. At frequencies above cutoff, the feedback again goes negative, and becomes less important because of the filter's low gain.
Fig. 1 shows the gain-versus-frequency of a lowpass filter with feedback, and Fig. 2 shows the corresponding phase shift. As the emphasis control is turned up, the gain response goes down at low frequencies and `peaks' at the cutoff point. The change in phase shift also becomes steeper. The two changes greatly enhance the 'true vibrato' aspect of filter modulation: when the emphasis control is turned up during filter modulation, the pitch shift at cutoff is greater, and the gain is peaked so that the pitch shift is more clearly audible.