Karplus–Strong string synthesis is a method of physical modelling synthesis that loops a short waveform through a filtered delay line to simulate the sound of a hammered or plucked string or some types of percussion.
At first glance, this technique can be viewed as subtractive synthesis based on a feedback loop similar to that of a comb filter for z-transform analysis. However, it can also be viewed as the simplest class of wavetable-modification algorithms now known as digital waveguide synthesis, because the delay line acts to store one period of the signal.
Alexander Strong invented the algorithm, and Kevin Karplus did the first analysis of how it worked. Together they developed software and hardware implementations of the algorithm, including a custom VLSI chip. They named the algorithm "Digitar" synthesis, as a portmanteau for "digital guitar".
How it works
A short excitation waveform (of length L samples) is generated. In the original algorithm, this was a burst of white noise, but it can also include any wideband signal, such as a rapid sine wave chirp or frequency sweep, or a single cycle of a sawtooth wave or square wave. This excitation is output and simultaneously fed back into a delay line L samples long. The output of the delay line is fed through a filter. The gain of the filter must be less than 1 at all frequencies, to maintain a stable positive feedback loop. The filter can be a first-order lowpass filter (as pictured). In the original algorithm, the filter consisted of averaging two adjacent samples, a particularly simple filter that can be implemented without a multiplier, requiring only shift and add operations. The filter characteristics are crucial in determining the harmonic structure of the decaying tone. The filtered output is simultaneously mixed into the output and fed back into the delay line.
Tuning the string
The fundamental frequency (specifically, the lowest nonzero resonant frequency) of the resulting signal is the lowest frequency at which the unwrapped phase response of the delay and filter in cascade is . The required phase delay D for a given fundamental frequency F0 is therefore calculated according to D = Fs/F0 where Fs is the sampling frequency.
The length of any digital delay line is a whole-number multiple of the sampling period. In order to obtain a fractional delay often needed for fine tuning the string below JND (Just Noticeable Difference), interpolating filters are used with parameters selected to obtain an appropriate phase delay at the fundamental frequency. Either IIR or FIR filters may be used, but FIR have the advantage that transients are suppressed if the fractional delay is changed over time. The most elementary fractional delay is the linear interpolation between two samples (e.g., s(4.2) = 0.8s(4) + 0.2s(5)). If the phase delay varies with frequency, harmonics may be sharpened or flattened relative to the fundamental frequency. The original algorithm used equal weighting on two adjacent samples, as this can be achieved without multiplication hardware, allowing extremely cheap implementations.
Z-transform analysis can be used to get the pitches and decay times of the harmonics more precisely, as explained in the 1983 paper that introduced the algorithm.
A demonstration of the Karplus-Strong algorithm can be heard in the following Vorbis file. The algorithm used a loop gain of 0.98 with increasingly attenuating first order lowpass filters. The pitch of the note was A2, or 220 Hz.
Holding the period (= length of the delay line) constant produces vibrations similar to those of a string or bell. Increasing the period sharply after the transient input produces drum-like sounds.