In 1983, Alex Strong and Kevin Karplus published a simple but effective algorithm for synthesizing the sound of a plucked string.

Pick the period $N$. Then:

If played at the frequency $f_s$, this sequence sounds like a string being plucked at frequency $f_s/(N+1/2)$

The Karplus-Strong algorithm is an example of digital waveguide synthesis. An instrument is physically modeled and simulated. In this case, the random samples crudely represents the initial pluck: each part of the string is in a random position moving at a random velocity.

The delay and feedback cause the waveform to repeat itself, oscillating as a string would. If we just had $y[n]=y[n-N]$, we would have a waveform that repeats with frequency $f_s/N$.

Instead, taking the average of two consecutive samples acts as a one-zero low-pass filter, mimicking dampening effects of a real string as it vibrates. Higher frequency oscillations lose energy quicker than lower frequency oscillations.

The filter $y[n]=(x[n]+x[n-1])/2$ has the transfer function $H(z)=(1+z^{-1})/2$. When $z=e^{ia}$, this is $e^{-ia/2}(e^{ia/2}+e^{-ia/2})/2=e^{-ia/2}\cos a/2$.

Thus an input $e^{ian}$ comes out as $e^{ia(n-1/2)}$, explaining why we divide the sampling frequency by $N+1/2$ to arrive at the frequency of the plucked string.

Although the basic algorithm produces surprisingly good results, we can do better.

At higher frequencies, rounding $f_s/(N+1/2)$ to the nearest integer is too crude. We can correct for the error by introducing an allpass filter in the loop: $y[n]=Cx[n]+x[n-1]-Cy[n-1]$.

At lower frequencies, the sound decays too slowly. We can shorten the decay by introducing a loss factor $\rho <1$, and set $y[n]=\rho (y[n-N]+y[n-(N+1)])/2$.

At higher frequencies, we have the opposite problem. We can stretch the decay by weighting the average. Pick some $0<S<1$ and set $y[n]=((1-S)y[n-N]+Sy[n-(N+1)])/2$. This changes the phase delay; see Jaffe and Smith for the exact formula (or derive it yourself).

When a real string is plucked harder, the waveform contains more high frequency components. Thus by putting the output through an appropriate low-pass filter we change the loudness of the output. One possible dynamics filter is $y[n]=(1-R)x[n]+Ry[n-1]$ for some $0<R<1$ that depends on the frequency and desired loudness.

To simulate string muting, we can introduce a loss factor when a note ends.

Slurs can be simulated by using a new value of N on the fly. Similarly, glissandi can be simulated by changing N gradually.