Explain this DSP notation

Question

I'm trying to implement this extenstion of the Karplus-Strong plucked string algorithm, but I don't understand the notation there used. Maybe it will take years of study, but maybe it won't - maybe you can tell me.

I think the equations below are in the frequency domain or something. Just starting with the first equation, H_p(z), the pick direction lowpass filter. For one direction you use p = 0, for the other, perhaps 0.9. This boils down to to 1 in the first case, or 0.1 / (1 - 0.9 z^-1) in the second.

alt text http://www.dsprelated.com/josimages/pasp/img902.png

Now, I feel like this might mean, in coding terms, something towards:

H_p(float* input, int time) {
  if (downpick) {
    return input[time];
  } else {
    return some_function_of(input[t], input[t-1]);
  }
}

Can someone give me a hint? Or is this futile and I really need all the DSP background to implement this? I was a mathematician once...but this ain't my domain.

Answer 1

So the z^-1 just means a one-unit delay.

Let's take H_p = (1-p)/(1-pz^-1).

If we follow the convention of "x" for input and "y" for output, the transfer function H = y/x (=output/input)

so we get y/x = (1-p)/(1-pz^-1)

or (1-p)x = (1-pz^-1)y

(1-p)x[n] = y[n] - py[n-1]

or: y[n] = py[n-1] + (1-p)x[n]

In C code this can be implemented

y += (1-p)*(x-y);

without any additional state beyond using the output "y" as a state variable itself. Or you can go for the more literal approach:

y_delayed_1 = y;
y = p*y_delayed_1 + (1-p)*x;

As far as the other equations go, they're all typical equations except for that second equation which looks like maybe it's a way of selecting either H_Β = 1-z^-1 OR 1-z^-2. (what's N?)

The filters are kind of vague and they'll be tougher for you to deal with unless you can find some prepackaged filters. In general they're of the form

H = H0*(1+az^-1+bz^-2+cz^-3...)/(1+rz^-1+sz^-2+tz^-3...)

and all you do is write down H = y/x, cross multiply to get

H0 * (1+az^-1+bz^-2+cz^-3...) * x = (1+rz^-1+sz^-2+tz^-3...) * y

and then isolate "y" by itself, making the output "y" a linear function of various delays of itself and of the input.

But designing filters (picking the a,b,c,etc.) is tougher than implementing them, for the most part.