So,

Andrew prodded me a bit with the whole "You know, you could use a linear 
filter rather than an fft" bit. It got me thinking about IIR (infinite 
impulse response) filters.

I made some offhand remark about IIR filters being exceedingly difficult 
to design.

So. I first read a primer on how IIR works. OK, seems simple enough. 
Multiply some of the input stream by a few coefficients and multiply 
some of the output stream by a few coefficients add it all together, and 
you get a filter.

Good. Remarkably simple and very easy to calculate. But how does one 
design the coefficients? Poles and Zeros and something called the z 
transform, the discrete cousin of the Laplace transform. Laplace 
transform.... ooh, isn't that complicated?

I was surprised when I realized just how simple it really is. 
Essentially a Fourier transform, but adding exponential curves to the 
mix. So you have increasing and decaying waveforms. So, after reading 
about that and gaining an understanding on that, I turned to the Z 
transform. Which is a bit different. Foremost, it's discrete, of course, 
but also uses polar coordinates rather than rectangular.

Now, my understanding of how poles and zeros relate to the way a filter 
will respond is this:

The poles and zeros in the s domain are arranged along the imaginary 
axis, and their position on this axis determines the frequency they will 
affect. The higher the imaginary portion, the higher the frequency, the 
real portion affects the exponential attack or decay. Obviously, the 
magnitude of the effect is related to their distance from the axis. 
Poles cannot be on the positive side of this axis. (The exponent 
increases on the positive side, therefore any filter built on that would 
be unstable and oscillate)

Now, that's all fine and great for doing nice manipulations to equations 
and designing analog filters, but it doesn't quite fit into the discrete 
world. Enter the z domain.

The z domain has similar properties to the s domain, only polar. So the 
magnitude relates to how the exponent will attack or decay, and the 
phase relates to the frequency. The polar orientation of the z domain 
actually makes sense when you think about it. There is only a finite 
number of frequencies available, and like the DFT, they will reflect at 
the halfway point. This is also why poles and zeros are symmetrical 
about the Real axis.

So, the polar coordinates will need to be translated to rectangular 
before they can be plugged into an equation as complex numbers.

What's the frequency response? In the s domain, it's the slice of the 
complex plane along the imaginary axis. In the z domain, it's the 
cylinder of the complex plane defined by the unit circle, and only 
really the top half of that.

So once you have the coordinates, they need to be plugged into something 
to get the coefficients for the filter. That's where the z transform 
comes in. Which for each unit of the summation does this:

X[z] = x[0]*z^0 + x[1]*z^-1 + ... + x[n]*z^-n

So then,

The equation for an IIR filter:

y[n] = a0*x[n] + a1*x[n-1] + ... + b1*y[n-1] + ...

becomes, through the Z transform:

H[z] = (a0 + a1*z^-1  + a2*z^-2 + a3*z^-3 ...)/(- b1*z^-1  - b2*z^-2 + 
-b3*z^-3 ...)

While I understand what the Z transform does, I'm not quite sure exactly 
what steps were taken to arrive at the form presented above.

But with that, it becomes rather easy to translate the locations of 
poles and zeros to the coefficients:

for(n = 0; n < n_poles; n++)
{
z[n] = z[n-1]

for(i = n-1; i > 0; i--)
{
  z[i] = -p[n]*z[i] + z[i-1]
}

z[0] = -p[n]*z[0]
}

The code may be a bit buggy.... I grabbed that from my notes from the 
other day.

(Keeping in mind, of course, complex numbers are used in the code above)

Going the other way, however is a different story....
-- 
~Mike

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